Research ArticleDetermining Bounds on Assumption Errors inOperational Analysis

Neal M Bengtson

School of Business Barton College Wilson NC 27893 USA

Correspondence should be addressed to Neal M Bengtson bengtsonbartonedu

Received 21 June 2013 Revised 18 November 2013 Accepted 4 December 2013 Published 12 January 2014

Academic Editor Zne-Jung Lee

Copyright copy 2014 Neal M Bengtson This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The technique of operational analysis (OA) is used in the study of systems performance mainly for estimating mean values ofvarious measures of interest such as number of jobs at a device and response times The basic principles of operational analysisallow errors in assumptions to be quantified over a time period The assumptions which are used to derive the operational analysisrelationships are studied Using Karush-Kuhn-Tucker (KKT) conditions bounds on error measures of these OA relationships arefound Examples of these bounds are used for representative performance measures to show limits on the difference betweentrue performance values and those estimated by operational analysis relationships A technique for finding tolerance limits onthe bounds is demonstrated with a simulation example

1 Introduction

The analysis of the performance of a network of devices isimportant in many areas Computer systems and industrialmanufacturing systems are two examples The types of net-works considered in this paper are operationally connectedqueue and server devices That is each device is connectedin some way with every other device in the network andeach device may have a queue assigned to it Certaininformation about these types of networks may be obtainedusing a technique known as operational analysis (OA)Relationships used to estimate performance measures (PMs)of networks may be derived in operational analysis undera few restrictive assumptions OA is a technique which wasoriginally defined as an aid in computer system performanceanalysis [1ndash6] It can be an aid in the understanding ofsystem performance in general [7] and is a complementaryapproach to stochastic analysis used in many networks ofservers performance analyzes and in computer programs[8ndash14] Other used or suggested applications for the OAapproach include telecommunications [15] E-commerce [1617] flexible manufacturing systems [18] and Petri nets [19ndash21] The performance measures derived are such things asaverage number of units at a device average response time

and throughput The behavior of a single arbitrary device ina network will be considered

Two basic principles define the OA approach [2]

(1) All assumptions that are made in analyzing theperformance of a real system should be subject todirect verification

(2) All variables that appear in any equation whichcharacterize the performance of a real system shouldbe verifiable by direct measurement

The validity of PMequations developed using these principlescan be shown for a particular set of data because they arebased on assumptions which can be directly tested by theobservation of data produced by the system of interest overa finite period of time

The most widely used assumption about the data is thatof job flow balance that is the number of arrivals to anetwork (for global flow balance) or to a device (for local flowbalance)must be equal to the number of departures from thatnetwork or device Also assumed is one step behavior onlyone unit may arrive or depart the network or device at a timeArrivals and completions do not occur simultaneously TheOA approach assumes that devices must have homogeneous

Hindawi Publishing CorporationJournal of OptimizationVolume 2014 Article ID 460570 10 pageshttpdxdoiorg1011552014460570

2 Journal of Optimization

service That is the service time of a device in a network isindependent of the queue length at any deviceHomogeneousarrivals are the corresponding condition for the arrival timesHomogeneous routing holds when the routing frequencies ofjobs leaving a device are independent of the queue lengthsat other devices in the network Device homogeneity existswhen the rate of output from a device is determined only byits queue length Other assumptions can be invoked as theneed arises to derive OA relations [22]The only requirementis that these assumptions meet the two basic principles oftestability given above

OA assumptions allow for the development of relation-ships which enable us to determine PMs by collecting onlya few types of data namely the number of arrivals anddepartures for each device state and the total time spentin each state [23] These PMs will be accurate only if theOA assumptions are met and only for the finite time periodobserved The accuracy of the assumptions can be measuredA device state is the number of items (customers jobsentities etc) both waiting and in service at a device

While OA research was originally proposed as an aid incomputer performance analysis it is more general in thatdevelopments can be applicable to any system that generatestime series data This would include computer simulation

The Abbreviations section gives definitions of variableused in this paper Errormeasures of variousOA assumptionshave been defined and are summarized in Table 1 for job flowbalance homogeneous service and homogeneous arrival[24] The limit over time of the expected value and varianceof the job flow balance error is zero [25] so that over timethis error is not significant for data runs of reasonable lengthThe expected value over time of other error measures suchas for the assumptions of homogeneous service and arrivalmay not in general tend to zero [25]

Different error measures in the form of relative errorshave been defined by Brumfield [22] By presenting a set ofnew assumptions formulas for the calculation of responsetime and average queue length in terms of the averageand coefficient of variation of service times are developedTwo examples of these new assumptions are homogeneityof queueing and service and homogeneity of residuals Forarrival 119895 ldquoforward residual is either the time remaining inthe service period during which 119895 arrives or zero if arrival119895 begins a service period similarly backward residual iseither the time since the beginning of the service periodduring which 119895 arrives or zero if arrival 119895 begins a serviceperiodrdquo [22] Relative error formulas for response time aredetermined with these new assumptions in addition to theold assumptions of homogeneous service and homogeneousarrivals Unfortunately the error terms are quite complexsince there are more assumptions with which to deal

If we are using a relationship to determine a PM derivedunder OA assumptions then the resulting value of the PM isin error if the founding assumptions do not holdThis can bechecked from the data because of the way OA assumptionsare defined The degree of error in the PM calculated is afunction of the assumption error measures Correction termshave been developed using the assumption error measures[24]When added to the PMs these correction terms produce

exact results It is these correction terms which are studied inthis paper

As an example assume we are interested in obtaining avalue for the average number of jobs in a computer systemthat a new job sees upon arrival If wemake the homogeneousservice assumption then this averagemay be estimated by [4]

119899119878

119860=119899

119880minus 1 (1)

where 119899119878119860is average number of jobs at a device seen by an

arriver assuming homogeneous service 119899 is average numberof jobs at the device and 119880 is device utilization

We are interested in finding a correction term such that

119899119860= 119899119878

119860+ 119862119899119860 (2)

The correction term is equal to

119862119899119860= 119890119865minus 119890lowast

119878 (3)

where

119890119865=119873minus1

sum119899=0

(119899 + 1) (119901119860 (119899) minus 119901119862 (119899))

119890lowast

119878=119873

sum119899=1

119899119901119862 (119899 minus 1) 119890119878 (119899)

(4)

Check the Abbreviations section for all symbol definitions119890119865 is the error measure for the job flow balance assumptionand 119890lowast

119878is a weak form of the homogeneous service error

measure because it may be equal to 0 even when some or allthe individual 119890

119878(119899) values are not

There are a couple of problems with using the errormeasures to derive correction terms One problem is thatthe amount of data needed to calculate an exact value foran assumption error measure is the same as to find theperformance measure of interest directly The process ofdetermining the error measure for each assumption used toderive a relation the correction term and the PM estimateto which the correction term is added is a way of gettingsomething which may be observed more directly Findingexact values for performance measures in this indirect wayover a finite time period may be worthwhile only if a numberof PMs are desired In this case a single assumption errormeasure is determined for each assumption and applied toall the PM correction terms of interest

Another problemwith the errormeasure technique is thatthese measures apply only to the data observed For anotherrun of data new error measure values need to be foundThis limitationmay be acceptable if PMs cannot be measureddirectly without changing the nature of the system forexample in a complex computer systemWewould like a wayto extend assumption error measures over longer sets of dataand thereby say something about the system that generatedthe data As Sevcik and Klawe [26] stated shortly after OAwas introduced ldquoBecause operational analysis is based onassumptions that can be tested but that are very unlikely to besatisfied exactly in any finite time period it is very important

Journal of Optimization 3

Table 1 Operational analysis error measures

Assumption State dependent error Overall errorJob-flow balance 119890

119865 (119899) = (119899 + 1) (119901119860 (119899) minus 119901119862 (119899)) 119899 isin 119868 119890119865= sum119873minus1

119899=0119890119865(119899)

Homogeneous service 119890119878(119899) =

1

119901119862(119899 minus 1)

(119879(119899)

119879 minus 119879(0)) minus 1 119899 isin 119868+ 119890lowast

119878= sum119873

119899=1119899119901119888 (119899 minus 1) 119890119878 (119899)

Homogeneous arrival 119890119860(119899) = 119901

119860(119899) (

119879 minus 119879(119873)

119879(119899)) minus 1 119899 isin 119868 119890lowast

119860= sum119873minus1

119899=1119890119860 (119899)

119899119879 (119899)

119879 minus 119879 (119873)

to develop a means of dealing with lsquofuzzy homogeneityrsquo orsituations in which the various independence assumptionsare satisfied within some tolerancerdquoThis paper addresses thisneed to define these assumption bounds

The next section will illustrate how OA relations may beused to reduce data collection while estimating performancemeasures This will be followed by a discussion of thedetermination of bounds on the OA assumption measure-ment errors for homogeneous service and homogeneousarrival Sample calculations of these bounds will be presentedafterward An illustration of the use of bounds in a simulationwill then be given

2 Simplifying Data Collection

Calculating performance measures with OA relationshipsthat are derived under one or more of the system behav-ior assumptions is usually simpler than using more directrelationships This is because by making the assumptions amodel has been created which reduces perhaps artificiallythe complexity of the behavior of the systemThe result is thatless information is needed tomake an estimate of the PM thanwould be needed for a direct measurement

There are situations where it is impractical if not impos-sible to collect sufficient data to determine exact values forPMs over a finite period of time In some cases only anestimate of a PM is needed and it is not worthwhile to go tothe trouble of determining the precise PM values Any PMvalue obtained for a behavior sequence is only an estimate ofthe underlying system PM With this realization in mind itmay seem unwise to spend a great deal of effort to obtain anexact value for a sequence which is in turn only an estimateof some other value A good approximation of the sequenceestimate may be sufficient

If we want the average response time R of a behaviorsequence we could accumulate the response times of allthe jobs that go through the system and get the exact R bydividing by the number of jobs A simpler procedure wouldbe to say that response time is

119877 =119878

1 minus 119880 (5)

where 119878 is mean time between completions during busyperiods and U is utilization

This equation will give the exact R value if we have abehavior sequence for a single server queue which is in flowbalance and has homogeneous arrivals and services If these

conditions do not hold the equation will not give R exactlybut an estimate of R call it 1198771015840 If we collect only the idle time119879(0) and the number of completions C we can use the sameequation to find 1198771015840 If the behavior sequence lasts for time 119879then

119880 = 1 minus119879 (0)

119879

119878 =119879 minus 119879 (0)

119862

(6)

Another example calculates the average number of jobsat a device With the same assumptions as for estimatingresponse time the average number of jobs in the queueserversystem is

119899 =119880

1 minus 119880 (7)

This value takes even less data to calculate than does 1198771015840 Thedirect calculation of 119899 requires accumulating data every timethere is an arrival or completion or requires keeping track ofthe total amount of time spent at each of the states

Using these equations for predicting future values for Rand 119899 of a system presents certain problems For exampleover future time will the assumptions of the system behave inthe sameway Sincewe candetermine anduse errormeasuresof the assumptions in order to correct assumption derivedPM estimate it is not necessary that assumptions hold in thefuture if they have not in the past With the determinationof correction terms all that is really necessary is for thecorrection terms to remain relatively constant that is for thesystemrsquos violations of assumptions to remain the same overfuture time periods

Without knowledge of the assumption error measuresand through them the correction terms the performancemeasure estimates may be quite bad for any particularbehavior sequence [23] As stated before in a stable systemthe job flow balance assumption errormeasure will go to zeroas time increases but as shown in [24] this is not necessarilytrue for other assumption error measures For any behaviorsequence it is important tomake some assessment if possibleof the behavior of the PM correction terms

3 Performance Measure Bounds

One approach to use the simplified OA formulas for PMs isto determine bounds on the maximum PM error That is we

4 Journal of Optimization

are interested in defining bounds on the difference betweentrue values of various PMs at a device for particular statesequences and those PMs estimated by using relationshipsderived under operational analysis assumptions We willassume the network is in steady state

In the following bounds are found for the assumptionsof homogeneous services and homogeneous arrivals In thecase of the job-flow balance assumption we know that theexpected value of the errormeasure and its variance go to zero[25]

lim119905rarrinfin

119864 [119890119865] = 0 lim

119905rarrinfin119881 [119890119865] = 0 (8)

Therefore as the length of the sequence increases the 119890119865 can

be expected to become insignificantWewill need to assume that amaximumvalue of the error

for services and arrivals for any state is known or can be setCall these values 120575119878 and 120575119860 for the maximum service errorand maximum arrival error respectively

31 Bounds on Homogeneous Service Assumption Error If120575119878is the maximum error for the homogeneous service

assumption then1003816100381610038161003816119890119878 (119899)

1003816100381610038161003816 le 120575119878 119899 isin 119868+ (9)

A more useful bound would be on the weak overall homoge-neous service error

119890lowast

119878le 120575lowast

119878 (10)

But this limit may be harder to know beforehand Equation(9) may be used to find an upper bound on 119890lowast

119878by using the

definition

119890lowast

119878=119873

sum119899=1

119899119901119862 (119899 minus 1) 119890119878 (119899) (11)

Substituting (9) yields

119890lowast

119878=119873

sum119899=1

119899119901119862 (119899 minus 1)

1003816100381610038161003816119890119878 (119899)1003816100381610038161003816 le 120575119878

119873

sum119899=1

119899119901119862 (119899 minus 1) (12)

The term119873

sum119898=1

119899119901119862 (119899 minus 1) = 119899119862 (13)

is the average number at the device seen by a completerTherefore

119890lowast

119878le 120575119878119899119862 (14)

The bound given by (14) does not take into considerationthe fact that the 119890

119878(119899) values are not independent In fact they

are related by the expression

119873

sum119899=1

119901119862 (119899 minus 1) 119890119878 (119899) = 0 (15)

We can get a tighter bound of the 119890lowast119878values by taking this

dependence into consideration Equation (15) can be shownby substituting the definition

119890119878 (119899) =

119878 (119899)

119878=

1

119901119862 (119899 minus 1)

[119879 (119899)

119879 minus 119879 (0)] minus 1 (16)

as follows119873

sum119899=1

119901119862 (119899 minus 1) 119890119878 (119899)

=119873

sum119899=1

119901119862 (119899 minus 1) [1

119901119862 (119899 minus 1)

119879 (119899)

119879 minus 119879 (0)minus 1]

=119873

sum119899=1

119879 (119899)

119879 minus 119879 (0)minus119873

sum119899=1

119901119862 (119899 minus 1)

=119879 minus 119879 (0)

119879 minus 119879 (0)minus 1 = 0

(17)

Sincewhat is desired is an upper bound on 119890lowast119878a solution to

the optimization problem below will give the desired result

Max 119890lowast

119878=119873

sum119898=1

119899119901119862 (119899 minus 1) 119890119878 (119899)

st119873

sum119899=1

119901119862 (119899 minus 1) 119890119878 (119899) = 0

1003816100381610038161003816119890119878 (119899)1003816100381610038161003816 le 120575119878 119899 isin 119868

+

(18)

In order to show the optimal solution first put this problemin primal and dual forms

primal

Max119873

sum119899=1

119899119901119862 (119899 minus 1) 119890119878 (119899)

st119873

sum119899=1

119901119862 (119899 minus 1) 119890119878 (119899) = 0 119906

minus 119890119878 (119899) le 120575119878 119899 isin 119868

+ 119910 (119899)

119890119878 (119899) le 120575119878 119899 isin 119868

+ 119908 (119899)

(19)

dual

Min 120575119873

sum119899=1

(119910 (119899) + 119908 (119899))

st 119901119862 (119899 minus 1) 119906 minus 119910 (119899) + 119908 (119899)

= 119899119901119862 (119899 minus 1) 119899 isin 119868

+

119910 (119899) ge 0 119899 isin 119868+

119908 (119899) ge 0 119899 isin 119868+

(20)

Journal of Optimization 5

The optimal solution to the problem will have to satisfytheKarush-Kuhn-Tucker (KKT) conditions that is feasibilityof the Primal and Dual as well as complementary slackness[27] The KKT conditions give the necessary conditions foroptimality of the general constrained problem

Checking dual feasibility for any u the constraints can besatisfied by construction such that if (119899minus119906) gt 0 then119910(119899) = 0119908(119899) = (119899 minus 119906)119901119862(119899 minus 1) and if (119899 minus 119906) lt 0 then 119910(119899) =minus(119899 minus 119906)119901119862(119899 minus 1) 119908(119899) = 0 This is because the main dualconstraint is

(119899 minus 119906) 119901119862 (119899 minus 1) = 119908 (119899) minus 119910 (119899) (21)

In order to satisfy complementary slackness if 119890119878(119899) gt

minus120575119878 then it must be true that 119910(119899) = 0 and if 119890

119878lt 120575119878 then it

must be true that119908(119899) = 0 In terms of the above primal-dualconstruction if 119890

119878(119899) gt minus120575

119878 then (119899minus119906) gt 0 and if 119890

119878(119899) lt 120575

119878

then (119899minus119906) lt 0 Any solution 119890119878(119899) 119899 isin 119868+ andu satisfying the

primal constraints and the above two conditions is optimalConsider the solution

119890119878 (119899) = minus120575119878 if 119899 lt 119899

119862

119890119878 (119899) = 120575119878 if 119899 gt 119899

119862

119906 = 119899119862

(22)

where 119899119862is the median state at completions

Assume for simplicity that there is an even number ofstates so that 119899

119862= 119899 for any n This solution is dual feasible

since we showed above that any n is a solution to the dualChecking primal feasibility the solution satisfies minus119890119878(119899) le

120575119878 and 119890119878(119899) le 120575119878The main constraint is

119873

sum119899=1

119901119862 (119899 minus 1) 119890119878 (119899)

= minus120575119878 sum119899lt119899119862

119901119862 (119899 minus 1) + 120575119878 sum119899gt119899119862

119901119862 (119899 minus 1)

= 120575119878[

[

minus sum119899lt119899119862

119901119862 (119899 minus 1) + sum

119899gt119899119862

119901119862 (119899 minus 1)

]

]

= 120575119878 [0]

(23)

since 119899119862is median

Lastly we check for complementary slackness Now119890119878(119899) gt minus120575

119878when 119899 gt 119899

119862 then 119899 minus 119910 gt 0 and 119890

119878(119899) lt 120575

119878

when 119899 lt 119899119862 then 119899 minus 119910 lt 0 Therefore complementary

slackness holds and the solution is an optimal oneThe solution value is

119873

sum119899=1

119899119901119862 (119899 minus 1) 119890119878 (119899)

= sum119899lt119899119862

119899119901119862 (119899 minus 1) (minus120575119878) + sum

119899gt119899119862

119899119901119862 (119899 minus 1) (120575119878)

(24)

Set this value equal to Δ119878 which is the overall completerrsquos

average minus the average of the set truncated at the medianThis can be shown by first taking

Δ 119878 = sum119899lt119899119862

119899119901119862 (119899 minus 1) (minus120575119878) + sum119899gt119899119862

119899119901119862 (119899 minus 1) (120575119878) (25)

120575119878times the completerrsquos average 119899

119862 is

120575119878119899119862= sum119899lt119899119862

119899119901119862 (119899 minus 1) (120575119878) + sum

119899gt119899119862

119899119901119862 (119899 minus 1) (120575119878) (26)

Subtracting Δ119878yields

120575119878119899119862 minus Δ 119878 = 2 sum119899lt119899119862

119899119901119862 (119899 minus 1) (120575119878)

Δ119878 = 120575119878

[

[

119899119862 minus 2 sum119899lt119899119862

119899119901119862 (119899 minus 1)]

]

(27)

We know that

sum119899lt119899119862

119901119862 (119899 minus 1) =

1

2 (28)

since 119899119862is a median Therefore

sum119899lt119899119862

2119901119862 (119899 minus 1) = 1

sum119899lt119899119862

1198992119901119862 (119899 minus 1) = 119899119862119879

(29)

is the average of the set of states truncated at the median Sothe bound on 119890lowast

119878is

Δ119878= 120575119878 [119899119862 minus 119899119862119879] (30)

As an example take the behavior sequence in Figure 1 Ifwe want to use the OA equation [24]

119899119860119878=

119880

1 minus 119880 minus 119901 (119873)(1 minus (119873 + 1) 119901 (119873)) (31)

to calculate 119899119860119878 which is the average number assuming flowbalance homogeneous arrival and services we would beinterested in the bound of the difference between 119899 and 119899119860119878We can calculate

119880 =119879 minus 119879 (0)

119879=5

6(32)

p(n)=13 and 119899119862

= 74 Assume the maximum error is120575119878= 35 If the other assumption errors are zero then the

difference 119899 minus 119899119860119878 is equal to the correction term

119862119899=

119880

1 minus 119880 minus 119901 (119873)(1 minus 119901 (119873)) 119890

lowast

119878 (33)

The upper bound on this correction term using (14) is

119880

1 minus 119880 minus 119901 (119873)(1 minus 119901 (119873)) 120575119878119899119862 = minus35 (34)

6 Journal of Optimization

n

321

3

2

4 5 6

Time

1

Figure 1 Example Behavior Sequence for estimating Homoge-neous service error bound

Using the tighter bound Δ119878 we get

119880

1 minus 119880 minus 119901 (119873)(1 minus 119901 (119873)) Δ 119878

= minus10

3(120575119878 (119899119862 minus 119899119862119879))

= minus10

3(3

5(7

4minus 1)) = minus15

(35)

This is a reduction of 5714 for this example of the differencebound

32 Bounds onHomogeneous Arrival Assumption Error As inthe previous section we can assume that a maximum error120575119860 for any state is known beforehand That is we assume

1003816100381610038161003816119890119860 (119899)1003816100381610038161003816 le 120575119860 119899 isin 119868 (36)

Then the weak overall homogeneous arrival error is boundedby

119890lowast

119860le119873minus1

sum119899=0

119899119879 (119899)

119879 minus 119879 (119873)

1003816100381610038161003816119890119860 (119899)1003816100381610038161003816 le 120575119860

119873minus1

sum119899=0

119899119879 (119899)

119879 minus 119879 (119873)le 120575119860119899119873

(37)

where 119899119873is the average number at the device excluding the

maximum stateAs with the service errors 119890

119878(119899) the 119890

119860(119899) are not

independent The dependency is

119873minus1

sum119899=0

119879 (119899)

119879 minus 119879 (119873)119890119860 (119899) = 0 (38)

This can be shown by substituting the definition of 119890119860(119899) into

the equation As with the homogeneous service assumptionswe set up the following optimization problem

Max 119890lowast

119860=119873minus1

sum119899=0

119899119879 (119899)

119879 minus 119879 (119873)119890119860 (119899)

st119873minus1

sum119899=0

119879 (119899)

119879 minus 119879 (119873)119890119860 (119899) = 0

1003816100381610038161003816119890119860 (119899)1003816100381610038161003816 le 120575119860 119899 isin 119868

(39)

Using the Karush-Kuhn-Tucker conditions as before we canshow that with 119899 as the median state

119890119860 (119899) = minus120575119860 if 119899 lt 119899

119890119860 (119899) = 120575119860 if 119899 gt 119899

119906 = 119899

(40)

is the solution to the optimization problem The value of thesolution is found by substituting into the primal objectivefunction to get

Δ119860= max 119890lowast

119860

= sum119899lt119899

119899119879 (119899)

119879 minus 119879 (119873)(minus120575119860) +119873minus1

sum119899gt119899

119899119879 (119899)

119879 minus 119879 (119873)(120575119860)

(41)

120575119860times the average excluding the maximum 119899

119873 is

120575119860119899119873= sum119899lt119899

119899119879 (119899)

119879 minus 119879 (119873)(120575119860) +119873minus1

sum119899gt119899

119899119879 (119899)

119879 minus 119879 (119873)(120575119860) (42)

Subtracting

120575119860119899119873 minus Δ119860 = 2 sum119899lt119899

119899119879 (119899)

119879 minus 119879 (119873)120575119860 (43)

therefore

Δ119860= 120575119860[119899119873minus 2 sum119899lt119899

119899119879 (119899)

119879 minus 119879 (119873)]

= 120575119860[119899119873minus 2 sum119899lt119899

119899119901 (119899)

1 minus 119901 (119873)]

(44)

The expression

2 sum119899lt119899

119899119901 (119899) (45)

is the average of the values truncated at the median Call thisvalue 119899

119879 Then substitution yields an upper bound on the

error measure due to violations in homogeneous arrivals of

Δ119860= 120575119860 [119899119873 minus

119899119879

1 minus 119901 (119873)] (46)

This bound may not be as useful as the bound on thehomogeneous service assumption error because the value ofΔ119860is based on knowing the119879(119899) values whereas forΔ

119878only

completion counts are necessary

Journal of Optimization 7

4 Example Performance MeasureError Bounds

Some examples are given next of howΔ119860andΔ

119878may be used

to determine bounds on the difference between exact valuesof various PMs and the OA estimated values for particularbehavior sequences

41 Arriverrsquos Average Queue Length Using the example fromthe introduction for a behavior sequence the average queuelength seen by an arriving job may be calculated by

119899119860= 119899119878

119860+ 119862119899119878

119860

(47)

where 119862119899119878

119860

= 119890119865minus119890lowast119878and 119899119878119860= (119899119880)minus1 is the average arriverrsquos

queue length assuming homogeneous servicesRearranging and using the Δ119860 bound give

119899119860minus 119899119878

119860= 119890119865minus 119890lowast

119878

119899119878

119860minus 119899119860 = 119890

lowast

119878minus 119890119865 le Δ 119878 minus 119890119865

(48)

Since 119890119865rarr 0 for any sequence of data in steady state we

assume flow balance holds Then

119899119878

119860minus 119899119860le Δ119878 (49)

This expression shows us that the difference between estimat-ing the average length seen by arrivers with the relationshipthat assumes homogeneous servers and the true value of 119899119860is less than or equal to the bound on 119890lowast

119878

42 Response Time The exact response time for a behaviorsequence can be found by

119877 = 119877119878+ 119862119877119878 (50)

where 119862119877119878 = minus119878119862

119899119878

119860

= minus119878(119890119865minus 119890lowast119878) and 119877119878 = 119878(119899

119860+ 1)

Since as before we are interested in the differencebetween a possible observed value (119877) and a calculated value(119877119878) we should do some rearranging and get 119877 minus 119877119878 = 119878(119890lowast

119878minus

119890119865) Again assuming 119890

119865is small and substituting 119890lowast

119878le Δ119878

yield

119877 minus 119877119878le 119878Δ 119878

119877 minus 119877119878le 119878 (120575

119878(119899119862minus 119899119862119879))

(51)

43 Average Number at Device If homogeneous arrivalshomogeneous services and job flow balance hold then theaverage number at a device (ie those both in queue and inservice) can be calculated by

119899119860119878=

119880

1 minus 119880 minus 119901 (119873)(1 minus (119873 + 1) 119901 (119873)) (52)

The exact average number if these assumptions do not hold is

119899 = 119899119860119878+ 119862119899119860119878 (53)

where

119862119899119860119878 =

119880

1 minus 119880 minus 119901 (119873)(1 minus 119901 (119873)) (119890

lowast

119860+ 119890lowast

119878minus 119890119865) (54)

Rearranging again

119899 minus 119899119860119878= 119862119899119860119878

=119880

1 minus 119880 minus 119901 (119873)(1 minus 119901 (119873)) (119890

lowast

119860+ 119890lowast

119878minus 119890119865)

(55)

and substituting 119890lowast119860le Δ119860 119890lowast119878le Δ119878 and 119890

119865= 0 we get

119899 minus 119899119860119878le

119880

1 minus 119880 minus 119901 (119873)(1 minus 119901 (119873)) (Δ119860 + Δ 119878) (56)

if 1 gt 119880 + 119901(119873) or

119899 minus 119899119860119878ge

119880

1 minus 119880 minus 119901 (119873)(1 minus 119901 (119873)) (Δ119860 + Δ 119878) (57)

if 1 lt 119880 + 119901(119873)Because the calculation ofΔ119860 requires knowledge of T(n)

values there may be little benefit in using the right hand sideof the above relationship to find this error bound If howeverthe behavior sequence can be assumed to have homogeneousarrivals then we can get a bound on the average numberat a device error without the knowledge of the individualT(n) values In that case the only time statistics we need areutilizationU and the fraction of time spent at the maximumstate p(N)

44 Throughput Using the OA version of Littlersquos Law [1] wecan say that the difference between a behavior sequencersquosactual throughput and that calculated assuming both homo-geneous arrivals and services will be

119883 minus 119883119860119878=119899

119877minus119899119860119878

119877119860119878 (58)

For 119877119860119878 we can substitute 119877119878 which is the more generalexpression since it does not need the homogeneous arrivalassumption From previous developments we know the fol-lowing relations hold

119899 = 119899119860119878+ 119862119899119860119878 (59)

where

119862119899119860119878 le

10038161003816100381610038161003816100381610038161003816

119880

1 minus 119880 minus 119901 (119873)(1 minus 119901 (119873)) (Δ119860 + Δ 119878)

10038161003816100381610038161003816100381610038161003816(60)

and 119877 = 119877119878 + 119862119877119878 where

119862119877119878 le 119878Δ

119878 (61)

Substituting into (58) gives

119883 minus 119883119860119878=119899119860119878 + 119862

119899119860119878

119877119878 + 119862119877119878minus119899119860119878

119877119878=119877119878119862119899119860119878 minus 119899119860119878119862119877119878

119877119878 (119877119878 + 119862119877119878)

(62)

8 Journal of Optimization

If the bound of (60) is substituted into (62) then it must bethat

1198771198781003816100381610038161003816(119880 (1 minus 119880 minus 119901 (119873))) (1 minus 119901 (119873)) (Δ119860 + Δ 119878)

1003816100381610038161003816 minus 119899119860119878119862119877119878

119877119878 (119877119878 + 119862119877119878)

ge119877119878119862119899119860119878 minus 119899119860119878119862119877119878

119877119878 (119877119878 + 119862119877119878)

(63)

If instead of bound (60) bound (61) is substituted thenumerator can only decrease and the denominator onlyincrease therefore

119877119878119862119899119860119878 minus 119899119860119878119878Δ

119878

119877119878 (119877119878 + 119878Δ119878)

le119877119878119862119899119860119878 minus 119899119860119878119862

119877119878

119877119878 (119877119878 + 119862119877119878)

(64)

If both (60) and (61) are substituted then the value of theexpression must fall within the limits defined by (63) and(64) Therefore the bounds on our throughput error are

119877119878119862119899119860119878 minus 119899119860119878119878Δ

119878

119877119878 (119877119878 + 119878Δ119878)

le 119883 minus 119883119860119878

le119877119878

1003816100381610038161003816(119880 (1 minus 119880 minus 119901 (119873))) (1 minus 119901 (119873)) (Δ119860 + Δ 119878)1003816100381610038161003816

119877119878 (119877119878 + 119862119877119878)

minus119899119860119878119862119877119878

119877119878 (119877119878 + 119862119877119878)

(65)

5 Using the PM Bounds

The bounds derived in the previous sections are actuallylimits on PM correction terms Assuming we know the 120575

119860

and 120575119878values a priori and that we can say something about

the homogeneous arrival assumption then these bounds canbe found without knowledge of all the 119879(119899) values for each nThis is the same simplification of data collection that we havein using the OA formulas instead of direct calculations

As an example of using the bounds in a simulation studyassume we are interested in finding an estimate for averagenumber at a device A series of 10 runs is made and the boundfor the correction term 119862119899 is calculated If it is positive wecan call this an upper bound If it is negative let this be alower bound In both cases the other bound is 0 Assumein the simulation runs these bounds always fall between minus6and 33 The correction term for each run is approximatedby taking the average value between the upper and lowerbounds We would like to be able to say something about theprobability that future runs will fall within the [minus6 33] limitsthat have already appeared We can do this using tolerancelimit calculations

Assume the 10 runs produced the results given in Table 2The average and standard deviation for these observations areminus134 and 155 respectively

Table 2 Example estimated correction terms for average number ata device

Run 119862119899

1 minus1752 minus3003 minus1254 1655 minus2506 minus1907 minus1758 1159 minus26010 minus145

Since the tolerance limits are going to be set at theobserved limits we can say

[minus6 33] = [119862119899 minus 119870119904 119862119899 + 119870119904]

119870 =119862119899minus (minus6)

119904=(33) minus 119862119899

119904= 30

(66)

From tolerance limit tables [28] we can say with 95confidence that at least 91 of future observations of 119862

119899will

fall within the interval [minus6 33] That is if we use 119899119860119878 valueswe have 95 confidence that the correct 119899 values for each runwill fall within these limits 91 of the time

6 Conclusion

In this paper bounds were developed for the operationalanalysis error measures of homogeneous service and arrivalassumptions These bounds allow us to take advantage of thesimplified data collection made possible by the use of opera-tional analysis relationships evenwhen the assumptions usedto derive those relationships are violated A tolerance limitbased method was given in order to be able to say somethingabout the confidence that future correction term values in atime series would be within certain limits

Abbreviations

119862 Total number of job completions at adevice

119862(119899) Total number of completions when n jobsare at a device

119862119909 Correction term makes the OA term 119909

equal to the directly calculatedperformance measure

119868 The set of states 0 1 2 119873 minus 1

119868+ The set of states 1 2 119873119870 Tolerance limit multiplier119873 The maximum number of jobs seen at a

device both in queue and in service

Journal of Optimization 9

119899 State of a device number of jobs both inqueue and in service

119899 The median state seen by a device119899119862 The median state at a device at job

completions119899 Average number of jobs at a device119899119860119878 The average number of jobs at a device

assuming flow balance homogeneousarrivals and homogeneous services

119899119862 The average number at the device seen bya completing job

119899119862119879 The average of the set of states seen by

completing job truncated at the medianstate value

119899119873 The average state at a device excluding the

maximum state119899119879 Time-average of the set of states truncated

at the median119899119878119860 Average number of jobs at a device seen by

an arriver assuming homogeneous service119901(119899) Proportion of time spent in state 119899119901119860(119899) The proportion of arrivals when 119899 jobs are

at a device 119860(119899)119860119901119862(119899) The proportion of completions that leave 119899

jobs at a device 119862(119899 + 1)119862119877 Average response time119877119878 Average response time assuming

homogeneous services119877119860119878 Average response time assuming

homogeneous arrivals and homogeneousservices

119878 Mean time between completions duringbusy periods

119904 Standard deviation119879 Total time of the period of observation119879(119899) Total time device was in state 119899119880 Utilization 1minusp(0)119883 Throughput119883119860119878 Throughput assuming homogeneous

arrivals and homogeneous services120575119860 Maximum error in homogeneous arrival

assumption error measure120575119878 Maximum error in homogeneous service

assumption error measureΔ119860 A bound on the homogeneous arrival

assumption error measure le 120575119860

Δ119878 A bound on the homogeneous service

assumption error measure le 120575119878

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] J P Buzen ldquoFundamental operational laws of computer systemperformancerdquo Acta Informatica vol 7 no 2 pp 167ndash182 1976

[2] J P Buzen ldquoOperational analysis an alternative to stochasticmodelingrdquo in Performance of Computer Installations pp 175ndash194 North-Holland Amsterdam The Netherlands 1978

[3] J P Buzen and P J Denning ldquoOperational treatment of queuedistributions andmean value analysisrdquo Tech Rep CSD-TR 309Department of Computer Science Purdue University 1979

[4] J P Buzen and P J Denning ldquoMeasuring and calculating queuelength distributionsrdquo Computer vol 13 no 4 pp 33ndash44 1980

[5] P J Denning and J P Buzen ldquoThe operational analysis ofqueueing network modelsrdquo Computer Surveys vol 10 no 3 pp225ndash261 1978

[6] J P Buzen ldquoFrom stochastic modeling to operational analysisthe journey beginsrdquo in Fundamental Concepts in ComputerScience E Gelenbe and J Kahane Eds pp 141ndash149 ImperialCollege Press London UK 2009

[7] E D Lazowska J Zahorjan G S Graham and K C SevcikQuantitative System Performance Prentice-Hall EnglewoodCliffs NJ USA 1984

[8] R Suri ldquoRobustness of queueing network formulasrdquo Journal ofthe ACM vol 30 no 3 pp 564ndash594 1983

[9] E D Lazowska J Zahorjan and E C Sevcik ldquoComputer sys-tem performance evaluation using queueing network modelsrdquoAnnual Reviews in Computer Science vol 1 pp 107ndash137 1986

[10] B V Ivanovskii ldquoOperational analysis of communication net-works with lockupsrdquoAutomatic Control and Computer Sciencesvol 22 no 3 pp 26ndash32 1988

[11] P J Denning ldquoQueueing in networks of computersrdquo AmericanScientist vol 79 pp 206ndash209 1991

[12] P J Denning ldquoIn the queue mean valuesrdquo American Scientistvol 79 pp 402ndash403 1991

[13] Y Dallery and X-R Cao ldquoOperational analysis of stochasticclosed queueing networksrdquo Performance Evaluation vol 14 no1 pp 43ndash61 1992

[14] M El-Taha and S Stidham Jr ldquoDeterministic analysis of queue-ing systems with heterogeneous serversrdquo Theoretical ComputerScience vol 106 no 2 pp 243ndash264 1992

[15] J P Buzen ldquoAn overview of performance prediction in MVSsystems and SNAnetworksrdquo inProceedings of the ACMFall JointComputer Conference pp 751ndash759 Dallas Tex USA 1986

[16] D A Menasce and V A Almeida Scaling for E-Business Tech-nologies Models Performance and Capacity Planning Prentice-Hall Englewood Cliffs NJ USA 2000

[17] D K Buch and V M Pentkovski ldquoPerformance characteriza-tion experience of multi-tier E-business systems using queuingoperational analysisrdquo in Proceedings of the IEEE InternationalSymposium on Performance Analysis of Systems and SoftwareTucson Ariz USA 2001

[18] M P Fanti B Maione G Piscitelli and B Turchiano ldquoTwomethod for real-time routing selection in flexible manufactur-ing systemsrdquo inProceedings of the IEEE International Conferenceon Robotics and Automation pp 1158ndash1166 May 1992

[19] W M Zuberek ldquoThroughput analysis in timed Petri netsrdquo inProceedings of the 35th Midwest Symposium on Circuits andSystems pp 1576ndash1580 Washington DC USA 1992

[20] W M Zuberek ldquoThroughput analysis of simple closed timedPetri netmodelsrdquo in Proceedings of the 36thMidwest Symposiumon Circuits and Systems pp 930ndash933 Detroit Mich USAAugust 1993

[21] G Chiola C Anglano J Campos J M Colom and M SilvaldquoOperational analysis of timed Petri nets and application to the

10 Journal of Optimization

computation of performance boundsrdquo in Quantitative Methodsin Parallel Systems F Baccelli A Jean-Marie and I MitraniEds Esprit Basic Research Series pp 161ndash174 Springer BerlinGermany 1995

[22] J A Brumfield Operational Analysis of Queuing Phenomena[PhD thesis] Department of Computer Science Purdue Uni-versity 1982

[23] N M Bengtson ldquoUsing operational analysis in simulation aqueuing network examplerdquo Journal of the Operational ResearchSociety vol 39 no 12 pp 1125ndash1136 1988

[24] N M Bengtson ldquoMeasuring errors in operational analysisassumptionsrdquo IEEE Transactions on Software Engineering vol13 no 7 pp 767ndash776 1987

[25] N M Bengtson ldquoOperational analysis revisited error measurelimits of assumptionsrdquo GSTF International Journal on Comput-ing vol 3 no 2 2013

[26] K C Sevcik and M Klawe ldquoOperational analysis versusstochastic modeling of computer systemsrdquo in Proceedings of theComputer Science and Statistics 12th Annual Symposium on theInterface pp 177ndash184 Waterloo Canada 1979

[27] F S Hillier and G J Lieberman Introduction to OperationsResearch McGraw-Hill New York NY USA 8th edition 2005

[28] C Eisenhart M W Hastay and W A Wallis Eds SelectedTechniques of Statistical Analysis McGraw-Hill New York NYUSA 1947

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Journal of Optimization

service That is the service time of a device in a network isindependent of the queue length at any deviceHomogeneousarrivals are the corresponding condition for the arrival timesHomogeneous routing holds when the routing frequencies ofjobs leaving a device are independent of the queue lengthsat other devices in the network Device homogeneity existswhen the rate of output from a device is determined only byits queue length Other assumptions can be invoked as theneed arises to derive OA relations [22]The only requirementis that these assumptions meet the two basic principles oftestability given above

OA assumptions allow for the development of relation-ships which enable us to determine PMs by collecting onlya few types of data namely the number of arrivals anddepartures for each device state and the total time spentin each state [23] These PMs will be accurate only if theOA assumptions are met and only for the finite time periodobserved The accuracy of the assumptions can be measuredA device state is the number of items (customers jobsentities etc) both waiting and in service at a device

While OA research was originally proposed as an aid incomputer performance analysis it is more general in thatdevelopments can be applicable to any system that generatestime series data This would include computer simulation

The Abbreviations section gives definitions of variableused in this paper Errormeasures of variousOA assumptionshave been defined and are summarized in Table 1 for job flowbalance homogeneous service and homogeneous arrival[24] The limit over time of the expected value and varianceof the job flow balance error is zero [25] so that over timethis error is not significant for data runs of reasonable lengthThe expected value over time of other error measures suchas for the assumptions of homogeneous service and arrivalmay not in general tend to zero [25]

Different error measures in the form of relative errorshave been defined by Brumfield [22] By presenting a set ofnew assumptions formulas for the calculation of responsetime and average queue length in terms of the averageand coefficient of variation of service times are developedTwo examples of these new assumptions are homogeneityof queueing and service and homogeneity of residuals Forarrival 119895 ldquoforward residual is either the time remaining inthe service period during which 119895 arrives or zero if arrival119895 begins a service period similarly backward residual iseither the time since the beginning of the service periodduring which 119895 arrives or zero if arrival 119895 begins a serviceperiodrdquo [22] Relative error formulas for response time aredetermined with these new assumptions in addition to theold assumptions of homogeneous service and homogeneousarrivals Unfortunately the error terms are quite complexsince there are more assumptions with which to deal

If we are using a relationship to determine a PM derivedunder OA assumptions then the resulting value of the PM isin error if the founding assumptions do not holdThis can bechecked from the data because of the way OA assumptionsare defined The degree of error in the PM calculated is afunction of the assumption error measures Correction termshave been developed using the assumption error measures[24]When added to the PMs these correction terms produce

exact results It is these correction terms which are studied inthis paper

As an example assume we are interested in obtaining avalue for the average number of jobs in a computer systemthat a new job sees upon arrival If wemake the homogeneousservice assumption then this averagemay be estimated by [4]

119899119878

119860=119899

119880minus 1 (1)

where 119899119878119860is average number of jobs at a device seen by an

arriver assuming homogeneous service 119899 is average numberof jobs at the device and 119880 is device utilization

We are interested in finding a correction term such that

119899119860= 119899119878

119860+ 119862119899119860 (2)

The correction term is equal to

119862119899119860= 119890119865minus 119890lowast

119878 (3)

where

119890119865=119873minus1

sum119899=0

(119899 + 1) (119901119860 (119899) minus 119901119862 (119899))

119890lowast

119878=119873

sum119899=1

119899119901119862 (119899 minus 1) 119890119878 (119899)

(4)

Check the Abbreviations section for all symbol definitions119890119865 is the error measure for the job flow balance assumptionand 119890lowast

119878is a weak form of the homogeneous service error

measure because it may be equal to 0 even when some or allthe individual 119890

119878(119899) values are not

There are a couple of problems with using the errormeasures to derive correction terms One problem is thatthe amount of data needed to calculate an exact value foran assumption error measure is the same as to find theperformance measure of interest directly The process ofdetermining the error measure for each assumption used toderive a relation the correction term and the PM estimateto which the correction term is added is a way of gettingsomething which may be observed more directly Findingexact values for performance measures in this indirect wayover a finite time period may be worthwhile only if a numberof PMs are desired In this case a single assumption errormeasure is determined for each assumption and applied toall the PM correction terms of interest

Another problemwith the errormeasure technique is thatthese measures apply only to the data observed For anotherrun of data new error measure values need to be foundThis limitationmay be acceptable if PMs cannot be measureddirectly without changing the nature of the system forexample in a complex computer systemWewould like a wayto extend assumption error measures over longer sets of dataand thereby say something about the system that generatedthe data As Sevcik and Klawe [26] stated shortly after OAwas introduced ldquoBecause operational analysis is based onassumptions that can be tested but that are very unlikely to besatisfied exactly in any finite time period it is very important

Journal of Optimization 3

Table 1 Operational analysis error measures

Assumption State dependent error Overall errorJob-flow balance 119890

119865 (119899) = (119899 + 1) (119901119860 (119899) minus 119901119862 (119899)) 119899 isin 119868 119890119865= sum119873minus1

119899=0119890119865(119899)

Homogeneous service 119890119878(119899) =

1

119901119862(119899 minus 1)

(119879(119899)

119879 minus 119879(0)) minus 1 119899 isin 119868+ 119890lowast

119878= sum119873

119899=1119899119901119888 (119899 minus 1) 119890119878 (119899)

Homogeneous arrival 119890119860(119899) = 119901

119860(119899) (

119879 minus 119879(119873)

119879(119899)) minus 1 119899 isin 119868 119890lowast

119860= sum119873minus1

119899=1119890119860 (119899)

119899119879 (119899)

119879 minus 119879 (119873)

to develop a means of dealing with lsquofuzzy homogeneityrsquo orsituations in which the various independence assumptionsare satisfied within some tolerancerdquoThis paper addresses thisneed to define these assumption bounds

The next section will illustrate how OA relations may beused to reduce data collection while estimating performancemeasures This will be followed by a discussion of thedetermination of bounds on the OA assumption measure-ment errors for homogeneous service and homogeneousarrival Sample calculations of these bounds will be presentedafterward An illustration of the use of bounds in a simulationwill then be given

2 Simplifying Data Collection

Calculating performance measures with OA relationshipsthat are derived under one or more of the system behav-ior assumptions is usually simpler than using more directrelationships This is because by making the assumptions amodel has been created which reduces perhaps artificiallythe complexity of the behavior of the systemThe result is thatless information is needed tomake an estimate of the PM thanwould be needed for a direct measurement

There are situations where it is impractical if not impos-sible to collect sufficient data to determine exact values forPMs over a finite period of time In some cases only anestimate of a PM is needed and it is not worthwhile to go tothe trouble of determining the precise PM values Any PMvalue obtained for a behavior sequence is only an estimate ofthe underlying system PM With this realization in mind itmay seem unwise to spend a great deal of effort to obtain anexact value for a sequence which is in turn only an estimateof some other value A good approximation of the sequenceestimate may be sufficient

If we want the average response time R of a behaviorsequence we could accumulate the response times of allthe jobs that go through the system and get the exact R bydividing by the number of jobs A simpler procedure wouldbe to say that response time is

119877 =119878

1 minus 119880 (5)

where 119878 is mean time between completions during busyperiods and U is utilization

This equation will give the exact R value if we have abehavior sequence for a single server queue which is in flowbalance and has homogeneous arrivals and services If these

conditions do not hold the equation will not give R exactlybut an estimate of R call it 1198771015840 If we collect only the idle time119879(0) and the number of completions C we can use the sameequation to find 1198771015840 If the behavior sequence lasts for time 119879then

119880 = 1 minus119879 (0)

119879

119878 =119879 minus 119879 (0)

119862

(6)

Another example calculates the average number of jobsat a device With the same assumptions as for estimatingresponse time the average number of jobs in the queueserversystem is

119899 =119880

1 minus 119880 (7)

This value takes even less data to calculate than does 1198771015840 Thedirect calculation of 119899 requires accumulating data every timethere is an arrival or completion or requires keeping track ofthe total amount of time spent at each of the states

Using these equations for predicting future values for Rand 119899 of a system presents certain problems For exampleover future time will the assumptions of the system behave inthe sameway Sincewe candetermine anduse errormeasuresof the assumptions in order to correct assumption derivedPM estimate it is not necessary that assumptions hold in thefuture if they have not in the past With the determinationof correction terms all that is really necessary is for thecorrection terms to remain relatively constant that is for thesystemrsquos violations of assumptions to remain the same overfuture time periods

Without knowledge of the assumption error measuresand through them the correction terms the performancemeasure estimates may be quite bad for any particularbehavior sequence [23] As stated before in a stable systemthe job flow balance assumption errormeasure will go to zeroas time increases but as shown in [24] this is not necessarilytrue for other assumption error measures For any behaviorsequence it is important tomake some assessment if possibleof the behavior of the PM correction terms

3 Performance Measure Bounds

One approach to use the simplified OA formulas for PMs isto determine bounds on the maximum PM error That is we

4 Journal of Optimization

are interested in defining bounds on the difference betweentrue values of various PMs at a device for particular statesequences and those PMs estimated by using relationshipsderived under operational analysis assumptions We willassume the network is in steady state

In the following bounds are found for the assumptionsof homogeneous services and homogeneous arrivals In thecase of the job-flow balance assumption we know that theexpected value of the errormeasure and its variance go to zero[25]

lim119905rarrinfin

119864 [119890119865] = 0 lim

119905rarrinfin119881 [119890119865] = 0 (8)

Therefore as the length of the sequence increases the 119890119865 can

be expected to become insignificantWewill need to assume that amaximumvalue of the error

for services and arrivals for any state is known or can be setCall these values 120575119878 and 120575119860 for the maximum service errorand maximum arrival error respectively

31 Bounds on Homogeneous Service Assumption Error If120575119878is the maximum error for the homogeneous service

assumption then1003816100381610038161003816119890119878 (119899)

1003816100381610038161003816 le 120575119878 119899 isin 119868+ (9)

A more useful bound would be on the weak overall homoge-neous service error

119890lowast

119878le 120575lowast

119878 (10)

But this limit may be harder to know beforehand Equation(9) may be used to find an upper bound on 119890lowast

119878by using the

definition

119890lowast

119878=119873

sum119899=1

119899119901119862 (119899 minus 1) 119890119878 (119899) (11)

Substituting (9) yields

119890lowast

119878=119873

sum119899=1

119899119901119862 (119899 minus 1)

1003816100381610038161003816119890119878 (119899)1003816100381610038161003816 le 120575119878

119873

sum119899=1

119899119901119862 (119899 minus 1) (12)

The term119873

sum119898=1

119899119901119862 (119899 minus 1) = 119899119862 (13)

is the average number at the device seen by a completerTherefore

119890lowast

119878le 120575119878119899119862 (14)

The bound given by (14) does not take into considerationthe fact that the 119890

119878(119899) values are not independent In fact they

are related by the expression

119873

sum119899=1

119901119862 (119899 minus 1) 119890119878 (119899) = 0 (15)

We can get a tighter bound of the 119890lowast119878values by taking this

dependence into consideration Equation (15) can be shownby substituting the definition

119890119878 (119899) =

119878 (119899)

119878=

1

119901119862 (119899 minus 1)

[119879 (119899)

119879 minus 119879 (0)] minus 1 (16)

as follows119873

sum119899=1

119901119862 (119899 minus 1) 119890119878 (119899)

=119873

sum119899=1

119901119862 (119899 minus 1) [1

119901119862 (119899 minus 1)

119879 (119899)

119879 minus 119879 (0)minus 1]

=119873

sum119899=1

119879 (119899)

119879 minus 119879 (0)minus119873

sum119899=1

119901119862 (119899 minus 1)

=119879 minus 119879 (0)

119879 minus 119879 (0)minus 1 = 0

(17)

Sincewhat is desired is an upper bound on 119890lowast119878a solution to

the optimization problem below will give the desired result

Max 119890lowast

119878=119873

sum119898=1

119899119901119862 (119899 minus 1) 119890119878 (119899)

st119873

sum119899=1

119901119862 (119899 minus 1) 119890119878 (119899) = 0

1003816100381610038161003816119890119878 (119899)1003816100381610038161003816 le 120575119878 119899 isin 119868

+

(18)

In order to show the optimal solution first put this problemin primal and dual forms

primal

Max119873

sum119899=1

119899119901119862 (119899 minus 1) 119890119878 (119899)

st119873

sum119899=1

119901119862 (119899 minus 1) 119890119878 (119899) = 0 119906

minus 119890119878 (119899) le 120575119878 119899 isin 119868

+ 119910 (119899)

119890119878 (119899) le 120575119878 119899 isin 119868

+ 119908 (119899)

(19)

dual

Min 120575119873

sum119899=1

(119910 (119899) + 119908 (119899))

st 119901119862 (119899 minus 1) 119906 minus 119910 (119899) + 119908 (119899)

= 119899119901119862 (119899 minus 1) 119899 isin 119868

+

119910 (119899) ge 0 119899 isin 119868+

119908 (119899) ge 0 119899 isin 119868+

(20)

Journal of Optimization 5

The optimal solution to the problem will have to satisfytheKarush-Kuhn-Tucker (KKT) conditions that is feasibilityof the Primal and Dual as well as complementary slackness[27] The KKT conditions give the necessary conditions foroptimality of the general constrained problem

Checking dual feasibility for any u the constraints can besatisfied by construction such that if (119899minus119906) gt 0 then119910(119899) = 0119908(119899) = (119899 minus 119906)119901119862(119899 minus 1) and if (119899 minus 119906) lt 0 then 119910(119899) =minus(119899 minus 119906)119901119862(119899 minus 1) 119908(119899) = 0 This is because the main dualconstraint is

(119899 minus 119906) 119901119862 (119899 minus 1) = 119908 (119899) minus 119910 (119899) (21)

In order to satisfy complementary slackness if 119890119878(119899) gt

minus120575119878 then it must be true that 119910(119899) = 0 and if 119890

119878lt 120575119878 then it

must be true that119908(119899) = 0 In terms of the above primal-dualconstruction if 119890

119878(119899) gt minus120575

119878 then (119899minus119906) gt 0 and if 119890

119878(119899) lt 120575

119878

then (119899minus119906) lt 0 Any solution 119890119878(119899) 119899 isin 119868+ andu satisfying the

primal constraints and the above two conditions is optimalConsider the solution

119890119878 (119899) = minus120575119878 if 119899 lt 119899

119862

119890119878 (119899) = 120575119878 if 119899 gt 119899

119862

119906 = 119899119862

(22)

where 119899119862is the median state at completions

Assume for simplicity that there is an even number ofstates so that 119899

119862= 119899 for any n This solution is dual feasible

since we showed above that any n is a solution to the dualChecking primal feasibility the solution satisfies minus119890119878(119899) le

120575119878 and 119890119878(119899) le 120575119878The main constraint is

119873

sum119899=1

119901119862 (119899 minus 1) 119890119878 (119899)

= minus120575119878 sum119899lt119899119862

119901119862 (119899 minus 1) + 120575119878 sum119899gt119899119862

119901119862 (119899 minus 1)

= 120575119878[

[

minus sum119899lt119899119862

119901119862 (119899 minus 1) + sum

119899gt119899119862

119901119862 (119899 minus 1)

]

]

= 120575119878 [0]

(23)

since 119899119862is median

Lastly we check for complementary slackness Now119890119878(119899) gt minus120575

119878when 119899 gt 119899

119862 then 119899 minus 119910 gt 0 and 119890

119878(119899) lt 120575

119878

when 119899 lt 119899119862 then 119899 minus 119910 lt 0 Therefore complementary

slackness holds and the solution is an optimal oneThe solution value is

119873

sum119899=1

119899119901119862 (119899 minus 1) 119890119878 (119899)

= sum119899lt119899119862

119899119901119862 (119899 minus 1) (minus120575119878) + sum

119899gt119899119862

119899119901119862 (119899 minus 1) (120575119878)

(24)

Set this value equal to Δ119878 which is the overall completerrsquos

average minus the average of the set truncated at the medianThis can be shown by first taking

Δ 119878 = sum119899lt119899119862

119899119901119862 (119899 minus 1) (minus120575119878) + sum119899gt119899119862

119899119901119862 (119899 minus 1) (120575119878) (25)

120575119878times the completerrsquos average 119899

119862 is

120575119878119899119862= sum119899lt119899119862

119899119901119862 (119899 minus 1) (120575119878) + sum

119899gt119899119862

119899119901119862 (119899 minus 1) (120575119878) (26)

Subtracting Δ119878yields

120575119878119899119862 minus Δ 119878 = 2 sum119899lt119899119862

119899119901119862 (119899 minus 1) (120575119878)

Δ119878 = 120575119878

[

[

119899119862 minus 2 sum119899lt119899119862

119899119901119862 (119899 minus 1)]

]

(27)

We know that

sum119899lt119899119862

119901119862 (119899 minus 1) =

1

2 (28)

since 119899119862is a median Therefore

sum119899lt119899119862

2119901119862 (119899 minus 1) = 1

sum119899lt119899119862

1198992119901119862 (119899 minus 1) = 119899119862119879

(29)

is the average of the set of states truncated at the median Sothe bound on 119890lowast

119878is

Δ119878= 120575119878 [119899119862 minus 119899119862119879] (30)

As an example take the behavior sequence in Figure 1 Ifwe want to use the OA equation [24]

119899119860119878=

119880

1 minus 119880 minus 119901 (119873)(1 minus (119873 + 1) 119901 (119873)) (31)

to calculate 119899119860119878 which is the average number assuming flowbalance homogeneous arrival and services we would beinterested in the bound of the difference between 119899 and 119899119860119878We can calculate

119880 =119879 minus 119879 (0)

119879=5

6(32)

p(n)=13 and 119899119862

= 74 Assume the maximum error is120575119878= 35 If the other assumption errors are zero then the

difference 119899 minus 119899119860119878 is equal to the correction term

119862119899=

119880

1 minus 119880 minus 119901 (119873)(1 minus 119901 (119873)) 119890

lowast

119878 (33)

The upper bound on this correction term using (14) is

119880

1 minus 119880 minus 119901 (119873)(1 minus 119901 (119873)) 120575119878119899119862 = minus35 (34)

6 Journal of Optimization

n

321

3

2

4 5 6

Time

1

Figure 1 Example Behavior Sequence for estimating Homoge-neous service error bound

Using the tighter bound Δ119878 we get

119880

1 minus 119880 minus 119901 (119873)(1 minus 119901 (119873)) Δ 119878

= minus10

3(120575119878 (119899119862 minus 119899119862119879))

= minus10

3(3

5(7

4minus 1)) = minus15

(35)

This is a reduction of 5714 for this example of the differencebound

32 Bounds onHomogeneous Arrival Assumption Error As inthe previous section we can assume that a maximum error120575119860 for any state is known beforehand That is we assume

1003816100381610038161003816119890119860 (119899)1003816100381610038161003816 le 120575119860 119899 isin 119868 (36)

Then the weak overall homogeneous arrival error is boundedby

119890lowast

119860le119873minus1

sum119899=0

119899119879 (119899)

119879 minus 119879 (119873)

1003816100381610038161003816119890119860 (119899)1003816100381610038161003816 le 120575119860

119873minus1

sum119899=0

119899119879 (119899)

119879 minus 119879 (119873)le 120575119860119899119873

(37)

where 119899119873is the average number at the device excluding the

maximum stateAs with the service errors 119890

119878(119899) the 119890

119860(119899) are not

independent The dependency is

119873minus1

sum119899=0

119879 (119899)

119879 minus 119879 (119873)119890119860 (119899) = 0 (38)

This can be shown by substituting the definition of 119890119860(119899) into

the equation As with the homogeneous service assumptionswe set up the following optimization problem

Max 119890lowast

119860=119873minus1

sum119899=0

119899119879 (119899)

119879 minus 119879 (119873)119890119860 (119899)

st119873minus1

sum119899=0

119879 (119899)

119879 minus 119879 (119873)119890119860 (119899) = 0

1003816100381610038161003816119890119860 (119899)1003816100381610038161003816 le 120575119860 119899 isin 119868

(39)

Using the Karush-Kuhn-Tucker conditions as before we canshow that with 119899 as the median state

119890119860 (119899) = minus120575119860 if 119899 lt 119899

119890119860 (119899) = 120575119860 if 119899 gt 119899

119906 = 119899

(40)

is the solution to the optimization problem The value of thesolution is found by substituting into the primal objectivefunction to get

Δ119860= max 119890lowast

119860

= sum119899lt119899

119899119879 (119899)

119879 minus 119879 (119873)(minus120575119860) +119873minus1

sum119899gt119899

119899119879 (119899)

119879 minus 119879 (119873)(120575119860)

(41)

120575119860times the average excluding the maximum 119899

119873 is

120575119860119899119873= sum119899lt119899

119899119879 (119899)

119879 minus 119879 (119873)(120575119860) +119873minus1

sum119899gt119899

119899119879 (119899)

119879 minus 119879 (119873)(120575119860) (42)

Subtracting

120575119860119899119873 minus Δ119860 = 2 sum119899lt119899

119899119879 (119899)

119879 minus 119879 (119873)120575119860 (43)

therefore

Δ119860= 120575119860[119899119873minus 2 sum119899lt119899

119899119879 (119899)

119879 minus 119879 (119873)]

= 120575119860[119899119873minus 2 sum119899lt119899

119899119901 (119899)

1 minus 119901 (119873)]

(44)

The expression

2 sum119899lt119899

119899119901 (119899) (45)

is the average of the values truncated at the median Call thisvalue 119899

119879 Then substitution yields an upper bound on the

error measure due to violations in homogeneous arrivals of

Δ119860= 120575119860 [119899119873 minus

119899119879

1 minus 119901 (119873)] (46)

This bound may not be as useful as the bound on thehomogeneous service assumption error because the value ofΔ119860is based on knowing the119879(119899) values whereas forΔ

119878only

completion counts are necessary

Journal of Optimization 7

4 Example Performance MeasureError Bounds

Some examples are given next of howΔ119860andΔ

119878may be used

to determine bounds on the difference between exact valuesof various PMs and the OA estimated values for particularbehavior sequences

41 Arriverrsquos Average Queue Length Using the example fromthe introduction for a behavior sequence the average queuelength seen by an arriving job may be calculated by

119899119860= 119899119878

119860+ 119862119899119878

119860

(47)

where 119862119899119878

119860

= 119890119865minus119890lowast119878and 119899119878119860= (119899119880)minus1 is the average arriverrsquos

queue length assuming homogeneous servicesRearranging and using the Δ119860 bound give

119899119860minus 119899119878

119860= 119890119865minus 119890lowast

119878

119899119878

119860minus 119899119860 = 119890

lowast

119878minus 119890119865 le Δ 119878 minus 119890119865

(48)

Since 119890119865rarr 0 for any sequence of data in steady state we

assume flow balance holds Then

119899119878

119860minus 119899119860le Δ119878 (49)

This expression shows us that the difference between estimat-ing the average length seen by arrivers with the relationshipthat assumes homogeneous servers and the true value of 119899119860is less than or equal to the bound on 119890lowast

119878

42 Response Time The exact response time for a behaviorsequence can be found by

119877 = 119877119878+ 119862119877119878 (50)

where 119862119877119878 = minus119878119862

119899119878

119860

= minus119878(119890119865minus 119890lowast119878) and 119877119878 = 119878(119899

119860+ 1)

Since as before we are interested in the differencebetween a possible observed value (119877) and a calculated value(119877119878) we should do some rearranging and get 119877 minus 119877119878 = 119878(119890lowast

119878minus

119890119865) Again assuming 119890

119865is small and substituting 119890lowast

119878le Δ119878

yield

119877 minus 119877119878le 119878Δ 119878

119877 minus 119877119878le 119878 (120575

119878(119899119862minus 119899119862119879))

(51)

43 Average Number at Device If homogeneous arrivalshomogeneous services and job flow balance hold then theaverage number at a device (ie those both in queue and inservice) can be calculated by

119899119860119878=

119880

1 minus 119880 minus 119901 (119873)(1 minus (119873 + 1) 119901 (119873)) (52)

The exact average number if these assumptions do not hold is

119899 = 119899119860119878+ 119862119899119860119878 (53)

where

119862119899119860119878 =

119880

1 minus 119880 minus 119901 (119873)(1 minus 119901 (119873)) (119890

lowast

119860+ 119890lowast

119878minus 119890119865) (54)

Rearranging again

119899 minus 119899119860119878= 119862119899119860119878

=119880

1 minus 119880 minus 119901 (119873)(1 minus 119901 (119873)) (119890

lowast

119860+ 119890lowast

119878minus 119890119865)

(55)

and substituting 119890lowast119860le Δ119860 119890lowast119878le Δ119878 and 119890

119865= 0 we get

119899 minus 119899119860119878le

119880

1 minus 119880 minus 119901 (119873)(1 minus 119901 (119873)) (Δ119860 + Δ 119878) (56)

if 1 gt 119880 + 119901(119873) or

119899 minus 119899119860119878ge

119880

1 minus 119880 minus 119901 (119873)(1 minus 119901 (119873)) (Δ119860 + Δ 119878) (57)

if 1 lt 119880 + 119901(119873)Because the calculation ofΔ119860 requires knowledge of T(n)

values there may be little benefit in using the right hand sideof the above relationship to find this error bound If howeverthe behavior sequence can be assumed to have homogeneousarrivals then we can get a bound on the average numberat a device error without the knowledge of the individualT(n) values In that case the only time statistics we need areutilizationU and the fraction of time spent at the maximumstate p(N)

44 Throughput Using the OA version of Littlersquos Law [1] wecan say that the difference between a behavior sequencersquosactual throughput and that calculated assuming both homo-geneous arrivals and services will be

119883 minus 119883119860119878=119899

119877minus119899119860119878

119877119860119878 (58)

For 119877119860119878 we can substitute 119877119878 which is the more generalexpression since it does not need the homogeneous arrivalassumption From previous developments we know the fol-lowing relations hold

119899 = 119899119860119878+ 119862119899119860119878 (59)

where

119862119899119860119878 le

10038161003816100381610038161003816100381610038161003816

119880

1 minus 119880 minus 119901 (119873)(1 minus 119901 (119873)) (Δ119860 + Δ 119878)

10038161003816100381610038161003816100381610038161003816(60)

and 119877 = 119877119878 + 119862119877119878 where

119862119877119878 le 119878Δ

119878 (61)

Substituting into (58) gives

119883 minus 119883119860119878=119899119860119878 + 119862

119899119860119878

119877119878 + 119862119877119878minus119899119860119878

119877119878=119877119878119862119899119860119878 minus 119899119860119878119862119877119878

119877119878 (119877119878 + 119862119877119878)

(62)

8 Journal of Optimization

If the bound of (60) is substituted into (62) then it must bethat

1198771198781003816100381610038161003816(119880 (1 minus 119880 minus 119901 (119873))) (1 minus 119901 (119873)) (Δ119860 + Δ 119878)

1003816100381610038161003816 minus 119899119860119878119862119877119878

119877119878 (119877119878 + 119862119877119878)

ge119877119878119862119899119860119878 minus 119899119860119878119862119877119878

119877119878 (119877119878 + 119862119877119878)

(63)

If instead of bound (60) bound (61) is substituted thenumerator can only decrease and the denominator onlyincrease therefore

119877119878119862119899119860119878 minus 119899119860119878119878Δ

119878

119877119878 (119877119878 + 119878Δ119878)

le119877119878119862119899119860119878 minus 119899119860119878119862

119877119878

119877119878 (119877119878 + 119862119877119878)

(64)

If both (60) and (61) are substituted then the value of theexpression must fall within the limits defined by (63) and(64) Therefore the bounds on our throughput error are

119877119878119862119899119860119878 minus 119899119860119878119878Δ

119878

119877119878 (119877119878 + 119878Δ119878)

le 119883 minus 119883119860119878

le119877119878

1003816100381610038161003816(119880 (1 minus 119880 minus 119901 (119873))) (1 minus 119901 (119873)) (Δ119860 + Δ 119878)1003816100381610038161003816

119877119878 (119877119878 + 119862119877119878)

minus119899119860119878119862119877119878

119877119878 (119877119878 + 119862119877119878)

(65)

5 Using the PM Bounds

The bounds derived in the previous sections are actuallylimits on PM correction terms Assuming we know the 120575

119860

and 120575119878values a priori and that we can say something about

the homogeneous arrival assumption then these bounds canbe found without knowledge of all the 119879(119899) values for each nThis is the same simplification of data collection that we havein using the OA formulas instead of direct calculations

As an example of using the bounds in a simulation studyassume we are interested in finding an estimate for averagenumber at a device A series of 10 runs is made and the boundfor the correction term 119862119899 is calculated If it is positive wecan call this an upper bound If it is negative let this be alower bound In both cases the other bound is 0 Assumein the simulation runs these bounds always fall between minus6and 33 The correction term for each run is approximatedby taking the average value between the upper and lowerbounds We would like to be able to say something about theprobability that future runs will fall within the [minus6 33] limitsthat have already appeared We can do this using tolerancelimit calculations

Assume the 10 runs produced the results given in Table 2The average and standard deviation for these observations areminus134 and 155 respectively

Table 2 Example estimated correction terms for average number ata device

Run 119862119899

1 minus1752 minus3003 minus1254 1655 minus2506 minus1907 minus1758 1159 minus26010 minus145

Since the tolerance limits are going to be set at theobserved limits we can say

[minus6 33] = [119862119899 minus 119870119904 119862119899 + 119870119904]

119870 =119862119899minus (minus6)

119904=(33) minus 119862119899

119904= 30

(66)

From tolerance limit tables [28] we can say with 95confidence that at least 91 of future observations of 119862

119899will

fall within the interval [minus6 33] That is if we use 119899119860119878 valueswe have 95 confidence that the correct 119899 values for each runwill fall within these limits 91 of the time

6 Conclusion

In this paper bounds were developed for the operationalanalysis error measures of homogeneous service and arrivalassumptions These bounds allow us to take advantage of thesimplified data collection made possible by the use of opera-tional analysis relationships evenwhen the assumptions usedto derive those relationships are violated A tolerance limitbased method was given in order to be able to say somethingabout the confidence that future correction term values in atime series would be within certain limits

Abbreviations

119862 Total number of job completions at adevice

119862(119899) Total number of completions when n jobsare at a device

119862119909 Correction term makes the OA term 119909

equal to the directly calculatedperformance measure

119868 The set of states 0 1 2 119873 minus 1

119868+ The set of states 1 2 119873119870 Tolerance limit multiplier119873 The maximum number of jobs seen at a

device both in queue and in service

Journal of Optimization 9

119899 State of a device number of jobs both inqueue and in service

119899 The median state seen by a device119899119862 The median state at a device at job

completions119899 Average number of jobs at a device119899119860119878 The average number of jobs at a device

assuming flow balance homogeneousarrivals and homogeneous services

119899119862 The average number at the device seen bya completing job

119899119862119879 The average of the set of states seen by

completing job truncated at the medianstate value

119899119873 The average state at a device excluding the

maximum state119899119879 Time-average of the set of states truncated

at the median119899119878119860 Average number of jobs at a device seen by

an arriver assuming homogeneous service119901(119899) Proportion of time spent in state 119899119901119860(119899) The proportion of arrivals when 119899 jobs are

at a device 119860(119899)119860119901119862(119899) The proportion of completions that leave 119899

jobs at a device 119862(119899 + 1)119862119877 Average response time119877119878 Average response time assuming

homogeneous services119877119860119878 Average response time assuming

homogeneous arrivals and homogeneousservices

119878 Mean time between completions duringbusy periods

119904 Standard deviation119879 Total time of the period of observation119879(119899) Total time device was in state 119899119880 Utilization 1minusp(0)119883 Throughput119883119860119878 Throughput assuming homogeneous

arrivals and homogeneous services120575119860 Maximum error in homogeneous arrival

assumption error measure120575119878 Maximum error in homogeneous service

assumption error measureΔ119860 A bound on the homogeneous arrival

assumption error measure le 120575119860

Δ119878 A bound on the homogeneous service

assumption error measure le 120575119878

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] J P Buzen ldquoFundamental operational laws of computer systemperformancerdquo Acta Informatica vol 7 no 2 pp 167ndash182 1976

[2] J P Buzen ldquoOperational analysis an alternative to stochasticmodelingrdquo in Performance of Computer Installations pp 175ndash194 North-Holland Amsterdam The Netherlands 1978

[3] J P Buzen and P J Denning ldquoOperational treatment of queuedistributions andmean value analysisrdquo Tech Rep CSD-TR 309Department of Computer Science Purdue University 1979

[4] J P Buzen and P J Denning ldquoMeasuring and calculating queuelength distributionsrdquo Computer vol 13 no 4 pp 33ndash44 1980

[5] P J Denning and J P Buzen ldquoThe operational analysis ofqueueing network modelsrdquo Computer Surveys vol 10 no 3 pp225ndash261 1978

[6] J P Buzen ldquoFrom stochastic modeling to operational analysisthe journey beginsrdquo in Fundamental Concepts in ComputerScience E Gelenbe and J Kahane Eds pp 141ndash149 ImperialCollege Press London UK 2009

[7] E D Lazowska J Zahorjan G S Graham and K C SevcikQuantitative System Performance Prentice-Hall EnglewoodCliffs NJ USA 1984

[8] R Suri ldquoRobustness of queueing network formulasrdquo Journal ofthe ACM vol 30 no 3 pp 564ndash594 1983

[9] E D Lazowska J Zahorjan and E C Sevcik ldquoComputer sys-tem performance evaluation using queueing network modelsrdquoAnnual Reviews in Computer Science vol 1 pp 107ndash137 1986

[10] B V Ivanovskii ldquoOperational analysis of communication net-works with lockupsrdquoAutomatic Control and Computer Sciencesvol 22 no 3 pp 26ndash32 1988

[11] P J Denning ldquoQueueing in networks of computersrdquo AmericanScientist vol 79 pp 206ndash209 1991

[12] P J Denning ldquoIn the queue mean valuesrdquo American Scientistvol 79 pp 402ndash403 1991

[13] Y Dallery and X-R Cao ldquoOperational analysis of stochasticclosed queueing networksrdquo Performance Evaluation vol 14 no1 pp 43ndash61 1992

[14] M El-Taha and S Stidham Jr ldquoDeterministic analysis of queue-ing systems with heterogeneous serversrdquo Theoretical ComputerScience vol 106 no 2 pp 243ndash264 1992

[15] J P Buzen ldquoAn overview of performance prediction in MVSsystems and SNAnetworksrdquo inProceedings of the ACMFall JointComputer Conference pp 751ndash759 Dallas Tex USA 1986

[16] D A Menasce and V A Almeida Scaling for E-Business Tech-nologies Models Performance and Capacity Planning Prentice-Hall Englewood Cliffs NJ USA 2000

[17] D K Buch and V M Pentkovski ldquoPerformance characteriza-tion experience of multi-tier E-business systems using queuingoperational analysisrdquo in Proceedings of the IEEE InternationalSymposium on Performance Analysis of Systems and SoftwareTucson Ariz USA 2001

[18] M P Fanti B Maione G Piscitelli and B Turchiano ldquoTwomethod for real-time routing selection in flexible manufactur-ing systemsrdquo inProceedings of the IEEE International Conferenceon Robotics and Automation pp 1158ndash1166 May 1992

[19] W M Zuberek ldquoThroughput analysis in timed Petri netsrdquo inProceedings of the 35th Midwest Symposium on Circuits andSystems pp 1576ndash1580 Washington DC USA 1992

[20] W M Zuberek ldquoThroughput analysis of simple closed timedPetri netmodelsrdquo in Proceedings of the 36thMidwest Symposiumon Circuits and Systems pp 930ndash933 Detroit Mich USAAugust 1993

[21] G Chiola C Anglano J Campos J M Colom and M SilvaldquoOperational analysis of timed Petri nets and application to the

10 Journal of Optimization

computation of performance boundsrdquo in Quantitative Methodsin Parallel Systems F Baccelli A Jean-Marie and I MitraniEds Esprit Basic Research Series pp 161ndash174 Springer BerlinGermany 1995

[22] J A Brumfield Operational Analysis of Queuing Phenomena[PhD thesis] Department of Computer Science Purdue Uni-versity 1982

[23] N M Bengtson ldquoUsing operational analysis in simulation aqueuing network examplerdquo Journal of the Operational ResearchSociety vol 39 no 12 pp 1125ndash1136 1988

[24] N M Bengtson ldquoMeasuring errors in operational analysisassumptionsrdquo IEEE Transactions on Software Engineering vol13 no 7 pp 767ndash776 1987

[25] N M Bengtson ldquoOperational analysis revisited error measurelimits of assumptionsrdquo GSTF International Journal on Comput-ing vol 3 no 2 2013

[26] K C Sevcik and M Klawe ldquoOperational analysis versusstochastic modeling of computer systemsrdquo in Proceedings of theComputer Science and Statistics 12th Annual Symposium on theInterface pp 177ndash184 Waterloo Canada 1979

[27] F S Hillier and G J Lieberman Introduction to OperationsResearch McGraw-Hill New York NY USA 8th edition 2005

[28] C Eisenhart M W Hastay and W A Wallis Eds SelectedTechniques of Statistical Analysis McGraw-Hill New York NYUSA 1947

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Optimization 3

Table 1 Operational analysis error measures

Assumption State dependent error Overall errorJob-flow balance 119890

119865 (119899) = (119899 + 1) (119901119860 (119899) minus 119901119862 (119899)) 119899 isin 119868 119890119865= sum119873minus1

119899=0119890119865(119899)

Homogeneous service 119890119878(119899) =

1

119901119862(119899 minus 1)

(119879(119899)

119879 minus 119879(0)) minus 1 119899 isin 119868+ 119890lowast

119878= sum119873

119899=1119899119901119888 (119899 minus 1) 119890119878 (119899)

Homogeneous arrival 119890119860(119899) = 119901

119860(119899) (

119879 minus 119879(119873)

119879(119899)) minus 1 119899 isin 119868 119890lowast

119860= sum119873minus1

119899=1119890119860 (119899)

119899119879 (119899)

119879 minus 119879 (119873)

to develop a means of dealing with lsquofuzzy homogeneityrsquo orsituations in which the various independence assumptionsare satisfied within some tolerancerdquoThis paper addresses thisneed to define these assumption bounds

The next section will illustrate how OA relations may beused to reduce data collection while estimating performancemeasures This will be followed by a discussion of thedetermination of bounds on the OA assumption measure-ment errors for homogeneous service and homogeneousarrival Sample calculations of these bounds will be presentedafterward An illustration of the use of bounds in a simulationwill then be given

2 Simplifying Data Collection

Calculating performance measures with OA relationshipsthat are derived under one or more of the system behav-ior assumptions is usually simpler than using more directrelationships This is because by making the assumptions amodel has been created which reduces perhaps artificiallythe complexity of the behavior of the systemThe result is thatless information is needed tomake an estimate of the PM thanwould be needed for a direct measurement

There are situations where it is impractical if not impos-sible to collect sufficient data to determine exact values forPMs over a finite period of time In some cases only anestimate of a PM is needed and it is not worthwhile to go tothe trouble of determining the precise PM values Any PMvalue obtained for a behavior sequence is only an estimate ofthe underlying system PM With this realization in mind itmay seem unwise to spend a great deal of effort to obtain anexact value for a sequence which is in turn only an estimateof some other value A good approximation of the sequenceestimate may be sufficient

If we want the average response time R of a behaviorsequence we could accumulate the response times of allthe jobs that go through the system and get the exact R bydividing by the number of jobs A simpler procedure wouldbe to say that response time is

119877 =119878

1 minus 119880 (5)

where 119878 is mean time between completions during busyperiods and U is utilization

This equation will give the exact R value if we have abehavior sequence for a single server queue which is in flowbalance and has homogeneous arrivals and services If these

conditions do not hold the equation will not give R exactlybut an estimate of R call it 1198771015840 If we collect only the idle time119879(0) and the number of completions C we can use the sameequation to find 1198771015840 If the behavior sequence lasts for time 119879then

119880 = 1 minus119879 (0)

119879

119878 =119879 minus 119879 (0)

119862

(6)

Another example calculates the average number of jobsat a device With the same assumptions as for estimatingresponse time the average number of jobs in the queueserversystem is

119899 =119880

1 minus 119880 (7)

This value takes even less data to calculate than does 1198771015840 Thedirect calculation of 119899 requires accumulating data every timethere is an arrival or completion or requires keeping track ofthe total amount of time spent at each of the states

Using these equations for predicting future values for Rand 119899 of a system presents certain problems For exampleover future time will the assumptions of the system behave inthe sameway Sincewe candetermine anduse errormeasuresof the assumptions in order to correct assumption derivedPM estimate it is not necessary that assumptions hold in thefuture if they have not in the past With the determinationof correction terms all that is really necessary is for thecorrection terms to remain relatively constant that is for thesystemrsquos violations of assumptions to remain the same overfuture time periods

Without knowledge of the assumption error measuresand through them the correction terms the performancemeasure estimates may be quite bad for any particularbehavior sequence [23] As stated before in a stable systemthe job flow balance assumption errormeasure will go to zeroas time increases but as shown in [24] this is not necessarilytrue for other assumption error measures For any behaviorsequence it is important tomake some assessment if possibleof the behavior of the PM correction terms

3 Performance Measure Bounds

One approach to use the simplified OA formulas for PMs isto determine bounds on the maximum PM error That is we

4 Journal of Optimization

are interested in defining bounds on the difference betweentrue values of various PMs at a device for particular statesequences and those PMs estimated by using relationshipsderived under operational analysis assumptions We willassume the network is in steady state

In the following bounds are found for the assumptionsof homogeneous services and homogeneous arrivals In thecase of the job-flow balance assumption we know that theexpected value of the errormeasure and its variance go to zero[25]

lim119905rarrinfin

119864 [119890119865] = 0 lim

119905rarrinfin119881 [119890119865] = 0 (8)

Therefore as the length of the sequence increases the 119890119865 can

be expected to become insignificantWewill need to assume that amaximumvalue of the error

for services and arrivals for any state is known or can be setCall these values 120575119878 and 120575119860 for the maximum service errorand maximum arrival error respectively

31 Bounds on Homogeneous Service Assumption Error If120575119878is the maximum error for the homogeneous service

assumption then1003816100381610038161003816119890119878 (119899)

1003816100381610038161003816 le 120575119878 119899 isin 119868+ (9)

A more useful bound would be on the weak overall homoge-neous service error

119890lowast

119878le 120575lowast

119878 (10)

But this limit may be harder to know beforehand Equation(9) may be used to find an upper bound on 119890lowast

119878by using the

definition

119890lowast

119878=119873

sum119899=1

119899119901119862 (119899 minus 1) 119890119878 (119899) (11)

Substituting (9) yields

119890lowast

119878=119873

sum119899=1

119899119901119862 (119899 minus 1)

1003816100381610038161003816119890119878 (119899)1003816100381610038161003816 le 120575119878

119873

sum119899=1

119899119901119862 (119899 minus 1) (12)

The term119873

sum119898=1

119899119901119862 (119899 minus 1) = 119899119862 (13)

is the average number at the device seen by a completerTherefore

119890lowast

119878le 120575119878119899119862 (14)

The bound given by (14) does not take into considerationthe fact that the 119890

119878(119899) values are not independent In fact they

are related by the expression

119873

sum119899=1

119901119862 (119899 minus 1) 119890119878 (119899) = 0 (15)

We can get a tighter bound of the 119890lowast119878values by taking this

dependence into consideration Equation (15) can be shownby substituting the definition

119890119878 (119899) =

119878 (119899)

119878=

1

119901119862 (119899 minus 1)

[119879 (119899)

119879 minus 119879 (0)] minus 1 (16)

as follows119873

sum119899=1

119901119862 (119899 minus 1) 119890119878 (119899)

=119873

sum119899=1

119901119862 (119899 minus 1) [1

119901119862 (119899 minus 1)

119879 (119899)

119879 minus 119879 (0)minus 1]

=119873

sum119899=1

119879 (119899)

119879 minus 119879 (0)minus119873

sum119899=1

119901119862 (119899 minus 1)

=119879 minus 119879 (0)

119879 minus 119879 (0)minus 1 = 0

(17)

Sincewhat is desired is an upper bound on 119890lowast119878a solution to

the optimization problem below will give the desired result

Max 119890lowast

119878=119873

sum119898=1

119899119901119862 (119899 minus 1) 119890119878 (119899)

st119873

sum119899=1

119901119862 (119899 minus 1) 119890119878 (119899) = 0

1003816100381610038161003816119890119878 (119899)1003816100381610038161003816 le 120575119878 119899 isin 119868

+

(18)

In order to show the optimal solution first put this problemin primal and dual forms

primal

Max119873

sum119899=1

119899119901119862 (119899 minus 1) 119890119878 (119899)

st119873

sum119899=1

119901119862 (119899 minus 1) 119890119878 (119899) = 0 119906

minus 119890119878 (119899) le 120575119878 119899 isin 119868

+ 119910 (119899)

119890119878 (119899) le 120575119878 119899 isin 119868

+ 119908 (119899)

(19)

dual

Min 120575119873

sum119899=1

(119910 (119899) + 119908 (119899))

st 119901119862 (119899 minus 1) 119906 minus 119910 (119899) + 119908 (119899)

= 119899119901119862 (119899 minus 1) 119899 isin 119868

+

119910 (119899) ge 0 119899 isin 119868+

119908 (119899) ge 0 119899 isin 119868+

(20)

Journal of Optimization 5

The optimal solution to the problem will have to satisfytheKarush-Kuhn-Tucker (KKT) conditions that is feasibilityof the Primal and Dual as well as complementary slackness[27] The KKT conditions give the necessary conditions foroptimality of the general constrained problem

Checking dual feasibility for any u the constraints can besatisfied by construction such that if (119899minus119906) gt 0 then119910(119899) = 0119908(119899) = (119899 minus 119906)119901119862(119899 minus 1) and if (119899 minus 119906) lt 0 then 119910(119899) =minus(119899 minus 119906)119901119862(119899 minus 1) 119908(119899) = 0 This is because the main dualconstraint is

(119899 minus 119906) 119901119862 (119899 minus 1) = 119908 (119899) minus 119910 (119899) (21)

In order to satisfy complementary slackness if 119890119878(119899) gt

minus120575119878 then it must be true that 119910(119899) = 0 and if 119890

119878lt 120575119878 then it

must be true that119908(119899) = 0 In terms of the above primal-dualconstruction if 119890

119878(119899) gt minus120575

119878 then (119899minus119906) gt 0 and if 119890

119878(119899) lt 120575

119878

then (119899minus119906) lt 0 Any solution 119890119878(119899) 119899 isin 119868+ andu satisfying the

primal constraints and the above two conditions is optimalConsider the solution

119890119878 (119899) = minus120575119878 if 119899 lt 119899

119862

119890119878 (119899) = 120575119878 if 119899 gt 119899

119862

119906 = 119899119862

(22)

where 119899119862is the median state at completions

Assume for simplicity that there is an even number ofstates so that 119899

119862= 119899 for any n This solution is dual feasible

since we showed above that any n is a solution to the dualChecking primal feasibility the solution satisfies minus119890119878(119899) le

120575119878 and 119890119878(119899) le 120575119878The main constraint is

119873

sum119899=1

119901119862 (119899 minus 1) 119890119878 (119899)

= minus120575119878 sum119899lt119899119862

119901119862 (119899 minus 1) + 120575119878 sum119899gt119899119862

119901119862 (119899 minus 1)

= 120575119878[

[

minus sum119899lt119899119862

119901119862 (119899 minus 1) + sum

119899gt119899119862

119901119862 (119899 minus 1)

]

]

= 120575119878 [0]

(23)

since 119899119862is median

Lastly we check for complementary slackness Now119890119878(119899) gt minus120575

119878when 119899 gt 119899

119862 then 119899 minus 119910 gt 0 and 119890

119878(119899) lt 120575

119878

when 119899 lt 119899119862 then 119899 minus 119910 lt 0 Therefore complementary

slackness holds and the solution is an optimal oneThe solution value is

119873

sum119899=1

119899119901119862 (119899 minus 1) 119890119878 (119899)

= sum119899lt119899119862

119899119901119862 (119899 minus 1) (minus120575119878) + sum

119899gt119899119862

119899119901119862 (119899 minus 1) (120575119878)

(24)

Set this value equal to Δ119878 which is the overall completerrsquos

average minus the average of the set truncated at the medianThis can be shown by first taking

Δ 119878 = sum119899lt119899119862

119899119901119862 (119899 minus 1) (minus120575119878) + sum119899gt119899119862

119899119901119862 (119899 minus 1) (120575119878) (25)

120575119878times the completerrsquos average 119899

119862 is

120575119878119899119862= sum119899lt119899119862

119899119901119862 (119899 minus 1) (120575119878) + sum

119899gt119899119862

119899119901119862 (119899 minus 1) (120575119878) (26)

Subtracting Δ119878yields

120575119878119899119862 minus Δ 119878 = 2 sum119899lt119899119862

119899119901119862 (119899 minus 1) (120575119878)

Δ119878 = 120575119878

[

[

119899119862 minus 2 sum119899lt119899119862

119899119901119862 (119899 minus 1)]

]

(27)

We know that

sum119899lt119899119862

119901119862 (119899 minus 1) =

1

2 (28)

since 119899119862is a median Therefore

sum119899lt119899119862

2119901119862 (119899 minus 1) = 1

sum119899lt119899119862

1198992119901119862 (119899 minus 1) = 119899119862119879

(29)

is the average of the set of states truncated at the median Sothe bound on 119890lowast

119878is

Δ119878= 120575119878 [119899119862 minus 119899119862119879] (30)

As an example take the behavior sequence in Figure 1 Ifwe want to use the OA equation [24]

119899119860119878=

119880

1 minus 119880 minus 119901 (119873)(1 minus (119873 + 1) 119901 (119873)) (31)

to calculate 119899119860119878 which is the average number assuming flowbalance homogeneous arrival and services we would beinterested in the bound of the difference between 119899 and 119899119860119878We can calculate

119880 =119879 minus 119879 (0)

119879=5

6(32)

p(n)=13 and 119899119862

= 74 Assume the maximum error is120575119878= 35 If the other assumption errors are zero then the

difference 119899 minus 119899119860119878 is equal to the correction term

119862119899=

119880

1 minus 119880 minus 119901 (119873)(1 minus 119901 (119873)) 119890

lowast

119878 (33)

The upper bound on this correction term using (14) is

119880

1 minus 119880 minus 119901 (119873)(1 minus 119901 (119873)) 120575119878119899119862 = minus35 (34)

6 Journal of Optimization

n

321

3

2

4 5 6

Time

1

Figure 1 Example Behavior Sequence for estimating Homoge-neous service error bound

Using the tighter bound Δ119878 we get

119880

1 minus 119880 minus 119901 (119873)(1 minus 119901 (119873)) Δ 119878

= minus10

3(120575119878 (119899119862 minus 119899119862119879))

= minus10

3(3

5(7

4minus 1)) = minus15

(35)

This is a reduction of 5714 for this example of the differencebound

32 Bounds onHomogeneous Arrival Assumption Error As inthe previous section we can assume that a maximum error120575119860 for any state is known beforehand That is we assume

1003816100381610038161003816119890119860 (119899)1003816100381610038161003816 le 120575119860 119899 isin 119868 (36)

Then the weak overall homogeneous arrival error is boundedby

119890lowast

119860le119873minus1

sum119899=0

119899119879 (119899)

119879 minus 119879 (119873)

1003816100381610038161003816119890119860 (119899)1003816100381610038161003816 le 120575119860

119873minus1

sum119899=0

119899119879 (119899)

119879 minus 119879 (119873)le 120575119860119899119873

(37)

where 119899119873is the average number at the device excluding the

maximum stateAs with the service errors 119890

119878(119899) the 119890

119860(119899) are not

independent The dependency is

119873minus1

sum119899=0

119879 (119899)

119879 minus 119879 (119873)119890119860 (119899) = 0 (38)

This can be shown by substituting the definition of 119890119860(119899) into

the equation As with the homogeneous service assumptionswe set up the following optimization problem

Max 119890lowast

119860=119873minus1

sum119899=0

119899119879 (119899)

119879 minus 119879 (119873)119890119860 (119899)

st119873minus1

sum119899=0

119879 (119899)

119879 minus 119879 (119873)119890119860 (119899) = 0

1003816100381610038161003816119890119860 (119899)1003816100381610038161003816 le 120575119860 119899 isin 119868

(39)

Using the Karush-Kuhn-Tucker conditions as before we canshow that with 119899 as the median state

119890119860 (119899) = minus120575119860 if 119899 lt 119899

119890119860 (119899) = 120575119860 if 119899 gt 119899

119906 = 119899

(40)

is the solution to the optimization problem The value of thesolution is found by substituting into the primal objectivefunction to get

Δ119860= max 119890lowast

119860

= sum119899lt119899

119899119879 (119899)

119879 minus 119879 (119873)(minus120575119860) +119873minus1

sum119899gt119899

119899119879 (119899)

119879 minus 119879 (119873)(120575119860)

(41)

120575119860times the average excluding the maximum 119899

119873 is

120575119860119899119873= sum119899lt119899

119899119879 (119899)

119879 minus 119879 (119873)(120575119860) +119873minus1

sum119899gt119899

119899119879 (119899)

119879 minus 119879 (119873)(120575119860) (42)

Subtracting

120575119860119899119873 minus Δ119860 = 2 sum119899lt119899

119899119879 (119899)

119879 minus 119879 (119873)120575119860 (43)

therefore

Δ119860= 120575119860[119899119873minus 2 sum119899lt119899

119899119879 (119899)

119879 minus 119879 (119873)]

= 120575119860[119899119873minus 2 sum119899lt119899

119899119901 (119899)

1 minus 119901 (119873)]

(44)

The expression

2 sum119899lt119899

119899119901 (119899) (45)

is the average of the values truncated at the median Call thisvalue 119899

119879 Then substitution yields an upper bound on the

error measure due to violations in homogeneous arrivals of

Δ119860= 120575119860 [119899119873 minus

119899119879

1 minus 119901 (119873)] (46)

This bound may not be as useful as the bound on thehomogeneous service assumption error because the value ofΔ119860is based on knowing the119879(119899) values whereas forΔ

119878only

completion counts are necessary

Journal of Optimization 7

4 Example Performance MeasureError Bounds

Some examples are given next of howΔ119860andΔ

119878may be used

to determine bounds on the difference between exact valuesof various PMs and the OA estimated values for particularbehavior sequences

41 Arriverrsquos Average Queue Length Using the example fromthe introduction for a behavior sequence the average queuelength seen by an arriving job may be calculated by

119899119860= 119899119878

119860+ 119862119899119878

119860

(47)

where 119862119899119878

119860

= 119890119865minus119890lowast119878and 119899119878119860= (119899119880)minus1 is the average arriverrsquos

queue length assuming homogeneous servicesRearranging and using the Δ119860 bound give

119899119860minus 119899119878

119860= 119890119865minus 119890lowast

119878

119899119878

119860minus 119899119860 = 119890

lowast

119878minus 119890119865 le Δ 119878 minus 119890119865

(48)

Since 119890119865rarr 0 for any sequence of data in steady state we

assume flow balance holds Then

119899119878

119860minus 119899119860le Δ119878 (49)

This expression shows us that the difference between estimat-ing the average length seen by arrivers with the relationshipthat assumes homogeneous servers and the true value of 119899119860is less than or equal to the bound on 119890lowast

119878

42 Response Time The exact response time for a behaviorsequence can be found by

119877 = 119877119878+ 119862119877119878 (50)

where 119862119877119878 = minus119878119862

119899119878

119860

= minus119878(119890119865minus 119890lowast119878) and 119877119878 = 119878(119899

119860+ 1)

Since as before we are interested in the differencebetween a possible observed value (119877) and a calculated value(119877119878) we should do some rearranging and get 119877 minus 119877119878 = 119878(119890lowast

119878minus

119890119865) Again assuming 119890

119865is small and substituting 119890lowast

119878le Δ119878

yield

119877 minus 119877119878le 119878Δ 119878

119877 minus 119877119878le 119878 (120575

119878(119899119862minus 119899119862119879))

(51)

43 Average Number at Device If homogeneous arrivalshomogeneous services and job flow balance hold then theaverage number at a device (ie those both in queue and inservice) can be calculated by

119899119860119878=

119880

1 minus 119880 minus 119901 (119873)(1 minus (119873 + 1) 119901 (119873)) (52)

The exact average number if these assumptions do not hold is

119899 = 119899119860119878+ 119862119899119860119878 (53)

where

119862119899119860119878 =

119880

1 minus 119880 minus 119901 (119873)(1 minus 119901 (119873)) (119890

lowast

119860+ 119890lowast

119878minus 119890119865) (54)

Rearranging again

119899 minus 119899119860119878= 119862119899119860119878

=119880

1 minus 119880 minus 119901 (119873)(1 minus 119901 (119873)) (119890

lowast

119860+ 119890lowast

119878minus 119890119865)

(55)

and substituting 119890lowast119860le Δ119860 119890lowast119878le Δ119878 and 119890

119865= 0 we get

119899 minus 119899119860119878le

119880

1 minus 119880 minus 119901 (119873)(1 minus 119901 (119873)) (Δ119860 + Δ 119878) (56)

if 1 gt 119880 + 119901(119873) or

119899 minus 119899119860119878ge

119880

1 minus 119880 minus 119901 (119873)(1 minus 119901 (119873)) (Δ119860 + Δ 119878) (57)

if 1 lt 119880 + 119901(119873)Because the calculation ofΔ119860 requires knowledge of T(n)

values there may be little benefit in using the right hand sideof the above relationship to find this error bound If howeverthe behavior sequence can be assumed to have homogeneousarrivals then we can get a bound on the average numberat a device error without the knowledge of the individualT(n) values In that case the only time statistics we need areutilizationU and the fraction of time spent at the maximumstate p(N)

44 Throughput Using the OA version of Littlersquos Law [1] wecan say that the difference between a behavior sequencersquosactual throughput and that calculated assuming both homo-geneous arrivals and services will be

119883 minus 119883119860119878=119899

119877minus119899119860119878

119877119860119878 (58)

For 119877119860119878 we can substitute 119877119878 which is the more generalexpression since it does not need the homogeneous arrivalassumption From previous developments we know the fol-lowing relations hold

119899 = 119899119860119878+ 119862119899119860119878 (59)

where

119862119899119860119878 le

10038161003816100381610038161003816100381610038161003816

119880

1 minus 119880 minus 119901 (119873)(1 minus 119901 (119873)) (Δ119860 + Δ 119878)

10038161003816100381610038161003816100381610038161003816(60)

and 119877 = 119877119878 + 119862119877119878 where

119862119877119878 le 119878Δ

119878 (61)

Substituting into (58) gives

119883 minus 119883119860119878=119899119860119878 + 119862

119899119860119878

119877119878 + 119862119877119878minus119899119860119878

119877119878=119877119878119862119899119860119878 minus 119899119860119878119862119877119878

119877119878 (119877119878 + 119862119877119878)

(62)

8 Journal of Optimization

If the bound of (60) is substituted into (62) then it must bethat

1198771198781003816100381610038161003816(119880 (1 minus 119880 minus 119901 (119873))) (1 minus 119901 (119873)) (Δ119860 + Δ 119878)

1003816100381610038161003816 minus 119899119860119878119862119877119878

119877119878 (119877119878 + 119862119877119878)

ge119877119878119862119899119860119878 minus 119899119860119878119862119877119878

119877119878 (119877119878 + 119862119877119878)

(63)

If instead of bound (60) bound (61) is substituted thenumerator can only decrease and the denominator onlyincrease therefore

119877119878119862119899119860119878 minus 119899119860119878119878Δ

119878

119877119878 (119877119878 + 119878Δ119878)

le119877119878119862119899119860119878 minus 119899119860119878119862

119877119878

119877119878 (119877119878 + 119862119877119878)

(64)

If both (60) and (61) are substituted then the value of theexpression must fall within the limits defined by (63) and(64) Therefore the bounds on our throughput error are

119877119878119862119899119860119878 minus 119899119860119878119878Δ

119878

119877119878 (119877119878 + 119878Δ119878)

le 119883 minus 119883119860119878

le119877119878

1003816100381610038161003816(119880 (1 minus 119880 minus 119901 (119873))) (1 minus 119901 (119873)) (Δ119860 + Δ 119878)1003816100381610038161003816

119877119878 (119877119878 + 119862119877119878)

minus119899119860119878119862119877119878

119877119878 (119877119878 + 119862119877119878)

(65)

5 Using the PM Bounds

The bounds derived in the previous sections are actuallylimits on PM correction terms Assuming we know the 120575

119860

and 120575119878values a priori and that we can say something about

the homogeneous arrival assumption then these bounds canbe found without knowledge of all the 119879(119899) values for each nThis is the same simplification of data collection that we havein using the OA formulas instead of direct calculations

As an example of using the bounds in a simulation studyassume we are interested in finding an estimate for averagenumber at a device A series of 10 runs is made and the boundfor the correction term 119862119899 is calculated If it is positive wecan call this an upper bound If it is negative let this be alower bound In both cases the other bound is 0 Assumein the simulation runs these bounds always fall between minus6and 33 The correction term for each run is approximatedby taking the average value between the upper and lowerbounds We would like to be able to say something about theprobability that future runs will fall within the [minus6 33] limitsthat have already appeared We can do this using tolerancelimit calculations

Assume the 10 runs produced the results given in Table 2The average and standard deviation for these observations areminus134 and 155 respectively

Table 2 Example estimated correction terms for average number ata device

Run 119862119899

1 minus1752 minus3003 minus1254 1655 minus2506 minus1907 minus1758 1159 minus26010 minus145

Since the tolerance limits are going to be set at theobserved limits we can say

[minus6 33] = [119862119899 minus 119870119904 119862119899 + 119870119904]

119870 =119862119899minus (minus6)

119904=(33) minus 119862119899

119904= 30

(66)

From tolerance limit tables [28] we can say with 95confidence that at least 91 of future observations of 119862

119899will

fall within the interval [minus6 33] That is if we use 119899119860119878 valueswe have 95 confidence that the correct 119899 values for each runwill fall within these limits 91 of the time

6 Conclusion

In this paper bounds were developed for the operationalanalysis error measures of homogeneous service and arrivalassumptions These bounds allow us to take advantage of thesimplified data collection made possible by the use of opera-tional analysis relationships evenwhen the assumptions usedto derive those relationships are violated A tolerance limitbased method was given in order to be able to say somethingabout the confidence that future correction term values in atime series would be within certain limits

Abbreviations

119862 Total number of job completions at adevice

119862(119899) Total number of completions when n jobsare at a device

119862119909 Correction term makes the OA term 119909

equal to the directly calculatedperformance measure

119868 The set of states 0 1 2 119873 minus 1

119868+ The set of states 1 2 119873119870 Tolerance limit multiplier119873 The maximum number of jobs seen at a

device both in queue and in service

Journal of Optimization 9

119899 State of a device number of jobs both inqueue and in service

119899 The median state seen by a device119899119862 The median state at a device at job

completions119899 Average number of jobs at a device119899119860119878 The average number of jobs at a device

assuming flow balance homogeneousarrivals and homogeneous services

119899119862 The average number at the device seen bya completing job

119899119862119879 The average of the set of states seen by

completing job truncated at the medianstate value

119899119873 The average state at a device excluding the

maximum state119899119879 Time-average of the set of states truncated

at the median119899119878119860 Average number of jobs at a device seen by

an arriver assuming homogeneous service119901(119899) Proportion of time spent in state 119899119901119860(119899) The proportion of arrivals when 119899 jobs are

at a device 119860(119899)119860119901119862(119899) The proportion of completions that leave 119899

jobs at a device 119862(119899 + 1)119862119877 Average response time119877119878 Average response time assuming

homogeneous services119877119860119878 Average response time assuming

homogeneous arrivals and homogeneousservices

119878 Mean time between completions duringbusy periods

119904 Standard deviation119879 Total time of the period of observation119879(119899) Total time device was in state 119899119880 Utilization 1minusp(0)119883 Throughput119883119860119878 Throughput assuming homogeneous

arrivals and homogeneous services120575119860 Maximum error in homogeneous arrival

assumption error measure120575119878 Maximum error in homogeneous service

assumption error measureΔ119860 A bound on the homogeneous arrival

assumption error measure le 120575119860

Δ119878 A bound on the homogeneous service

assumption error measure le 120575119878

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] J P Buzen ldquoFundamental operational laws of computer systemperformancerdquo Acta Informatica vol 7 no 2 pp 167ndash182 1976

[2] J P Buzen ldquoOperational analysis an alternative to stochasticmodelingrdquo in Performance of Computer Installations pp 175ndash194 North-Holland Amsterdam The Netherlands 1978

[3] J P Buzen and P J Denning ldquoOperational treatment of queuedistributions andmean value analysisrdquo Tech Rep CSD-TR 309Department of Computer Science Purdue University 1979

[4] J P Buzen and P J Denning ldquoMeasuring and calculating queuelength distributionsrdquo Computer vol 13 no 4 pp 33ndash44 1980

[5] P J Denning and J P Buzen ldquoThe operational analysis ofqueueing network modelsrdquo Computer Surveys vol 10 no 3 pp225ndash261 1978

[6] J P Buzen ldquoFrom stochastic modeling to operational analysisthe journey beginsrdquo in Fundamental Concepts in ComputerScience E Gelenbe and J Kahane Eds pp 141ndash149 ImperialCollege Press London UK 2009

[7] E D Lazowska J Zahorjan G S Graham and K C SevcikQuantitative System Performance Prentice-Hall EnglewoodCliffs NJ USA 1984

[8] R Suri ldquoRobustness of queueing network formulasrdquo Journal ofthe ACM vol 30 no 3 pp 564ndash594 1983

[9] E D Lazowska J Zahorjan and E C Sevcik ldquoComputer sys-tem performance evaluation using queueing network modelsrdquoAnnual Reviews in Computer Science vol 1 pp 107ndash137 1986

[10] B V Ivanovskii ldquoOperational analysis of communication net-works with lockupsrdquoAutomatic Control and Computer Sciencesvol 22 no 3 pp 26ndash32 1988

[11] P J Denning ldquoQueueing in networks of computersrdquo AmericanScientist vol 79 pp 206ndash209 1991

[12] P J Denning ldquoIn the queue mean valuesrdquo American Scientistvol 79 pp 402ndash403 1991

[13] Y Dallery and X-R Cao ldquoOperational analysis of stochasticclosed queueing networksrdquo Performance Evaluation vol 14 no1 pp 43ndash61 1992

[14] M El-Taha and S Stidham Jr ldquoDeterministic analysis of queue-ing systems with heterogeneous serversrdquo Theoretical ComputerScience vol 106 no 2 pp 243ndash264 1992

[15] J P Buzen ldquoAn overview of performance prediction in MVSsystems and SNAnetworksrdquo inProceedings of the ACMFall JointComputer Conference pp 751ndash759 Dallas Tex USA 1986

[16] D A Menasce and V A Almeida Scaling for E-Business Tech-nologies Models Performance and Capacity Planning Prentice-Hall Englewood Cliffs NJ USA 2000

[17] D K Buch and V M Pentkovski ldquoPerformance characteriza-tion experience of multi-tier E-business systems using queuingoperational analysisrdquo in Proceedings of the IEEE InternationalSymposium on Performance Analysis of Systems and SoftwareTucson Ariz USA 2001

[18] M P Fanti B Maione G Piscitelli and B Turchiano ldquoTwomethod for real-time routing selection in flexible manufactur-ing systemsrdquo inProceedings of the IEEE International Conferenceon Robotics and Automation pp 1158ndash1166 May 1992

[19] W M Zuberek ldquoThroughput analysis in timed Petri netsrdquo inProceedings of the 35th Midwest Symposium on Circuits andSystems pp 1576ndash1580 Washington DC USA 1992

[20] W M Zuberek ldquoThroughput analysis of simple closed timedPetri netmodelsrdquo in Proceedings of the 36thMidwest Symposiumon Circuits and Systems pp 930ndash933 Detroit Mich USAAugust 1993

[21] G Chiola C Anglano J Campos J M Colom and M SilvaldquoOperational analysis of timed Petri nets and application to the

10 Journal of Optimization

computation of performance boundsrdquo in Quantitative Methodsin Parallel Systems F Baccelli A Jean-Marie and I MitraniEds Esprit Basic Research Series pp 161ndash174 Springer BerlinGermany 1995

[22] J A Brumfield Operational Analysis of Queuing Phenomena[PhD thesis] Department of Computer Science Purdue Uni-versity 1982

[23] N M Bengtson ldquoUsing operational analysis in simulation aqueuing network examplerdquo Journal of the Operational ResearchSociety vol 39 no 12 pp 1125ndash1136 1988

[24] N M Bengtson ldquoMeasuring errors in operational analysisassumptionsrdquo IEEE Transactions on Software Engineering vol13 no 7 pp 767ndash776 1987

[25] N M Bengtson ldquoOperational analysis revisited error measurelimits of assumptionsrdquo GSTF International Journal on Comput-ing vol 3 no 2 2013

[26] K C Sevcik and M Klawe ldquoOperational analysis versusstochastic modeling of computer systemsrdquo in Proceedings of theComputer Science and Statistics 12th Annual Symposium on theInterface pp 177ndash184 Waterloo Canada 1979

[27] F S Hillier and G J Lieberman Introduction to OperationsResearch McGraw-Hill New York NY USA 8th edition 2005

[28] C Eisenhart M W Hastay and W A Wallis Eds SelectedTechniques of Statistical Analysis McGraw-Hill New York NYUSA 1947

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Journal of Optimization

are interested in defining bounds on the difference betweentrue values of various PMs at a device for particular statesequences and those PMs estimated by using relationshipsderived under operational analysis assumptions We willassume the network is in steady state

In the following bounds are found for the assumptionsof homogeneous services and homogeneous arrivals In thecase of the job-flow balance assumption we know that theexpected value of the errormeasure and its variance go to zero[25]

lim119905rarrinfin

119864 [119890119865] = 0 lim

119905rarrinfin119881 [119890119865] = 0 (8)

Therefore as the length of the sequence increases the 119890119865 can

be expected to become insignificantWewill need to assume that amaximumvalue of the error

for services and arrivals for any state is known or can be setCall these values 120575119878 and 120575119860 for the maximum service errorand maximum arrival error respectively

31 Bounds on Homogeneous Service Assumption Error If120575119878is the maximum error for the homogeneous service

assumption then1003816100381610038161003816119890119878 (119899)

1003816100381610038161003816 le 120575119878 119899 isin 119868+ (9)

A more useful bound would be on the weak overall homoge-neous service error

119890lowast

119878le 120575lowast

119878 (10)

But this limit may be harder to know beforehand Equation(9) may be used to find an upper bound on 119890lowast

119878by using the

definition

119890lowast

119878=119873

sum119899=1

119899119901119862 (119899 minus 1) 119890119878 (119899) (11)

Substituting (9) yields

119890lowast

119878=119873

sum119899=1

119899119901119862 (119899 minus 1)

1003816100381610038161003816119890119878 (119899)1003816100381610038161003816 le 120575119878

119873

sum119899=1

119899119901119862 (119899 minus 1) (12)

The term119873

sum119898=1

119899119901119862 (119899 minus 1) = 119899119862 (13)

is the average number at the device seen by a completerTherefore

119890lowast

119878le 120575119878119899119862 (14)

The bound given by (14) does not take into considerationthe fact that the 119890

119878(119899) values are not independent In fact they

are related by the expression

119873

sum119899=1

119901119862 (119899 minus 1) 119890119878 (119899) = 0 (15)

We can get a tighter bound of the 119890lowast119878values by taking this

dependence into consideration Equation (15) can be shownby substituting the definition

119890119878 (119899) =

119878 (119899)

119878=

1

119901119862 (119899 minus 1)

[119879 (119899)

119879 minus 119879 (0)] minus 1 (16)

as follows119873

sum119899=1

119901119862 (119899 minus 1) 119890119878 (119899)

=119873

sum119899=1

119901119862 (119899 minus 1) [1

119901119862 (119899 minus 1)

119879 (119899)

119879 minus 119879 (0)minus 1]

=119873

sum119899=1

119879 (119899)

119879 minus 119879 (0)minus119873

sum119899=1

119901119862 (119899 minus 1)

=119879 minus 119879 (0)

119879 minus 119879 (0)minus 1 = 0

(17)

Sincewhat is desired is an upper bound on 119890lowast119878a solution to

the optimization problem below will give the desired result

Max 119890lowast

119878=119873

sum119898=1

119899119901119862 (119899 minus 1) 119890119878 (119899)

st119873

sum119899=1

119901119862 (119899 minus 1) 119890119878 (119899) = 0

1003816100381610038161003816119890119878 (119899)1003816100381610038161003816 le 120575119878 119899 isin 119868

+

(18)

In order to show the optimal solution first put this problemin primal and dual forms

primal

Max119873

sum119899=1

119899119901119862 (119899 minus 1) 119890119878 (119899)

st119873

sum119899=1

119901119862 (119899 minus 1) 119890119878 (119899) = 0 119906

minus 119890119878 (119899) le 120575119878 119899 isin 119868

+ 119910 (119899)

119890119878 (119899) le 120575119878 119899 isin 119868

+ 119908 (119899)

(19)

dual

Min 120575119873

sum119899=1

(119910 (119899) + 119908 (119899))

st 119901119862 (119899 minus 1) 119906 minus 119910 (119899) + 119908 (119899)

= 119899119901119862 (119899 minus 1) 119899 isin 119868

+

119910 (119899) ge 0 119899 isin 119868+

119908 (119899) ge 0 119899 isin 119868+

(20)

Journal of Optimization 5

The optimal solution to the problem will have to satisfytheKarush-Kuhn-Tucker (KKT) conditions that is feasibilityof the Primal and Dual as well as complementary slackness[27] The KKT conditions give the necessary conditions foroptimality of the general constrained problem

Checking dual feasibility for any u the constraints can besatisfied by construction such that if (119899minus119906) gt 0 then119910(119899) = 0119908(119899) = (119899 minus 119906)119901119862(119899 minus 1) and if (119899 minus 119906) lt 0 then 119910(119899) =minus(119899 minus 119906)119901119862(119899 minus 1) 119908(119899) = 0 This is because the main dualconstraint is

(119899 minus 119906) 119901119862 (119899 minus 1) = 119908 (119899) minus 119910 (119899) (21)

In order to satisfy complementary slackness if 119890119878(119899) gt

minus120575119878 then it must be true that 119910(119899) = 0 and if 119890

119878lt 120575119878 then it

must be true that119908(119899) = 0 In terms of the above primal-dualconstruction if 119890

119878(119899) gt minus120575

119878 then (119899minus119906) gt 0 and if 119890

119878(119899) lt 120575

119878

then (119899minus119906) lt 0 Any solution 119890119878(119899) 119899 isin 119868+ andu satisfying the

primal constraints and the above two conditions is optimalConsider the solution

119890119878 (119899) = minus120575119878 if 119899 lt 119899

119862

119890119878 (119899) = 120575119878 if 119899 gt 119899

119862

119906 = 119899119862

(22)

where 119899119862is the median state at completions

Assume for simplicity that there is an even number ofstates so that 119899

119862= 119899 for any n This solution is dual feasible

since we showed above that any n is a solution to the dualChecking primal feasibility the solution satisfies minus119890119878(119899) le

120575119878 and 119890119878(119899) le 120575119878The main constraint is

119873

sum119899=1

119901119862 (119899 minus 1) 119890119878 (119899)

= minus120575119878 sum119899lt119899119862

119901119862 (119899 minus 1) + 120575119878 sum119899gt119899119862

119901119862 (119899 minus 1)

= 120575119878[

[

minus sum119899lt119899119862

119901119862 (119899 minus 1) + sum

119899gt119899119862

119901119862 (119899 minus 1)

]

]

= 120575119878 [0]

(23)

since 119899119862is median

Lastly we check for complementary slackness Now119890119878(119899) gt minus120575

119878when 119899 gt 119899

119862 then 119899 minus 119910 gt 0 and 119890

119878(119899) lt 120575

119878

when 119899 lt 119899119862 then 119899 minus 119910 lt 0 Therefore complementary

slackness holds and the solution is an optimal oneThe solution value is

119873

sum119899=1

119899119901119862 (119899 minus 1) 119890119878 (119899)

= sum119899lt119899119862

119899119901119862 (119899 minus 1) (minus120575119878) + sum

119899gt119899119862

119899119901119862 (119899 minus 1) (120575119878)

(24)

Set this value equal to Δ119878 which is the overall completerrsquos

average minus the average of the set truncated at the medianThis can be shown by first taking

Δ 119878 = sum119899lt119899119862

119899119901119862 (119899 minus 1) (minus120575119878) + sum119899gt119899119862

119899119901119862 (119899 minus 1) (120575119878) (25)

120575119878times the completerrsquos average 119899

119862 is

120575119878119899119862= sum119899lt119899119862

119899119901119862 (119899 minus 1) (120575119878) + sum

119899gt119899119862

119899119901119862 (119899 minus 1) (120575119878) (26)

Subtracting Δ119878yields

120575119878119899119862 minus Δ 119878 = 2 sum119899lt119899119862

119899119901119862 (119899 minus 1) (120575119878)

Δ119878 = 120575119878

[

[

119899119862 minus 2 sum119899lt119899119862

119899119901119862 (119899 minus 1)]

]

(27)

We know that

sum119899lt119899119862

119901119862 (119899 minus 1) =

1

2 (28)

since 119899119862is a median Therefore

sum119899lt119899119862

2119901119862 (119899 minus 1) = 1

sum119899lt119899119862

1198992119901119862 (119899 minus 1) = 119899119862119879

(29)

is the average of the set of states truncated at the median Sothe bound on 119890lowast

119878is

Δ119878= 120575119878 [119899119862 minus 119899119862119879] (30)

As an example take the behavior sequence in Figure 1 Ifwe want to use the OA equation [24]

119899119860119878=

119880

1 minus 119880 minus 119901 (119873)(1 minus (119873 + 1) 119901 (119873)) (31)

to calculate 119899119860119878 which is the average number assuming flowbalance homogeneous arrival and services we would beinterested in the bound of the difference between 119899 and 119899119860119878We can calculate

119880 =119879 minus 119879 (0)

119879=5

6(32)

p(n)=13 and 119899119862

= 74 Assume the maximum error is120575119878= 35 If the other assumption errors are zero then the

difference 119899 minus 119899119860119878 is equal to the correction term

119862119899=

119880

1 minus 119880 minus 119901 (119873)(1 minus 119901 (119873)) 119890

lowast

119878 (33)

The upper bound on this correction term using (14) is

119880

1 minus 119880 minus 119901 (119873)(1 minus 119901 (119873)) 120575119878119899119862 = minus35 (34)

6 Journal of Optimization

n

321

3

2

4 5 6

Time

1

Figure 1 Example Behavior Sequence for estimating Homoge-neous service error bound

Using the tighter bound Δ119878 we get

119880

1 minus 119880 minus 119901 (119873)(1 minus 119901 (119873)) Δ 119878

= minus10

3(120575119878 (119899119862 minus 119899119862119879))

= minus10

3(3

5(7

4minus 1)) = minus15

(35)

This is a reduction of 5714 for this example of the differencebound

32 Bounds onHomogeneous Arrival Assumption Error As inthe previous section we can assume that a maximum error120575119860 for any state is known beforehand That is we assume

1003816100381610038161003816119890119860 (119899)1003816100381610038161003816 le 120575119860 119899 isin 119868 (36)

Then the weak overall homogeneous arrival error is boundedby

119890lowast

119860le119873minus1

sum119899=0

119899119879 (119899)

119879 minus 119879 (119873)

1003816100381610038161003816119890119860 (119899)1003816100381610038161003816 le 120575119860

119873minus1

sum119899=0

119899119879 (119899)

119879 minus 119879 (119873)le 120575119860119899119873

(37)

where 119899119873is the average number at the device excluding the

maximum stateAs with the service errors 119890

119878(119899) the 119890

119860(119899) are not

independent The dependency is

119873minus1

sum119899=0

119879 (119899)

119879 minus 119879 (119873)119890119860 (119899) = 0 (38)

This can be shown by substituting the definition of 119890119860(119899) into

the equation As with the homogeneous service assumptionswe set up the following optimization problem

Max 119890lowast

119860=119873minus1

sum119899=0

119899119879 (119899)

119879 minus 119879 (119873)119890119860 (119899)

st119873minus1

sum119899=0

119879 (119899)

119879 minus 119879 (119873)119890119860 (119899) = 0

1003816100381610038161003816119890119860 (119899)1003816100381610038161003816 le 120575119860 119899 isin 119868

(39)

Using the Karush-Kuhn-Tucker conditions as before we canshow that with 119899 as the median state

119890119860 (119899) = minus120575119860 if 119899 lt 119899

119890119860 (119899) = 120575119860 if 119899 gt 119899

119906 = 119899

(40)

is the solution to the optimization problem The value of thesolution is found by substituting into the primal objectivefunction to get

Δ119860= max 119890lowast

119860

= sum119899lt119899

119899119879 (119899)

119879 minus 119879 (119873)(minus120575119860) +119873minus1

sum119899gt119899

119899119879 (119899)

119879 minus 119879 (119873)(120575119860)

(41)

120575119860times the average excluding the maximum 119899

119873 is

120575119860119899119873= sum119899lt119899

119899119879 (119899)

119879 minus 119879 (119873)(120575119860) +119873minus1

sum119899gt119899

119899119879 (119899)

119879 minus 119879 (119873)(120575119860) (42)

Subtracting

120575119860119899119873 minus Δ119860 = 2 sum119899lt119899

119899119879 (119899)

119879 minus 119879 (119873)120575119860 (43)

therefore

Δ119860= 120575119860[119899119873minus 2 sum119899lt119899

119899119879 (119899)

119879 minus 119879 (119873)]

= 120575119860[119899119873minus 2 sum119899lt119899

119899119901 (119899)

1 minus 119901 (119873)]

(44)

The expression

2 sum119899lt119899

119899119901 (119899) (45)

is the average of the values truncated at the median Call thisvalue 119899

119879 Then substitution yields an upper bound on the

error measure due to violations in homogeneous arrivals of

Δ119860= 120575119860 [119899119873 minus

119899119879

1 minus 119901 (119873)] (46)

This bound may not be as useful as the bound on thehomogeneous service assumption error because the value ofΔ119860is based on knowing the119879(119899) values whereas forΔ

119878only

completion counts are necessary

Journal of Optimization 7

4 Example Performance MeasureError Bounds

Some examples are given next of howΔ119860andΔ

119878may be used

to determine bounds on the difference between exact valuesof various PMs and the OA estimated values for particularbehavior sequences

41 Arriverrsquos Average Queue Length Using the example fromthe introduction for a behavior sequence the average queuelength seen by an arriving job may be calculated by

119899119860= 119899119878

119860+ 119862119899119878

119860

(47)

where 119862119899119878

119860

= 119890119865minus119890lowast119878and 119899119878119860= (119899119880)minus1 is the average arriverrsquos

queue length assuming homogeneous servicesRearranging and using the Δ119860 bound give

119899119860minus 119899119878

119860= 119890119865minus 119890lowast

119878

119899119878

119860minus 119899119860 = 119890

lowast

119878minus 119890119865 le Δ 119878 minus 119890119865

(48)

Since 119890119865rarr 0 for any sequence of data in steady state we

assume flow balance holds Then

119899119878

119860minus 119899119860le Δ119878 (49)

This expression shows us that the difference between estimat-ing the average length seen by arrivers with the relationshipthat assumes homogeneous servers and the true value of 119899119860is less than or equal to the bound on 119890lowast

119878

42 Response Time The exact response time for a behaviorsequence can be found by

119877 = 119877119878+ 119862119877119878 (50)

where 119862119877119878 = minus119878119862

119899119878

119860

= minus119878(119890119865minus 119890lowast119878) and 119877119878 = 119878(119899

119860+ 1)

Since as before we are interested in the differencebetween a possible observed value (119877) and a calculated value(119877119878) we should do some rearranging and get 119877 minus 119877119878 = 119878(119890lowast

119878minus

119890119865) Again assuming 119890

119865is small and substituting 119890lowast

119878le Δ119878

yield

119877 minus 119877119878le 119878Δ 119878

119877 minus 119877119878le 119878 (120575

119878(119899119862minus 119899119862119879))

(51)

43 Average Number at Device If homogeneous arrivalshomogeneous services and job flow balance hold then theaverage number at a device (ie those both in queue and inservice) can be calculated by

119899119860119878=

119880

1 minus 119880 minus 119901 (119873)(1 minus (119873 + 1) 119901 (119873)) (52)

The exact average number if these assumptions do not hold is

119899 = 119899119860119878+ 119862119899119860119878 (53)

where

119862119899119860119878 =

119880

1 minus 119880 minus 119901 (119873)(1 minus 119901 (119873)) (119890

lowast

119860+ 119890lowast

119878minus 119890119865) (54)

Rearranging again

119899 minus 119899119860119878= 119862119899119860119878

=119880

1 minus 119880 minus 119901 (119873)(1 minus 119901 (119873)) (119890

lowast

119860+ 119890lowast

119878minus 119890119865)

(55)

and substituting 119890lowast119860le Δ119860 119890lowast119878le Δ119878 and 119890

119865= 0 we get

119899 minus 119899119860119878le

119880

1 minus 119880 minus 119901 (119873)(1 minus 119901 (119873)) (Δ119860 + Δ 119878) (56)

if 1 gt 119880 + 119901(119873) or

119899 minus 119899119860119878ge

119880

1 minus 119880 minus 119901 (119873)(1 minus 119901 (119873)) (Δ119860 + Δ 119878) (57)

if 1 lt 119880 + 119901(119873)Because the calculation ofΔ119860 requires knowledge of T(n)

values there may be little benefit in using the right hand sideof the above relationship to find this error bound If howeverthe behavior sequence can be assumed to have homogeneousarrivals then we can get a bound on the average numberat a device error without the knowledge of the individualT(n) values In that case the only time statistics we need areutilizationU and the fraction of time spent at the maximumstate p(N)

44 Throughput Using the OA version of Littlersquos Law [1] wecan say that the difference between a behavior sequencersquosactual throughput and that calculated assuming both homo-geneous arrivals and services will be

119883 minus 119883119860119878=119899

119877minus119899119860119878

119877119860119878 (58)

For 119877119860119878 we can substitute 119877119878 which is the more generalexpression since it does not need the homogeneous arrivalassumption From previous developments we know the fol-lowing relations hold

119899 = 119899119860119878+ 119862119899119860119878 (59)

where

119862119899119860119878 le

10038161003816100381610038161003816100381610038161003816

119880

1 minus 119880 minus 119901 (119873)(1 minus 119901 (119873)) (Δ119860 + Δ 119878)

10038161003816100381610038161003816100381610038161003816(60)

and 119877 = 119877119878 + 119862119877119878 where

119862119877119878 le 119878Δ

119878 (61)

Substituting into (58) gives

119883 minus 119883119860119878=119899119860119878 + 119862

119899119860119878

119877119878 + 119862119877119878minus119899119860119878

119877119878=119877119878119862119899119860119878 minus 119899119860119878119862119877119878

119877119878 (119877119878 + 119862119877119878)

(62)

8 Journal of Optimization

If the bound of (60) is substituted into (62) then it must bethat

1198771198781003816100381610038161003816(119880 (1 minus 119880 minus 119901 (119873))) (1 minus 119901 (119873)) (Δ119860 + Δ 119878)

1003816100381610038161003816 minus 119899119860119878119862119877119878

119877119878 (119877119878 + 119862119877119878)

ge119877119878119862119899119860119878 minus 119899119860119878119862119877119878

119877119878 (119877119878 + 119862119877119878)

(63)

If instead of bound (60) bound (61) is substituted thenumerator can only decrease and the denominator onlyincrease therefore

119877119878119862119899119860119878 minus 119899119860119878119878Δ

119878

119877119878 (119877119878 + 119878Δ119878)

le119877119878119862119899119860119878 minus 119899119860119878119862

119877119878

119877119878 (119877119878 + 119862119877119878)

(64)

If both (60) and (61) are substituted then the value of theexpression must fall within the limits defined by (63) and(64) Therefore the bounds on our throughput error are

119877119878119862119899119860119878 minus 119899119860119878119878Δ

119878

119877119878 (119877119878 + 119878Δ119878)

le 119883 minus 119883119860119878

le119877119878

1003816100381610038161003816(119880 (1 minus 119880 minus 119901 (119873))) (1 minus 119901 (119873)) (Δ119860 + Δ 119878)1003816100381610038161003816

119877119878 (119877119878 + 119862119877119878)

minus119899119860119878119862119877119878

119877119878 (119877119878 + 119862119877119878)

(65)

5 Using the PM Bounds

The bounds derived in the previous sections are actuallylimits on PM correction terms Assuming we know the 120575

119860

and 120575119878values a priori and that we can say something about

the homogeneous arrival assumption then these bounds canbe found without knowledge of all the 119879(119899) values for each nThis is the same simplification of data collection that we havein using the OA formulas instead of direct calculations

As an example of using the bounds in a simulation studyassume we are interested in finding an estimate for averagenumber at a device A series of 10 runs is made and the boundfor the correction term 119862119899 is calculated If it is positive wecan call this an upper bound If it is negative let this be alower bound In both cases the other bound is 0 Assumein the simulation runs these bounds always fall between minus6and 33 The correction term for each run is approximatedby taking the average value between the upper and lowerbounds We would like to be able to say something about theprobability that future runs will fall within the [minus6 33] limitsthat have already appeared We can do this using tolerancelimit calculations

Assume the 10 runs produced the results given in Table 2The average and standard deviation for these observations areminus134 and 155 respectively

Table 2 Example estimated correction terms for average number ata device

Run 119862119899

1 minus1752 minus3003 minus1254 1655 minus2506 minus1907 minus1758 1159 minus26010 minus145

Since the tolerance limits are going to be set at theobserved limits we can say

[minus6 33] = [119862119899 minus 119870119904 119862119899 + 119870119904]

119870 =119862119899minus (minus6)

119904=(33) minus 119862119899

119904= 30

(66)

From tolerance limit tables [28] we can say with 95confidence that at least 91 of future observations of 119862

119899will

fall within the interval [minus6 33] That is if we use 119899119860119878 valueswe have 95 confidence that the correct 119899 values for each runwill fall within these limits 91 of the time

6 Conclusion

In this paper bounds were developed for the operationalanalysis error measures of homogeneous service and arrivalassumptions These bounds allow us to take advantage of thesimplified data collection made possible by the use of opera-tional analysis relationships evenwhen the assumptions usedto derive those relationships are violated A tolerance limitbased method was given in order to be able to say somethingabout the confidence that future correction term values in atime series would be within certain limits

Abbreviations

119862 Total number of job completions at adevice

119862(119899) Total number of completions when n jobsare at a device

119862119909 Correction term makes the OA term 119909

equal to the directly calculatedperformance measure

119868 The set of states 0 1 2 119873 minus 1

119868+ The set of states 1 2 119873119870 Tolerance limit multiplier119873 The maximum number of jobs seen at a

device both in queue and in service

Journal of Optimization 9

119899 State of a device number of jobs both inqueue and in service

119899 The median state seen by a device119899119862 The median state at a device at job

completions119899 Average number of jobs at a device119899119860119878 The average number of jobs at a device

assuming flow balance homogeneousarrivals and homogeneous services

119899119862 The average number at the device seen bya completing job

119899119862119879 The average of the set of states seen by

completing job truncated at the medianstate value

119899119873 The average state at a device excluding the

maximum state119899119879 Time-average of the set of states truncated

at the median119899119878119860 Average number of jobs at a device seen by

an arriver assuming homogeneous service119901(119899) Proportion of time spent in state 119899119901119860(119899) The proportion of arrivals when 119899 jobs are

at a device 119860(119899)119860119901119862(119899) The proportion of completions that leave 119899

jobs at a device 119862(119899 + 1)119862119877 Average response time119877119878 Average response time assuming

homogeneous services119877119860119878 Average response time assuming

homogeneous arrivals and homogeneousservices

119878 Mean time between completions duringbusy periods

119904 Standard deviation119879 Total time of the period of observation119879(119899) Total time device was in state 119899119880 Utilization 1minusp(0)119883 Throughput119883119860119878 Throughput assuming homogeneous

arrivals and homogeneous services120575119860 Maximum error in homogeneous arrival

assumption error measure120575119878 Maximum error in homogeneous service

assumption error measureΔ119860 A bound on the homogeneous arrival

assumption error measure le 120575119860

Δ119878 A bound on the homogeneous service

assumption error measure le 120575119878

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] J P Buzen ldquoFundamental operational laws of computer systemperformancerdquo Acta Informatica vol 7 no 2 pp 167ndash182 1976

[2] J P Buzen ldquoOperational analysis an alternative to stochasticmodelingrdquo in Performance of Computer Installations pp 175ndash194 North-Holland Amsterdam The Netherlands 1978

[3] J P Buzen and P J Denning ldquoOperational treatment of queuedistributions andmean value analysisrdquo Tech Rep CSD-TR 309Department of Computer Science Purdue University 1979

[4] J P Buzen and P J Denning ldquoMeasuring and calculating queuelength distributionsrdquo Computer vol 13 no 4 pp 33ndash44 1980

[5] P J Denning and J P Buzen ldquoThe operational analysis ofqueueing network modelsrdquo Computer Surveys vol 10 no 3 pp225ndash261 1978

[6] J P Buzen ldquoFrom stochastic modeling to operational analysisthe journey beginsrdquo in Fundamental Concepts in ComputerScience E Gelenbe and J Kahane Eds pp 141ndash149 ImperialCollege Press London UK 2009

[7] E D Lazowska J Zahorjan G S Graham and K C SevcikQuantitative System Performance Prentice-Hall EnglewoodCliffs NJ USA 1984

[8] R Suri ldquoRobustness of queueing network formulasrdquo Journal ofthe ACM vol 30 no 3 pp 564ndash594 1983

[9] E D Lazowska J Zahorjan and E C Sevcik ldquoComputer sys-tem performance evaluation using queueing network modelsrdquoAnnual Reviews in Computer Science vol 1 pp 107ndash137 1986

[10] B V Ivanovskii ldquoOperational analysis of communication net-works with lockupsrdquoAutomatic Control and Computer Sciencesvol 22 no 3 pp 26ndash32 1988

[11] P J Denning ldquoQueueing in networks of computersrdquo AmericanScientist vol 79 pp 206ndash209 1991

[12] P J Denning ldquoIn the queue mean valuesrdquo American Scientistvol 79 pp 402ndash403 1991

[13] Y Dallery and X-R Cao ldquoOperational analysis of stochasticclosed queueing networksrdquo Performance Evaluation vol 14 no1 pp 43ndash61 1992

[14] M El-Taha and S Stidham Jr ldquoDeterministic analysis of queue-ing systems with heterogeneous serversrdquo Theoretical ComputerScience vol 106 no 2 pp 243ndash264 1992

[15] J P Buzen ldquoAn overview of performance prediction in MVSsystems and SNAnetworksrdquo inProceedings of the ACMFall JointComputer Conference pp 751ndash759 Dallas Tex USA 1986

[16] D A Menasce and V A Almeida Scaling for E-Business Tech-nologies Models Performance and Capacity Planning Prentice-Hall Englewood Cliffs NJ USA 2000

[17] D K Buch and V M Pentkovski ldquoPerformance characteriza-tion experience of multi-tier E-business systems using queuingoperational analysisrdquo in Proceedings of the IEEE InternationalSymposium on Performance Analysis of Systems and SoftwareTucson Ariz USA 2001

[18] M P Fanti B Maione G Piscitelli and B Turchiano ldquoTwomethod for real-time routing selection in flexible manufactur-ing systemsrdquo inProceedings of the IEEE International Conferenceon Robotics and Automation pp 1158ndash1166 May 1992

[19] W M Zuberek ldquoThroughput analysis in timed Petri netsrdquo inProceedings of the 35th Midwest Symposium on Circuits andSystems pp 1576ndash1580 Washington DC USA 1992

[20] W M Zuberek ldquoThroughput analysis of simple closed timedPetri netmodelsrdquo in Proceedings of the 36thMidwest Symposiumon Circuits and Systems pp 930ndash933 Detroit Mich USAAugust 1993

[21] G Chiola C Anglano J Campos J M Colom and M SilvaldquoOperational analysis of timed Petri nets and application to the

10 Journal of Optimization

computation of performance boundsrdquo in Quantitative Methodsin Parallel Systems F Baccelli A Jean-Marie and I MitraniEds Esprit Basic Research Series pp 161ndash174 Springer BerlinGermany 1995

[22] J A Brumfield Operational Analysis of Queuing Phenomena[PhD thesis] Department of Computer Science Purdue Uni-versity 1982

[23] N M Bengtson ldquoUsing operational analysis in simulation aqueuing network examplerdquo Journal of the Operational ResearchSociety vol 39 no 12 pp 1125ndash1136 1988

[24] N M Bengtson ldquoMeasuring errors in operational analysisassumptionsrdquo IEEE Transactions on Software Engineering vol13 no 7 pp 767ndash776 1987

[25] N M Bengtson ldquoOperational analysis revisited error measurelimits of assumptionsrdquo GSTF International Journal on Comput-ing vol 3 no 2 2013

[26] K C Sevcik and M Klawe ldquoOperational analysis versusstochastic modeling of computer systemsrdquo in Proceedings of theComputer Science and Statistics 12th Annual Symposium on theInterface pp 177ndash184 Waterloo Canada 1979

[27] F S Hillier and G J Lieberman Introduction to OperationsResearch McGraw-Hill New York NY USA 8th edition 2005

[28] C Eisenhart M W Hastay and W A Wallis Eds SelectedTechniques of Statistical Analysis McGraw-Hill New York NYUSA 1947

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Optimization 5

The optimal solution to the problem will have to satisfytheKarush-Kuhn-Tucker (KKT) conditions that is feasibilityof the Primal and Dual as well as complementary slackness[27] The KKT conditions give the necessary conditions foroptimality of the general constrained problem

Checking dual feasibility for any u the constraints can besatisfied by construction such that if (119899minus119906) gt 0 then119910(119899) = 0119908(119899) = (119899 minus 119906)119901119862(119899 minus 1) and if (119899 minus 119906) lt 0 then 119910(119899) =minus(119899 minus 119906)119901119862(119899 minus 1) 119908(119899) = 0 This is because the main dualconstraint is

(119899 minus 119906) 119901119862 (119899 minus 1) = 119908 (119899) minus 119910 (119899) (21)

In order to satisfy complementary slackness if 119890119878(119899) gt

minus120575119878 then it must be true that 119910(119899) = 0 and if 119890

119878lt 120575119878 then it

must be true that119908(119899) = 0 In terms of the above primal-dualconstruction if 119890

119878(119899) gt minus120575

119878 then (119899minus119906) gt 0 and if 119890

119878(119899) lt 120575

119878

then (119899minus119906) lt 0 Any solution 119890119878(119899) 119899 isin 119868+ andu satisfying the

primal constraints and the above two conditions is optimalConsider the solution

119890119878 (119899) = minus120575119878 if 119899 lt 119899

119862

119890119878 (119899) = 120575119878 if 119899 gt 119899

119862

119906 = 119899119862

(22)

where 119899119862is the median state at completions

Assume for simplicity that there is an even number ofstates so that 119899

119862= 119899 for any n This solution is dual feasible

since we showed above that any n is a solution to the dualChecking primal feasibility the solution satisfies minus119890119878(119899) le

120575119878 and 119890119878(119899) le 120575119878The main constraint is

119873

sum119899=1

119901119862 (119899 minus 1) 119890119878 (119899)

= minus120575119878 sum119899lt119899119862

119901119862 (119899 minus 1) + 120575119878 sum119899gt119899119862

119901119862 (119899 minus 1)

= 120575119878[

[

minus sum119899lt119899119862

119901119862 (119899 minus 1) + sum

119899gt119899119862

119901119862 (119899 minus 1)

]

]

= 120575119878 [0]

(23)

since 119899119862is median

Lastly we check for complementary slackness Now119890119878(119899) gt minus120575

119878when 119899 gt 119899

119862 then 119899 minus 119910 gt 0 and 119890

119878(119899) lt 120575

119878

when 119899 lt 119899119862 then 119899 minus 119910 lt 0 Therefore complementary

slackness holds and the solution is an optimal oneThe solution value is

119873

sum119899=1

119899119901119862 (119899 minus 1) 119890119878 (119899)

= sum119899lt119899119862

119899119901119862 (119899 minus 1) (minus120575119878) + sum

119899gt119899119862

119899119901119862 (119899 minus 1) (120575119878)

(24)

Set this value equal to Δ119878 which is the overall completerrsquos

average minus the average of the set truncated at the medianThis can be shown by first taking

Δ 119878 = sum119899lt119899119862

119899119901119862 (119899 minus 1) (minus120575119878) + sum119899gt119899119862

119899119901119862 (119899 minus 1) (120575119878) (25)

120575119878times the completerrsquos average 119899

119862 is

120575119878119899119862= sum119899lt119899119862

119899119901119862 (119899 minus 1) (120575119878) + sum

119899gt119899119862

119899119901119862 (119899 minus 1) (120575119878) (26)

Subtracting Δ119878yields

120575119878119899119862 minus Δ 119878 = 2 sum119899lt119899119862

119899119901119862 (119899 minus 1) (120575119878)

Δ119878 = 120575119878

[

[

119899119862 minus 2 sum119899lt119899119862

119899119901119862 (119899 minus 1)]

]

(27)

We know that

sum119899lt119899119862

119901119862 (119899 minus 1) =

1

2 (28)

since 119899119862is a median Therefore

sum119899lt119899119862

2119901119862 (119899 minus 1) = 1

sum119899lt119899119862

1198992119901119862 (119899 minus 1) = 119899119862119879

(29)

is the average of the set of states truncated at the median Sothe bound on 119890lowast

119878is

Δ119878= 120575119878 [119899119862 minus 119899119862119879] (30)

As an example take the behavior sequence in Figure 1 Ifwe want to use the OA equation [24]

119899119860119878=

119880

1 minus 119880 minus 119901 (119873)(1 minus (119873 + 1) 119901 (119873)) (31)

to calculate 119899119860119878 which is the average number assuming flowbalance homogeneous arrival and services we would beinterested in the bound of the difference between 119899 and 119899119860119878We can calculate

119880 =119879 minus 119879 (0)

119879=5

6(32)

p(n)=13 and 119899119862

= 74 Assume the maximum error is120575119878= 35 If the other assumption errors are zero then the

difference 119899 minus 119899119860119878 is equal to the correction term

119862119899=

119880

1 minus 119880 minus 119901 (119873)(1 minus 119901 (119873)) 119890

lowast

119878 (33)

The upper bound on this correction term using (14) is

119880

1 minus 119880 minus 119901 (119873)(1 minus 119901 (119873)) 120575119878119899119862 = minus35 (34)

6 Journal of Optimization

n

321

3

2

4 5 6

Time

1

Figure 1 Example Behavior Sequence for estimating Homoge-neous service error bound

Using the tighter bound Δ119878 we get

119880

1 minus 119880 minus 119901 (119873)(1 minus 119901 (119873)) Δ 119878

= minus10

3(120575119878 (119899119862 minus 119899119862119879))

= minus10

3(3

5(7

4minus 1)) = minus15

(35)

This is a reduction of 5714 for this example of the differencebound

32 Bounds onHomogeneous Arrival Assumption Error As inthe previous section we can assume that a maximum error120575119860 for any state is known beforehand That is we assume

1003816100381610038161003816119890119860 (119899)1003816100381610038161003816 le 120575119860 119899 isin 119868 (36)

Then the weak overall homogeneous arrival error is boundedby

119890lowast

119860le119873minus1

sum119899=0

119899119879 (119899)

119879 minus 119879 (119873)

1003816100381610038161003816119890119860 (119899)1003816100381610038161003816 le 120575119860

119873minus1

sum119899=0

119899119879 (119899)

119879 minus 119879 (119873)le 120575119860119899119873

(37)

where 119899119873is the average number at the device excluding the

maximum stateAs with the service errors 119890

119878(119899) the 119890

119860(119899) are not

independent The dependency is

119873minus1

sum119899=0

119879 (119899)

119879 minus 119879 (119873)119890119860 (119899) = 0 (38)

This can be shown by substituting the definition of 119890119860(119899) into

the equation As with the homogeneous service assumptionswe set up the following optimization problem

Max 119890lowast

119860=119873minus1

sum119899=0

119899119879 (119899)

119879 minus 119879 (119873)119890119860 (119899)

st119873minus1

sum119899=0

119879 (119899)

119879 minus 119879 (119873)119890119860 (119899) = 0

1003816100381610038161003816119890119860 (119899)1003816100381610038161003816 le 120575119860 119899 isin 119868

(39)

Using the Karush-Kuhn-Tucker conditions as before we canshow that with 119899 as the median state

119890119860 (119899) = minus120575119860 if 119899 lt 119899

119890119860 (119899) = 120575119860 if 119899 gt 119899

119906 = 119899

(40)

is the solution to the optimization problem The value of thesolution is found by substituting into the primal objectivefunction to get

Δ119860= max 119890lowast

119860

= sum119899lt119899

119899119879 (119899)

119879 minus 119879 (119873)(minus120575119860) +119873minus1

sum119899gt119899

119899119879 (119899)

119879 minus 119879 (119873)(120575119860)

(41)

120575119860times the average excluding the maximum 119899

119873 is

120575119860119899119873= sum119899lt119899

119899119879 (119899)

119879 minus 119879 (119873)(120575119860) +119873minus1

sum119899gt119899

119899119879 (119899)

119879 minus 119879 (119873)(120575119860) (42)

Subtracting

120575119860119899119873 minus Δ119860 = 2 sum119899lt119899

119899119879 (119899)

119879 minus 119879 (119873)120575119860 (43)

therefore

Δ119860= 120575119860[119899119873minus 2 sum119899lt119899

119899119879 (119899)

119879 minus 119879 (119873)]

= 120575119860[119899119873minus 2 sum119899lt119899

119899119901 (119899)

1 minus 119901 (119873)]

(44)

The expression

2 sum119899lt119899

119899119901 (119899) (45)

is the average of the values truncated at the median Call thisvalue 119899

119879 Then substitution yields an upper bound on the

error measure due to violations in homogeneous arrivals of

Δ119860= 120575119860 [119899119873 minus

119899119879

1 minus 119901 (119873)] (46)

This bound may not be as useful as the bound on thehomogeneous service assumption error because the value ofΔ119860is based on knowing the119879(119899) values whereas forΔ

119878only

completion counts are necessary

Journal of Optimization 7

4 Example Performance MeasureError Bounds

Some examples are given next of howΔ119860andΔ

119878may be used

to determine bounds on the difference between exact valuesof various PMs and the OA estimated values for particularbehavior sequences

41 Arriverrsquos Average Queue Length Using the example fromthe introduction for a behavior sequence the average queuelength seen by an arriving job may be calculated by

119899119860= 119899119878

119860+ 119862119899119878

119860

(47)

where 119862119899119878

119860

= 119890119865minus119890lowast119878and 119899119878119860= (119899119880)minus1 is the average arriverrsquos

queue length assuming homogeneous servicesRearranging and using the Δ119860 bound give

119899119860minus 119899119878

119860= 119890119865minus 119890lowast

119878

119899119878

119860minus 119899119860 = 119890

lowast

119878minus 119890119865 le Δ 119878 minus 119890119865

(48)

Since 119890119865rarr 0 for any sequence of data in steady state we

assume flow balance holds Then

119899119878

119860minus 119899119860le Δ119878 (49)

This expression shows us that the difference between estimat-ing the average length seen by arrivers with the relationshipthat assumes homogeneous servers and the true value of 119899119860is less than or equal to the bound on 119890lowast

119878

42 Response Time The exact response time for a behaviorsequence can be found by

119877 = 119877119878+ 119862119877119878 (50)

where 119862119877119878 = minus119878119862

119899119878

119860

= minus119878(119890119865minus 119890lowast119878) and 119877119878 = 119878(119899

119860+ 1)

Since as before we are interested in the differencebetween a possible observed value (119877) and a calculated value(119877119878) we should do some rearranging and get 119877 minus 119877119878 = 119878(119890lowast

119878minus

119890119865) Again assuming 119890

119865is small and substituting 119890lowast

119878le Δ119878

yield

119877 minus 119877119878le 119878Δ 119878

119877 minus 119877119878le 119878 (120575

119878(119899119862minus 119899119862119879))

(51)

43 Average Number at Device If homogeneous arrivalshomogeneous services and job flow balance hold then theaverage number at a device (ie those both in queue and inservice) can be calculated by

119899119860119878=

119880

1 minus 119880 minus 119901 (119873)(1 minus (119873 + 1) 119901 (119873)) (52)

The exact average number if these assumptions do not hold is

119899 = 119899119860119878+ 119862119899119860119878 (53)

where

119862119899119860119878 =

119880

1 minus 119880 minus 119901 (119873)(1 minus 119901 (119873)) (119890

lowast

119860+ 119890lowast

119878minus 119890119865) (54)

Rearranging again

119899 minus 119899119860119878= 119862119899119860119878

=119880

1 minus 119880 minus 119901 (119873)(1 minus 119901 (119873)) (119890

lowast

119860+ 119890lowast

119878minus 119890119865)

(55)

and substituting 119890lowast119860le Δ119860 119890lowast119878le Δ119878 and 119890

119865= 0 we get

119899 minus 119899119860119878le

119880

1 minus 119880 minus 119901 (119873)(1 minus 119901 (119873)) (Δ119860 + Δ 119878) (56)

if 1 gt 119880 + 119901(119873) or

119899 minus 119899119860119878ge

119880

1 minus 119880 minus 119901 (119873)(1 minus 119901 (119873)) (Δ119860 + Δ 119878) (57)

if 1 lt 119880 + 119901(119873)Because the calculation ofΔ119860 requires knowledge of T(n)

values there may be little benefit in using the right hand sideof the above relationship to find this error bound If howeverthe behavior sequence can be assumed to have homogeneousarrivals then we can get a bound on the average numberat a device error without the knowledge of the individualT(n) values In that case the only time statistics we need areutilizationU and the fraction of time spent at the maximumstate p(N)

44 Throughput Using the OA version of Littlersquos Law [1] wecan say that the difference between a behavior sequencersquosactual throughput and that calculated assuming both homo-geneous arrivals and services will be

119883 minus 119883119860119878=119899

119877minus119899119860119878

119877119860119878 (58)

For 119877119860119878 we can substitute 119877119878 which is the more generalexpression since it does not need the homogeneous arrivalassumption From previous developments we know the fol-lowing relations hold

119899 = 119899119860119878+ 119862119899119860119878 (59)

where

119862119899119860119878 le

10038161003816100381610038161003816100381610038161003816

119880

1 minus 119880 minus 119901 (119873)(1 minus 119901 (119873)) (Δ119860 + Δ 119878)

10038161003816100381610038161003816100381610038161003816(60)

and 119877 = 119877119878 + 119862119877119878 where

119862119877119878 le 119878Δ

119878 (61)

Substituting into (58) gives

119883 minus 119883119860119878=119899119860119878 + 119862

119899119860119878

119877119878 + 119862119877119878minus119899119860119878

119877119878=119877119878119862119899119860119878 minus 119899119860119878119862119877119878

119877119878 (119877119878 + 119862119877119878)

(62)

8 Journal of Optimization

If the bound of (60) is substituted into (62) then it must bethat

1198771198781003816100381610038161003816(119880 (1 minus 119880 minus 119901 (119873))) (1 minus 119901 (119873)) (Δ119860 + Δ 119878)

1003816100381610038161003816 minus 119899119860119878119862119877119878

119877119878 (119877119878 + 119862119877119878)

ge119877119878119862119899119860119878 minus 119899119860119878119862119877119878

119877119878 (119877119878 + 119862119877119878)

(63)

If instead of bound (60) bound (61) is substituted thenumerator can only decrease and the denominator onlyincrease therefore

119877119878119862119899119860119878 minus 119899119860119878119878Δ

119878

119877119878 (119877119878 + 119878Δ119878)

le119877119878119862119899119860119878 minus 119899119860119878119862

119877119878

119877119878 (119877119878 + 119862119877119878)

(64)

If both (60) and (61) are substituted then the value of theexpression must fall within the limits defined by (63) and(64) Therefore the bounds on our throughput error are

119877119878119862119899119860119878 minus 119899119860119878119878Δ

119878

119877119878 (119877119878 + 119878Δ119878)

le 119883 minus 119883119860119878

le119877119878

1003816100381610038161003816(119880 (1 minus 119880 minus 119901 (119873))) (1 minus 119901 (119873)) (Δ119860 + Δ 119878)1003816100381610038161003816

119877119878 (119877119878 + 119862119877119878)

minus119899119860119878119862119877119878

119877119878 (119877119878 + 119862119877119878)

(65)

5 Using the PM Bounds

The bounds derived in the previous sections are actuallylimits on PM correction terms Assuming we know the 120575

119860

and 120575119878values a priori and that we can say something about

the homogeneous arrival assumption then these bounds canbe found without knowledge of all the 119879(119899) values for each nThis is the same simplification of data collection that we havein using the OA formulas instead of direct calculations

As an example of using the bounds in a simulation studyassume we are interested in finding an estimate for averagenumber at a device A series of 10 runs is made and the boundfor the correction term 119862119899 is calculated If it is positive wecan call this an upper bound If it is negative let this be alower bound In both cases the other bound is 0 Assumein the simulation runs these bounds always fall between minus6and 33 The correction term for each run is approximatedby taking the average value between the upper and lowerbounds We would like to be able to say something about theprobability that future runs will fall within the [minus6 33] limitsthat have already appeared We can do this using tolerancelimit calculations

Assume the 10 runs produced the results given in Table 2The average and standard deviation for these observations areminus134 and 155 respectively

Table 2 Example estimated correction terms for average number ata device

Run 119862119899

1 minus1752 minus3003 minus1254 1655 minus2506 minus1907 minus1758 1159 minus26010 minus145

Since the tolerance limits are going to be set at theobserved limits we can say

[minus6 33] = [119862119899 minus 119870119904 119862119899 + 119870119904]

119870 =119862119899minus (minus6)

119904=(33) minus 119862119899

119904= 30

(66)

From tolerance limit tables [28] we can say with 95confidence that at least 91 of future observations of 119862

119899will

fall within the interval [minus6 33] That is if we use 119899119860119878 valueswe have 95 confidence that the correct 119899 values for each runwill fall within these limits 91 of the time

6 Conclusion

In this paper bounds were developed for the operationalanalysis error measures of homogeneous service and arrivalassumptions These bounds allow us to take advantage of thesimplified data collection made possible by the use of opera-tional analysis relationships evenwhen the assumptions usedto derive those relationships are violated A tolerance limitbased method was given in order to be able to say somethingabout the confidence that future correction term values in atime series would be within certain limits

Abbreviations

119862 Total number of job completions at adevice

119862(119899) Total number of completions when n jobsare at a device

119862119909 Correction term makes the OA term 119909

equal to the directly calculatedperformance measure

119868 The set of states 0 1 2 119873 minus 1

119868+ The set of states 1 2 119873119870 Tolerance limit multiplier119873 The maximum number of jobs seen at a

device both in queue and in service

Journal of Optimization 9

119899 State of a device number of jobs both inqueue and in service

119899 The median state seen by a device119899119862 The median state at a device at job

completions119899 Average number of jobs at a device119899119860119878 The average number of jobs at a device

assuming flow balance homogeneousarrivals and homogeneous services

119899119862 The average number at the device seen bya completing job

119899119862119879 The average of the set of states seen by

completing job truncated at the medianstate value

119899119873 The average state at a device excluding the

maximum state119899119879 Time-average of the set of states truncated

at the median119899119878119860 Average number of jobs at a device seen by

an arriver assuming homogeneous service119901(119899) Proportion of time spent in state 119899119901119860(119899) The proportion of arrivals when 119899 jobs are

at a device 119860(119899)119860119901119862(119899) The proportion of completions that leave 119899

jobs at a device 119862(119899 + 1)119862119877 Average response time119877119878 Average response time assuming

homogeneous services119877119860119878 Average response time assuming

homogeneous arrivals and homogeneousservices

119878 Mean time between completions duringbusy periods

119904 Standard deviation119879 Total time of the period of observation119879(119899) Total time device was in state 119899119880 Utilization 1minusp(0)119883 Throughput119883119860119878 Throughput assuming homogeneous

arrivals and homogeneous services120575119860 Maximum error in homogeneous arrival

assumption error measure120575119878 Maximum error in homogeneous service

assumption error measureΔ119860 A bound on the homogeneous arrival

assumption error measure le 120575119860

Δ119878 A bound on the homogeneous service

assumption error measure le 120575119878

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] J P Buzen ldquoFundamental operational laws of computer systemperformancerdquo Acta Informatica vol 7 no 2 pp 167ndash182 1976

[2] J P Buzen ldquoOperational analysis an alternative to stochasticmodelingrdquo in Performance of Computer Installations pp 175ndash194 North-Holland Amsterdam The Netherlands 1978

[3] J P Buzen and P J Denning ldquoOperational treatment of queuedistributions andmean value analysisrdquo Tech Rep CSD-TR 309Department of Computer Science Purdue University 1979

[4] J P Buzen and P J Denning ldquoMeasuring and calculating queuelength distributionsrdquo Computer vol 13 no 4 pp 33ndash44 1980

[5] P J Denning and J P Buzen ldquoThe operational analysis ofqueueing network modelsrdquo Computer Surveys vol 10 no 3 pp225ndash261 1978

[6] J P Buzen ldquoFrom stochastic modeling to operational analysisthe journey beginsrdquo in Fundamental Concepts in ComputerScience E Gelenbe and J Kahane Eds pp 141ndash149 ImperialCollege Press London UK 2009

[7] E D Lazowska J Zahorjan G S Graham and K C SevcikQuantitative System Performance Prentice-Hall EnglewoodCliffs NJ USA 1984

[8] R Suri ldquoRobustness of queueing network formulasrdquo Journal ofthe ACM vol 30 no 3 pp 564ndash594 1983

[9] E D Lazowska J Zahorjan and E C Sevcik ldquoComputer sys-tem performance evaluation using queueing network modelsrdquoAnnual Reviews in Computer Science vol 1 pp 107ndash137 1986

[10] B V Ivanovskii ldquoOperational analysis of communication net-works with lockupsrdquoAutomatic Control and Computer Sciencesvol 22 no 3 pp 26ndash32 1988

[11] P J Denning ldquoQueueing in networks of computersrdquo AmericanScientist vol 79 pp 206ndash209 1991

[12] P J Denning ldquoIn the queue mean valuesrdquo American Scientistvol 79 pp 402ndash403 1991

[13] Y Dallery and X-R Cao ldquoOperational analysis of stochasticclosed queueing networksrdquo Performance Evaluation vol 14 no1 pp 43ndash61 1992

[14] M El-Taha and S Stidham Jr ldquoDeterministic analysis of queue-ing systems with heterogeneous serversrdquo Theoretical ComputerScience vol 106 no 2 pp 243ndash264 1992

[15] J P Buzen ldquoAn overview of performance prediction in MVSsystems and SNAnetworksrdquo inProceedings of the ACMFall JointComputer Conference pp 751ndash759 Dallas Tex USA 1986

[16] D A Menasce and V A Almeida Scaling for E-Business Tech-nologies Models Performance and Capacity Planning Prentice-Hall Englewood Cliffs NJ USA 2000

[17] D K Buch and V M Pentkovski ldquoPerformance characteriza-tion experience of multi-tier E-business systems using queuingoperational analysisrdquo in Proceedings of the IEEE InternationalSymposium on Performance Analysis of Systems and SoftwareTucson Ariz USA 2001

[18] M P Fanti B Maione G Piscitelli and B Turchiano ldquoTwomethod for real-time routing selection in flexible manufactur-ing systemsrdquo inProceedings of the IEEE International Conferenceon Robotics and Automation pp 1158ndash1166 May 1992

[19] W M Zuberek ldquoThroughput analysis in timed Petri netsrdquo inProceedings of the 35th Midwest Symposium on Circuits andSystems pp 1576ndash1580 Washington DC USA 1992

[20] W M Zuberek ldquoThroughput analysis of simple closed timedPetri netmodelsrdquo in Proceedings of the 36thMidwest Symposiumon Circuits and Systems pp 930ndash933 Detroit Mich USAAugust 1993

[21] G Chiola C Anglano J Campos J M Colom and M SilvaldquoOperational analysis of timed Petri nets and application to the

10 Journal of Optimization

computation of performance boundsrdquo in Quantitative Methodsin Parallel Systems F Baccelli A Jean-Marie and I MitraniEds Esprit Basic Research Series pp 161ndash174 Springer BerlinGermany 1995

[22] J A Brumfield Operational Analysis of Queuing Phenomena[PhD thesis] Department of Computer Science Purdue Uni-versity 1982

[23] N M Bengtson ldquoUsing operational analysis in simulation aqueuing network examplerdquo Journal of the Operational ResearchSociety vol 39 no 12 pp 1125ndash1136 1988

[24] N M Bengtson ldquoMeasuring errors in operational analysisassumptionsrdquo IEEE Transactions on Software Engineering vol13 no 7 pp 767ndash776 1987

[25] N M Bengtson ldquoOperational analysis revisited error measurelimits of assumptionsrdquo GSTF International Journal on Comput-ing vol 3 no 2 2013

[26] K C Sevcik and M Klawe ldquoOperational analysis versusstochastic modeling of computer systemsrdquo in Proceedings of theComputer Science and Statistics 12th Annual Symposium on theInterface pp 177ndash184 Waterloo Canada 1979

[27] F S Hillier and G J Lieberman Introduction to OperationsResearch McGraw-Hill New York NY USA 8th edition 2005

[28] C Eisenhart M W Hastay and W A Wallis Eds SelectedTechniques of Statistical Analysis McGraw-Hill New York NYUSA 1947

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Journal of Optimization

n

321

3

2

4 5 6

Time

1

Figure 1 Example Behavior Sequence for estimating Homoge-neous service error bound

Using the tighter bound Δ119878 we get

119880

1 minus 119880 minus 119901 (119873)(1 minus 119901 (119873)) Δ 119878

= minus10

3(120575119878 (119899119862 minus 119899119862119879))

= minus10

3(3

5(7

4minus 1)) = minus15

(35)

This is a reduction of 5714 for this example of the differencebound

32 Bounds onHomogeneous Arrival Assumption Error As inthe previous section we can assume that a maximum error120575119860 for any state is known beforehand That is we assume

1003816100381610038161003816119890119860 (119899)1003816100381610038161003816 le 120575119860 119899 isin 119868 (36)

Then the weak overall homogeneous arrival error is boundedby

119890lowast

119860le119873minus1

sum119899=0

119899119879 (119899)

119879 minus 119879 (119873)

1003816100381610038161003816119890119860 (119899)1003816100381610038161003816 le 120575119860

119873minus1

sum119899=0

119899119879 (119899)

119879 minus 119879 (119873)le 120575119860119899119873

(37)

where 119899119873is the average number at the device excluding the

maximum stateAs with the service errors 119890

119878(119899) the 119890

119860(119899) are not

independent The dependency is

119873minus1

sum119899=0

119879 (119899)

119879 minus 119879 (119873)119890119860 (119899) = 0 (38)

This can be shown by substituting the definition of 119890119860(119899) into

the equation As with the homogeneous service assumptionswe set up the following optimization problem

Max 119890lowast

119860=119873minus1

sum119899=0

119899119879 (119899)

119879 minus 119879 (119873)119890119860 (119899)

st119873minus1

sum119899=0

119879 (119899)

119879 minus 119879 (119873)119890119860 (119899) = 0

1003816100381610038161003816119890119860 (119899)1003816100381610038161003816 le 120575119860 119899 isin 119868

(39)

Using the Karush-Kuhn-Tucker conditions as before we canshow that with 119899 as the median state

119890119860 (119899) = minus120575119860 if 119899 lt 119899

119890119860 (119899) = 120575119860 if 119899 gt 119899

119906 = 119899

(40)

is the solution to the optimization problem The value of thesolution is found by substituting into the primal objectivefunction to get

Δ119860= max 119890lowast

119860

= sum119899lt119899

119899119879 (119899)

119879 minus 119879 (119873)(minus120575119860) +119873minus1

sum119899gt119899

119899119879 (119899)

119879 minus 119879 (119873)(120575119860)

(41)

120575119860times the average excluding the maximum 119899

119873 is

120575119860119899119873= sum119899lt119899

119899119879 (119899)

119879 minus 119879 (119873)(120575119860) +119873minus1

sum119899gt119899

119899119879 (119899)

119879 minus 119879 (119873)(120575119860) (42)

Subtracting

120575119860119899119873 minus Δ119860 = 2 sum119899lt119899

119899119879 (119899)

119879 minus 119879 (119873)120575119860 (43)

therefore

Δ119860= 120575119860[119899119873minus 2 sum119899lt119899

119899119879 (119899)

119879 minus 119879 (119873)]

= 120575119860[119899119873minus 2 sum119899lt119899

119899119901 (119899)

1 minus 119901 (119873)]

(44)

The expression

2 sum119899lt119899

119899119901 (119899) (45)

is the average of the values truncated at the median Call thisvalue 119899

119879 Then substitution yields an upper bound on the

error measure due to violations in homogeneous arrivals of

Δ119860= 120575119860 [119899119873 minus

119899119879

1 minus 119901 (119873)] (46)

This bound may not be as useful as the bound on thehomogeneous service assumption error because the value ofΔ119860is based on knowing the119879(119899) values whereas forΔ

119878only

completion counts are necessary

Journal of Optimization 7

4 Example Performance MeasureError Bounds

Some examples are given next of howΔ119860andΔ

119878may be used

to determine bounds on the difference between exact valuesof various PMs and the OA estimated values for particularbehavior sequences

41 Arriverrsquos Average Queue Length Using the example fromthe introduction for a behavior sequence the average queuelength seen by an arriving job may be calculated by

119899119860= 119899119878

119860+ 119862119899119878

119860

(47)

where 119862119899119878

119860

= 119890119865minus119890lowast119878and 119899119878119860= (119899119880)minus1 is the average arriverrsquos

queue length assuming homogeneous servicesRearranging and using the Δ119860 bound give

119899119860minus 119899119878

119860= 119890119865minus 119890lowast

119878

119899119878

119860minus 119899119860 = 119890

lowast

119878minus 119890119865 le Δ 119878 minus 119890119865

(48)

Since 119890119865rarr 0 for any sequence of data in steady state we

assume flow balance holds Then

119899119878

119860minus 119899119860le Δ119878 (49)

This expression shows us that the difference between estimat-ing the average length seen by arrivers with the relationshipthat assumes homogeneous servers and the true value of 119899119860is less than or equal to the bound on 119890lowast

119878

42 Response Time The exact response time for a behaviorsequence can be found by

119877 = 119877119878+ 119862119877119878 (50)

where 119862119877119878 = minus119878119862

119899119878

119860

= minus119878(119890119865minus 119890lowast119878) and 119877119878 = 119878(119899

119860+ 1)

Since as before we are interested in the differencebetween a possible observed value (119877) and a calculated value(119877119878) we should do some rearranging and get 119877 minus 119877119878 = 119878(119890lowast

119878minus

119890119865) Again assuming 119890

119865is small and substituting 119890lowast

119878le Δ119878

yield

119877 minus 119877119878le 119878Δ 119878

119877 minus 119877119878le 119878 (120575

119878(119899119862minus 119899119862119879))

(51)

43 Average Number at Device If homogeneous arrivalshomogeneous services and job flow balance hold then theaverage number at a device (ie those both in queue and inservice) can be calculated by

119899119860119878=

119880

1 minus 119880 minus 119901 (119873)(1 minus (119873 + 1) 119901 (119873)) (52)

The exact average number if these assumptions do not hold is

119899 = 119899119860119878+ 119862119899119860119878 (53)

where

119862119899119860119878 =

119880

1 minus 119880 minus 119901 (119873)(1 minus 119901 (119873)) (119890

lowast

119860+ 119890lowast

119878minus 119890119865) (54)

Rearranging again

119899 minus 119899119860119878= 119862119899119860119878

=119880

1 minus 119880 minus 119901 (119873)(1 minus 119901 (119873)) (119890

lowast

119860+ 119890lowast

119878minus 119890119865)

(55)

and substituting 119890lowast119860le Δ119860 119890lowast119878le Δ119878 and 119890

119865= 0 we get

119899 minus 119899119860119878le

119880

1 minus 119880 minus 119901 (119873)(1 minus 119901 (119873)) (Δ119860 + Δ 119878) (56)

if 1 gt 119880 + 119901(119873) or

119899 minus 119899119860119878ge

119880

1 minus 119880 minus 119901 (119873)(1 minus 119901 (119873)) (Δ119860 + Δ 119878) (57)

if 1 lt 119880 + 119901(119873)Because the calculation ofΔ119860 requires knowledge of T(n)

values there may be little benefit in using the right hand sideof the above relationship to find this error bound If howeverthe behavior sequence can be assumed to have homogeneousarrivals then we can get a bound on the average numberat a device error without the knowledge of the individualT(n) values In that case the only time statistics we need areutilizationU and the fraction of time spent at the maximumstate p(N)

44 Throughput Using the OA version of Littlersquos Law [1] wecan say that the difference between a behavior sequencersquosactual throughput and that calculated assuming both homo-geneous arrivals and services will be

119883 minus 119883119860119878=119899

119877minus119899119860119878

119877119860119878 (58)

For 119877119860119878 we can substitute 119877119878 which is the more generalexpression since it does not need the homogeneous arrivalassumption From previous developments we know the fol-lowing relations hold

119899 = 119899119860119878+ 119862119899119860119878 (59)

where

119862119899119860119878 le

10038161003816100381610038161003816100381610038161003816

119880

1 minus 119880 minus 119901 (119873)(1 minus 119901 (119873)) (Δ119860 + Δ 119878)

10038161003816100381610038161003816100381610038161003816(60)

and 119877 = 119877119878 + 119862119877119878 where

119862119877119878 le 119878Δ

119878 (61)

Substituting into (58) gives

119883 minus 119883119860119878=119899119860119878 + 119862

119899119860119878

119877119878 + 119862119877119878minus119899119860119878

119877119878=119877119878119862119899119860119878 minus 119899119860119878119862119877119878

119877119878 (119877119878 + 119862119877119878)

(62)

8 Journal of Optimization

If the bound of (60) is substituted into (62) then it must bethat

1198771198781003816100381610038161003816(119880 (1 minus 119880 minus 119901 (119873))) (1 minus 119901 (119873)) (Δ119860 + Δ 119878)

1003816100381610038161003816 minus 119899119860119878119862119877119878

119877119878 (119877119878 + 119862119877119878)

ge119877119878119862119899119860119878 minus 119899119860119878119862119877119878

119877119878 (119877119878 + 119862119877119878)

(63)

If instead of bound (60) bound (61) is substituted thenumerator can only decrease and the denominator onlyincrease therefore

119877119878119862119899119860119878 minus 119899119860119878119878Δ

119878

119877119878 (119877119878 + 119878Δ119878)

le119877119878119862119899119860119878 minus 119899119860119878119862

119877119878

119877119878 (119877119878 + 119862119877119878)

(64)

If both (60) and (61) are substituted then the value of theexpression must fall within the limits defined by (63) and(64) Therefore the bounds on our throughput error are

119877119878119862119899119860119878 minus 119899119860119878119878Δ

119878

119877119878 (119877119878 + 119878Δ119878)

le 119883 minus 119883119860119878

le119877119878

1003816100381610038161003816(119880 (1 minus 119880 minus 119901 (119873))) (1 minus 119901 (119873)) (Δ119860 + Δ 119878)1003816100381610038161003816

119877119878 (119877119878 + 119862119877119878)

minus119899119860119878119862119877119878

119877119878 (119877119878 + 119862119877119878)

(65)

5 Using the PM Bounds

The bounds derived in the previous sections are actuallylimits on PM correction terms Assuming we know the 120575

119860

and 120575119878values a priori and that we can say something about

the homogeneous arrival assumption then these bounds canbe found without knowledge of all the 119879(119899) values for each nThis is the same simplification of data collection that we havein using the OA formulas instead of direct calculations

As an example of using the bounds in a simulation studyassume we are interested in finding an estimate for averagenumber at a device A series of 10 runs is made and the boundfor the correction term 119862119899 is calculated If it is positive wecan call this an upper bound If it is negative let this be alower bound In both cases the other bound is 0 Assumein the simulation runs these bounds always fall between minus6and 33 The correction term for each run is approximatedby taking the average value between the upper and lowerbounds We would like to be able to say something about theprobability that future runs will fall within the [minus6 33] limitsthat have already appeared We can do this using tolerancelimit calculations

Assume the 10 runs produced the results given in Table 2The average and standard deviation for these observations areminus134 and 155 respectively

Table 2 Example estimated correction terms for average number ata device

Run 119862119899

1 minus1752 minus3003 minus1254 1655 minus2506 minus1907 minus1758 1159 minus26010 minus145

Since the tolerance limits are going to be set at theobserved limits we can say

[minus6 33] = [119862119899 minus 119870119904 119862119899 + 119870119904]

119870 =119862119899minus (minus6)

119904=(33) minus 119862119899

119904= 30

(66)

From tolerance limit tables [28] we can say with 95confidence that at least 91 of future observations of 119862

119899will

fall within the interval [minus6 33] That is if we use 119899119860119878 valueswe have 95 confidence that the correct 119899 values for each runwill fall within these limits 91 of the time

6 Conclusion

In this paper bounds were developed for the operationalanalysis error measures of homogeneous service and arrivalassumptions These bounds allow us to take advantage of thesimplified data collection made possible by the use of opera-tional analysis relationships evenwhen the assumptions usedto derive those relationships are violated A tolerance limitbased method was given in order to be able to say somethingabout the confidence that future correction term values in atime series would be within certain limits

Abbreviations

119862 Total number of job completions at adevice

119862(119899) Total number of completions when n jobsare at a device

119862119909 Correction term makes the OA term 119909

equal to the directly calculatedperformance measure

119868 The set of states 0 1 2 119873 minus 1

119868+ The set of states 1 2 119873119870 Tolerance limit multiplier119873 The maximum number of jobs seen at a

device both in queue and in service

Journal of Optimization 9

119899 State of a device number of jobs both inqueue and in service

119899 The median state seen by a device119899119862 The median state at a device at job

completions119899 Average number of jobs at a device119899119860119878 The average number of jobs at a device

assuming flow balance homogeneousarrivals and homogeneous services

119899119862 The average number at the device seen bya completing job

119899119862119879 The average of the set of states seen by

completing job truncated at the medianstate value

119899119873 The average state at a device excluding the

maximum state119899119879 Time-average of the set of states truncated

at the median119899119878119860 Average number of jobs at a device seen by

an arriver assuming homogeneous service119901(119899) Proportion of time spent in state 119899119901119860(119899) The proportion of arrivals when 119899 jobs are

at a device 119860(119899)119860119901119862(119899) The proportion of completions that leave 119899

jobs at a device 119862(119899 + 1)119862119877 Average response time119877119878 Average response time assuming

homogeneous services119877119860119878 Average response time assuming

homogeneous arrivals and homogeneousservices

119878 Mean time between completions duringbusy periods

119904 Standard deviation119879 Total time of the period of observation119879(119899) Total time device was in state 119899119880 Utilization 1minusp(0)119883 Throughput119883119860119878 Throughput assuming homogeneous

arrivals and homogeneous services120575119860 Maximum error in homogeneous arrival

assumption error measure120575119878 Maximum error in homogeneous service

assumption error measureΔ119860 A bound on the homogeneous arrival

assumption error measure le 120575119860

Δ119878 A bound on the homogeneous service

assumption error measure le 120575119878

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] J P Buzen ldquoFundamental operational laws of computer systemperformancerdquo Acta Informatica vol 7 no 2 pp 167ndash182 1976

[2] J P Buzen ldquoOperational analysis an alternative to stochasticmodelingrdquo in Performance of Computer Installations pp 175ndash194 North-Holland Amsterdam The Netherlands 1978

[3] J P Buzen and P J Denning ldquoOperational treatment of queuedistributions andmean value analysisrdquo Tech Rep CSD-TR 309Department of Computer Science Purdue University 1979

[4] J P Buzen and P J Denning ldquoMeasuring and calculating queuelength distributionsrdquo Computer vol 13 no 4 pp 33ndash44 1980

[5] P J Denning and J P Buzen ldquoThe operational analysis ofqueueing network modelsrdquo Computer Surveys vol 10 no 3 pp225ndash261 1978

[6] J P Buzen ldquoFrom stochastic modeling to operational analysisthe journey beginsrdquo in Fundamental Concepts in ComputerScience E Gelenbe and J Kahane Eds pp 141ndash149 ImperialCollege Press London UK 2009

[7] E D Lazowska J Zahorjan G S Graham and K C SevcikQuantitative System Performance Prentice-Hall EnglewoodCliffs NJ USA 1984

[8] R Suri ldquoRobustness of queueing network formulasrdquo Journal ofthe ACM vol 30 no 3 pp 564ndash594 1983

[9] E D Lazowska J Zahorjan and E C Sevcik ldquoComputer sys-tem performance evaluation using queueing network modelsrdquoAnnual Reviews in Computer Science vol 1 pp 107ndash137 1986

[10] B V Ivanovskii ldquoOperational analysis of communication net-works with lockupsrdquoAutomatic Control and Computer Sciencesvol 22 no 3 pp 26ndash32 1988

[11] P J Denning ldquoQueueing in networks of computersrdquo AmericanScientist vol 79 pp 206ndash209 1991

[12] P J Denning ldquoIn the queue mean valuesrdquo American Scientistvol 79 pp 402ndash403 1991

[13] Y Dallery and X-R Cao ldquoOperational analysis of stochasticclosed queueing networksrdquo Performance Evaluation vol 14 no1 pp 43ndash61 1992

[14] M El-Taha and S Stidham Jr ldquoDeterministic analysis of queue-ing systems with heterogeneous serversrdquo Theoretical ComputerScience vol 106 no 2 pp 243ndash264 1992

[15] J P Buzen ldquoAn overview of performance prediction in MVSsystems and SNAnetworksrdquo inProceedings of the ACMFall JointComputer Conference pp 751ndash759 Dallas Tex USA 1986

[16] D A Menasce and V A Almeida Scaling for E-Business Tech-nologies Models Performance and Capacity Planning Prentice-Hall Englewood Cliffs NJ USA 2000

[17] D K Buch and V M Pentkovski ldquoPerformance characteriza-tion experience of multi-tier E-business systems using queuingoperational analysisrdquo in Proceedings of the IEEE InternationalSymposium on Performance Analysis of Systems and SoftwareTucson Ariz USA 2001

[18] M P Fanti B Maione G Piscitelli and B Turchiano ldquoTwomethod for real-time routing selection in flexible manufactur-ing systemsrdquo inProceedings of the IEEE International Conferenceon Robotics and Automation pp 1158ndash1166 May 1992

[19] W M Zuberek ldquoThroughput analysis in timed Petri netsrdquo inProceedings of the 35th Midwest Symposium on Circuits andSystems pp 1576ndash1580 Washington DC USA 1992

[20] W M Zuberek ldquoThroughput analysis of simple closed timedPetri netmodelsrdquo in Proceedings of the 36thMidwest Symposiumon Circuits and Systems pp 930ndash933 Detroit Mich USAAugust 1993

[21] G Chiola C Anglano J Campos J M Colom and M SilvaldquoOperational analysis of timed Petri nets and application to the

10 Journal of Optimization

computation of performance boundsrdquo in Quantitative Methodsin Parallel Systems F Baccelli A Jean-Marie and I MitraniEds Esprit Basic Research Series pp 161ndash174 Springer BerlinGermany 1995

[22] J A Brumfield Operational Analysis of Queuing Phenomena[PhD thesis] Department of Computer Science Purdue Uni-versity 1982

[23] N M Bengtson ldquoUsing operational analysis in simulation aqueuing network examplerdquo Journal of the Operational ResearchSociety vol 39 no 12 pp 1125ndash1136 1988

[24] N M Bengtson ldquoMeasuring errors in operational analysisassumptionsrdquo IEEE Transactions on Software Engineering vol13 no 7 pp 767ndash776 1987

[25] N M Bengtson ldquoOperational analysis revisited error measurelimits of assumptionsrdquo GSTF International Journal on Comput-ing vol 3 no 2 2013

[26] K C Sevcik and M Klawe ldquoOperational analysis versusstochastic modeling of computer systemsrdquo in Proceedings of theComputer Science and Statistics 12th Annual Symposium on theInterface pp 177ndash184 Waterloo Canada 1979

[27] F S Hillier and G J Lieberman Introduction to OperationsResearch McGraw-Hill New York NY USA 8th edition 2005

[28] C Eisenhart M W Hastay and W A Wallis Eds SelectedTechniques of Statistical Analysis McGraw-Hill New York NYUSA 1947

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Optimization 7

4 Example Performance MeasureError Bounds

Some examples are given next of howΔ119860andΔ

119878may be used

to determine bounds on the difference between exact valuesof various PMs and the OA estimated values for particularbehavior sequences

41 Arriverrsquos Average Queue Length Using the example fromthe introduction for a behavior sequence the average queuelength seen by an arriving job may be calculated by

119899119860= 119899119878

119860+ 119862119899119878

119860

(47)

where 119862119899119878

119860

= 119890119865minus119890lowast119878and 119899119878119860= (119899119880)minus1 is the average arriverrsquos

queue length assuming homogeneous servicesRearranging and using the Δ119860 bound give

119899119860minus 119899119878

119860= 119890119865minus 119890lowast

119878

119899119878

119860minus 119899119860 = 119890

lowast

119878minus 119890119865 le Δ 119878 minus 119890119865

(48)

Since 119890119865rarr 0 for any sequence of data in steady state we

assume flow balance holds Then

119899119878

119860minus 119899119860le Δ119878 (49)

This expression shows us that the difference between estimat-ing the average length seen by arrivers with the relationshipthat assumes homogeneous servers and the true value of 119899119860is less than or equal to the bound on 119890lowast

119878

42 Response Time The exact response time for a behaviorsequence can be found by

119877 = 119877119878+ 119862119877119878 (50)

where 119862119877119878 = minus119878119862

119899119878

119860

= minus119878(119890119865minus 119890lowast119878) and 119877119878 = 119878(119899

119860+ 1)

Since as before we are interested in the differencebetween a possible observed value (119877) and a calculated value(119877119878) we should do some rearranging and get 119877 minus 119877119878 = 119878(119890lowast

119878minus

119890119865) Again assuming 119890

119865is small and substituting 119890lowast

119878le Δ119878

yield

119877 minus 119877119878le 119878Δ 119878

119877 minus 119877119878le 119878 (120575

119878(119899119862minus 119899119862119879))

(51)

43 Average Number at Device If homogeneous arrivalshomogeneous services and job flow balance hold then theaverage number at a device (ie those both in queue and inservice) can be calculated by

119899119860119878=

119880

1 minus 119880 minus 119901 (119873)(1 minus (119873 + 1) 119901 (119873)) (52)

The exact average number if these assumptions do not hold is

119899 = 119899119860119878+ 119862119899119860119878 (53)

where

119862119899119860119878 =

119880

1 minus 119880 minus 119901 (119873)(1 minus 119901 (119873)) (119890

lowast

119860+ 119890lowast

119878minus 119890119865) (54)

Rearranging again

119899 minus 119899119860119878= 119862119899119860119878

=119880

1 minus 119880 minus 119901 (119873)(1 minus 119901 (119873)) (119890

lowast

119860+ 119890lowast

119878minus 119890119865)

(55)

and substituting 119890lowast119860le Δ119860 119890lowast119878le Δ119878 and 119890

119865= 0 we get

119899 minus 119899119860119878le

119880

1 minus 119880 minus 119901 (119873)(1 minus 119901 (119873)) (Δ119860 + Δ 119878) (56)

if 1 gt 119880 + 119901(119873) or

119899 minus 119899119860119878ge

119880

1 minus 119880 minus 119901 (119873)(1 minus 119901 (119873)) (Δ119860 + Δ 119878) (57)

if 1 lt 119880 + 119901(119873)Because the calculation ofΔ119860 requires knowledge of T(n)

values there may be little benefit in using the right hand sideof the above relationship to find this error bound If howeverthe behavior sequence can be assumed to have homogeneousarrivals then we can get a bound on the average numberat a device error without the knowledge of the individualT(n) values In that case the only time statistics we need areutilizationU and the fraction of time spent at the maximumstate p(N)

44 Throughput Using the OA version of Littlersquos Law [1] wecan say that the difference between a behavior sequencersquosactual throughput and that calculated assuming both homo-geneous arrivals and services will be

119883 minus 119883119860119878=119899

119877minus119899119860119878

119877119860119878 (58)

For 119877119860119878 we can substitute 119877119878 which is the more generalexpression since it does not need the homogeneous arrivalassumption From previous developments we know the fol-lowing relations hold

119899 = 119899119860119878+ 119862119899119860119878 (59)

where

119862119899119860119878 le

10038161003816100381610038161003816100381610038161003816

119880

1 minus 119880 minus 119901 (119873)(1 minus 119901 (119873)) (Δ119860 + Δ 119878)

10038161003816100381610038161003816100381610038161003816(60)

and 119877 = 119877119878 + 119862119877119878 where

119862119877119878 le 119878Δ

119878 (61)

Substituting into (58) gives

119883 minus 119883119860119878=119899119860119878 + 119862

119899119860119878

119877119878 + 119862119877119878minus119899119860119878

119877119878=119877119878119862119899119860119878 minus 119899119860119878119862119877119878

119877119878 (119877119878 + 119862119877119878)

(62)

8 Journal of Optimization

If the bound of (60) is substituted into (62) then it must bethat

1198771198781003816100381610038161003816(119880 (1 minus 119880 minus 119901 (119873))) (1 minus 119901 (119873)) (Δ119860 + Δ 119878)

1003816100381610038161003816 minus 119899119860119878119862119877119878

119877119878 (119877119878 + 119862119877119878)

ge119877119878119862119899119860119878 minus 119899119860119878119862119877119878

119877119878 (119877119878 + 119862119877119878)

(63)

If instead of bound (60) bound (61) is substituted thenumerator can only decrease and the denominator onlyincrease therefore

119877119878119862119899119860119878 minus 119899119860119878119878Δ

119878

119877119878 (119877119878 + 119878Δ119878)

le119877119878119862119899119860119878 minus 119899119860119878119862

119877119878

119877119878 (119877119878 + 119862119877119878)

(64)

If both (60) and (61) are substituted then the value of theexpression must fall within the limits defined by (63) and(64) Therefore the bounds on our throughput error are

119877119878119862119899119860119878 minus 119899119860119878119878Δ

119878

119877119878 (119877119878 + 119878Δ119878)

le 119883 minus 119883119860119878

le119877119878

1003816100381610038161003816(119880 (1 minus 119880 minus 119901 (119873))) (1 minus 119901 (119873)) (Δ119860 + Δ 119878)1003816100381610038161003816

119877119878 (119877119878 + 119862119877119878)

minus119899119860119878119862119877119878

119877119878 (119877119878 + 119862119877119878)

(65)

5 Using the PM Bounds

The bounds derived in the previous sections are actuallylimits on PM correction terms Assuming we know the 120575

119860

and 120575119878values a priori and that we can say something about

the homogeneous arrival assumption then these bounds canbe found without knowledge of all the 119879(119899) values for each nThis is the same simplification of data collection that we havein using the OA formulas instead of direct calculations

As an example of using the bounds in a simulation studyassume we are interested in finding an estimate for averagenumber at a device A series of 10 runs is made and the boundfor the correction term 119862119899 is calculated If it is positive wecan call this an upper bound If it is negative let this be alower bound In both cases the other bound is 0 Assumein the simulation runs these bounds always fall between minus6and 33 The correction term for each run is approximatedby taking the average value between the upper and lowerbounds We would like to be able to say something about theprobability that future runs will fall within the [minus6 33] limitsthat have already appeared We can do this using tolerancelimit calculations

Assume the 10 runs produced the results given in Table 2The average and standard deviation for these observations areminus134 and 155 respectively

Table 2 Example estimated correction terms for average number ata device

Run 119862119899

1 minus1752 minus3003 minus1254 1655 minus2506 minus1907 minus1758 1159 minus26010 minus145

Since the tolerance limits are going to be set at theobserved limits we can say

[minus6 33] = [119862119899 minus 119870119904 119862119899 + 119870119904]

119870 =119862119899minus (minus6)

119904=(33) minus 119862119899

119904= 30

(66)

From tolerance limit tables [28] we can say with 95confidence that at least 91 of future observations of 119862

119899will

fall within the interval [minus6 33] That is if we use 119899119860119878 valueswe have 95 confidence that the correct 119899 values for each runwill fall within these limits 91 of the time

6 Conclusion

In this paper bounds were developed for the operationalanalysis error measures of homogeneous service and arrivalassumptions These bounds allow us to take advantage of thesimplified data collection made possible by the use of opera-tional analysis relationships evenwhen the assumptions usedto derive those relationships are violated A tolerance limitbased method was given in order to be able to say somethingabout the confidence that future correction term values in atime series would be within certain limits

Abbreviations

119862 Total number of job completions at adevice

119862(119899) Total number of completions when n jobsare at a device

119862119909 Correction term makes the OA term 119909

equal to the directly calculatedperformance measure

119868 The set of states 0 1 2 119873 minus 1

119868+ The set of states 1 2 119873119870 Tolerance limit multiplier119873 The maximum number of jobs seen at a

device both in queue and in service

Journal of Optimization 9

119899 State of a device number of jobs both inqueue and in service

119899 The median state seen by a device119899119862 The median state at a device at job

completions119899 Average number of jobs at a device119899119860119878 The average number of jobs at a device

assuming flow balance homogeneousarrivals and homogeneous services

119899119862 The average number at the device seen bya completing job

119899119862119879 The average of the set of states seen by

completing job truncated at the medianstate value

119899119873 The average state at a device excluding the

maximum state119899119879 Time-average of the set of states truncated

at the median119899119878119860 Average number of jobs at a device seen by

an arriver assuming homogeneous service119901(119899) Proportion of time spent in state 119899119901119860(119899) The proportion of arrivals when 119899 jobs are

at a device 119860(119899)119860119901119862(119899) The proportion of completions that leave 119899

jobs at a device 119862(119899 + 1)119862119877 Average response time119877119878 Average response time assuming

homogeneous services119877119860119878 Average response time assuming

homogeneous arrivals and homogeneousservices

119878 Mean time between completions duringbusy periods

119904 Standard deviation119879 Total time of the period of observation119879(119899) Total time device was in state 119899119880 Utilization 1minusp(0)119883 Throughput119883119860119878 Throughput assuming homogeneous

arrivals and homogeneous services120575119860 Maximum error in homogeneous arrival

assumption error measure120575119878 Maximum error in homogeneous service

assumption error measureΔ119860 A bound on the homogeneous arrival

assumption error measure le 120575119860

Δ119878 A bound on the homogeneous service

assumption error measure le 120575119878

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] J P Buzen ldquoFundamental operational laws of computer systemperformancerdquo Acta Informatica vol 7 no 2 pp 167ndash182 1976

[2] J P Buzen ldquoOperational analysis an alternative to stochasticmodelingrdquo in Performance of Computer Installations pp 175ndash194 North-Holland Amsterdam The Netherlands 1978

[3] J P Buzen and P J Denning ldquoOperational treatment of queuedistributions andmean value analysisrdquo Tech Rep CSD-TR 309Department of Computer Science Purdue University 1979

[4] J P Buzen and P J Denning ldquoMeasuring and calculating queuelength distributionsrdquo Computer vol 13 no 4 pp 33ndash44 1980

[5] P J Denning and J P Buzen ldquoThe operational analysis ofqueueing network modelsrdquo Computer Surveys vol 10 no 3 pp225ndash261 1978

[6] J P Buzen ldquoFrom stochastic modeling to operational analysisthe journey beginsrdquo in Fundamental Concepts in ComputerScience E Gelenbe and J Kahane Eds pp 141ndash149 ImperialCollege Press London UK 2009

[7] E D Lazowska J Zahorjan G S Graham and K C SevcikQuantitative System Performance Prentice-Hall EnglewoodCliffs NJ USA 1984

[8] R Suri ldquoRobustness of queueing network formulasrdquo Journal ofthe ACM vol 30 no 3 pp 564ndash594 1983

[9] E D Lazowska J Zahorjan and E C Sevcik ldquoComputer sys-tem performance evaluation using queueing network modelsrdquoAnnual Reviews in Computer Science vol 1 pp 107ndash137 1986

[10] B V Ivanovskii ldquoOperational analysis of communication net-works with lockupsrdquoAutomatic Control and Computer Sciencesvol 22 no 3 pp 26ndash32 1988

[11] P J Denning ldquoQueueing in networks of computersrdquo AmericanScientist vol 79 pp 206ndash209 1991

[12] P J Denning ldquoIn the queue mean valuesrdquo American Scientistvol 79 pp 402ndash403 1991

[13] Y Dallery and X-R Cao ldquoOperational analysis of stochasticclosed queueing networksrdquo Performance Evaluation vol 14 no1 pp 43ndash61 1992

[14] M El-Taha and S Stidham Jr ldquoDeterministic analysis of queue-ing systems with heterogeneous serversrdquo Theoretical ComputerScience vol 106 no 2 pp 243ndash264 1992

[15] J P Buzen ldquoAn overview of performance prediction in MVSsystems and SNAnetworksrdquo inProceedings of the ACMFall JointComputer Conference pp 751ndash759 Dallas Tex USA 1986

[16] D A Menasce and V A Almeida Scaling for E-Business Tech-nologies Models Performance and Capacity Planning Prentice-Hall Englewood Cliffs NJ USA 2000

[17] D K Buch and V M Pentkovski ldquoPerformance characteriza-tion experience of multi-tier E-business systems using queuingoperational analysisrdquo in Proceedings of the IEEE InternationalSymposium on Performance Analysis of Systems and SoftwareTucson Ariz USA 2001

[18] M P Fanti B Maione G Piscitelli and B Turchiano ldquoTwomethod for real-time routing selection in flexible manufactur-ing systemsrdquo inProceedings of the IEEE International Conferenceon Robotics and Automation pp 1158ndash1166 May 1992

[19] W M Zuberek ldquoThroughput analysis in timed Petri netsrdquo inProceedings of the 35th Midwest Symposium on Circuits andSystems pp 1576ndash1580 Washington DC USA 1992

[20] W M Zuberek ldquoThroughput analysis of simple closed timedPetri netmodelsrdquo in Proceedings of the 36thMidwest Symposiumon Circuits and Systems pp 930ndash933 Detroit Mich USAAugust 1993

[21] G Chiola C Anglano J Campos J M Colom and M SilvaldquoOperational analysis of timed Petri nets and application to the

10 Journal of Optimization

computation of performance boundsrdquo in Quantitative Methodsin Parallel Systems F Baccelli A Jean-Marie and I MitraniEds Esprit Basic Research Series pp 161ndash174 Springer BerlinGermany 1995

[22] J A Brumfield Operational Analysis of Queuing Phenomena[PhD thesis] Department of Computer Science Purdue Uni-versity 1982

[23] N M Bengtson ldquoUsing operational analysis in simulation aqueuing network examplerdquo Journal of the Operational ResearchSociety vol 39 no 12 pp 1125ndash1136 1988

[24] N M Bengtson ldquoMeasuring errors in operational analysisassumptionsrdquo IEEE Transactions on Software Engineering vol13 no 7 pp 767ndash776 1987

[25] N M Bengtson ldquoOperational analysis revisited error measurelimits of assumptionsrdquo GSTF International Journal on Comput-ing vol 3 no 2 2013

[26] K C Sevcik and M Klawe ldquoOperational analysis versusstochastic modeling of computer systemsrdquo in Proceedings of theComputer Science and Statistics 12th Annual Symposium on theInterface pp 177ndash184 Waterloo Canada 1979

[27] F S Hillier and G J Lieberman Introduction to OperationsResearch McGraw-Hill New York NY USA 8th edition 2005

[28] C Eisenhart M W Hastay and W A Wallis Eds SelectedTechniques of Statistical Analysis McGraw-Hill New York NYUSA 1947

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Journal of Optimization

If the bound of (60) is substituted into (62) then it must bethat

1198771198781003816100381610038161003816(119880 (1 minus 119880 minus 119901 (119873))) (1 minus 119901 (119873)) (Δ119860 + Δ 119878)

1003816100381610038161003816 minus 119899119860119878119862119877119878

119877119878 (119877119878 + 119862119877119878)

ge119877119878119862119899119860119878 minus 119899119860119878119862119877119878

119877119878 (119877119878 + 119862119877119878)

(63)

If instead of bound (60) bound (61) is substituted thenumerator can only decrease and the denominator onlyincrease therefore

119877119878119862119899119860119878 minus 119899119860119878119878Δ

119878

119877119878 (119877119878 + 119878Δ119878)

le119877119878119862119899119860119878 minus 119899119860119878119862

119877119878

119877119878 (119877119878 + 119862119877119878)

(64)

If both (60) and (61) are substituted then the value of theexpression must fall within the limits defined by (63) and(64) Therefore the bounds on our throughput error are

119877119878119862119899119860119878 minus 119899119860119878119878Δ

119878

119877119878 (119877119878 + 119878Δ119878)

le 119883 minus 119883119860119878

le119877119878

1003816100381610038161003816(119880 (1 minus 119880 minus 119901 (119873))) (1 minus 119901 (119873)) (Δ119860 + Δ 119878)1003816100381610038161003816

119877119878 (119877119878 + 119862119877119878)

minus119899119860119878119862119877119878

119877119878 (119877119878 + 119862119877119878)

(65)

5 Using the PM Bounds

The bounds derived in the previous sections are actuallylimits on PM correction terms Assuming we know the 120575

119860

and 120575119878values a priori and that we can say something about

the homogeneous arrival assumption then these bounds canbe found without knowledge of all the 119879(119899) values for each nThis is the same simplification of data collection that we havein using the OA formulas instead of direct calculations

As an example of using the bounds in a simulation studyassume we are interested in finding an estimate for averagenumber at a device A series of 10 runs is made and the boundfor the correction term 119862119899 is calculated If it is positive wecan call this an upper bound If it is negative let this be alower bound In both cases the other bound is 0 Assumein the simulation runs these bounds always fall between minus6and 33 The correction term for each run is approximatedby taking the average value between the upper and lowerbounds We would like to be able to say something about theprobability that future runs will fall within the [minus6 33] limitsthat have already appeared We can do this using tolerancelimit calculations

Assume the 10 runs produced the results given in Table 2The average and standard deviation for these observations areminus134 and 155 respectively

Table 2 Example estimated correction terms for average number ata device

Run 119862119899

1 minus1752 minus3003 minus1254 1655 minus2506 minus1907 minus1758 1159 minus26010 minus145

Since the tolerance limits are going to be set at theobserved limits we can say

[minus6 33] = [119862119899 minus 119870119904 119862119899 + 119870119904]

119870 =119862119899minus (minus6)

119904=(33) minus 119862119899

119904= 30

(66)

From tolerance limit tables [28] we can say with 95confidence that at least 91 of future observations of 119862

119899will

fall within the interval [minus6 33] That is if we use 119899119860119878 valueswe have 95 confidence that the correct 119899 values for each runwill fall within these limits 91 of the time

6 Conclusion

In this paper bounds were developed for the operationalanalysis error measures of homogeneous service and arrivalassumptions These bounds allow us to take advantage of thesimplified data collection made possible by the use of opera-tional analysis relationships evenwhen the assumptions usedto derive those relationships are violated A tolerance limitbased method was given in order to be able to say somethingabout the confidence that future correction term values in atime series would be within certain limits

Abbreviations

119862 Total number of job completions at adevice

119862(119899) Total number of completions when n jobsare at a device

119862119909 Correction term makes the OA term 119909

equal to the directly calculatedperformance measure

119868 The set of states 0 1 2 119873 minus 1

119868+ The set of states 1 2 119873119870 Tolerance limit multiplier119873 The maximum number of jobs seen at a

device both in queue and in service

Journal of Optimization 9

119899 State of a device number of jobs both inqueue and in service

119899 The median state seen by a device119899119862 The median state at a device at job

completions119899 Average number of jobs at a device119899119860119878 The average number of jobs at a device

assuming flow balance homogeneousarrivals and homogeneous services

119899119862 The average number at the device seen bya completing job

119899119862119879 The average of the set of states seen by

completing job truncated at the medianstate value

119899119873 The average state at a device excluding the

maximum state119899119879 Time-average of the set of states truncated

at the median119899119878119860 Average number of jobs at a device seen by

an arriver assuming homogeneous service119901(119899) Proportion of time spent in state 119899119901119860(119899) The proportion of arrivals when 119899 jobs are

at a device 119860(119899)119860119901119862(119899) The proportion of completions that leave 119899

jobs at a device 119862(119899 + 1)119862119877 Average response time119877119878 Average response time assuming

homogeneous services119877119860119878 Average response time assuming

homogeneous arrivals and homogeneousservices

119878 Mean time between completions duringbusy periods

119904 Standard deviation119879 Total time of the period of observation119879(119899) Total time device was in state 119899119880 Utilization 1minusp(0)119883 Throughput119883119860119878 Throughput assuming homogeneous

arrivals and homogeneous services120575119860 Maximum error in homogeneous arrival

assumption error measure120575119878 Maximum error in homogeneous service

assumption error measureΔ119860 A bound on the homogeneous arrival

assumption error measure le 120575119860

Δ119878 A bound on the homogeneous service

assumption error measure le 120575119878

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] J P Buzen ldquoFundamental operational laws of computer systemperformancerdquo Acta Informatica vol 7 no 2 pp 167ndash182 1976

[2] J P Buzen ldquoOperational analysis an alternative to stochasticmodelingrdquo in Performance of Computer Installations pp 175ndash194 North-Holland Amsterdam The Netherlands 1978

[3] J P Buzen and P J Denning ldquoOperational treatment of queuedistributions andmean value analysisrdquo Tech Rep CSD-TR 309Department of Computer Science Purdue University 1979

[4] J P Buzen and P J Denning ldquoMeasuring and calculating queuelength distributionsrdquo Computer vol 13 no 4 pp 33ndash44 1980

[5] P J Denning and J P Buzen ldquoThe operational analysis ofqueueing network modelsrdquo Computer Surveys vol 10 no 3 pp225ndash261 1978

[6] J P Buzen ldquoFrom stochastic modeling to operational analysisthe journey beginsrdquo in Fundamental Concepts in ComputerScience E Gelenbe and J Kahane Eds pp 141ndash149 ImperialCollege Press London UK 2009

[7] E D Lazowska J Zahorjan G S Graham and K C SevcikQuantitative System Performance Prentice-Hall EnglewoodCliffs NJ USA 1984

[8] R Suri ldquoRobustness of queueing network formulasrdquo Journal ofthe ACM vol 30 no 3 pp 564ndash594 1983

[9] E D Lazowska J Zahorjan and E C Sevcik ldquoComputer sys-tem performance evaluation using queueing network modelsrdquoAnnual Reviews in Computer Science vol 1 pp 107ndash137 1986

[10] B V Ivanovskii ldquoOperational analysis of communication net-works with lockupsrdquoAutomatic Control and Computer Sciencesvol 22 no 3 pp 26ndash32 1988

[11] P J Denning ldquoQueueing in networks of computersrdquo AmericanScientist vol 79 pp 206ndash209 1991

[12] P J Denning ldquoIn the queue mean valuesrdquo American Scientistvol 79 pp 402ndash403 1991

[13] Y Dallery and X-R Cao ldquoOperational analysis of stochasticclosed queueing networksrdquo Performance Evaluation vol 14 no1 pp 43ndash61 1992

[14] M El-Taha and S Stidham Jr ldquoDeterministic analysis of queue-ing systems with heterogeneous serversrdquo Theoretical ComputerScience vol 106 no 2 pp 243ndash264 1992

[15] J P Buzen ldquoAn overview of performance prediction in MVSsystems and SNAnetworksrdquo inProceedings of the ACMFall JointComputer Conference pp 751ndash759 Dallas Tex USA 1986

[16] D A Menasce and V A Almeida Scaling for E-Business Tech-nologies Models Performance and Capacity Planning Prentice-Hall Englewood Cliffs NJ USA 2000

[17] D K Buch and V M Pentkovski ldquoPerformance characteriza-tion experience of multi-tier E-business systems using queuingoperational analysisrdquo in Proceedings of the IEEE InternationalSymposium on Performance Analysis of Systems and SoftwareTucson Ariz USA 2001

[18] M P Fanti B Maione G Piscitelli and B Turchiano ldquoTwomethod for real-time routing selection in flexible manufactur-ing systemsrdquo inProceedings of the IEEE International Conferenceon Robotics and Automation pp 1158ndash1166 May 1992

[19] W M Zuberek ldquoThroughput analysis in timed Petri netsrdquo inProceedings of the 35th Midwest Symposium on Circuits andSystems pp 1576ndash1580 Washington DC USA 1992

[20] W M Zuberek ldquoThroughput analysis of simple closed timedPetri netmodelsrdquo in Proceedings of the 36thMidwest Symposiumon Circuits and Systems pp 930ndash933 Detroit Mich USAAugust 1993

[21] G Chiola C Anglano J Campos J M Colom and M SilvaldquoOperational analysis of timed Petri nets and application to the

10 Journal of Optimization

computation of performance boundsrdquo in Quantitative Methodsin Parallel Systems F Baccelli A Jean-Marie and I MitraniEds Esprit Basic Research Series pp 161ndash174 Springer BerlinGermany 1995

[22] J A Brumfield Operational Analysis of Queuing Phenomena[PhD thesis] Department of Computer Science Purdue Uni-versity 1982

[23] N M Bengtson ldquoUsing operational analysis in simulation aqueuing network examplerdquo Journal of the Operational ResearchSociety vol 39 no 12 pp 1125ndash1136 1988

[24] N M Bengtson ldquoMeasuring errors in operational analysisassumptionsrdquo IEEE Transactions on Software Engineering vol13 no 7 pp 767ndash776 1987

[25] N M Bengtson ldquoOperational analysis revisited error measurelimits of assumptionsrdquo GSTF International Journal on Comput-ing vol 3 no 2 2013

[26] K C Sevcik and M Klawe ldquoOperational analysis versusstochastic modeling of computer systemsrdquo in Proceedings of theComputer Science and Statistics 12th Annual Symposium on theInterface pp 177ndash184 Waterloo Canada 1979

[27] F S Hillier and G J Lieberman Introduction to OperationsResearch McGraw-Hill New York NY USA 8th edition 2005

[28] C Eisenhart M W Hastay and W A Wallis Eds SelectedTechniques of Statistical Analysis McGraw-Hill New York NYUSA 1947

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Optimization 9

119899 State of a device number of jobs both inqueue and in service

119899 The median state seen by a device119899119862 The median state at a device at job

completions119899 Average number of jobs at a device119899119860119878 The average number of jobs at a device

assuming flow balance homogeneousarrivals and homogeneous services

119899119862 The average number at the device seen bya completing job

119899119862119879 The average of the set of states seen by

completing job truncated at the medianstate value

119899119873 The average state at a device excluding the

maximum state119899119879 Time-average of the set of states truncated

at the median119899119878119860 Average number of jobs at a device seen by

an arriver assuming homogeneous service119901(119899) Proportion of time spent in state 119899119901119860(119899) The proportion of arrivals when 119899 jobs are

at a device 119860(119899)119860119901119862(119899) The proportion of completions that leave 119899

jobs at a device 119862(119899 + 1)119862119877 Average response time119877119878 Average response time assuming

homogeneous services119877119860119878 Average response time assuming

homogeneous arrivals and homogeneousservices

119878 Mean time between completions duringbusy periods

119904 Standard deviation119879 Total time of the period of observation119879(119899) Total time device was in state 119899119880 Utilization 1minusp(0)119883 Throughput119883119860119878 Throughput assuming homogeneous

arrivals and homogeneous services120575119860 Maximum error in homogeneous arrival

assumption error measure120575119878 Maximum error in homogeneous service

assumption error measureΔ119860 A bound on the homogeneous arrival

assumption error measure le 120575119860

Δ119878 A bound on the homogeneous service

assumption error measure le 120575119878

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] J P Buzen ldquoFundamental operational laws of computer systemperformancerdquo Acta Informatica vol 7 no 2 pp 167ndash182 1976

[2] J P Buzen ldquoOperational analysis an alternative to stochasticmodelingrdquo in Performance of Computer Installations pp 175ndash194 North-Holland Amsterdam The Netherlands 1978

[3] J P Buzen and P J Denning ldquoOperational treatment of queuedistributions andmean value analysisrdquo Tech Rep CSD-TR 309Department of Computer Science Purdue University 1979

[4] J P Buzen and P J Denning ldquoMeasuring and calculating queuelength distributionsrdquo Computer vol 13 no 4 pp 33ndash44 1980

[5] P J Denning and J P Buzen ldquoThe operational analysis ofqueueing network modelsrdquo Computer Surveys vol 10 no 3 pp225ndash261 1978

[6] J P Buzen ldquoFrom stochastic modeling to operational analysisthe journey beginsrdquo in Fundamental Concepts in ComputerScience E Gelenbe and J Kahane Eds pp 141ndash149 ImperialCollege Press London UK 2009

[7] E D Lazowska J Zahorjan G S Graham and K C SevcikQuantitative System Performance Prentice-Hall EnglewoodCliffs NJ USA 1984

[8] R Suri ldquoRobustness of queueing network formulasrdquo Journal ofthe ACM vol 30 no 3 pp 564ndash594 1983

[9] E D Lazowska J Zahorjan and E C Sevcik ldquoComputer sys-tem performance evaluation using queueing network modelsrdquoAnnual Reviews in Computer Science vol 1 pp 107ndash137 1986

[10] B V Ivanovskii ldquoOperational analysis of communication net-works with lockupsrdquoAutomatic Control and Computer Sciencesvol 22 no 3 pp 26ndash32 1988

[11] P J Denning ldquoQueueing in networks of computersrdquo AmericanScientist vol 79 pp 206ndash209 1991

[12] P J Denning ldquoIn the queue mean valuesrdquo American Scientistvol 79 pp 402ndash403 1991

[13] Y Dallery and X-R Cao ldquoOperational analysis of stochasticclosed queueing networksrdquo Performance Evaluation vol 14 no1 pp 43ndash61 1992

[14] M El-Taha and S Stidham Jr ldquoDeterministic analysis of queue-ing systems with heterogeneous serversrdquo Theoretical ComputerScience vol 106 no 2 pp 243ndash264 1992

[15] J P Buzen ldquoAn overview of performance prediction in MVSsystems and SNAnetworksrdquo inProceedings of the ACMFall JointComputer Conference pp 751ndash759 Dallas Tex USA 1986

[16] D A Menasce and V A Almeida Scaling for E-Business Tech-nologies Models Performance and Capacity Planning Prentice-Hall Englewood Cliffs NJ USA 2000

[17] D K Buch and V M Pentkovski ldquoPerformance characteriza-tion experience of multi-tier E-business systems using queuingoperational analysisrdquo in Proceedings of the IEEE InternationalSymposium on Performance Analysis of Systems and SoftwareTucson Ariz USA 2001

[18] M P Fanti B Maione G Piscitelli and B Turchiano ldquoTwomethod for real-time routing selection in flexible manufactur-ing systemsrdquo inProceedings of the IEEE International Conferenceon Robotics and Automation pp 1158ndash1166 May 1992

[19] W M Zuberek ldquoThroughput analysis in timed Petri netsrdquo inProceedings of the 35th Midwest Symposium on Circuits andSystems pp 1576ndash1580 Washington DC USA 1992

[20] W M Zuberek ldquoThroughput analysis of simple closed timedPetri netmodelsrdquo in Proceedings of the 36thMidwest Symposiumon Circuits and Systems pp 930ndash933 Detroit Mich USAAugust 1993

[21] G Chiola C Anglano J Campos J M Colom and M SilvaldquoOperational analysis of timed Petri nets and application to the

10 Journal of Optimization

computation of performance boundsrdquo in Quantitative Methodsin Parallel Systems F Baccelli A Jean-Marie and I MitraniEds Esprit Basic Research Series pp 161ndash174 Springer BerlinGermany 1995

[22] J A Brumfield Operational Analysis of Queuing Phenomena[PhD thesis] Department of Computer Science Purdue Uni-versity 1982

[23] N M Bengtson ldquoUsing operational analysis in simulation aqueuing network examplerdquo Journal of the Operational ResearchSociety vol 39 no 12 pp 1125ndash1136 1988

[24] N M Bengtson ldquoMeasuring errors in operational analysisassumptionsrdquo IEEE Transactions on Software Engineering vol13 no 7 pp 767ndash776 1987

[25] N M Bengtson ldquoOperational analysis revisited error measurelimits of assumptionsrdquo GSTF International Journal on Comput-ing vol 3 no 2 2013

[26] K C Sevcik and M Klawe ldquoOperational analysis versusstochastic modeling of computer systemsrdquo in Proceedings of theComputer Science and Statistics 12th Annual Symposium on theInterface pp 177ndash184 Waterloo Canada 1979

[27] F S Hillier and G J Lieberman Introduction to OperationsResearch McGraw-Hill New York NY USA 8th edition 2005

[28] C Eisenhart M W Hastay and W A Wallis Eds SelectedTechniques of Statistical Analysis McGraw-Hill New York NYUSA 1947

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Journal of Optimization

computation of performance boundsrdquo in Quantitative Methodsin Parallel Systems F Baccelli A Jean-Marie and I MitraniEds Esprit Basic Research Series pp 161ndash174 Springer BerlinGermany 1995

[22] J A Brumfield Operational Analysis of Queuing Phenomena[PhD thesis] Department of Computer Science Purdue Uni-versity 1982

[23] N M Bengtson ldquoUsing operational analysis in simulation aqueuing network examplerdquo Journal of the Operational ResearchSociety vol 39 no 12 pp 1125ndash1136 1988

[24] N M Bengtson ldquoMeasuring errors in operational analysisassumptionsrdquo IEEE Transactions on Software Engineering vol13 no 7 pp 767ndash776 1987

[25] N M Bengtson ldquoOperational analysis revisited error measurelimits of assumptionsrdquo GSTF International Journal on Comput-ing vol 3 no 2 2013

[26] K C Sevcik and M Klawe ldquoOperational analysis versusstochastic modeling of computer systemsrdquo in Proceedings of theComputer Science and Statistics 12th Annual Symposium on theInterface pp 177ndash184 Waterloo Canada 1979

[27] F S Hillier and G J Lieberman Introduction to OperationsResearch McGraw-Hill New York NY USA 8th edition 2005

[28] C Eisenhart M W Hastay and W A Wallis Eds SelectedTechniques of Statistical Analysis McGraw-Hill New York NYUSA 1947

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of