Research Article Phase-Type Arrivals and Impatient ...downloads.hindawi.com/journals/aor/2016/4024950.pdf · Chen et al. [] studied M/M/m/k queue with preemptive resume and impatience

Research ArticlePhase-Type Arrivals and Impatient Customers inMultiserver Queue with Multiple Working Vacations

Cosmika Goswami and N Selvaraju

Department of Mathematics Indian Institute of Technology Guwahati Guwahati 781 039 India

Correspondence should be addressed to Cosmika Goswami cosmikagoswamigmailcom

Received 27 October 2015 Accepted 14 February 2016

Academic Editor Viliam Makis

Copyright copy 2016 C Goswami and N SelvarajuThis is an open access article distributed under theCreative CommonsAttributionLicense which permits unrestricted use distribution and reproduction in anymedium provided the originalwork is properly cited

We consider a PHMc queue with multiple working vacations where the customers waiting in queue for service are impatientTheworking vacation policy is the one in which the servers serve at a lower rate during the vacation period rather than completelyceasing the service Customerrsquos impatience is due to its arrival during the period where all the servers are in working vacations andthe arriving customer has to join the queue We formulate the system as a nonhomogeneous quasi-birth-death process and usefinite truncation method to find the stationary probability vector Various performance measures like the average number of busyservers in the system during a vacation as well as during a nonvacation period server availability blocking probability and averagenumber of lost customers are given Numerical examples are provided to illustrate the effects of various parameters and interarrivaldistributions on system performance

1 Introduction

In communication networks multiple servers are used toreduce the traffic congestion and improve the system per-formance Multiple services are also used in highly effi-cient bandwidth-intensive applications Different servicesmay require different channel capacities and capacity of achannel depends upon the number of resources allocatedto it To understand the network behavior and to makeintelligent decisions in their management these systems canbe modelled as multiserver queueing systems with servervacations Levy and Yechiali [1] first discussed an MMcqueue with exponentially distributed vacations Tian and Li[2] Tian et al [3] and Tian and Zhang [4] studied a varietyof vacation models with multiple servers They establishedthe conditional stochastic decomposition properties on thesteady-state queue length and the waiting time when all theservers are busy and obtained the stationary distributionsfor queue length and waiting times Tian and Zhang [5]considered a two-threshold vacation policy in the contextof a multiserver queueing model MMc A multiserver

queueing system with identical unreliable servers with PH-distributed service times is considered by Yang and Alfa[6] Chakravarthy [7] studied an MAPMc queueing sys-tem in which a group of servers take a simultaneous PHvacation

The phenomenon of customer impatience is commonlyobserved in queueing systems where customers leave aservice system before receiving service due to the longwaiting time or due to uncertainty of receiving serviceCustomer impatience or reneging represents loss in revenuesand customer goodwill to the service provider The problemof queues with impatient customers was first analyzed byPalm [8] A bibliography can be found in Gross et al[9] Perel and Yechiali [10] considered a two-phase serviceimpatient model where the customers become impatientif the server is in slow service phase There are situationswhere customerrsquos impatience is due to the absence of theserver more precisely due to the server being on vacationand is independent of the customers in system Altman andYechiali [11 12] studied the customer impatience in a classicalvacation model and system with additional task respectively

Hindawi Publishing CorporationAdvances in Operations ResearchVolume 2016 Article ID 4024950 17 pageshttpdxdoiorg10115520164024950

2 Advances in Operations Research

Economou and Kapodistria [13] considered an unreliablequeue where the customers leave the system at system failuretimes

Multiserver queues with impatience however haveattracted much attention in queueing literature possiblybecause of explosive demands to efficiently design andmanage call or contact centres Baccelli et al [14] studiedthe waiting time distribution in MMc queue with generalimpatience bound on queueing times by constructing asimple Markov process and also gave the waiting timedistribution in the MG1 queue with general impatienceon queueing times Yechiali [15] considered an MMcsystem which as a whole suffers occasionally a disastrousbreakdown upon which all present customers (waiting andserved) are cleared from the system and lost Stationarydistribution of a multiserver vacation queue with con-stant impatient times is studied by Sakuma and Inoie [16]Chen et al [17] studied MMmk queue with preemptiveresume and impatience of the prioritised customers andderived the queue length distraction in stationary stateand performance measures using the method of matrixanalysis

In communication systems wavelength division multi-plexing (WDM) is a method of transmitting packets fromdifferent sources over the same fiber optic link to thedestination A WDM network divides the available fiberbandwidth into WDM channels details in Ho and Woei [18]and also inWang [19] This division of bandwidth or channelallocation is based on the capacities required for variousservices For a high performance system WDM channelallocation should lead to optimized resource utilization in agiven network which is physically feasible and cost-effectiveA reconfigurable WDM system can be modelled as a queuewith working vacations (WVs) as explained in Goswami andSelvaraju [20] This vacation cannot be put in a classicalvacation framework because here unless the system is emptythe service does not cease completely Servi and Finn [21]were the first to model such a WDM network into a WVqueueing model Liu et al [22] studied the MM1WVmodel with multiple WVs whereas the single WV model isanalyzed by Tian et al [23] The same model is studied byXu et al [24] and also by Xiu et al [25] with single WVand setup times Wang et al [26] presented the MM1WVmodel using Newtonrsquos method to compute the steady-stateprobabilities and system performance measures Wu andTakagi [27] extended Servi and Finnrsquos work to MG1WVmodel with generally distributed service times and vacationduration times Baba [28] considered the GIM1WV systemwith general independent arrival process where the distri-butions of the vacation duration times and service timesare exponential Chen et al [29 30] proposed an N-policyWV and a cyclic polling system for WDM taking the servicetimes as exponential and PH distribution respectively Linand Ke [31] considered a multiserver MMc queue and acost model is derived to determine the optimal values ofthe number of servers and the WV rate simultaneouslyin order to minimize the total expected cost per unittime

Short distance networks like local area networks (LANs)mostly usemultimodeWDMlinksMultimode link is a singlefiber link that supports many propagation paths or transversemodes through it Aronson et al [32] explained how thebandwidth of the fiber is multiplied by the number of pathsused by using WDM in multimode fiber LAN over InternetProtocol (IP) allows the forwarding of LAN packets over theInternet or an intranet network One of the most criticalperformance measures in LAN over IP is the percentageof packets that are transmitted within hard delay bound ortime constraint If quality of service requirements is not metwithin the time bound end users may terminate the Internetconnections A connection is terminated by pressing the stopbutton refreshing the connection or following a differentlink This behavior can be termed the impatience of a userin LANs To study the effect of multiple servers and userimpatience on the performance in a WDM network we con-sider in this paper a multiserver model with asynchronousmultiple working vacation (AMWV) policy and impatientcustomers In an AMWV policy the servers take vacationsindividually and continue taking vacations till they do notfind any customer in the system An MM1WV impatientmodel with single and multiple WV policies is studied bySelvaraju andGoswami [33] Analysis of a finite bufferMM2working vacations queue with balking and reneging whereinthe servers operate under a triadic (0QNM) policy isdone recently by Laxmi and Jyothsna [34] Lin and Ke[31] presented a multiserver WV queue with exponentialinterarrivals but none of these models represent systems withnonexponential arrivals or state-dependent systems To studythe role of arrival processes in a multiserver model havingimpatient customers we consider here the PHarrival processPH distribution is a general nonexponential distributioncharacterized by a Markov chain Importance of consideringPH interarrivals is the fact that PH distribution is able tocapture the nonexponential effects on arrivals while infor-mation flows in modern communication systems are rarelyexponential PH distribution is able to capture the profoundeffect of arrivals in system performance measures and makesthe mathematical model more convincing to fit a real worldscenario

The paper is organized as follows In Section 2 weformulate the system as a three-dimensional continuous-time Markov chain whose generator matrix is a level-dependent quasi-birth-death (QBD) process Section 3 givesthe finite truncation method used to find the station-ary probability vector of the level-dependent process Thevarious performance measures are listed in Section 4 andin Section 5 the numerical illustrations of the system arepresented

2 Model Description

We consider a PHMc queue with multiple WVs and impa-tient customers The interarrival times of customers follow aPH distribution PH(120572 119879) of dimension 119899 and with arrivalrate 120582 A PH distribution denotes the distribution of time

Advances in Operations Research 3

until absorption in a finite Markov chain whose transitionrate matrix is of type

119875 = [119879 1198790

0 0] (1)

and 120572 is the initial probability vector satisfying 120572e119899= 1 and

1198790= minus119879e

119899 where e

119899is the column vector of dimension 119899

with all the entries equal to oneThematrix119879 is a nonsingularsquare matrix with (119879)

119894119894lt 0 1 le 119894 le 119899 and (119879)

119894119895ge 0

1 le 119894 = 119895 le 119899 The matrix 1198790 is a nonnegative 119899-dimensionalcolumn vector grouping the absorption rates from any stateto the absorbing one The matrix 1198790120572 gives the transitionfromone phase to another with an arrival of a customer to thesystem

The customers are served according to FCFS basis Anarriving customer who finds all the 119888 servers busy has towait in queue that is when the number of customers inthe system is more than 119888 a queue begins to form Theservers work independently of each other The service timesof each server during the nonvacation period follow anexponential distribution with rate 120583

119887 denoted by Exp(120583

119887) A

server goes to a WV as soon as it completes a service andfinds no customer to serve in the system For each serverthe duration of WVs follows Exp(120579) distribution During aWV period of a server if a customer arrives to that serverit will serve the customer with Exp(120583V) distribution where120583V lt 120583119887 that is the customer will be served at a lowerservice rate When a server returns from its vacation if itfinds at least one customer in queue waiting for service orfinds an ongoing service in that server the server switchesits service rate from 120583V to 120583119887 and a nonvacation period startsOtherwise if the server finds an empty queue after return-ing from one vacation it immediately leaves for anotherWV

An arriving customer gets service immediately upon itsarrival if it finds any of the 119888 servers empty But if all theservers are busy the customer has to wait in a queue Awaiting customer becomes impatient when it finds all theservers serving at rate 120583V that is if the waiting customer findsall the servers in their WV period the customer activates an

impatient timer 119883 This impatient timer 119883 follows Exp(120585)distribution and is independent of the number of customersin the queue at thatmoment If no server returns from itsWVperiod by the time119883 expires the customer leaves the systemand never returns Otherwise if any of the servers returnsfrom its vacation before the time 119883 expires the customerstops the timer and stays in the system until its serviceis completed Here the customerrsquos impatience depends notonly on waiting time in a queue but also on the number ofservers that are inWVsThe interarrival times service timesvacation duration times and the impatient times all are takento be mutually independent

To model this system we define a continuous-timeMarkov chain

Δ = (119873119905 119876119905 119869119905) 119905 ge 0 (2)

where 119873119905denotes the total number of customers in the

system119876119905denotes the number of busy servers in nonvacation

state and 119869119905gives the phase of the arrival process The state

space of this Markov chain is

119864 = (119894 119895 119896) 0 le 119894 lt 119888 119895 le 119894 1 le 119896 le 119899

cup (119894 119895 119896) 119894 ge 119888 0 le 119895 le 119888 1 le 119896 le 119899

(3)

The lexicographical order of the states that is (0 01) (0 0 119899) (1 0 1) (1 0 119899) (1 1 1) (1 1 119899)(2 0 1) (2 0 119899) (2 1 1) (2 1 119899) (2 2 1) (2 2 119899) (119888 0 1) (119888 119888 119899) (119888 + 1 0 1) (119888 +1 119888 119899) gives the infinitesimal generator matrix of theMarkov chain as with

119876 =

(((((

(

119860(0)

1119860(0)

0

119860(1)

2119860(1)

1119860(1)

0

119860(2)

2119860(2)

1119860(2)

0

119860(3)

2119860(3)

1119860(3)

0

d d d

)))))

)

(4)

where for 1 le 119894 lt 119888

119860(119894)

2=

[[[[[[[[[[[

[

119894120583V119868

(120583119887+ (119894 minus 1) 120583V) 119868

(2120583119887+ (119894 minus 2) 120583V) 119868

d

((119894 minus 1) 120583119887+ 120583V) 119868

119894120583119887119868

]]]]]]]]]]]

](119894+1)119899times(119894+1)119899

(5)

and for 0 le 119894 lt 119888


119860(119894)

1=

[[[[[[[

[

119879 minus (119888120579 + 119894120583V) 119868 119888120579119868

119879 minus ((119888 minus 1) 120579 + 120583119887+ (119894 minus 1) 120583V) 119868 (119888 minus 1) 120579119868

d

119879 minus 119894120583119887119868

]]]]]]]

](119894+1)119899times(119894+1)119899

119860(119894)

0=

[[[[[[

[

1198790120572

1198790120572

d

1198790120572

]]]]]]

](119894+1)119899times(119894+1)119899

(6)

For 119894 ge 119888

119860(119888+119897)

2=

[[[[[[[[[[

[

(119888120583V + 119897120585) 119868

(120583119887+ (119888 minus 1) 120583V) 119868

(2120583119887+ (119888 minus 2) 120583V) 119868

d

119888120583119887119868

]]]]]]]]]]

](119888+1)119899times(119888+1)119899

119860(119888+119897)

1=

[[[[[[[

[

119879 minus (119888 (120579 + 120583V) + 119897120585) 119868 119888120579119868

d d

119879 minus (120579 + 120583V + (119888 minus 1) 120583119887) 119868 120579119868

119879 minus (119888120583119887) 119868

]]]]]]]

](119888+1)119899times(119888+1)119899

119860(119888+119897)

0=

[[[[[[

[

1198790120572

1198790120572

d

1198790120572

]]]]]]

](119888+1)119899times(119888+1)119899

(7)

The matrix 119868 is an identity matrix of dimension 119899 Herefor 0 le 119894 le 119888 minus 1 dimension of the matrices 119860(119894)

0

119860(119894)

1 and 119860(119894)

2increases with the levels and for 119894 ge 119888 the

matrices are of dimensions (119888 + 1)119899 times (119888 + 1)119899 each Itcan be observed that 119876 given above is the generator of anonhomogeneous QBD process which we assume to be irre-ducible with levels denoting the number of customers in thesystem

3 Stationary Distribution

The queueing system under study is stable for 120588 = 120582119888120583119887lt 1

[28 35]

Let 119909 be the stationary probability vector associated with119876 satisfying

119909119876 = 0

119909e = 1(8)

Aggregating terms depending on levels we get 119909 =[1199090 1199091 1199092 sdot sdot sdot] Further depending on number of busyservers in nonvacation we get for 0 le 119894 le 119888 minus 1 119909

119894=

[1199091198941 1199091198942 119909

119894119894] which are row (119894 +1)119899-vectors and for 119894 ge 119888

119909119894are row (119888+1)119899-vectors Each119909

119894119895vector is an 119899-dimensional

row vector 119909119894119895= [1199091198941198951 119909

119894119895119899] for 119895 le 119894 depending on the

phases of arrivals


In this model the generator matrix 119876 is spatially non-homogeneous and a closed-form analytical solution or adirect algorithmic computation of the stationary probabilityvector 119909 is quite difficult if not impossible For such level-dependent QBDs (LDQBDs) the stationary vectors are usu-ally approximated by using various numerical approximationmethods like finite truncation method (Artalejo et al [36]and Chakravarthy et al [37]) generalized truncationmethod(Falin [38] and Artalejo and Pozo [39]) truncation methodusing LDQBD processes (Bright and Taylor [40] and Krish-namoorthy et al [41]) andmatrix-geometric approximations(Neuts and Rao [42])

Different methods have different levels of computableefficiency but it is expected that whichever method is usedthe general behavior of the performance measures of asystem with a change in system parameters is not affectedby the method used Since the finite truncation method iscomparatively tractable compared to the others we choosethis method to derive the stationary distributions of thenonhomogeneous QBD with the generator matrix given by(4)

In the finite truncation method the infinite generatormatrix is truncated at a finite level 119870 That is the systemof equations given by 119909119876 = 0 and 119909e = 1 is truncatedat a sufficiently large value say 119870 and the resulting finitesystem is solved for the equilibrium probability vector Thelevel 119870 is arbitrary but fixed and it is chosen such thatcustomer loss probability due to truncation is small As forhigher dimension generator matrices the level 119870 is difficultto find analytically a trial-and-error approach needs to beadopted An appropriate level say 119870

119891 is determined by

starting with a reasonable initial value for119870 and increasing itprogressively until an appropriately chosen cut-off criterionis met Stationary probability vector 119909 can then be evaluatedby an iterative method such as that by Gauss-Seidel [43]which takes advantage of the sparsity and structure of 119876For each new value of 119870 the previously computed vector119909 is used as the initial solution to reduce the number ofiterations required [42]Thus the numerical implementationof the approximation based on finite truncation implies thedetermination of an appropriate cut-off level119870

119891 Here we use

the algorithm given by Artalejo et al [36] the steps of whichare described below

For119870 as the cut-off point the modified generator will be

(119870)

=

(((((((((

(

119860(0)

1119860(0)

0

119860(1)

2119860(1)

1119860(1)

0

119860(2)

2119860(2)

1119860(2)

0

d d d

119860(119870minus1)

2119860(119870minus1)

1119860(119870minus1)

0

119860(119870)

2120601(119870)

1

)))))))))

)

(9)

where 120601(119870)1= 119860(119870)

1+119860(119870)

0 Let 120587 be the stationary distribution

of (119870) which satisfies

120587 (119870) = 0

120587e = 1(10)

where 120587 = [120587(0) 120587(1) 120587(119870)] by aggregating terms of theQBD (119870) depending on levels Define 119911 = [119911

0(119870) 119911

1(119870)]

with

1199110(119870) = [120587 (0) 120587 (1) 120587 (119870 minus 1)]

1199111(119870) = 120587 (119870)

(11)

And 119911(119870 119894) = 120587(119894) 0 le 119894 le 119870 Here 1199110(119870) is a row vector

of dimension119898 = 119899119888(119888 + 1)2 + 119899(119888 + 1)(119870 minus 119888) and 1199111(119870) is

a row vector with dimension 119899(119888 + 1) By partitioning (119870)according to 119911

0(119870) and 119911

1(119870) we have

(1199110(119870) 119911

1(119870)) (

11986100(119870) 119861

01(119870)

11986110 (119870) 11986111 (119870)

) = (0119898 0119899(119888+1)) (12)

where 11986100(119870) is the matrix obtained by deleting the last

column matrices and last row matrices from (119870) 11986101(119870) =

trans[0 0 0 119860(119870minus1)0] 11986110(119870) = [0 0 0 119860

(119870)

2] and

11986111(119870) = 120601

(119870)

1 These are block structured matrices with

(119870 times 119870) (119870 times 1) (1 times 119870) and (1 times 1) blocks respectively0119898and 0119899(119888+1)

are row vectors of dimensions 119898 and 119899(119888 + 1)respectively with all entries equal to zero From (12) we findthat

1199111 (119870) 11986110 (119870) 119861

minus1

00(119870) = minus1199110 (119870) (13)

1199111(119870) [119861

11(119870) minus 119861

10(119870) 119861minus1

00(119870) 11986101(119870)] = 0

119899(119888+1) (14)

Further we can have

11986100(119870) = (

11986100(119870 minus 1) 119861

01(119870 minus 1)

1198620(119870 minus 1) 119862

1(119870 minus 1)

) (15)

where

1198620 (119870 minus 1) = [0119899 02119899 0 119860

(119870minus1)

2]

1198621(119870 minus 1) = minus119860

(119870minus1)

1

(16)

The inverse of matrix 11986100(119870) can be determined using

methods given in Hunter [44] as

119861minus1

00(119870) = (

11986300 (119870) 11986301 (119870)

11986310(119870) 119863

11(119870)) (17)


where

11986300 (119870) = [11986100 (119870 minus 1)

minus 11986110(119870 minus 1) 119862

minus1

1(119870 minus 1) 119862

0(119870 minus 1)]

minus1

11986310(119870) = minus119862

minus1

1(119870 minus 1) 119862

0(119870 minus 1)119863

00(119870)

11986311 (119870) = [1198621 (119870 minus 1)

minus 1198620(119870 minus 1) 119861

minus1

00(119870 minus 1) 119861

01(119870 minus 1)]

minus1

11986301(119870) = minus119861

minus1

00(119870 minus 1) 119861

01(119870 minus 1)119863

11(119870)

(18)

As the dimensions of the matrices increase in each iterationthe calculations to compute the above matrices involvemultiplications and inversion of increasingly large matricesIf we exploit the structure of matrices in the above equationswe notice that the sparse blocks of 119861

01(119870 minus 1) and C

0(119870 minus 1)

simplify the calculations 11986101(119870 minus 1) has only one nonzero

square matrix 119860(119870minus1)0

of dimension 119899(119888 + 1) in the last rowsand 119862

0(119870minus1) has one119860(119870minus1)

2in the last columns So 119861minus1

00(119870minus

1)11986101(119870 minus 1) can be written in simplified form as [119863

10(119870 minus

1) 11986311(119870minus1)]119860

(119870minus1)

0 Further119862

0(119870minus1)119861

minus1

00(119870minus1)119861

01(119870minus1)

becomes 119860(119870minus1)211986311(119870 minus 1)119860

(119870minus1)

0 These substitutions make

the remaining operations in 11986301(119870) and 119863

11(119870) simple as

they involve multiplications and inversion of only knownsimple matrices of size 119899(119888 + 1) The key step is to computethe matrix 119863

00(119870) The inverse in the definition of 119863

00(119870)

can be computed by using small-rank adjustment that is ifwe have the inverse of a matrix 119860 and we want the inverse ofits adjustment 119861 = 119860+119883119882119884 where119882 is a matrix of smallerorder than 119860 then we have

119861minus1= [119868 minus 119860

minus1119883(119882

minus1+ 119884119860minus1119883)minus1

119884]119860minus1 (19)

Here we have 119860 = 11986100(119870 minus 1) 119883 = minus119861

01(119870 minus 1) 119882minus1 =

119862minus1

1(119870 minus 1) and 119884 = 119862

0(119870 minus 1) Thus we obtain that

11986300(119870) = 119861

minus1

= [119868 minus 11986301 (119870)1198620 (119870 minus 1)] 119861

minus1

00(119870 minus 1)

(20)

so11986300is obtained bymultiplications and additions of already

computed matrices Finally we have

11986311(119870)

= [1198621(119870 minus 1) minus 119860

(119870minus1)

211986311(119870 minus 1)119860

(119870minus1)

0]minus1

11986301(119870)

= minus [11986310(119870 minus 1) 119863

11(119870 minus 1)] 119860

(119870minus1)

011986311(119870)

(1) 119870 = 119888 + 1(2) compute 119861minus1

00(119870)

(3) compute 1199111(119870) by (14) and (22)

(4) compute 1199110(119870) by (13)

(5) store 11986100(119870) 119861

minus1

00(119870) 119861

01(119870) and 119861

10(119870)

(6) 119870 = 119870 + 1(7) while 119870 le 119870

119891

(8) do(9) compute 119861minus1

00(119870) by (17)

(10) compute 1199111(119870) by (14) and (22)

(11) compute 1199110(119870) by (13)

(12) update 11986100(119870) 119861minus1

00(119870) 119861

01(119870) and 119861

10(119870)

(13) if max0le119894le119870119911(119870 119894) minus 119911(119870 minus 1 119894)

infinlt 120598

(14) then 119870119891= 119870

(15) break(16) else 119870 = 119870 + 1

Algorithm 1 Finite truncation method

11986300(119870) = [119868 minus 119863

01(119870)119862

0(119870 minus 1)] 119861

minus1

00(119870 minus 1)

11986310 (119870) = minus119862

minus1

1(119870 minus 1) 1198620 (119870 minus 1)11986300 (119870)

(21)

So the computation of vector 1199111(119870) is reduced to solving

system (14) subject to the normalization condition

120587 (119870) [e119899(119888+1) minus 11986110 (119870) 119861minus1

00(119870) e119899119870(119888+1)] = 1 (22)

where e119899(119888+1)

and 119890119899119870(119888+1)

are column vectors of dimensions119899(119888 + 1) and 119899119870(119888 + 1) respectively with all entries equal toone Finally the vector 119911

0(119870) can be solved substituting 119911

1(119870)

in (13) To get the cut-off value successive increments of 119870are made starting from 119870 = 119888 + 1 and we stop at the point119870 = 119870

119891when

max0le119894le119870119891

10038171003817100381710038171003817119911 (119870119891 119894) minus 119911 (119870

119891minus 1 119894)10038171003817100381710038171003817infinlt 120598 (23)

where 120598 is an infinitesimal quantity and sdot infin

is the infinitynormThe whole method of computing the stationary distri-bution using the finite truncation method is summarized inAlgorithm 1

4 Performance Measures

The performance measures give the qualitative behavior ofthe model under study In a multiserver queueing modelthe efficiency of the model depends upon the mean num-ber of busy servers the mean queue length the blockingprobability and the mean number of customers lost due toimpatience

In our model the server serves even during its vacationTherefore the number of busy servers will be 119894 0 le 119894 le 119888 if


there are 119894 customers in the system and when the system hasmore than 119888 customers all the servers will be busy servingcustomers either in WV or in nonvacation with rates 120583Vand 120583

119887 respectively The mean number of servers busy in

nonvacation is

119861119904=

119888minus1

sum

119894=0

119894

sum

119895=1

119895119909119894119895e119899+

infin

sum

119894=119888

119888

sum

119895=1

119895119909119894119895e119899 (24)

The mean queue length of the system under study is

119871 = 119864 (119873) =

119888minus1

sum

119894=1

119894119909119894e(119894+1)119899+

infin

sum

119894=119888

119894119909119894e(119888+1)119899 (25)

Availability of the server 119877 is the probability that an arrivalfinds a server free It can happen only if the number of totalcustomers in the system in less than 119888 and is given by

119877 = 119875 (119873 lt 119888) =

119888minus1

sum

119894=0

119909119894e(119894+1)119899 (26)

The blocking probability of a multiserver queue is the proba-bility of refraining a customer from service In our model acustomer is kept waiting in the queue for service when all theservers are in busy state either inWV or in nonvacation thatis when the number of customers in the system is more than119888

119861119901= 119875 (119873 gt 119888) = 1 minus

119888minus1

sum

119894=0

119909119894e(119894+1)119899= 1 minus 119877 (27)

The mean number of customers lost by the system is theaverage of customers who have abandoned the system as aresult of waiting in a queue (119894 gt 119888) with all servers in WV(119895 = 0) therefore

119873119888=

infin

sum

119894=119888+1

1198941199091198940e119899 (28)

5 Numerical Examples

Let us illustrate the behavior of our PHMcWV queuewith the help of some numerical examples Algorithm 1 iscoded in MATLABcopy The algorithm computes the stationarydistribution and its main objective is to find the terminationcriteria of the level 119870

119891 We start with an initial value 119870 ge

119888 + 1 and progressively increase the value of119870 until a changein the stationary probability 119911 is sufficiently small due toincreased 119870 We choose the smallest value of 119870

119891such that

max0le119894le119870119891119911(119870119891 119894) minus 119911(119870

119891minus 1 119894)

infinlt 120598 for 120598 = 10minus6 With

this selection criterion we find the values of 119870119891 the mean

queue lengths and the blocking probabilities for various setsof parameter values and for different arrival processes Herewe take some examples (from Chakravarthy et al [37]) of

well known distributions and give their PH representationsbelow

(1) Exponential (Exp)

119879 = minus1

1198790= 1

120572 = 1

(29)

(2) Erlang-2 (Erl)

119879 = (

minus2 2

0 minus2)

1198790= (

0

2)

120572 = [1 0]

(30)

(3) Hyperexponential-2 (Hyp)

119879 = (

minus19 0

0 minus019)

1198790= (

19

019)

120572 = [09 01]

(31)

All these PH distributions have the same mean arrival rate120582 = 1 The standard deviations of the three distributions are10 070711 and 224472 respectivelyThe service rate duringnonvacation period 120583

119887 is calculated for specific values of

120588 using the formula 120588 = 120582119888120583119887 We have chosen 120588 =

01 05 and 09 for given values of 119888 (119888 = 1 3 6) Theeffect of parameters on system performance is illustratedhere We will mention the models having interarrivals asexponential Erlang and hyperexponentially distributed asexponential model Erlang model and hyperexponentialmodel respectively

51 Effect on Cut-Off Value 119870119891 We have illustrated here the

effect of the parameters namely traffic intensity (120588) rate ofvacation duration (120579) service rate during WV (120583V) and thetype of arrival process on the truncation cut-off value 119870

119891

For three different values of 119888 (119888 = 1 3 6) different tablesare presented The impatient rate is fixed at 120585 = 01 for thetables We have the following observations from Tables 1 2and 3

(1) The cut-off value increases with the increase in thevariance of the distribution of the interarrival timesFor Erlang model the termination is the fastestwhereas for the hyperexponential one it is the slowestThis behavior seems to be the same for all sets ofparameter values and for all 119888


Table 1 Multiserver model with 119888 = 1

120588 120579 120583V119870119891

119871 119861119901

Erl Exp Hyp Erl Exp Hyp Erl Exp Hyp

01

01

00 25 28 36 64229 64127 63354 09354 09143 0795530 10 13 18 13951 14544 15802 03211 03173 0308860 7 9 13 11771 11963 12238 01652 01647 0163990 6 8 11 11149 11244 11358 01109 01108 01107

1

00 14 16 22 20525 20533 20481 06062 05479 0432530 8 11 15 13040 13342 13793 02650 02562 0242160 6 8 12 11656 11811 12011 01561 01545 0152490 6 7 11 11137 11229 11337 01099 01097 01094

100

00 5 6 10 11131 11215 11305 01097 01089 0108230 5 6 10 11099 11184 11273 01066 01061 0105660 5 6 10 11068 11153 11243 01037 01034 0103190 5 6 10 11040 11124 11214 01009 01008 01008

05

01

00 25 28 42 64099 65109 69481 09579 09412 0823605 21 25 41 44283 45761 52829 09062 08788 0745710 17 21 40 29028 31115 39708 07716 07451 0652015 15 20 38 21227 23112 30542 06121 05987 05579

1

00 17 19 38 27191 28952 36542 07863 07442 0627005 16 21 38 23956 25839 33851 07113 06784 0594910 16 21 38 21386 23326 31440 06331 06121 0561615 15 20 38 19441 21339 29316 05605 05505 05280

100

00 14 18 37 18203 20117 28092 05061 05049 0502705 14 20 37 18178 20080 28066 05046 05037 0502110 15 20 37 18146 20055 28041 05030 05025 0501415 15 20 37 18121 20030 28016 05015 05012 05007

09

01

00 74 99 246 112890 132254 285672 09893 09846 0950903 73 98 253 100678 119600 271965 09805 09735 0935006 71 97 260 87709 106206 257831 09593 09495 0913909 75 102 265 74716 92882 243478 09142 09057 08880

1

00 73 96 231 87160 108465 272102 09574 09479 0918703 72 96 233 84580 105880 269545 09445 09361 0913506 72 96 234 82028 103341 267009 09289 09228 0908009 71 95 235 79573 100868 264490 09111 09080 09022

100

00 81 110 265 78679 100268 268945 09013 09010 0900403 81 110 265 78651 100240 268905 09010 09007 0900306 81 110 265 78623 100213 268865 09006 09005 0900209 81 110 265 78596 100185 268825 09003 09002 09001

(2) For a particular arrival process when the traffic loadis small the value of 119870

119891decreases with increase in

120583V and also with the increase in 120579 But for high 120588 (gt05) and high 120579 it shows the reverse property for allarrival processes and for any number of servers 119888 thatis when the system load is heavy and the system hassmall vacation duration the cut-off value seems to behigh for all types of arrival processes and any numberof servers

(3) When the vacation duration rate 120579 is too high (=100)119870119891value remains unaffected by vacation-service rate120583V for any number of servers

These observations show that the cut-off value depends onthe system parameters and also on the arrival process butbecomes independent of vacation-service rates when we havesystems with small vacation duration



120588 120579 120583V119870119891

119871 119861119901


01

01

00 22 25 33 41401 41552 41155 04953 04909 0445210 10 13 18 18312 18449 18849 00376 00606 0106420 8 10 13 14721 14743 14805 00050 00133 0029430 7 8 11 13301 13307 13326 00013 00049 00117

1

00 9 12 16 16336 16535 16721 00203 00421 0073010 7 10 13 14723 14792 14875 00056 00163 0034320 7 9 12 13787 13811 13847 00022 00078 0018130 6 8 11 13165 13172 13189 00011 00044 00107

100

00 6 7 11 13037 13051 13062 00009 00039 0009610 6 7 11 13026 13039 13049 00009 00039 0009520 6 7 11 13015 13027 13036 00009 00038 0009330 6 7 11 13005 13016 13023 00009 00037 00092

05

01

00 36 41 58 61463 65540 81834 07544 07535 0709402 26 30 48 44310 47592 61809 05820 05952 0606204 20 24 43 33472 35687 46449 03792 04116 0487806 16 20 39 27521 28807 35933 02308 02694 03736

1

00 17 19 37 31041 32843 40155 03311 03663 0441402 16 19 37 29373 30968 38141 02832 03217 0411604 17 21 37 27875 29234 36269 02421 02821 0382506 16 21 37 26620 27814 34534 02074 02473 03544

100

00 16 19 37 26304 27513 34302 01989 02385 0346802 16 21 37 26286 27437 34280 01984 02380 0346304 16 21 37 26267 27418 34257 01978 02375 0345906 16 21 37 26249 27399 34234 01973 02370 03455

09

01

00 78 100 224 140668 163843 336740 09618 09582 0940301 75 98 224 127466 149913 319614 09424 09390 0925102 73 95 229 113801 135544 301618 09089 09077 0905203 71 93 234 99871 120651 282970 08541 08593 08796

1

00 76 101 239 97276 117783 283667 08612 08643 0880601 76 100 238 95526 116045 281630 08466 08523 0875802 75 100 237 93873 114217 279598 08312 08397 0870803 75 99 237 92075 112421 277421 08149 08265 08656

100

00 82 111 269 91219 111422 293701 08041 08176 0862201 82 111 269 91194 111398 293614 08038 08175 0862102 82 111 269 91169 111375 293527 08036 08173 0862103 82 111 269 91144 111351 293440 08033 08171 08620

52 Effect on Mean Queue Length

(1) The mean queue length of the system depends uponthe arrival process Tables 1 2 and 3 show that systemswith interarrival distributions of high variance havehigher number of customers in the queue for anynumber of serversWe have fixed the impatient rate at120585 = 01 For 119888 = 1 3 6 with 120588 = 05 we have Figures1 2 and 3 respectively and with 120588 = 09 we have

Figures 4 5 and 6 where the changes in mean queuelengths are given for increasing vacation-service ratesA hyperexponential model always has the highestmean queue length compared to the correspondingErlang and exponential models irrespective of thenumber of servers

(2) When the traffic load is heavy 120588 = 09 increasein vacation-service rate does not affect the mean



120588 120579 120583V119870119891

119871 119861119901


01

01

00 20 23 32 32118 32976 34094 00679 00887 0140505 11 13 18 21893 22065 22152 00008 00034 0015910 9 11 14 18365 18407 18401 00000 00003 0002415 8 9 12 16458 16464 16462 00000 00001 00005

1

00 8 11 15 17766 17958 18184 00000 00005 0003505 8 10 14 17120 17219 17334 00000 00002 0001610 8 9 13 16587 16631 16680 00000 00001 0000815 8 9 12 16136 16145 16155 00000 00000 00004

100

00 8 8 12 16022 16050 16039 00000 00000 0000405 8 8 12 16016 16041 16028 00000 00000 0000410 8 8 12 16009 16032 16016 00000 00000 0000415 8 8 12 16002 16024 16005 00000 00000 00004

05

01

00 30 34 52 63770 68052 84877 04072 04526 0545601 24 30 46 53485 56091 69287 02546 03073 0441102 21 25 41 46431 48028 57143 01445 01924 0337603 18 22 39 41697 42487 48192 00797 01170 02476

1

00 18 20 39 45180 46593 51256 01208 01617 0288901 18 20 39 43618 44869 49560 01007 01399 0267902 19 23 38 42119 42803 48001 00837 01208 0247703 18 22 38 40862 41494 46449 00696 01042 02284

100

00 17 20 38 40659 41500 46956 00662 01001 0223301 17 20 38 40633 41474 46923 00659 00998 0223002 18 23 38 40514 41099 46889 00657 00995 0222603 18 23 38 40489 41074 46856 00655 00993 02223

09

01

000 78 101 229 151572 173703 354516 09045 09054 09082005 76 99 227 142268 163514 340044 08719 08759 08903010 75 98 225 131999 152452 324807 08265 08360 08684015 73 96 222 121491 140835 309017 07663 07840 08418

1

000 79 105 246 120928 139233 312097 07748 07881 08391005 79 104 245 119039 137614 309647 07595 07758 08343010 78 103 245 117512 135946 306805 07436 07629 08292015 78 103 244 115535 134025 304553 07270 07496 08241

100

000 84 114 272 115312 133305 356288 07160 07407 08207005 84 114 272 115273 133271 356049 07157 07405 08206010 84 114 272 115235 133237 355810 07155 07403 08205015 84 113 272 115196 133446 355572 07152 07401 08204

queue lengths regardless of the arrival process or thenumber of servers (Figures 4 5 and 6)

(3) For 120588 = 01 and 119888 = 1 3 6 we plot Figures 7 8 and9 respectively Here the hyperexponential model hasthe least queue length compared to the correspondingErlang and exponential models For 119888 = 6 thequeue lengths are the same and all of them are shown

in Figure 9 Also it can be seen that for increasedvacation-service rates arrival processes do not havemuch influence on mean queue lengths

(4) The impatient rate 120585 affects the queue lengths sig-nificantly especially when 120588 = 01 In Figures10 11 and 12 the change in queue lengths withthe increase in impatient rate is shown When the


0 05 1 15 21

2

3

4

5

6

7M

ean

queu

e len

gth

Exp Erl Hyp

c = 1

Service rate 120583

Figure 1 Mean queue length versus vacation-service rate with 120588 =05 120585 = 01 120579 = 01 and 119888 = 1

0 05 1 15 2 25 31

2

3

4

5

6

7

8

9

Mea

n qu

eue l

engt

h

Exp Erl Hyp

c = 3

Service rate 120583


impatient rate is small the mean queue length forErlang model becomes minimum As the impatientrate increases it shows the reverse behaviorThe pointof inflection depends upon the service rate 120583V But theimpatient rate does not have much effect on queuelengths when the arrival process is Erlang whereasfor hyperexponential model the mean queue lengthdecreases significantly with the increase in impatientrate

Therefore systems with hyperexponential arrivals have thelongest queues compared to corresponding Erlang or expo-nential arrivals For light loaded systems (120588 = 01) andhighly impatient customers (1120585 lt 10) hyperexponentialarrivals give theminimumqueue lengthsWhen the customer

0 005 01 015 02 025 034

45

5

55

6

65

7

75

8

85

Mea

n qu

eue l

engt

h

Hyp Erl Exp

c = 6

Service rate 120583


0 02 04 06 08 15

10

15

20

25

30M

ean

queu

e len

gth c = 1

Service rate 120583

Exp Erl Hyp


impatient rates are small the system behavior depends on thevacation-service rates

53 Effect on Blocking Probability From the tables and thegraphs plotted for blocking probability we have seen thefollowing properties for the models under study

(1) Figure 13 gives that for a single-server system theblocking probability of a hyperexponential model isminimum and that for Erlang is maximum while120579 = 1 120585 = 1 and 120588 = 09 This behavior isalso observed in multiserver models when 120579 is toosmall For higher values of 120579 multiserver modelsfollow the reverse nature that is Erlang model gives


0 005 01 015 02 025 038

10

12

14

16

18

20

22

24

26

28

Mea

n qu

eue l

engt

h

c = 3

Service rate 120583

Exp Erl Hyp


0 002 004 006 008 01 012 014 01610

15

20

25

30

35

Mea

n qu

eue l

engt

h c = 6

Service rate 120583

Exp Erl Hyp


theminimumqueue length and the hyperexponentialone gives themaximumof those three different arrivalmodels That is the chance of blocking a customerwith Poisson arrival in a single-server as well as ina multiserver queue is always sandwiched betweenthose with Erlang and hyperexponential arrivals

(2) When we have a single-server Erlang model theblocking probabilities seem to reduce up to 6 withan increased rate of service during vacation Becauseof a single-server queue the server rarely goes tovacation (as the systems are rarely empty) and thecustomers are served at a higher rate most of thetimes And even when the system goes to vacation

0 2 4 6 8 1011

12

13

14

15

16

17

18

19

2

21

Mea

n qu

eue l

engt

h

c = 1

Service rate 120583

Exp Erl Hyp


0 05 1 15 2 25 3

14

16

18

2

22

24

26

28

3

Mea

n qu

eue l

engt

h

c = 3

Service rate 120583

Exp Erl Hyp


because of the single-server queue the queue startedto form rapidly making the customers impatientand leave the system more often than a multiservermodel These contribute significantly to dropping theblocking probability in a single-server queue as seenin the plot

(3) Figures 14 and 15 show that the hyperexponentialmodel is not much affected by the vacation-servicerate whereas the Erlang model can reduce the block-ing probability by up to 4 for increased vacation-service rates


0 05 1 1516

18

2

22

24

26

28

3

32

Mea

n qu

eue l

engt

h

c = 6

Service rate 120583

Exp Erl Hyp


0 2 4 6 8 10118

12

122

124

126

128

13

Mea

n qu

eue l

engt

h c = 1 120579 = 01

Impatient rate 120585

Exp Erl Hyp

Figure 10 Mean queue length versus impatient rate with 120588 = 01120583V = 05 and 119888 = 1

A model with 119888 = 1 and Erlang-2 arrival process has themaximum chance to make a customer wait in queue com-pared to exponential and hyperexponential arrivals but asthe number of servers is increased hyperexponential arrivalmodel has the highest blocking probability The exponentialarrival model always remains in between these two

54 Average Number of Servers Busy in Nonvacation Themean number of servers that are in working status duringnonvacation period is shown in subsequent figures

(1) Figure 16 is a plot of blocking probabilities withchanging 120579 Here for an increase in vacation durationthe number of servers that remain busy is highbecause the servers serve at a low rate but for longer

0 2 4 6 8 10146

1465

147

1475

148

1485

Mea

n qu

eue l

engt

h

c = 3 120579 = 01


Exp Erl Hyp


0 2 4 6 8 1026

265

27

275

Mea

n qu

eue l

engt

h

c = 6 120579 = 01


Exp Erl Hyp


time and a new arrival will be served by an idle serverif any Consequently it increases the number of busyservers in the system

(2) Figure 17 shows that if the service rate is fast cus-tomers are served at a faster rate which resultsin a lower number of busy servers in nonvacationperiod This is true for all the three types of arrivalmodels

(3) The impatience makes a customer leave the systemunserved and for high impatient rates more serversremain idle (Figure 18) But if the impatient rate isincreased beyond a certain value (120585 gt 6) the meannumber of busy servers remains unaffected


0 02 04 06 08 1088

089

09

091

092

093

094

095

Bloc

king

pro

babi

lity

c = 1

Service rate 120583

Exp Erl Hyp

Figure 13 Blocking probability versus vacation-service rate with120588 = 09 120579 = 1 120585 = 1 and 119888 = 1

0 005 01 015 02 025 03081

082

083

084

085

086

087

088

089

Bloc

king

pro

babi

lity

c = 3

Service rate 120583

Exp Erl Hyp


55 Average Customer Loss We plot the mean number ofcustomers who abandon the queue without getting served inFigures 19 and 20 The values of 120579 for these plots are 120579 = 01and 120579 = 1 respectively keeping the other parameters fixedfor both cases

When 120579 = 01 that is the system has longer vacations thenumber of lost customers is less compared to the correspond-ing model for 120579 = 1 In both cases the hyperexponentialmodels have themaximum customer loss which is up to 60more than the Erlang model Also the effect of impatientrates on customer loss is negligible for a system having smallvacation duration

For Figure 19 when vacation duration rate 120579 = 01 wecan see local maxima for Erlang and exponential models

0 002 004 006 008 01 012 014 016072

074

076

078

08

082

084

Bloc

king

pro

babi

lity

c = 6

Service rate 120583

Exp Erl Hyp


0 2 4 6 8 100

01

02

03

04

05

06M

ean

busy

serv

ers i

n no

nvac

atio

n

Vacation duration rate 120579

Exp Erl Hyp

Figure 16 Mean number of busy servers versus vacation durationrate with 119888 = 1 120588 = 05 120585 = 01 120583

119887= 10 and 120583V = 04

but minima for hyperexponential one at the point whereimpatient rate is equal to one From Figure 18 we cansee that when 119888 = 3 the number of busy servers innonvacation drops sharply until impatient rate becomes oneand then it is almost consistent thereafter This drop is moresignificant for hyperexponential model This suggests thatas impatient rates increase from zero to one the numberof busy servers in nonvacation period becomes less whichincreases the probability of losing more customers for Erlangand exponential models As the impatient rate increasesbeyond one the number of busy servers in nonvacationremains consistent and the loss of customer is influencedmainly by the increased rate of impatience But for thehyperexponential model this behaviour alters because of the


2 3 4 5 6 7 8 9 10005

01

015

02

025

03

035

04

045

05

Mea

n bu

sy se

rver

s in

nonv

acat

ion

Service rate 120583b

Exp Erl Hyp

Figure 17 Mean number of busy servers versus service rate with119888 = 1 120588 = 05 120585 = 01 and 120583V = 04

0 2 4 6 8 10078

08

082

084

086

088

09

092

094

096

098

Mea

n bu

sy se

rver

s in

nonv

acat

ion


Exp Erl Hyp

Figure 18 Mean number of busy servers versus impatient rate with119888 = 3 120588 = 05 120583V = 04 and 120579 = 01

change in mean queue lengths with the change of impatientrates (Section 52(4)) Its influence can be seen more towardsthe point of inflection where the blocking probability ofthe hyperexponential model becomes the same as that ofthe exponential one But as the impatient rate increasesthe customers leave the system increasing the customerloss

Thus we have seen the role of various parameters onsystem performances and we are now in a position to handlethem to enhance the system efficiency

6 Conclusion

In this paper we have analyzed the nonhomogeneous QBDmodel of a PHMc queue with impatient customers and

0 2 4 6 8 10064

065

066

067

068

069

07

071

072

073

Mea

n cu

stom

er lo

ss

Exp Erl Hyp


Figure 19 Mean customer loss versus impatient rate with 119888 = 3120588 = 05 120583V = 04 and 120579 = 01

0 2 4 6 8 1001

015

02

025

03

035

04

045

Mea

n cu

stom

er lo

ss

Exp Erl Hyp



multiple working vacations We have used the finite trun-cation method to determine the stationary distribution Theeffects of system parameters on the performance measuresof the model are illustrated with the help of some numericalexamples Comparisons are made for different interarrivaltime distributions and the effects of the parameters on thosedistributions are also presented

Disclosure

The present address of Cosmika Goswami is School of Engi-neering University of Glasgow Rankine Building GlasgowG12 8LT UK


Competing Interests

The authors declare that there are no competing interests

References

[1] Y Levy and U Yechiali ldquoAn MMc queue with serversvacationsrdquo INFOR vol 14 no 2 pp 153ndash163 1976

[2] N Tian and Q Li ldquoThe MMc queue with PH synchronousvacationsrdquo System in Mathematical Science vol 13 no 1 pp 7ndash16 2000

[3] N Tian Q-L Li and J Cao ldquoConditional stochastic decompo-sitions in the MMc queue with server vacationsrdquo Communi-cations in Statistics Stochastic Models vol 15 no 2 pp 367ndash3771999

[4] N Tian and Z Zhang ldquoMMc queue with synchronous vaca-tions of some servers and its application to electronic commerceoperationsrdquo Working Paper 2000

[5] N Tian and Z Zhang ldquoA two threshold vacation policy inmultiserver queueing systemsrdquo European Journal of OperationalResearch vol 168 pp 153ndash163 2006

[6] X Yang and A S Alfa ldquoA class of multi-server queueing systemwith server failuresrdquo Computers and Industrial Engineering vol56 no 1 pp 33ndash43 2009

[7] S R Chakravarthy ldquoAnalysis of a multi-server queue withMarkovian arrivals and synchronous phase type vacationsrdquoAsia-Pacific Journal of Operational Research vol 26 no 1 pp85ndash113 2009

[8] C Palm ldquoMethods of judging the annoyance caused by conges-tionrdquo Telecommunications vol 4 pp 153ndash163 1953

[9] D Gross J F Shortle J M Thompson and C M HarrisFundamentals of Queueing Theory John Wiley amp Sons 4thedition 2008

[10] N Perel and U Yechiali ldquoQueues with slow servers and impa-tient customersrdquo European Journal of Operational Research vol201 no 1 pp 247ndash258 2010

[11] E Altman and U Yechiali ldquoAnalysis of customersrsquo impatiencein queues with server vacationsrdquo Queueing Systems vol 52 no4 pp 261ndash279 2006

[12] E Altman and U Yechiali ldquoInfinite-server queues with systemrsquosadditional tasks and impatient customersrdquo Probability in theEngineering and Informational Sciences vol 22 no 4 pp 477ndash493 2008

[13] A Economou and S Kapodistria ldquoSynchronized abandon-ments in a single server unreliable queuerdquo European Journal ofOperational Research vol 203 no 1 pp 143ndash155 2010

[14] F Baccelli P Boyer and G Hebuterne ldquoSingle server queueswith impatient customersrdquoAdvances in Applied Probability vol16 no 4 pp 887ndash905 1984

[15] U Yechiali ldquoQueues with system disasters and impatient cus-tomers when system is downrdquo Queueing Systems vol 56 no3-4 pp 195ndash202 2007

[16] Y Sakuma and A Inoie ldquoStationary distribution of a multi-server vacation queue with constant impatient timesrdquo Opera-tions Research Letters vol 40 no 4 pp 239ndash243 2012

[17] P-S Chen Y-J Zhu and X Geng ldquoMMmk queue with pre-emptive resume and impatience of the prioritized customersrdquoSystems Engineering and Electronics vol 30 no 6 pp 1069ndash1073 2008

[18] C-S Ho and CWoei ldquoStudy of re-provisioningmechanism fordynamic traffic in WDM optical networksrdquo in Proceedings of

the 10th International Conference on Advanced CommunicationTechnology pp 288ndash291 Gangwon-Do Republic of KoreaFebruary 2008

[19] J Wang ldquoQueueing analysis of WDM-based access networkswith reconfiguration delayrdquo in Proceedings of the 4th Interna-tional Conference on Queueing Theory and Network Applica-tions Article No 13 Singapore July 2009

[20] C Goswami and N Selvaraju ldquoThe discrete-time MAPPH1queue with multiple working vacationsrdquo Applied MathematicalModelling vol 34 no 4 pp 931ndash946 2010

[21] L D Servi and S G Finn ldquoMM1 queues with workingvacations (MM1WV)rdquo Performance Evaluation vol 50 no1 pp 41ndash52 2002

[22] W-Y Liu X-L Xu and N-S Tian ldquoStochastic decomposi-tions in the MM1 queue with working vacationsrdquo OperationsResearch Letters vol 35 no 5 pp 595ndash600 2007

[23] N Tian X Zhao and K Wang ldquoThe MM1 queue with singleworking vacationrdquo International Journal of Information andManagement Sciences vol 19 no 4 pp 621ndash634 2008

[24] X Xu Z Zhang and N Tian ldquoThe MM1 queue with singleworking vacation and set-up timesrdquo International Journal ofOperational Research vol 6 no 3 pp 420ndash434 2009

[25] C Xiu N Tian and Y Liu ldquoThe MM1 queue with singleworking vacation serving at a slower rate during the start-upperiodrdquo Journal of Mathematics Research vol 2 no 1 pp 98ndash102 2010

[26] K-H Wang W-L Chen and D-Y Yang ldquoOptimal manage-ment of the machine repair problem with working vacationNewtonrsquos methodrdquo Journal of Computational and AppliedMath-ematics vol 233 no 2 pp 449ndash458 2009

[27] D-A Wu and H Takagi ldquoMG1 queue with multiple workingvacationsrdquo Performance Evaluation vol 63 no 7 pp 654ndash6812006

[28] Y Baba ldquoAnalysis of a GIM1 queue with multiple workingvacationsrdquo Operations Research Letters vol 33 no 2 pp 201ndash209 2005

[29] H Chen F Wang N Tian and D Lu ldquoStudy on N-policyworking vacation polling system for WDMrdquo in Proceedings ofIEEE International Conference on Communication Technologypp 394ndash397 Hangzhou China November 2008

[30] H Chen F Wang N Tian and J Qian ldquoStudy on workingvacation polling system for WDMwith PH distribution servicetimerdquo in Proceedings of the International Symposium on Com-puter Science and Computational Technology (ISCSCT rsquo08) pp426ndash429 Shanghai China December 2008

[31] C-H Lin and J-C Ke ldquoMulti-server system with singleworking vacationrdquo Applied Mathematical Modelling vol 33 no7 pp 2967ndash2977 2009

[32] L B Aronson B E Lemoff L A Buckman and D W DolfildquoLow-cost multimode WDM for local area networks up to10Gbsrdquo IEEE Photonics Technology Letters vol 10 no 10 pp1489ndash1491 1998

[33] N Selvaraju and C Goswami ldquoImpatient customers in anMM1 queue with single and multiple working vacationsrdquoComputers amp Industrial Engineering vol 65 no 2 pp 207ndash2152013

[34] P V Laxmi and K Jyothsna ldquoAnalysis of a working vacationsqueue with impatient customers operating under a triadicpolicyrdquo International Journal of Management Science and Engi-neering Management vol 9 no 3 pp 191ndash200 2014


[35] N Tian and Z G Zhang ldquoStationary distributions of GIMcqueue with PH type vacationsrdquo Queueing Systems vol 44 no2 pp 183ndash202 2003

[36] J R Artalejo A Economou and A Gomez-Corral ldquoAlgorith-mic analysis of the GeoGeoc retrial queuerdquo European Journalof Operational Research vol 189 no 3 pp 1042ndash1056 2008

[37] S R Chakravarthy A Krishnamoorthy andVC Joshua ldquoAnal-ysis of a multi-server retrial queue with search of customersfrom the orbitrdquo Performance Evaluation vol 63 no 8 pp 776ndash798 2006

[38] G I Falin ldquoCalculation of probability characteristics of amultiline system with repeated callsrdquo Moscow University Com-putational Mathematics and Cybernetics vol 1 pp 43ndash49 1983

[39] J R Artalejo and M Pozo ldquoNumerical calculation of thestationary distribution of the main multiserver retrial queuerdquoAnnals of Operations Research vol 116 pp 41ndash56 2002

[40] L Bright and P G Taylor ldquoCalculating the equilibrium dis-tribution in level dependent quasi-birth-and-death processesrdquoCommunications in Statistics StochasticModels vol 11 no 3 pp497ndash525 1995

[41] A Krishnamoorthy S Babu and V C Narayanan ldquoThe MAP(PHPH)1 queue with self-generation of priorities and non-preemptive servicerdquo European Journal of Operational Researchvol 195 no 1 pp 174ndash185 2009

[42] M F Neuts and B M Rao ldquoNumerical investigation of amultiserver retrial modelrdquo Queueing Systems vol 7 no 2 pp169ndash189 1990

[43] G H Golub and C F VanLoanMatrix ComputationsThe JohnHopkins University Press London UK 1996

[44] J Hunter Mathematical Techniques of Applied Probability Dis-crete Time Models Basic Theory vol 1 Academic Press NewYork NY USA 1983

Submit your manuscripts athttpwwwhindawicom

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Stochastic AnalysisInternational Journal of


Economou and Kapodistria [13] considered an unreliablequeue where the customers leave the system at system failuretimes

Multiserver queues with impatience however haveattracted much attention in queueing literature possiblybecause of explosive demands to efficiently design andmanage call or contact centres Baccelli et al [14] studiedthe waiting time distribution in MMc queue with generalimpatience bound on queueing times by constructing asimple Markov process and also gave the waiting timedistribution in the MG1 queue with general impatienceon queueing times Yechiali [15] considered an MMcsystem which as a whole suffers occasionally a disastrousbreakdown upon which all present customers (waiting andserved) are cleared from the system and lost Stationarydistribution of a multiserver vacation queue with con-stant impatient times is studied by Sakuma and Inoie [16]Chen et al [17] studied MMmk queue with preemptiveresume and impatience of the prioritised customers andderived the queue length distraction in stationary stateand performance measures using the method of matrixanalysis

In communication systems wavelength division multi-plexing (WDM) is a method of transmitting packets fromdifferent sources over the same fiber optic link to thedestination A WDM network divides the available fiberbandwidth into WDM channels details in Ho and Woei [18]and also inWang [19] This division of bandwidth or channelallocation is based on the capacities required for variousservices For a high performance system WDM channelallocation should lead to optimized resource utilization in agiven network which is physically feasible and cost-effectiveA reconfigurable WDM system can be modelled as a queuewith working vacations (WVs) as explained in Goswami andSelvaraju [20] This vacation cannot be put in a classicalvacation framework because here unless the system is emptythe service does not cease completely Servi and Finn [21]were the first to model such a WDM network into a WVqueueing model Liu et al [22] studied the MM1WVmodel with multiple WVs whereas the single WV model isanalyzed by Tian et al [23] The same model is studied byXu et al [24] and also by Xiu et al [25] with single WVand setup times Wang et al [26] presented the MM1WVmodel using Newtonrsquos method to compute the steady-stateprobabilities and system performance measures Wu andTakagi [27] extended Servi and Finnrsquos work to MG1WVmodel with generally distributed service times and vacationduration times Baba [28] considered the GIM1WV systemwith general independent arrival process where the distri-butions of the vacation duration times and service timesare exponential Chen et al [29 30] proposed an N-policyWV and a cyclic polling system for WDM taking the servicetimes as exponential and PH distribution respectively Linand Ke [31] considered a multiserver MMc queue and acost model is derived to determine the optimal values ofthe number of servers and the WV rate simultaneouslyin order to minimize the total expected cost per unittime

Short distance networks like local area networks (LANs)mostly usemultimodeWDMlinksMultimode link is a singlefiber link that supports many propagation paths or transversemodes through it Aronson et al [32] explained how thebandwidth of the fiber is multiplied by the number of pathsused by using WDM in multimode fiber LAN over InternetProtocol (IP) allows the forwarding of LAN packets over theInternet or an intranet network One of the most criticalperformance measures in LAN over IP is the percentageof packets that are transmitted within hard delay bound ortime constraint If quality of service requirements is not metwithin the time bound end users may terminate the Internetconnections A connection is terminated by pressing the stopbutton refreshing the connection or following a differentlink This behavior can be termed the impatience of a userin LANs To study the effect of multiple servers and userimpatience on the performance in a WDM network we con-sider in this paper a multiserver model with asynchronousmultiple working vacation (AMWV) policy and impatientcustomers In an AMWV policy the servers take vacationsindividually and continue taking vacations till they do notfind any customer in the system An MM1WV impatientmodel with single and multiple WV policies is studied bySelvaraju andGoswami [33] Analysis of a finite bufferMM2working vacations queue with balking and reneging whereinthe servers operate under a triadic (0QNM) policy isdone recently by Laxmi and Jyothsna [34] Lin and Ke[31] presented a multiserver WV queue with exponentialinterarrivals but none of these models represent systems withnonexponential arrivals or state-dependent systems To studythe role of arrival processes in a multiserver model havingimpatient customers we consider here the PHarrival processPH distribution is a general nonexponential distributioncharacterized by a Markov chain Importance of consideringPH interarrivals is the fact that PH distribution is able tocapture the nonexponential effects on arrivals while infor-mation flows in modern communication systems are rarelyexponential PH distribution is able to capture the profoundeffect of arrivals in system performance measures and makesthe mathematical model more convincing to fit a real worldscenario

The paper is organized as follows In Section 2 weformulate the system as a three-dimensional continuous-time Markov chain whose generator matrix is a level-dependent quasi-birth-death (QBD) process Section 3 givesthe finite truncation method used to find the station-ary probability vector of the level-dependent process Thevarious performance measures are listed in Section 4 andin Section 5 the numerical illustrations of the system arepresented

2 Model Description

We consider a PHMc queue with multiple WVs and impa-tient customers The interarrival times of customers follow aPH distribution PH(120572 119879) of dimension 119899 and with arrivalrate 120582 A PH distribution denotes the distribution of time



119875 = [119879 1198790

0 0] (1)


1198790= minus119879e

119899 where e



119894119894lt 0 1 le 119894 le 119899 and (119879)

119894119895ge 0




119887) A





Δ = (119873119905 119876119905 119869119905) 119905 ge 0 (2)





119864 = (119894 119895 119896) 0 le 119894 lt 119888 119895 le 119894 1 le 119896 le 119899

cup (119894 119895 119896) 119894 ge 119888 0 le 119895 le 119888 1 le 119896 le 119899

(3)


119876 =

(((((

(

119860(0)

1119860(0)

0

119860(1)

2119860(1)

1119860(1)

0

119860(2)

2119860(2)

1119860(2)

0

119860(3)

2119860(3)

1119860(3)

0

d d d

)))))

)

(4)

where for 1 le 119894 lt 119888

119860(119894)

2=

[[[[[[[[[[[

[

119894120583V119868

(120583119887+ (119894 minus 1) 120583V) 119868

(2120583119887+ (119894 minus 2) 120583V) 119868

d

((119894 minus 1) 120583119887+ 120583V) 119868

119894120583119887119868

]]]]]]]]]]]

](119894+1)119899times(119894+1)119899

(5)

and for 0 le 119894 lt 119888


119860(119894)

1=

[[[[[[[

[

119879 minus (119888120579 + 119894120583V) 119868 119888120579119868


d

119879 minus 119894120583119887119868

]]]]]]]

](119894+1)119899times(119894+1)119899

119860(119894)

0=

[[[[[[

[

1198790120572

1198790120572

d

1198790120572

]]]]]]

](119894+1)119899times(119894+1)119899

(6)

For 119894 ge 119888

119860(119888+119897)

2=

[[[[[[[[[[

[

(119888120583V + 119897120585) 119868

(120583119887+ (119888 minus 1) 120583V) 119868

(2120583119887+ (119888 minus 2) 120583V) 119868

d

119888120583119887119868

]]]]]]]]]]

](119888+1)119899times(119888+1)119899

119860(119888+119897)

1=

[[[[[[[

[

119879 minus (119888 (120579 + 120583V) + 119897120585) 119868 119888120579119868

d d

119879 minus (120579 + 120583V + (119888 minus 1) 120583119887) 119868 120579119868

119879 minus (119888120583119887) 119868

]]]]]]]

](119888+1)119899times(119888+1)119899

119860(119888+119897)

0=

[[[[[[

[

1198790120572

1198790120572

d

1198790120572

]]]]]]

](119888+1)119899times(119888+1)119899

(7)


0

119860(119894)

1 and 119860(119894)





[28 35]


119909119876 = 0

119909e = 1(8)


119894=

[1199091198941 1199091198942 119909




row vector 119909119894119895= [1199091198941198951 119909


phases of arrivals







119891 Here we use



(119870)

=

(((((((((

(

119860(0)

1119860(0)

0

119860(1)

2119860(1)

1119860(1)

0

119860(2)

2119860(2)

1119860(2)

0

d d d

119860(119870minus1)

2119860(119870minus1)

1119860(119870minus1)

0

119860(119870)

2120601(119870)

1

)))))))))

)

(9)

where 120601(119870)1= 119860(119870)

1+119860(119870)



120587 (119870) = 0

120587e = 1(10)


0(119870) 119911

1(119870)]

with

1199110(119870) = [120587 (0) 120587 (1) 120587 (119870 minus 1)]

1199111(119870) = 120587 (119870)

(11)




0(119870) and 119911

1(119870) we have

(1199110(119870) 119911

1(119870)) (

11986100(119870) 119861

01(119870)

11986110 (119870) 11986111 (119870)

) = (0119898 0119899(119888+1)) (12)



trans[0 0 0 119860(119870minus1)0] 11986110(119870) = [0 0 0 119860

(119870)

2] and

11986111(119870) = 120601

(119870)




1199111 (119870) 11986110 (119870) 119861

minus1

00(119870) = minus1199110 (119870) (13)

1199111(119870) [119861

11(119870) minus 119861

10(119870) 119861minus1

00(119870) 11986101(119870)] = 0

119899(119888+1) (14)

Further we can have

11986100(119870) = (

11986100(119870 minus 1) 119861

01(119870 minus 1)

1198620(119870 minus 1) 119862

1(119870 minus 1)

) (15)

where

1198620 (119870 minus 1) = [0119899 02119899 0 119860

(119870minus1)

2]

1198621(119870 minus 1) = minus119860

(119870minus1)

1

(16)



119861minus1

00(119870) = (

11986300 (119870) 11986301 (119870)

11986310(119870) 119863

11(119870)) (17)


where

11986300 (119870) = [11986100 (119870 minus 1)

minus 11986110(119870 minus 1) 119862

minus1

1(119870 minus 1) 119862

0(119870 minus 1)]

minus1

11986310(119870) = minus119862

minus1

1(119870 minus 1) 119862

0(119870 minus 1)119863

00(119870)

11986311 (119870) = [1198621 (119870 minus 1)

minus 1198620(119870 minus 1) 119861

minus1

00(119870 minus 1) 119861

01(119870 minus 1)]

minus1

11986301(119870) = minus119861

minus1

00(119870 minus 1) 119861

01(119870 minus 1)119863

11(119870)

(18)


01(119870 minus 1) and C

0(119870 minus 1)






00(119870minus


10(119870 minus

1) 11986311(119870minus1)]119860

(119870minus1)

0 Further119862

0(119870minus1)119861

minus1

00(119870minus1)119861

01(119870minus1)


(119870minus1)






00(119870)


119861minus1= [119868 minus 119860

minus1119883(119882


119884]119860minus1 (19)


01(119870 minus 1) 119882minus1 =

119862minus1

1(119870 minus 1) and 119884 = 119862


11986300(119870) = 119861

minus1

= [119868 minus 11986301 (119870)1198620 (119870 minus 1)] 119861

minus1

00(119870 minus 1)

(20)



11986311(119870)

= [1198621(119870 minus 1) minus 119860

(119870minus1)

211986311(119870 minus 1)119860

(119870minus1)

0]minus1

11986301(119870)

= minus [11986310(119870 minus 1) 119863

11(119870 minus 1)] 119860

(119870minus1)

011986311(119870)

(1) 119870 = 119888 + 1(2) compute 119861minus1

00(119870)

(3) compute 1199111(119870) by (14) and (22)

(4) compute 1199110(119870) by (13)

(5) store 11986100(119870) 119861

minus1

00(119870) 119861

01(119870) and 119861

10(119870)

(6) 119870 = 119870 + 1(7) while 119870 le 119870

119891


00(119870) by (17)

(10) compute 1199111(119870) by (14) and (22)

(11) compute 1199110(119870) by (13)

(12) update 11986100(119870) 119861minus1

00(119870) 119861

01(119870) and 119861

10(119870)


infinlt 120598

(14) then 119870119891= 119870

(15) break(16) else 119870 = 119870 + 1


11986300(119870) = [119868 minus 119863

01(119870)119862

0(119870 minus 1)] 119861

minus1

00(119870 minus 1)

11986310 (119870) = minus119862

minus1

1(119870 minus 1) 1198620 (119870 minus 1)11986300 (119870)

(21)



120587 (119870) [e119899(119888+1) minus 11986110 (119870) 119861minus1

00(119870) e119899119870(119888+1)] = 1 (22)

where e119899(119888+1)

and 119890119899119870(119888+1)



1(119870)


119891when

max0le119894le119870119891

10038171003817100381710038171003817119911 (119870119891 119894) minus 119911 (119870

119891minus 1 119894)10038171003817100381710038171003817infinlt 120598 (23)









nonvacation is

119861119904=

119888minus1

sum

119894=0

119894

sum

119895=1

119895119909119894119895e119899+

infin

sum

119894=119888

119888

sum

119895=1

119895119909119894119895e119899 (24)


119871 = 119864 (119873) =

119888minus1

sum

119894=1

119894119909119894e(119894+1)119899+

infin

sum

119894=119888

119894119909119894e(119888+1)119899 (25)


119877 = 119875 (119873 lt 119888) =

119888minus1

sum

119894=0

119909119894e(119894+1)119899 (26)


119861119901= 119875 (119873 gt 119888) = 1 minus

119888minus1

sum

119894=0

119909119894e(119894+1)119899= 1 minus 119877 (27)


119873119888=

infin

sum

119894=119888+1

1198941199091198940e119899 (28)





119891such that

max0le119894le119870119891119911(119870119891 119894) minus 119911(119870

119891minus 1 119894)






119879 = minus1

1198790= 1

120572 = 1

(29)

(2) Erlang-2 (Erl)

119879 = (

minus2 2

0 minus2)

1198790= (

0

2)

120572 = [1 0]

(30)


119879 = (

minus19 0

0 minus019)

1198790= (

19

019)

120572 = [09 01]

(31)







119891





120588 120579 120583V119870119891

119871 119861119901


01

01

00 25 28 36 64229 64127 63354 09354 09143 0795530 10 13 18 13951 14544 15802 03211 03173 0308860 7 9 13 11771 11963 12238 01652 01647 0163990 6 8 11 11149 11244 11358 01109 01108 01107

1

00 14 16 22 20525 20533 20481 06062 05479 0432530 8 11 15 13040 13342 13793 02650 02562 0242160 6 8 12 11656 11811 12011 01561 01545 0152490 6 7 11 11137 11229 11337 01099 01097 01094

100

00 5 6 10 11131 11215 11305 01097 01089 0108230 5 6 10 11099 11184 11273 01066 01061 0105660 5 6 10 11068 11153 11243 01037 01034 0103190 5 6 10 11040 11124 11214 01009 01008 01008

05

01

00 25 28 42 64099 65109 69481 09579 09412 0823605 21 25 41 44283 45761 52829 09062 08788 0745710 17 21 40 29028 31115 39708 07716 07451 0652015 15 20 38 21227 23112 30542 06121 05987 05579

1

00 17 19 38 27191 28952 36542 07863 07442 0627005 16 21 38 23956 25839 33851 07113 06784 0594910 16 21 38 21386 23326 31440 06331 06121 0561615 15 20 38 19441 21339 29316 05605 05505 05280

100

00 14 18 37 18203 20117 28092 05061 05049 0502705 14 20 37 18178 20080 28066 05046 05037 0502110 15 20 37 18146 20055 28041 05030 05025 0501415 15 20 37 18121 20030 28016 05015 05012 05007

09

01

00 74 99 246 112890 132254 285672 09893 09846 0950903 73 98 253 100678 119600 271965 09805 09735 0935006 71 97 260 87709 106206 257831 09593 09495 0913909 75 102 265 74716 92882 243478 09142 09057 08880

1

00 73 96 231 87160 108465 272102 09574 09479 0918703 72 96 233 84580 105880 269545 09445 09361 0913506 72 96 234 82028 103341 267009 09289 09228 0908009 71 95 235 79573 100868 264490 09111 09080 09022

100

00 81 110 265 78679 100268 268945 09013 09010 0900403 81 110 265 78651 100240 268905 09010 09007 0900306 81 110 265 78623 100213 268865 09006 09005 0900209 81 110 265 78596 100185 268825 09003 09002 09001








120588 120579 120583V119870119891

119871 119861119901


01

01

00 22 25 33 41401 41552 41155 04953 04909 0445210 10 13 18 18312 18449 18849 00376 00606 0106420 8 10 13 14721 14743 14805 00050 00133 0029430 7 8 11 13301 13307 13326 00013 00049 00117

1

00 9 12 16 16336 16535 16721 00203 00421 0073010 7 10 13 14723 14792 14875 00056 00163 0034320 7 9 12 13787 13811 13847 00022 00078 0018130 6 8 11 13165 13172 13189 00011 00044 00107

100

00 6 7 11 13037 13051 13062 00009 00039 0009610 6 7 11 13026 13039 13049 00009 00039 0009520 6 7 11 13015 13027 13036 00009 00038 0009330 6 7 11 13005 13016 13023 00009 00037 00092

05

01

00 36 41 58 61463 65540 81834 07544 07535 0709402 26 30 48 44310 47592 61809 05820 05952 0606204 20 24 43 33472 35687 46449 03792 04116 0487806 16 20 39 27521 28807 35933 02308 02694 03736

1

00 17 19 37 31041 32843 40155 03311 03663 0441402 16 19 37 29373 30968 38141 02832 03217 0411604 17 21 37 27875 29234 36269 02421 02821 0382506 16 21 37 26620 27814 34534 02074 02473 03544

100

00 16 19 37 26304 27513 34302 01989 02385 0346802 16 21 37 26286 27437 34280 01984 02380 0346304 16 21 37 26267 27418 34257 01978 02375 0345906 16 21 37 26249 27399 34234 01973 02370 03455

09

01

00 78 100 224 140668 163843 336740 09618 09582 0940301 75 98 224 127466 149913 319614 09424 09390 0925102 73 95 229 113801 135544 301618 09089 09077 0905203 71 93 234 99871 120651 282970 08541 08593 08796

1

00 76 101 239 97276 117783 283667 08612 08643 0880601 76 100 238 95526 116045 281630 08466 08523 0875802 75 100 237 93873 114217 279598 08312 08397 0870803 75 99 237 92075 112421 277421 08149 08265 08656

100

00 82 111 269 91219 111422 293701 08041 08176 0862201 82 111 269 91194 111398 293614 08038 08175 0862102 82 111 269 91169 111375 293527 08036 08173 0862103 82 111 269 91144 111351 293440 08033 08171 08620







120588 120579 120583V119870119891

119871 119861119901


01

01

00 20 23 32 32118 32976 34094 00679 00887 0140505 11 13 18 21893 22065 22152 00008 00034 0015910 9 11 14 18365 18407 18401 00000 00003 0002415 8 9 12 16458 16464 16462 00000 00001 00005

1

00 8 11 15 17766 17958 18184 00000 00005 0003505 8 10 14 17120 17219 17334 00000 00002 0001610 8 9 13 16587 16631 16680 00000 00001 0000815 8 9 12 16136 16145 16155 00000 00000 00004

100

00 8 8 12 16022 16050 16039 00000 00000 0000405 8 8 12 16016 16041 16028 00000 00000 0000410 8 8 12 16009 16032 16016 00000 00000 0000415 8 8 12 16002 16024 16005 00000 00000 00004

05

01

00 30 34 52 63770 68052 84877 04072 04526 0545601 24 30 46 53485 56091 69287 02546 03073 0441102 21 25 41 46431 48028 57143 01445 01924 0337603 18 22 39 41697 42487 48192 00797 01170 02476

1

00 18 20 39 45180 46593 51256 01208 01617 0288901 18 20 39 43618 44869 49560 01007 01399 0267902 19 23 38 42119 42803 48001 00837 01208 0247703 18 22 38 40862 41494 46449 00696 01042 02284

100

00 17 20 38 40659 41500 46956 00662 01001 0223301 17 20 38 40633 41474 46923 00659 00998 0223002 18 23 38 40514 41099 46889 00657 00995 0222603 18 23 38 40489 41074 46856 00655 00993 02223

09

01

000 78 101 229 151572 173703 354516 09045 09054 09082005 76 99 227 142268 163514 340044 08719 08759 08903010 75 98 225 131999 152452 324807 08265 08360 08684015 73 96 222 121491 140835 309017 07663 07840 08418

1

000 79 105 246 120928 139233 312097 07748 07881 08391005 79 104 245 119039 137614 309647 07595 07758 08343010 78 103 245 117512 135946 306805 07436 07629 08292015 78 103 244 115535 134025 304553 07270 07496 08241

100

000 84 114 272 115312 133305 356288 07160 07407 08207005 84 114 272 115273 133271 356049 07157 07405 08206010 84 114 272 115235 133237 355810 07155 07403 08205015 84 113 272 115196 133446 355572 07152 07401 08204






0 05 1 15 21

2

3

4

5

6

7M

ean

queu

e len

gth

Exp Erl Hyp

c = 1

Service rate 120583


0 05 1 15 2 25 31

2

3

4

5

6

7

8

9

Mea

n qu

eue l

engt

h

Exp Erl Hyp

c = 3

Service rate 120583




0 005 01 015 02 025 034

45

5

55

6

65

7

75

8

85

Mea

n qu

eue l

engt

h

Hyp Erl Exp

c = 6

Service rate 120583


0 02 04 06 08 15

10

15

20

25

30M

ean

queu

e len

gth c = 1

Service rate 120583

Exp Erl Hyp






0 005 01 015 02 025 038

10

12

14

16

18

20

22

24

26

28

Mea

n qu

eue l

engt

h

c = 3

Service rate 120583

Exp Erl Hyp


0 002 004 006 008 01 012 014 01610

15

20

25

30

35

Mea

n qu

eue l

engt

h c = 6

Service rate 120583

Exp Erl Hyp




0 2 4 6 8 1011

12

13

14

15

16

17

18

19

2

21

Mea

n qu

eue l

engt

h

c = 1

Service rate 120583

Exp Erl Hyp


0 05 1 15 2 25 3

14

16

18

2

22

24

26

28

3

Mea

n qu

eue l

engt

h

c = 3

Service rate 120583

Exp Erl Hyp





0 05 1 1516

18

2

22

24

26

28

3

32

Mea

n qu

eue l

engt

h

c = 6

Service rate 120583

Exp Erl Hyp


0 2 4 6 8 10118

12

122

124

126

128

13

Mea

n qu

eue l

engt

h c = 1 120579 = 01


Exp Erl Hyp





0 2 4 6 8 10146

1465

147

1475

148

1485

Mea

n qu

eue l

engt

h

c = 3 120579 = 01


Exp Erl Hyp


0 2 4 6 8 1026

265

27

275

Mea

n qu

eue l

engt

h

c = 6 120579 = 01


Exp Erl Hyp






0 02 04 06 08 1088

089

09

091

092

093

094

095

Bloc

king

pro

babi

lity

c = 1

Service rate 120583

Exp Erl Hyp


0 005 01 015 02 025 03081

082

083

084

085

086

087

088

089

Bloc

king

pro

babi

lity

c = 3

Service rate 120583

Exp Erl Hyp





0 002 004 006 008 01 012 014 016072

074

076

078

08

082

084

Bloc

king

pro

babi

lity

c = 6

Service rate 120583

Exp Erl Hyp


0 2 4 6 8 100

01

02

03

04

05

06M

ean

busy

serv

ers i

n no

nvac

atio

n


Exp Erl Hyp


119887= 10 and 120583V = 04



2 3 4 5 6 7 8 9 10005

01

015

02

025

03

035

04

045

05

Mea

n bu

sy se

rver

s in

nonv

acat

ion


Exp Erl Hyp


0 2 4 6 8 10078

08

082

084

086

088

09

092

094

096

098

Mea

n bu

sy se

rver

s in

nonv

acat

ion


Exp Erl Hyp




6 Conclusion


0 2 4 6 8 10064

065

066

067

068

069

07

071

072

073

Mea

n cu

stom

er lo

ss

Exp Erl Hyp



0 2 4 6 8 1001

015

02

025

03

035

04

045

Mea

n cu

stom

er lo

ss

Exp Erl Hyp




Disclosure



Competing Interests


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119875 = [119879 1198790

0 0] (1)


1198790= minus119879e

119899 where e



119894119894lt 0 1 le 119894 le 119899 and (119879)

119894119895ge 0




119887) A





Δ = (119873119905 119876119905 119869119905) 119905 ge 0 (2)





119864 = (119894 119895 119896) 0 le 119894 lt 119888 119895 le 119894 1 le 119896 le 119899

cup (119894 119895 119896) 119894 ge 119888 0 le 119895 le 119888 1 le 119896 le 119899

(3)


119876 =

(((((

(

119860(0)

1119860(0)

0

119860(1)

2119860(1)

1119860(1)

0

119860(2)

2119860(2)

1119860(2)

0

119860(3)

2119860(3)

1119860(3)

0

d d d

)))))

)

(4)

where for 1 le 119894 lt 119888

119860(119894)

2=

[[[[[[[[[[[

[

119894120583V119868

(120583119887+ (119894 minus 1) 120583V) 119868

(2120583119887+ (119894 minus 2) 120583V) 119868

d

((119894 minus 1) 120583119887+ 120583V) 119868

119894120583119887119868

]]]]]]]]]]]

](119894+1)119899times(119894+1)119899

(5)

and for 0 le 119894 lt 119888


119860(119894)

1=

[[[[[[[

[

119879 minus (119888120579 + 119894120583V) 119868 119888120579119868


d

119879 minus 119894120583119887119868

]]]]]]]

](119894+1)119899times(119894+1)119899

119860(119894)

0=

[[[[[[

[

1198790120572

1198790120572

d

1198790120572

]]]]]]

](119894+1)119899times(119894+1)119899

(6)

For 119894 ge 119888

119860(119888+119897)

2=

[[[[[[[[[[

[

(119888120583V + 119897120585) 119868

(120583119887+ (119888 minus 1) 120583V) 119868

(2120583119887+ (119888 minus 2) 120583V) 119868

d

119888120583119887119868

]]]]]]]]]]

](119888+1)119899times(119888+1)119899

119860(119888+119897)

1=

[[[[[[[

[

119879 minus (119888 (120579 + 120583V) + 119897120585) 119868 119888120579119868

d d

119879 minus (120579 + 120583V + (119888 minus 1) 120583119887) 119868 120579119868

119879 minus (119888120583119887) 119868

]]]]]]]

](119888+1)119899times(119888+1)119899

119860(119888+119897)

0=

[[[[[[

[

1198790120572

1198790120572

d

1198790120572

]]]]]]

](119888+1)119899times(119888+1)119899

(7)


0

119860(119894)

1 and 119860(119894)





[28 35]


119909119876 = 0

119909e = 1(8)


119894=

[1199091198941 1199091198942 119909




row vector 119909119894119895= [1199091198941198951 119909


phases of arrivals







119891 Here we use



(119870)

=

(((((((((

(

119860(0)

1119860(0)

0

119860(1)

2119860(1)

1119860(1)

0

119860(2)

2119860(2)

1119860(2)

0

d d d

119860(119870minus1)

2119860(119870minus1)

1119860(119870minus1)

0

119860(119870)

2120601(119870)

1

)))))))))

)

(9)

where 120601(119870)1= 119860(119870)

1+119860(119870)



120587 (119870) = 0

120587e = 1(10)


0(119870) 119911

1(119870)]

with

1199110(119870) = [120587 (0) 120587 (1) 120587 (119870 minus 1)]

1199111(119870) = 120587 (119870)

(11)




0(119870) and 119911

1(119870) we have

(1199110(119870) 119911

1(119870)) (

11986100(119870) 119861

01(119870)

11986110 (119870) 11986111 (119870)

) = (0119898 0119899(119888+1)) (12)



trans[0 0 0 119860(119870minus1)0] 11986110(119870) = [0 0 0 119860

(119870)

2] and

11986111(119870) = 120601

(119870)




1199111 (119870) 11986110 (119870) 119861

minus1

00(119870) = minus1199110 (119870) (13)

1199111(119870) [119861

11(119870) minus 119861

10(119870) 119861minus1

00(119870) 11986101(119870)] = 0

119899(119888+1) (14)

Further we can have

11986100(119870) = (

11986100(119870 minus 1) 119861

01(119870 minus 1)

1198620(119870 minus 1) 119862

1(119870 minus 1)

) (15)

where

1198620 (119870 minus 1) = [0119899 02119899 0 119860

(119870minus1)

2]

1198621(119870 minus 1) = minus119860

(119870minus1)

1

(16)



119861minus1

00(119870) = (

11986300 (119870) 11986301 (119870)

11986310(119870) 119863

11(119870)) (17)


where

11986300 (119870) = [11986100 (119870 minus 1)

minus 11986110(119870 minus 1) 119862

minus1

1(119870 minus 1) 119862

0(119870 minus 1)]

minus1

11986310(119870) = minus119862

minus1

1(119870 minus 1) 119862

0(119870 minus 1)119863

00(119870)

11986311 (119870) = [1198621 (119870 minus 1)

minus 1198620(119870 minus 1) 119861

minus1

00(119870 minus 1) 119861

01(119870 minus 1)]

minus1

11986301(119870) = minus119861

minus1

00(119870 minus 1) 119861

01(119870 minus 1)119863

11(119870)

(18)


01(119870 minus 1) and C

0(119870 minus 1)






00(119870minus


10(119870 minus

1) 11986311(119870minus1)]119860

(119870minus1)

0 Further119862

0(119870minus1)119861

minus1

00(119870minus1)119861

01(119870minus1)


(119870minus1)






00(119870)


119861minus1= [119868 minus 119860

minus1119883(119882


119884]119860minus1 (19)


01(119870 minus 1) 119882minus1 =

119862minus1

1(119870 minus 1) and 119884 = 119862


11986300(119870) = 119861

minus1

= [119868 minus 11986301 (119870)1198620 (119870 minus 1)] 119861

minus1

00(119870 minus 1)

(20)



11986311(119870)

= [1198621(119870 minus 1) minus 119860

(119870minus1)

211986311(119870 minus 1)119860

(119870minus1)

0]minus1

11986301(119870)

= minus [11986310(119870 minus 1) 119863

11(119870 minus 1)] 119860

(119870minus1)

011986311(119870)

(1) 119870 = 119888 + 1(2) compute 119861minus1

00(119870)

(3) compute 1199111(119870) by (14) and (22)

(4) compute 1199110(119870) by (13)

(5) store 11986100(119870) 119861

minus1

00(119870) 119861

01(119870) and 119861

10(119870)

(6) 119870 = 119870 + 1(7) while 119870 le 119870

119891


00(119870) by (17)

(10) compute 1199111(119870) by (14) and (22)

(11) compute 1199110(119870) by (13)

(12) update 11986100(119870) 119861minus1

00(119870) 119861

01(119870) and 119861

10(119870)


infinlt 120598

(14) then 119870119891= 119870

(15) break(16) else 119870 = 119870 + 1


11986300(119870) = [119868 minus 119863

01(119870)119862

0(119870 minus 1)] 119861

minus1

00(119870 minus 1)

11986310 (119870) = minus119862

minus1

1(119870 minus 1) 1198620 (119870 minus 1)11986300 (119870)

(21)



120587 (119870) [e119899(119888+1) minus 11986110 (119870) 119861minus1

00(119870) e119899119870(119888+1)] = 1 (22)

where e119899(119888+1)

and 119890119899119870(119888+1)



1(119870)


119891when

max0le119894le119870119891

10038171003817100381710038171003817119911 (119870119891 119894) minus 119911 (119870

119891minus 1 119894)10038171003817100381710038171003817infinlt 120598 (23)









nonvacation is

119861119904=

119888minus1

sum

119894=0

119894

sum

119895=1

119895119909119894119895e119899+

infin

sum

119894=119888

119888

sum

119895=1

119895119909119894119895e119899 (24)


119871 = 119864 (119873) =

119888minus1

sum

119894=1

119894119909119894e(119894+1)119899+

infin

sum

119894=119888

119894119909119894e(119888+1)119899 (25)


119877 = 119875 (119873 lt 119888) =

119888minus1

sum

119894=0

119909119894e(119894+1)119899 (26)


119861119901= 119875 (119873 gt 119888) = 1 minus

119888minus1

sum

119894=0

119909119894e(119894+1)119899= 1 minus 119877 (27)


119873119888=

infin

sum

119894=119888+1

1198941199091198940e119899 (28)





119891such that

max0le119894le119870119891119911(119870119891 119894) minus 119911(119870

119891minus 1 119894)






119879 = minus1

1198790= 1

120572 = 1

(29)

(2) Erlang-2 (Erl)

119879 = (

minus2 2

0 minus2)

1198790= (

0

2)

120572 = [1 0]

(30)


119879 = (

minus19 0

0 minus019)

1198790= (

19

019)

120572 = [09 01]

(31)







119891





120588 120579 120583V119870119891

119871 119861119901


01

01

00 25 28 36 64229 64127 63354 09354 09143 0795530 10 13 18 13951 14544 15802 03211 03173 0308860 7 9 13 11771 11963 12238 01652 01647 0163990 6 8 11 11149 11244 11358 01109 01108 01107

1

00 14 16 22 20525 20533 20481 06062 05479 0432530 8 11 15 13040 13342 13793 02650 02562 0242160 6 8 12 11656 11811 12011 01561 01545 0152490 6 7 11 11137 11229 11337 01099 01097 01094

100

00 5 6 10 11131 11215 11305 01097 01089 0108230 5 6 10 11099 11184 11273 01066 01061 0105660 5 6 10 11068 11153 11243 01037 01034 0103190 5 6 10 11040 11124 11214 01009 01008 01008

05

01

00 25 28 42 64099 65109 69481 09579 09412 0823605 21 25 41 44283 45761 52829 09062 08788 0745710 17 21 40 29028 31115 39708 07716 07451 0652015 15 20 38 21227 23112 30542 06121 05987 05579

1

00 17 19 38 27191 28952 36542 07863 07442 0627005 16 21 38 23956 25839 33851 07113 06784 0594910 16 21 38 21386 23326 31440 06331 06121 0561615 15 20 38 19441 21339 29316 05605 05505 05280

100

00 14 18 37 18203 20117 28092 05061 05049 0502705 14 20 37 18178 20080 28066 05046 05037 0502110 15 20 37 18146 20055 28041 05030 05025 0501415 15 20 37 18121 20030 28016 05015 05012 05007

09

01

00 74 99 246 112890 132254 285672 09893 09846 0950903 73 98 253 100678 119600 271965 09805 09735 0935006 71 97 260 87709 106206 257831 09593 09495 0913909 75 102 265 74716 92882 243478 09142 09057 08880

1

00 73 96 231 87160 108465 272102 09574 09479 0918703 72 96 233 84580 105880 269545 09445 09361 0913506 72 96 234 82028 103341 267009 09289 09228 0908009 71 95 235 79573 100868 264490 09111 09080 09022

100

00 81 110 265 78679 100268 268945 09013 09010 0900403 81 110 265 78651 100240 268905 09010 09007 0900306 81 110 265 78623 100213 268865 09006 09005 0900209 81 110 265 78596 100185 268825 09003 09002 09001








120588 120579 120583V119870119891

119871 119861119901


01

01

00 22 25 33 41401 41552 41155 04953 04909 0445210 10 13 18 18312 18449 18849 00376 00606 0106420 8 10 13 14721 14743 14805 00050 00133 0029430 7 8 11 13301 13307 13326 00013 00049 00117

1

00 9 12 16 16336 16535 16721 00203 00421 0073010 7 10 13 14723 14792 14875 00056 00163 0034320 7 9 12 13787 13811 13847 00022 00078 0018130 6 8 11 13165 13172 13189 00011 00044 00107

100

00 6 7 11 13037 13051 13062 00009 00039 0009610 6 7 11 13026 13039 13049 00009 00039 0009520 6 7 11 13015 13027 13036 00009 00038 0009330 6 7 11 13005 13016 13023 00009 00037 00092

05

01

00 36 41 58 61463 65540 81834 07544 07535 0709402 26 30 48 44310 47592 61809 05820 05952 0606204 20 24 43 33472 35687 46449 03792 04116 0487806 16 20 39 27521 28807 35933 02308 02694 03736

1

00 17 19 37 31041 32843 40155 03311 03663 0441402 16 19 37 29373 30968 38141 02832 03217 0411604 17 21 37 27875 29234 36269 02421 02821 0382506 16 21 37 26620 27814 34534 02074 02473 03544

100

00 16 19 37 26304 27513 34302 01989 02385 0346802 16 21 37 26286 27437 34280 01984 02380 0346304 16 21 37 26267 27418 34257 01978 02375 0345906 16 21 37 26249 27399 34234 01973 02370 03455

09

01

00 78 100 224 140668 163843 336740 09618 09582 0940301 75 98 224 127466 149913 319614 09424 09390 0925102 73 95 229 113801 135544 301618 09089 09077 0905203 71 93 234 99871 120651 282970 08541 08593 08796

1

00 76 101 239 97276 117783 283667 08612 08643 0880601 76 100 238 95526 116045 281630 08466 08523 0875802 75 100 237 93873 114217 279598 08312 08397 0870803 75 99 237 92075 112421 277421 08149 08265 08656

100

00 82 111 269 91219 111422 293701 08041 08176 0862201 82 111 269 91194 111398 293614 08038 08175 0862102 82 111 269 91169 111375 293527 08036 08173 0862103 82 111 269 91144 111351 293440 08033 08171 08620







120588 120579 120583V119870119891

119871 119861119901


01

01

00 20 23 32 32118 32976 34094 00679 00887 0140505 11 13 18 21893 22065 22152 00008 00034 0015910 9 11 14 18365 18407 18401 00000 00003 0002415 8 9 12 16458 16464 16462 00000 00001 00005

1

00 8 11 15 17766 17958 18184 00000 00005 0003505 8 10 14 17120 17219 17334 00000 00002 0001610 8 9 13 16587 16631 16680 00000 00001 0000815 8 9 12 16136 16145 16155 00000 00000 00004

100

00 8 8 12 16022 16050 16039 00000 00000 0000405 8 8 12 16016 16041 16028 00000 00000 0000410 8 8 12 16009 16032 16016 00000 00000 0000415 8 8 12 16002 16024 16005 00000 00000 00004

05

01

00 30 34 52 63770 68052 84877 04072 04526 0545601 24 30 46 53485 56091 69287 02546 03073 0441102 21 25 41 46431 48028 57143 01445 01924 0337603 18 22 39 41697 42487 48192 00797 01170 02476

1

00 18 20 39 45180 46593 51256 01208 01617 0288901 18 20 39 43618 44869 49560 01007 01399 0267902 19 23 38 42119 42803 48001 00837 01208 0247703 18 22 38 40862 41494 46449 00696 01042 02284

100

00 17 20 38 40659 41500 46956 00662 01001 0223301 17 20 38 40633 41474 46923 00659 00998 0223002 18 23 38 40514 41099 46889 00657 00995 0222603 18 23 38 40489 41074 46856 00655 00993 02223

09

01

000 78 101 229 151572 173703 354516 09045 09054 09082005 76 99 227 142268 163514 340044 08719 08759 08903010 75 98 225 131999 152452 324807 08265 08360 08684015 73 96 222 121491 140835 309017 07663 07840 08418

1

000 79 105 246 120928 139233 312097 07748 07881 08391005 79 104 245 119039 137614 309647 07595 07758 08343010 78 103 245 117512 135946 306805 07436 07629 08292015 78 103 244 115535 134025 304553 07270 07496 08241

100

000 84 114 272 115312 133305 356288 07160 07407 08207005 84 114 272 115273 133271 356049 07157 07405 08206010 84 114 272 115235 133237 355810 07155 07403 08205015 84 113 272 115196 133446 355572 07152 07401 08204






0 05 1 15 21

2

3

4

5

6

7M

ean

queu

e len

gth

Exp Erl Hyp

c = 1

Service rate 120583


0 05 1 15 2 25 31

2

3

4

5

6

7

8

9

Mea

n qu

eue l

engt

h

Exp Erl Hyp

c = 3

Service rate 120583




0 005 01 015 02 025 034

45

5

55

6

65

7

75

8

85

Mea

n qu

eue l

engt

h

Hyp Erl Exp

c = 6

Service rate 120583


0 02 04 06 08 15

10

15

20

25

30M

ean

queu

e len

gth c = 1

Service rate 120583

Exp Erl Hyp






0 005 01 015 02 025 038

10

12

14

16

18

20

22

24

26

28

Mea

n qu

eue l

engt

h

c = 3

Service rate 120583

Exp Erl Hyp


0 002 004 006 008 01 012 014 01610

15

20

25

30

35

Mea

n qu

eue l

engt

h c = 6

Service rate 120583

Exp Erl Hyp




0 2 4 6 8 1011

12

13

14

15

16

17

18

19

2

21

Mea

n qu

eue l

engt

h

c = 1

Service rate 120583

Exp Erl Hyp


0 05 1 15 2 25 3

14

16

18

2

22

24

26

28

3

Mea

n qu

eue l

engt

h

c = 3

Service rate 120583

Exp Erl Hyp





0 05 1 1516

18

2

22

24

26

28

3

32

Mea

n qu

eue l

engt

h

c = 6

Service rate 120583

Exp Erl Hyp


0 2 4 6 8 10118

12

122

124

126

128

13

Mea

n qu

eue l

engt

h c = 1 120579 = 01


Exp Erl Hyp





0 2 4 6 8 10146

1465

147

1475

148

1485

Mea

n qu

eue l

engt

h

c = 3 120579 = 01


Exp Erl Hyp


0 2 4 6 8 1026

265

27

275

Mea

n qu

eue l

engt

h

c = 6 120579 = 01


Exp Erl Hyp






0 02 04 06 08 1088

089

09

091

092

093

094

095

Bloc

king

pro

babi

lity

c = 1

Service rate 120583

Exp Erl Hyp


0 005 01 015 02 025 03081

082

083

084

085

086

087

088

089

Bloc

king

pro

babi

lity

c = 3

Service rate 120583

Exp Erl Hyp





0 002 004 006 008 01 012 014 016072

074

076

078

08

082

084

Bloc

king

pro

babi

lity

c = 6

Service rate 120583

Exp Erl Hyp


0 2 4 6 8 100

01

02

03

04

05

06M

ean

busy

serv

ers i

n no

nvac

atio

n


Exp Erl Hyp


119887= 10 and 120583V = 04



2 3 4 5 6 7 8 9 10005

01

015

02

025

03

035

04

045

05

Mea

n bu

sy se

rver

s in

nonv

acat

ion


Exp Erl Hyp


0 2 4 6 8 10078

08

082

084

086

088

09

092

094

096

098

Mea

n bu

sy se

rver

s in

nonv

acat

ion


Exp Erl Hyp




6 Conclusion


0 2 4 6 8 10064

065

066

067

068

069

07

071

072

073

Mea

n cu

stom

er lo

ss

Exp Erl Hyp



0 2 4 6 8 1001

015

02

025

03

035

04

045

Mea

n cu

stom

er lo

ss

Exp Erl Hyp




Disclosure



Competing Interests


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










119860(119894)

1=

[[[[[[[

[

119879 minus (119888120579 + 119894120583V) 119868 119888120579119868


d

119879 minus 119894120583119887119868

]]]]]]]

](119894+1)119899times(119894+1)119899

119860(119894)

0=

[[[[[[

[

1198790120572

1198790120572

d

1198790120572

]]]]]]

](119894+1)119899times(119894+1)119899

(6)

For 119894 ge 119888

119860(119888+119897)

2=

[[[[[[[[[[

[

(119888120583V + 119897120585) 119868

(120583119887+ (119888 minus 1) 120583V) 119868

(2120583119887+ (119888 minus 2) 120583V) 119868

d

119888120583119887119868

]]]]]]]]]]

](119888+1)119899times(119888+1)119899

119860(119888+119897)

1=

[[[[[[[

[

119879 minus (119888 (120579 + 120583V) + 119897120585) 119868 119888120579119868

d d

119879 minus (120579 + 120583V + (119888 minus 1) 120583119887) 119868 120579119868

119879 minus (119888120583119887) 119868

]]]]]]]

](119888+1)119899times(119888+1)119899

119860(119888+119897)

0=

[[[[[[

[

1198790120572

1198790120572

d

1198790120572

]]]]]]

](119888+1)119899times(119888+1)119899

(7)


0

119860(119894)

1 and 119860(119894)





[28 35]


119909119876 = 0

119909e = 1(8)


119894=

[1199091198941 1199091198942 119909




row vector 119909119894119895= [1199091198941198951 119909


phases of arrivals







119891 Here we use



(119870)

=

(((((((((

(

119860(0)

1119860(0)

0

119860(1)

2119860(1)

1119860(1)

0

119860(2)

2119860(2)

1119860(2)

0

d d d

119860(119870minus1)

2119860(119870minus1)

1119860(119870minus1)

0

119860(119870)

2120601(119870)

1

)))))))))

)

(9)

where 120601(119870)1= 119860(119870)

1+119860(119870)



120587 (119870) = 0

120587e = 1(10)


0(119870) 119911

1(119870)]

with

1199110(119870) = [120587 (0) 120587 (1) 120587 (119870 minus 1)]

1199111(119870) = 120587 (119870)

(11)




0(119870) and 119911

1(119870) we have

(1199110(119870) 119911

1(119870)) (

11986100(119870) 119861

01(119870)

11986110 (119870) 11986111 (119870)

) = (0119898 0119899(119888+1)) (12)



trans[0 0 0 119860(119870minus1)0] 11986110(119870) = [0 0 0 119860

(119870)

2] and

11986111(119870) = 120601

(119870)




1199111 (119870) 11986110 (119870) 119861

minus1

00(119870) = minus1199110 (119870) (13)

1199111(119870) [119861

11(119870) minus 119861

10(119870) 119861minus1

00(119870) 11986101(119870)] = 0

119899(119888+1) (14)

Further we can have

11986100(119870) = (

11986100(119870 minus 1) 119861

01(119870 minus 1)

1198620(119870 minus 1) 119862

1(119870 minus 1)

) (15)

where

1198620 (119870 minus 1) = [0119899 02119899 0 119860

(119870minus1)

2]

1198621(119870 minus 1) = minus119860

(119870minus1)

1

(16)



119861minus1

00(119870) = (

11986300 (119870) 11986301 (119870)

11986310(119870) 119863

11(119870)) (17)


where

11986300 (119870) = [11986100 (119870 minus 1)

minus 11986110(119870 minus 1) 119862

minus1

1(119870 minus 1) 119862

0(119870 minus 1)]

minus1

11986310(119870) = minus119862

minus1

1(119870 minus 1) 119862

0(119870 minus 1)119863

00(119870)

11986311 (119870) = [1198621 (119870 minus 1)

minus 1198620(119870 minus 1) 119861

minus1

00(119870 minus 1) 119861

01(119870 minus 1)]

minus1

11986301(119870) = minus119861

minus1

00(119870 minus 1) 119861

01(119870 minus 1)119863

11(119870)

(18)


01(119870 minus 1) and C

0(119870 minus 1)






00(119870minus


10(119870 minus

1) 11986311(119870minus1)]119860

(119870minus1)

0 Further119862

0(119870minus1)119861

minus1

00(119870minus1)119861

01(119870minus1)


(119870minus1)






00(119870)


119861minus1= [119868 minus 119860

minus1119883(119882


119884]119860minus1 (19)


01(119870 minus 1) 119882minus1 =

119862minus1

1(119870 minus 1) and 119884 = 119862


11986300(119870) = 119861

minus1

= [119868 minus 11986301 (119870)1198620 (119870 minus 1)] 119861

minus1

00(119870 minus 1)

(20)



11986311(119870)

= [1198621(119870 minus 1) minus 119860

(119870minus1)

211986311(119870 minus 1)119860

(119870minus1)

0]minus1

11986301(119870)

= minus [11986310(119870 minus 1) 119863

11(119870 minus 1)] 119860

(119870minus1)

011986311(119870)

(1) 119870 = 119888 + 1(2) compute 119861minus1

00(119870)

(3) compute 1199111(119870) by (14) and (22)

(4) compute 1199110(119870) by (13)

(5) store 11986100(119870) 119861

minus1

00(119870) 119861

01(119870) and 119861

10(119870)

(6) 119870 = 119870 + 1(7) while 119870 le 119870

119891


00(119870) by (17)

(10) compute 1199111(119870) by (14) and (22)

(11) compute 1199110(119870) by (13)

(12) update 11986100(119870) 119861minus1

00(119870) 119861

01(119870) and 119861

10(119870)


infinlt 120598

(14) then 119870119891= 119870

(15) break(16) else 119870 = 119870 + 1


11986300(119870) = [119868 minus 119863

01(119870)119862

0(119870 minus 1)] 119861

minus1

00(119870 minus 1)

11986310 (119870) = minus119862

minus1

1(119870 minus 1) 1198620 (119870 minus 1)11986300 (119870)

(21)



120587 (119870) [e119899(119888+1) minus 11986110 (119870) 119861minus1

00(119870) e119899119870(119888+1)] = 1 (22)

where e119899(119888+1)

and 119890119899119870(119888+1)



1(119870)


119891when

max0le119894le119870119891

10038171003817100381710038171003817119911 (119870119891 119894) minus 119911 (119870

119891minus 1 119894)10038171003817100381710038171003817infinlt 120598 (23)









nonvacation is

119861119904=

119888minus1

sum

119894=0

119894

sum

119895=1

119895119909119894119895e119899+

infin

sum

119894=119888

119888

sum

119895=1

119895119909119894119895e119899 (24)


119871 = 119864 (119873) =

119888minus1

sum

119894=1

119894119909119894e(119894+1)119899+

infin

sum

119894=119888

119894119909119894e(119888+1)119899 (25)


119877 = 119875 (119873 lt 119888) =

119888minus1

sum

119894=0

119909119894e(119894+1)119899 (26)


119861119901= 119875 (119873 gt 119888) = 1 minus

119888minus1

sum

119894=0

119909119894e(119894+1)119899= 1 minus 119877 (27)


119873119888=

infin

sum

119894=119888+1

1198941199091198940e119899 (28)





119891such that

max0le119894le119870119891119911(119870119891 119894) minus 119911(119870

119891minus 1 119894)






119879 = minus1

1198790= 1

120572 = 1

(29)

(2) Erlang-2 (Erl)

119879 = (

minus2 2

0 minus2)

1198790= (

0

2)

120572 = [1 0]

(30)


119879 = (

minus19 0

0 minus019)

1198790= (

19

019)

120572 = [09 01]

(31)







119891





120588 120579 120583V119870119891

119871 119861119901


01

01

00 25 28 36 64229 64127 63354 09354 09143 0795530 10 13 18 13951 14544 15802 03211 03173 0308860 7 9 13 11771 11963 12238 01652 01647 0163990 6 8 11 11149 11244 11358 01109 01108 01107

1

00 14 16 22 20525 20533 20481 06062 05479 0432530 8 11 15 13040 13342 13793 02650 02562 0242160 6 8 12 11656 11811 12011 01561 01545 0152490 6 7 11 11137 11229 11337 01099 01097 01094

100

00 5 6 10 11131 11215 11305 01097 01089 0108230 5 6 10 11099 11184 11273 01066 01061 0105660 5 6 10 11068 11153 11243 01037 01034 0103190 5 6 10 11040 11124 11214 01009 01008 01008

05

01

00 25 28 42 64099 65109 69481 09579 09412 0823605 21 25 41 44283 45761 52829 09062 08788 0745710 17 21 40 29028 31115 39708 07716 07451 0652015 15 20 38 21227 23112 30542 06121 05987 05579

1

00 17 19 38 27191 28952 36542 07863 07442 0627005 16 21 38 23956 25839 33851 07113 06784 0594910 16 21 38 21386 23326 31440 06331 06121 0561615 15 20 38 19441 21339 29316 05605 05505 05280

100

00 14 18 37 18203 20117 28092 05061 05049 0502705 14 20 37 18178 20080 28066 05046 05037 0502110 15 20 37 18146 20055 28041 05030 05025 0501415 15 20 37 18121 20030 28016 05015 05012 05007

09

01

00 74 99 246 112890 132254 285672 09893 09846 0950903 73 98 253 100678 119600 271965 09805 09735 0935006 71 97 260 87709 106206 257831 09593 09495 0913909 75 102 265 74716 92882 243478 09142 09057 08880

1

00 73 96 231 87160 108465 272102 09574 09479 0918703 72 96 233 84580 105880 269545 09445 09361 0913506 72 96 234 82028 103341 267009 09289 09228 0908009 71 95 235 79573 100868 264490 09111 09080 09022

100

00 81 110 265 78679 100268 268945 09013 09010 0900403 81 110 265 78651 100240 268905 09010 09007 0900306 81 110 265 78623 100213 268865 09006 09005 0900209 81 110 265 78596 100185 268825 09003 09002 09001








120588 120579 120583V119870119891

119871 119861119901


01

01

00 22 25 33 41401 41552 41155 04953 04909 0445210 10 13 18 18312 18449 18849 00376 00606 0106420 8 10 13 14721 14743 14805 00050 00133 0029430 7 8 11 13301 13307 13326 00013 00049 00117

1

00 9 12 16 16336 16535 16721 00203 00421 0073010 7 10 13 14723 14792 14875 00056 00163 0034320 7 9 12 13787 13811 13847 00022 00078 0018130 6 8 11 13165 13172 13189 00011 00044 00107

100

00 6 7 11 13037 13051 13062 00009 00039 0009610 6 7 11 13026 13039 13049 00009 00039 0009520 6 7 11 13015 13027 13036 00009 00038 0009330 6 7 11 13005 13016 13023 00009 00037 00092

05

01

00 36 41 58 61463 65540 81834 07544 07535 0709402 26 30 48 44310 47592 61809 05820 05952 0606204 20 24 43 33472 35687 46449 03792 04116 0487806 16 20 39 27521 28807 35933 02308 02694 03736

1

00 17 19 37 31041 32843 40155 03311 03663 0441402 16 19 37 29373 30968 38141 02832 03217 0411604 17 21 37 27875 29234 36269 02421 02821 0382506 16 21 37 26620 27814 34534 02074 02473 03544

100

00 16 19 37 26304 27513 34302 01989 02385 0346802 16 21 37 26286 27437 34280 01984 02380 0346304 16 21 37 26267 27418 34257 01978 02375 0345906 16 21 37 26249 27399 34234 01973 02370 03455

09

01

00 78 100 224 140668 163843 336740 09618 09582 0940301 75 98 224 127466 149913 319614 09424 09390 0925102 73 95 229 113801 135544 301618 09089 09077 0905203 71 93 234 99871 120651 282970 08541 08593 08796

1

00 76 101 239 97276 117783 283667 08612 08643 0880601 76 100 238 95526 116045 281630 08466 08523 0875802 75 100 237 93873 114217 279598 08312 08397 0870803 75 99 237 92075 112421 277421 08149 08265 08656

100

00 82 111 269 91219 111422 293701 08041 08176 0862201 82 111 269 91194 111398 293614 08038 08175 0862102 82 111 269 91169 111375 293527 08036 08173 0862103 82 111 269 91144 111351 293440 08033 08171 08620







120588 120579 120583V119870119891

119871 119861119901


01

01

00 20 23 32 32118 32976 34094 00679 00887 0140505 11 13 18 21893 22065 22152 00008 00034 0015910 9 11 14 18365 18407 18401 00000 00003 0002415 8 9 12 16458 16464 16462 00000 00001 00005

1

00 8 11 15 17766 17958 18184 00000 00005 0003505 8 10 14 17120 17219 17334 00000 00002 0001610 8 9 13 16587 16631 16680 00000 00001 0000815 8 9 12 16136 16145 16155 00000 00000 00004

100

00 8 8 12 16022 16050 16039 00000 00000 0000405 8 8 12 16016 16041 16028 00000 00000 0000410 8 8 12 16009 16032 16016 00000 00000 0000415 8 8 12 16002 16024 16005 00000 00000 00004

05

01

00 30 34 52 63770 68052 84877 04072 04526 0545601 24 30 46 53485 56091 69287 02546 03073 0441102 21 25 41 46431 48028 57143 01445 01924 0337603 18 22 39 41697 42487 48192 00797 01170 02476

1

00 18 20 39 45180 46593 51256 01208 01617 0288901 18 20 39 43618 44869 49560 01007 01399 0267902 19 23 38 42119 42803 48001 00837 01208 0247703 18 22 38 40862 41494 46449 00696 01042 02284

100

00 17 20 38 40659 41500 46956 00662 01001 0223301 17 20 38 40633 41474 46923 00659 00998 0223002 18 23 38 40514 41099 46889 00657 00995 0222603 18 23 38 40489 41074 46856 00655 00993 02223

09

01

000 78 101 229 151572 173703 354516 09045 09054 09082005 76 99 227 142268 163514 340044 08719 08759 08903010 75 98 225 131999 152452 324807 08265 08360 08684015 73 96 222 121491 140835 309017 07663 07840 08418

1

000 79 105 246 120928 139233 312097 07748 07881 08391005 79 104 245 119039 137614 309647 07595 07758 08343010 78 103 245 117512 135946 306805 07436 07629 08292015 78 103 244 115535 134025 304553 07270 07496 08241

100

000 84 114 272 115312 133305 356288 07160 07407 08207005 84 114 272 115273 133271 356049 07157 07405 08206010 84 114 272 115235 133237 355810 07155 07403 08205015 84 113 272 115196 133446 355572 07152 07401 08204






0 05 1 15 21

2

3

4

5

6

7M

ean

queu

e len

gth

Exp Erl Hyp

c = 1

Service rate 120583


0 05 1 15 2 25 31

2

3

4

5

6

7

8

9

Mea

n qu

eue l

engt

h

Exp Erl Hyp

c = 3

Service rate 120583




0 005 01 015 02 025 034

45

5

55

6

65

7

75

8

85

Mea

n qu

eue l

engt

h

Hyp Erl Exp

c = 6

Service rate 120583


0 02 04 06 08 15

10

15

20

25

30M

ean

queu

e len

gth c = 1

Service rate 120583

Exp Erl Hyp






0 005 01 015 02 025 038

10

12

14

16

18

20

22

24

26

28

Mea

n qu

eue l

engt

h

c = 3

Service rate 120583

Exp Erl Hyp


0 002 004 006 008 01 012 014 01610

15

20

25

30

35

Mea

n qu

eue l

engt

h c = 6

Service rate 120583

Exp Erl Hyp




0 2 4 6 8 1011

12

13

14

15

16

17

18

19

2

21

Mea

n qu

eue l

engt

h

c = 1

Service rate 120583

Exp Erl Hyp


0 05 1 15 2 25 3

14

16

18

2

22

24

26

28

3

Mea

n qu

eue l

engt

h

c = 3

Service rate 120583

Exp Erl Hyp





0 05 1 1516

18

2

22

24

26

28

3

32

Mea

n qu

eue l

engt

h

c = 6

Service rate 120583

Exp Erl Hyp


0 2 4 6 8 10118

12

122

124

126

128

13

Mea

n qu

eue l

engt

h c = 1 120579 = 01


Exp Erl Hyp





0 2 4 6 8 10146

1465

147

1475

148

1485

Mea

n qu

eue l

engt

h

c = 3 120579 = 01


Exp Erl Hyp


0 2 4 6 8 1026

265

27

275

Mea

n qu

eue l

engt

h

c = 6 120579 = 01


Exp Erl Hyp






0 02 04 06 08 1088

089

09

091

092

093

094

095

Bloc

king

pro

babi

lity

c = 1

Service rate 120583

Exp Erl Hyp


0 005 01 015 02 025 03081

082

083

084

085

086

087

088

089

Bloc

king

pro

babi

lity

c = 3

Service rate 120583

Exp Erl Hyp





0 002 004 006 008 01 012 014 016072

074

076

078

08

082

084

Bloc

king

pro

babi

lity

c = 6

Service rate 120583

Exp Erl Hyp


0 2 4 6 8 100

01

02

03

04

05

06M

ean

busy

serv

ers i

n no

nvac

atio

n


Exp Erl Hyp


119887= 10 and 120583V = 04



2 3 4 5 6 7 8 9 10005

01

015

02

025

03

035

04

045

05

Mea

n bu

sy se

rver

s in

nonv

acat

ion


Exp Erl Hyp


0 2 4 6 8 10078

08

082

084

086

088

09

092

094

096

098

Mea

n bu

sy se

rver

s in

nonv

acat

ion


Exp Erl Hyp




6 Conclusion


0 2 4 6 8 10064

065

066

067

068

069

07

071

072

073

Mea

n cu

stom

er lo

ss

Exp Erl Hyp



0 2 4 6 8 1001

015

02

025

03

035

04

045

Mea

n cu

stom

er lo

ss

Exp Erl Hyp




Disclosure



Competing Interests


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















119891 Here we use



(119870)

=

(((((((((

(

119860(0)

1119860(0)

0

119860(1)

2119860(1)

1119860(1)

0

119860(2)

2119860(2)

1119860(2)

0

d d d

119860(119870minus1)

2119860(119870minus1)

1119860(119870minus1)

0

119860(119870)

2120601(119870)

1

)))))))))

)

(9)

where 120601(119870)1= 119860(119870)

1+119860(119870)



120587 (119870) = 0

120587e = 1(10)


0(119870) 119911

1(119870)]

with

1199110(119870) = [120587 (0) 120587 (1) 120587 (119870 minus 1)]

1199111(119870) = 120587 (119870)

(11)




0(119870) and 119911

1(119870) we have

(1199110(119870) 119911

1(119870)) (

11986100(119870) 119861

01(119870)

11986110 (119870) 11986111 (119870)

) = (0119898 0119899(119888+1)) (12)



trans[0 0 0 119860(119870minus1)0] 11986110(119870) = [0 0 0 119860

(119870)

2] and

11986111(119870) = 120601

(119870)




1199111 (119870) 11986110 (119870) 119861

minus1

00(119870) = minus1199110 (119870) (13)

1199111(119870) [119861

11(119870) minus 119861

10(119870) 119861minus1

00(119870) 11986101(119870)] = 0

119899(119888+1) (14)

Further we can have

11986100(119870) = (

11986100(119870 minus 1) 119861

01(119870 minus 1)

1198620(119870 minus 1) 119862

1(119870 minus 1)

) (15)

where

1198620 (119870 minus 1) = [0119899 02119899 0 119860

(119870minus1)

2]

1198621(119870 minus 1) = minus119860

(119870minus1)

1

(16)



119861minus1

00(119870) = (

11986300 (119870) 11986301 (119870)

11986310(119870) 119863

11(119870)) (17)


where

11986300 (119870) = [11986100 (119870 minus 1)

minus 11986110(119870 minus 1) 119862

minus1

1(119870 minus 1) 119862

0(119870 minus 1)]

minus1

11986310(119870) = minus119862

minus1

1(119870 minus 1) 119862

0(119870 minus 1)119863

00(119870)

11986311 (119870) = [1198621 (119870 minus 1)

minus 1198620(119870 minus 1) 119861

minus1

00(119870 minus 1) 119861

01(119870 minus 1)]

minus1

11986301(119870) = minus119861

minus1

00(119870 minus 1) 119861

01(119870 minus 1)119863

11(119870)

(18)


01(119870 minus 1) and C

0(119870 minus 1)






00(119870minus


10(119870 minus

1) 11986311(119870minus1)]119860

(119870minus1)

0 Further119862

0(119870minus1)119861

minus1

00(119870minus1)119861

01(119870minus1)


(119870minus1)






00(119870)


119861minus1= [119868 minus 119860

minus1119883(119882


119884]119860minus1 (19)


01(119870 minus 1) 119882minus1 =

119862minus1

1(119870 minus 1) and 119884 = 119862


11986300(119870) = 119861

minus1

= [119868 minus 11986301 (119870)1198620 (119870 minus 1)] 119861

minus1

00(119870 minus 1)

(20)



11986311(119870)

= [1198621(119870 minus 1) minus 119860

(119870minus1)

211986311(119870 minus 1)119860

(119870minus1)

0]minus1

11986301(119870)

= minus [11986310(119870 minus 1) 119863

11(119870 minus 1)] 119860

(119870minus1)

011986311(119870)

(1) 119870 = 119888 + 1(2) compute 119861minus1

00(119870)

(3) compute 1199111(119870) by (14) and (22)

(4) compute 1199110(119870) by (13)

(5) store 11986100(119870) 119861

minus1

00(119870) 119861

01(119870) and 119861

10(119870)

(6) 119870 = 119870 + 1(7) while 119870 le 119870

119891


00(119870) by (17)

(10) compute 1199111(119870) by (14) and (22)

(11) compute 1199110(119870) by (13)

(12) update 11986100(119870) 119861minus1

00(119870) 119861

01(119870) and 119861

10(119870)


infinlt 120598

(14) then 119870119891= 119870

(15) break(16) else 119870 = 119870 + 1


11986300(119870) = [119868 minus 119863

01(119870)119862

0(119870 minus 1)] 119861

minus1

00(119870 minus 1)

11986310 (119870) = minus119862

minus1

1(119870 minus 1) 1198620 (119870 minus 1)11986300 (119870)

(21)



120587 (119870) [e119899(119888+1) minus 11986110 (119870) 119861minus1

00(119870) e119899119870(119888+1)] = 1 (22)

where e119899(119888+1)

and 119890119899119870(119888+1)



1(119870)


119891when

max0le119894le119870119891

10038171003817100381710038171003817119911 (119870119891 119894) minus 119911 (119870

119891minus 1 119894)10038171003817100381710038171003817infinlt 120598 (23)









nonvacation is

119861119904=

119888minus1

sum

119894=0

119894

sum

119895=1

119895119909119894119895e119899+

infin

sum

119894=119888

119888

sum

119895=1

119895119909119894119895e119899 (24)


119871 = 119864 (119873) =

119888minus1

sum

119894=1

119894119909119894e(119894+1)119899+

infin

sum

119894=119888

119894119909119894e(119888+1)119899 (25)


119877 = 119875 (119873 lt 119888) =

119888minus1

sum

119894=0

119909119894e(119894+1)119899 (26)


119861119901= 119875 (119873 gt 119888) = 1 minus

119888minus1

sum

119894=0

119909119894e(119894+1)119899= 1 minus 119877 (27)


119873119888=

infin

sum

119894=119888+1

1198941199091198940e119899 (28)





119891such that

max0le119894le119870119891119911(119870119891 119894) minus 119911(119870

119891minus 1 119894)






119879 = minus1

1198790= 1

120572 = 1

(29)

(2) Erlang-2 (Erl)

119879 = (

minus2 2

0 minus2)

1198790= (

0

2)

120572 = [1 0]

(30)


119879 = (

minus19 0

0 minus019)

1198790= (

19

019)

120572 = [09 01]

(31)







119891





120588 120579 120583V119870119891

119871 119861119901


01

01

00 25 28 36 64229 64127 63354 09354 09143 0795530 10 13 18 13951 14544 15802 03211 03173 0308860 7 9 13 11771 11963 12238 01652 01647 0163990 6 8 11 11149 11244 11358 01109 01108 01107

1

00 14 16 22 20525 20533 20481 06062 05479 0432530 8 11 15 13040 13342 13793 02650 02562 0242160 6 8 12 11656 11811 12011 01561 01545 0152490 6 7 11 11137 11229 11337 01099 01097 01094

100

00 5 6 10 11131 11215 11305 01097 01089 0108230 5 6 10 11099 11184 11273 01066 01061 0105660 5 6 10 11068 11153 11243 01037 01034 0103190 5 6 10 11040 11124 11214 01009 01008 01008

05

01

00 25 28 42 64099 65109 69481 09579 09412 0823605 21 25 41 44283 45761 52829 09062 08788 0745710 17 21 40 29028 31115 39708 07716 07451 0652015 15 20 38 21227 23112 30542 06121 05987 05579

1

00 17 19 38 27191 28952 36542 07863 07442 0627005 16 21 38 23956 25839 33851 07113 06784 0594910 16 21 38 21386 23326 31440 06331 06121 0561615 15 20 38 19441 21339 29316 05605 05505 05280

100

00 14 18 37 18203 20117 28092 05061 05049 0502705 14 20 37 18178 20080 28066 05046 05037 0502110 15 20 37 18146 20055 28041 05030 05025 0501415 15 20 37 18121 20030 28016 05015 05012 05007

09

01

00 74 99 246 112890 132254 285672 09893 09846 0950903 73 98 253 100678 119600 271965 09805 09735 0935006 71 97 260 87709 106206 257831 09593 09495 0913909 75 102 265 74716 92882 243478 09142 09057 08880

1

00 73 96 231 87160 108465 272102 09574 09479 0918703 72 96 233 84580 105880 269545 09445 09361 0913506 72 96 234 82028 103341 267009 09289 09228 0908009 71 95 235 79573 100868 264490 09111 09080 09022

100

00 81 110 265 78679 100268 268945 09013 09010 0900403 81 110 265 78651 100240 268905 09010 09007 0900306 81 110 265 78623 100213 268865 09006 09005 0900209 81 110 265 78596 100185 268825 09003 09002 09001








120588 120579 120583V119870119891

119871 119861119901


01

01

00 22 25 33 41401 41552 41155 04953 04909 0445210 10 13 18 18312 18449 18849 00376 00606 0106420 8 10 13 14721 14743 14805 00050 00133 0029430 7 8 11 13301 13307 13326 00013 00049 00117

1

00 9 12 16 16336 16535 16721 00203 00421 0073010 7 10 13 14723 14792 14875 00056 00163 0034320 7 9 12 13787 13811 13847 00022 00078 0018130 6 8 11 13165 13172 13189 00011 00044 00107

100

00 6 7 11 13037 13051 13062 00009 00039 0009610 6 7 11 13026 13039 13049 00009 00039 0009520 6 7 11 13015 13027 13036 00009 00038 0009330 6 7 11 13005 13016 13023 00009 00037 00092

05

01

00 36 41 58 61463 65540 81834 07544 07535 0709402 26 30 48 44310 47592 61809 05820 05952 0606204 20 24 43 33472 35687 46449 03792 04116 0487806 16 20 39 27521 28807 35933 02308 02694 03736

1

00 17 19 37 31041 32843 40155 03311 03663 0441402 16 19 37 29373 30968 38141 02832 03217 0411604 17 21 37 27875 29234 36269 02421 02821 0382506 16 21 37 26620 27814 34534 02074 02473 03544

100

00 16 19 37 26304 27513 34302 01989 02385 0346802 16 21 37 26286 27437 34280 01984 02380 0346304 16 21 37 26267 27418 34257 01978 02375 0345906 16 21 37 26249 27399 34234 01973 02370 03455

09

01

00 78 100 224 140668 163843 336740 09618 09582 0940301 75 98 224 127466 149913 319614 09424 09390 0925102 73 95 229 113801 135544 301618 09089 09077 0905203 71 93 234 99871 120651 282970 08541 08593 08796

1

00 76 101 239 97276 117783 283667 08612 08643 0880601 76 100 238 95526 116045 281630 08466 08523 0875802 75 100 237 93873 114217 279598 08312 08397 0870803 75 99 237 92075 112421 277421 08149 08265 08656

100

00 82 111 269 91219 111422 293701 08041 08176 0862201 82 111 269 91194 111398 293614 08038 08175 0862102 82 111 269 91169 111375 293527 08036 08173 0862103 82 111 269 91144 111351 293440 08033 08171 08620







120588 120579 120583V119870119891

119871 119861119901


01

01

00 20 23 32 32118 32976 34094 00679 00887 0140505 11 13 18 21893 22065 22152 00008 00034 0015910 9 11 14 18365 18407 18401 00000 00003 0002415 8 9 12 16458 16464 16462 00000 00001 00005

1

00 8 11 15 17766 17958 18184 00000 00005 0003505 8 10 14 17120 17219 17334 00000 00002 0001610 8 9 13 16587 16631 16680 00000 00001 0000815 8 9 12 16136 16145 16155 00000 00000 00004

100

00 8 8 12 16022 16050 16039 00000 00000 0000405 8 8 12 16016 16041 16028 00000 00000 0000410 8 8 12 16009 16032 16016 00000 00000 0000415 8 8 12 16002 16024 16005 00000 00000 00004

05

01

00 30 34 52 63770 68052 84877 04072 04526 0545601 24 30 46 53485 56091 69287 02546 03073 0441102 21 25 41 46431 48028 57143 01445 01924 0337603 18 22 39 41697 42487 48192 00797 01170 02476

1

00 18 20 39 45180 46593 51256 01208 01617 0288901 18 20 39 43618 44869 49560 01007 01399 0267902 19 23 38 42119 42803 48001 00837 01208 0247703 18 22 38 40862 41494 46449 00696 01042 02284

100

00 17 20 38 40659 41500 46956 00662 01001 0223301 17 20 38 40633 41474 46923 00659 00998 0223002 18 23 38 40514 41099 46889 00657 00995 0222603 18 23 38 40489 41074 46856 00655 00993 02223

09

01

000 78 101 229 151572 173703 354516 09045 09054 09082005 76 99 227 142268 163514 340044 08719 08759 08903010 75 98 225 131999 152452 324807 08265 08360 08684015 73 96 222 121491 140835 309017 07663 07840 08418

1

000 79 105 246 120928 139233 312097 07748 07881 08391005 79 104 245 119039 137614 309647 07595 07758 08343010 78 103 245 117512 135946 306805 07436 07629 08292015 78 103 244 115535 134025 304553 07270 07496 08241

100

000 84 114 272 115312 133305 356288 07160 07407 08207005 84 114 272 115273 133271 356049 07157 07405 08206010 84 114 272 115235 133237 355810 07155 07403 08205015 84 113 272 115196 133446 355572 07152 07401 08204






0 05 1 15 21

2

3

4

5

6

7M

ean

queu

e len

gth

Exp Erl Hyp

c = 1

Service rate 120583


0 05 1 15 2 25 31

2

3

4

5

6

7

8

9

Mea

n qu

eue l

engt

h

Exp Erl Hyp

c = 3

Service rate 120583




0 005 01 015 02 025 034

45

5

55

6

65

7

75

8

85

Mea

n qu

eue l

engt

h

Hyp Erl Exp

c = 6

Service rate 120583


0 02 04 06 08 15

10

15

20

25

30M

ean

queu

e len

gth c = 1

Service rate 120583

Exp Erl Hyp






0 005 01 015 02 025 038

10

12

14

16

18

20

22

24

26

28

Mea

n qu

eue l

engt

h

c = 3

Service rate 120583

Exp Erl Hyp


0 002 004 006 008 01 012 014 01610

15

20

25

30

35

Mea

n qu

eue l

engt

h c = 6

Service rate 120583

Exp Erl Hyp




0 2 4 6 8 1011

12

13

14

15

16

17

18

19

2

21

Mea

n qu

eue l

engt

h

c = 1

Service rate 120583

Exp Erl Hyp


0 05 1 15 2 25 3

14

16

18

2

22

24

26

28

3

Mea

n qu

eue l

engt

h

c = 3

Service rate 120583

Exp Erl Hyp





0 05 1 1516

18

2

22

24

26

28

3

32

Mea

n qu

eue l

engt

h

c = 6

Service rate 120583

Exp Erl Hyp


0 2 4 6 8 10118

12

122

124

126

128

13

Mea

n qu

eue l

engt

h c = 1 120579 = 01


Exp Erl Hyp





0 2 4 6 8 10146

1465

147

1475

148

1485

Mea

n qu

eue l

engt

h

c = 3 120579 = 01


Exp Erl Hyp


0 2 4 6 8 1026

265

27

275

Mea

n qu

eue l

engt

h

c = 6 120579 = 01


Exp Erl Hyp






0 02 04 06 08 1088

089

09

091

092

093

094

095

Bloc

king

pro

babi

lity

c = 1

Service rate 120583

Exp Erl Hyp


0 005 01 015 02 025 03081

082

083

084

085

086

087

088

089

Bloc

king

pro

babi

lity

c = 3

Service rate 120583

Exp Erl Hyp





0 002 004 006 008 01 012 014 016072

074

076

078

08

082

084

Bloc

king

pro

babi

lity

c = 6

Service rate 120583

Exp Erl Hyp


0 2 4 6 8 100

01

02

03

04

05

06M

ean

busy

serv

ers i

n no

nvac

atio

n


Exp Erl Hyp


119887= 10 and 120583V = 04



2 3 4 5 6 7 8 9 10005

01

015

02

025

03

035

04

045

05

Mea

n bu

sy se

rver

s in

nonv

acat

ion


Exp Erl Hyp


0 2 4 6 8 10078

08

082

084

086

088

09

092

094

096

098

Mea

n bu

sy se

rver

s in

nonv

acat

ion


Exp Erl Hyp




6 Conclusion


0 2 4 6 8 10064

065

066

067

068

069

07

071

072

073

Mea

n cu

stom

er lo

ss

Exp Erl Hyp



0 2 4 6 8 1001

015

02

025

03

035

04

045

Mea

n cu

stom

er lo

ss

Exp Erl Hyp




Disclosure



Competing Interests


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










where

11986300 (119870) = [11986100 (119870 minus 1)

minus 11986110(119870 minus 1) 119862

minus1

1(119870 minus 1) 119862

0(119870 minus 1)]

minus1

11986310(119870) = minus119862

minus1

1(119870 minus 1) 119862

0(119870 minus 1)119863

00(119870)

11986311 (119870) = [1198621 (119870 minus 1)

minus 1198620(119870 minus 1) 119861

minus1

00(119870 minus 1) 119861

01(119870 minus 1)]

minus1

11986301(119870) = minus119861

minus1

00(119870 minus 1) 119861

01(119870 minus 1)119863

11(119870)

(18)


01(119870 minus 1) and C

0(119870 minus 1)






00(119870minus


10(119870 minus

1) 11986311(119870minus1)]119860

(119870minus1)

0 Further119862

0(119870minus1)119861

minus1

00(119870minus1)119861

01(119870minus1)


(119870minus1)






00(119870)


119861minus1= [119868 minus 119860

minus1119883(119882


119884]119860minus1 (19)


01(119870 minus 1) 119882minus1 =

119862minus1

1(119870 minus 1) and 119884 = 119862


11986300(119870) = 119861

minus1

= [119868 minus 11986301 (119870)1198620 (119870 minus 1)] 119861

minus1

00(119870 minus 1)

(20)



11986311(119870)

= [1198621(119870 minus 1) minus 119860

(119870minus1)

211986311(119870 minus 1)119860

(119870minus1)

0]minus1

11986301(119870)

= minus [11986310(119870 minus 1) 119863

11(119870 minus 1)] 119860

(119870minus1)

011986311(119870)

(1) 119870 = 119888 + 1(2) compute 119861minus1

00(119870)

(3) compute 1199111(119870) by (14) and (22)

(4) compute 1199110(119870) by (13)

(5) store 11986100(119870) 119861

minus1

00(119870) 119861

01(119870) and 119861

10(119870)

(6) 119870 = 119870 + 1(7) while 119870 le 119870

119891


00(119870) by (17)

(10) compute 1199111(119870) by (14) and (22)

(11) compute 1199110(119870) by (13)

(12) update 11986100(119870) 119861minus1

00(119870) 119861

01(119870) and 119861

10(119870)


infinlt 120598

(14) then 119870119891= 119870

(15) break(16) else 119870 = 119870 + 1


11986300(119870) = [119868 minus 119863

01(119870)119862

0(119870 minus 1)] 119861

minus1

00(119870 minus 1)

11986310 (119870) = minus119862

minus1

1(119870 minus 1) 1198620 (119870 minus 1)11986300 (119870)

(21)



120587 (119870) [e119899(119888+1) minus 11986110 (119870) 119861minus1

00(119870) e119899119870(119888+1)] = 1 (22)

where e119899(119888+1)

and 119890119899119870(119888+1)



1(119870)


119891when

max0le119894le119870119891

10038171003817100381710038171003817119911 (119870119891 119894) minus 119911 (119870

119891minus 1 119894)10038171003817100381710038171003817infinlt 120598 (23)









nonvacation is

119861119904=

119888minus1

sum

119894=0

119894

sum

119895=1

119895119909119894119895e119899+

infin

sum

119894=119888

119888

sum

119895=1

119895119909119894119895e119899 (24)


119871 = 119864 (119873) =

119888minus1

sum

119894=1

119894119909119894e(119894+1)119899+

infin

sum

119894=119888

119894119909119894e(119888+1)119899 (25)


119877 = 119875 (119873 lt 119888) =

119888minus1

sum

119894=0

119909119894e(119894+1)119899 (26)


119861119901= 119875 (119873 gt 119888) = 1 minus

119888minus1

sum

119894=0

119909119894e(119894+1)119899= 1 minus 119877 (27)


119873119888=

infin

sum

119894=119888+1

1198941199091198940e119899 (28)





119891such that

max0le119894le119870119891119911(119870119891 119894) minus 119911(119870

119891minus 1 119894)






119879 = minus1

1198790= 1

120572 = 1

(29)

(2) Erlang-2 (Erl)

119879 = (

minus2 2

0 minus2)

1198790= (

0

2)

120572 = [1 0]

(30)


119879 = (

minus19 0

0 minus019)

1198790= (

19

019)

120572 = [09 01]

(31)







119891





120588 120579 120583V119870119891

119871 119861119901


01

01

00 25 28 36 64229 64127 63354 09354 09143 0795530 10 13 18 13951 14544 15802 03211 03173 0308860 7 9 13 11771 11963 12238 01652 01647 0163990 6 8 11 11149 11244 11358 01109 01108 01107

1

00 14 16 22 20525 20533 20481 06062 05479 0432530 8 11 15 13040 13342 13793 02650 02562 0242160 6 8 12 11656 11811 12011 01561 01545 0152490 6 7 11 11137 11229 11337 01099 01097 01094

100

00 5 6 10 11131 11215 11305 01097 01089 0108230 5 6 10 11099 11184 11273 01066 01061 0105660 5 6 10 11068 11153 11243 01037 01034 0103190 5 6 10 11040 11124 11214 01009 01008 01008

05

01

00 25 28 42 64099 65109 69481 09579 09412 0823605 21 25 41 44283 45761 52829 09062 08788 0745710 17 21 40 29028 31115 39708 07716 07451 0652015 15 20 38 21227 23112 30542 06121 05987 05579

1

00 17 19 38 27191 28952 36542 07863 07442 0627005 16 21 38 23956 25839 33851 07113 06784 0594910 16 21 38 21386 23326 31440 06331 06121 0561615 15 20 38 19441 21339 29316 05605 05505 05280

100

00 14 18 37 18203 20117 28092 05061 05049 0502705 14 20 37 18178 20080 28066 05046 05037 0502110 15 20 37 18146 20055 28041 05030 05025 0501415 15 20 37 18121 20030 28016 05015 05012 05007

09

01

00 74 99 246 112890 132254 285672 09893 09846 0950903 73 98 253 100678 119600 271965 09805 09735 0935006 71 97 260 87709 106206 257831 09593 09495 0913909 75 102 265 74716 92882 243478 09142 09057 08880

1

00 73 96 231 87160 108465 272102 09574 09479 0918703 72 96 233 84580 105880 269545 09445 09361 0913506 72 96 234 82028 103341 267009 09289 09228 0908009 71 95 235 79573 100868 264490 09111 09080 09022

100

00 81 110 265 78679 100268 268945 09013 09010 0900403 81 110 265 78651 100240 268905 09010 09007 0900306 81 110 265 78623 100213 268865 09006 09005 0900209 81 110 265 78596 100185 268825 09003 09002 09001








120588 120579 120583V119870119891

119871 119861119901


01

01

00 22 25 33 41401 41552 41155 04953 04909 0445210 10 13 18 18312 18449 18849 00376 00606 0106420 8 10 13 14721 14743 14805 00050 00133 0029430 7 8 11 13301 13307 13326 00013 00049 00117

1

00 9 12 16 16336 16535 16721 00203 00421 0073010 7 10 13 14723 14792 14875 00056 00163 0034320 7 9 12 13787 13811 13847 00022 00078 0018130 6 8 11 13165 13172 13189 00011 00044 00107

100

00 6 7 11 13037 13051 13062 00009 00039 0009610 6 7 11 13026 13039 13049 00009 00039 0009520 6 7 11 13015 13027 13036 00009 00038 0009330 6 7 11 13005 13016 13023 00009 00037 00092

05

01

00 36 41 58 61463 65540 81834 07544 07535 0709402 26 30 48 44310 47592 61809 05820 05952 0606204 20 24 43 33472 35687 46449 03792 04116 0487806 16 20 39 27521 28807 35933 02308 02694 03736

1

00 17 19 37 31041 32843 40155 03311 03663 0441402 16 19 37 29373 30968 38141 02832 03217 0411604 17 21 37 27875 29234 36269 02421 02821 0382506 16 21 37 26620 27814 34534 02074 02473 03544

100

00 16 19 37 26304 27513 34302 01989 02385 0346802 16 21 37 26286 27437 34280 01984 02380 0346304 16 21 37 26267 27418 34257 01978 02375 0345906 16 21 37 26249 27399 34234 01973 02370 03455

09

01

00 78 100 224 140668 163843 336740 09618 09582 0940301 75 98 224 127466 149913 319614 09424 09390 0925102 73 95 229 113801 135544 301618 09089 09077 0905203 71 93 234 99871 120651 282970 08541 08593 08796

1

00 76 101 239 97276 117783 283667 08612 08643 0880601 76 100 238 95526 116045 281630 08466 08523 0875802 75 100 237 93873 114217 279598 08312 08397 0870803 75 99 237 92075 112421 277421 08149 08265 08656

100

00 82 111 269 91219 111422 293701 08041 08176 0862201 82 111 269 91194 111398 293614 08038 08175 0862102 82 111 269 91169 111375 293527 08036 08173 0862103 82 111 269 91144 111351 293440 08033 08171 08620







120588 120579 120583V119870119891

119871 119861119901


01

01

00 20 23 32 32118 32976 34094 00679 00887 0140505 11 13 18 21893 22065 22152 00008 00034 0015910 9 11 14 18365 18407 18401 00000 00003 0002415 8 9 12 16458 16464 16462 00000 00001 00005

1

00 8 11 15 17766 17958 18184 00000 00005 0003505 8 10 14 17120 17219 17334 00000 00002 0001610 8 9 13 16587 16631 16680 00000 00001 0000815 8 9 12 16136 16145 16155 00000 00000 00004

100

00 8 8 12 16022 16050 16039 00000 00000 0000405 8 8 12 16016 16041 16028 00000 00000 0000410 8 8 12 16009 16032 16016 00000 00000 0000415 8 8 12 16002 16024 16005 00000 00000 00004

05

01

00 30 34 52 63770 68052 84877 04072 04526 0545601 24 30 46 53485 56091 69287 02546 03073 0441102 21 25 41 46431 48028 57143 01445 01924 0337603 18 22 39 41697 42487 48192 00797 01170 02476

1

00 18 20 39 45180 46593 51256 01208 01617 0288901 18 20 39 43618 44869 49560 01007 01399 0267902 19 23 38 42119 42803 48001 00837 01208 0247703 18 22 38 40862 41494 46449 00696 01042 02284

100

00 17 20 38 40659 41500 46956 00662 01001 0223301 17 20 38 40633 41474 46923 00659 00998 0223002 18 23 38 40514 41099 46889 00657 00995 0222603 18 23 38 40489 41074 46856 00655 00993 02223

09

01

000 78 101 229 151572 173703 354516 09045 09054 09082005 76 99 227 142268 163514 340044 08719 08759 08903010 75 98 225 131999 152452 324807 08265 08360 08684015 73 96 222 121491 140835 309017 07663 07840 08418

1

000 79 105 246 120928 139233 312097 07748 07881 08391005 79 104 245 119039 137614 309647 07595 07758 08343010 78 103 245 117512 135946 306805 07436 07629 08292015 78 103 244 115535 134025 304553 07270 07496 08241

100

000 84 114 272 115312 133305 356288 07160 07407 08207005 84 114 272 115273 133271 356049 07157 07405 08206010 84 114 272 115235 133237 355810 07155 07403 08205015 84 113 272 115196 133446 355572 07152 07401 08204






0 05 1 15 21

2

3

4

5

6

7M

ean

queu

e len

gth

Exp Erl Hyp

c = 1

Service rate 120583


0 05 1 15 2 25 31

2

3

4

5

6

7

8

9

Mea

n qu

eue l

engt

h

Exp Erl Hyp

c = 3

Service rate 120583




0 005 01 015 02 025 034

45

5

55

6

65

7

75

8

85

Mea

n qu

eue l

engt

h

Hyp Erl Exp

c = 6

Service rate 120583


0 02 04 06 08 15

10

15

20

25

30M

ean

queu

e len

gth c = 1

Service rate 120583

Exp Erl Hyp






0 005 01 015 02 025 038

10

12

14

16

18

20

22

24

26

28

Mea

n qu

eue l

engt

h

c = 3

Service rate 120583

Exp Erl Hyp


0 002 004 006 008 01 012 014 01610

15

20

25

30

35

Mea

n qu

eue l

engt

h c = 6

Service rate 120583

Exp Erl Hyp




0 2 4 6 8 1011

12

13

14

15

16

17

18

19

2

21

Mea

n qu

eue l

engt

h

c = 1

Service rate 120583

Exp Erl Hyp


0 05 1 15 2 25 3

14

16

18

2

22

24

26

28

3

Mea

n qu

eue l

engt

h

c = 3

Service rate 120583

Exp Erl Hyp





0 05 1 1516

18

2

22

24

26

28

3

32

Mea

n qu

eue l

engt

h

c = 6

Service rate 120583

Exp Erl Hyp


0 2 4 6 8 10118

12

122

124

126

128

13

Mea

n qu

eue l

engt

h c = 1 120579 = 01


Exp Erl Hyp





0 2 4 6 8 10146

1465

147

1475

148

1485

Mea

n qu

eue l

engt

h

c = 3 120579 = 01


Exp Erl Hyp


0 2 4 6 8 1026

265

27

275

Mea

n qu

eue l

engt

h

c = 6 120579 = 01


Exp Erl Hyp






0 02 04 06 08 1088

089

09

091

092

093

094

095

Bloc

king

pro

babi

lity

c = 1

Service rate 120583

Exp Erl Hyp


0 005 01 015 02 025 03081

082

083

084

085

086

087

088

089

Bloc

king

pro

babi

lity

c = 3

Service rate 120583

Exp Erl Hyp





0 002 004 006 008 01 012 014 016072

074

076

078

08

082

084

Bloc

king

pro

babi

lity

c = 6

Service rate 120583

Exp Erl Hyp


0 2 4 6 8 100

01

02

03

04

05

06M

ean

busy

serv

ers i

n no

nvac

atio

n


Exp Erl Hyp


119887= 10 and 120583V = 04



2 3 4 5 6 7 8 9 10005

01

015

02

025

03

035

04

045

05

Mea

n bu

sy se

rver

s in

nonv

acat

ion


Exp Erl Hyp


0 2 4 6 8 10078

08

082

084

086

088

09

092

094

096

098

Mea

n bu

sy se

rver

s in

nonv

acat

ion


Exp Erl Hyp




6 Conclusion


0 2 4 6 8 10064

065

066

067

068

069

07

071

072

073

Mea

n cu

stom

er lo

ss

Exp Erl Hyp



0 2 4 6 8 1001

015

02

025

03

035

04

045

Mea

n cu

stom

er lo

ss

Exp Erl Hyp




Disclosure



Competing Interests


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












nonvacation is

119861119904=

119888minus1

sum

119894=0

119894

sum

119895=1

119895119909119894119895e119899+

infin

sum

119894=119888

119888

sum

119895=1

119895119909119894119895e119899 (24)


119871 = 119864 (119873) =

119888minus1

sum

119894=1

119894119909119894e(119894+1)119899+

infin

sum

119894=119888

119894119909119894e(119888+1)119899 (25)


119877 = 119875 (119873 lt 119888) =

119888minus1

sum

119894=0

119909119894e(119894+1)119899 (26)


119861119901= 119875 (119873 gt 119888) = 1 minus

119888minus1

sum

119894=0

119909119894e(119894+1)119899= 1 minus 119877 (27)


119873119888=

infin

sum

119894=119888+1

1198941199091198940e119899 (28)





119891such that

max0le119894le119870119891119911(119870119891 119894) minus 119911(119870

119891minus 1 119894)






119879 = minus1

1198790= 1

120572 = 1

(29)

(2) Erlang-2 (Erl)

119879 = (

minus2 2

0 minus2)

1198790= (

0

2)

120572 = [1 0]

(30)


119879 = (

minus19 0

0 minus019)

1198790= (

19

019)

120572 = [09 01]

(31)







119891





120588 120579 120583V119870119891

119871 119861119901


01

01

00 25 28 36 64229 64127 63354 09354 09143 0795530 10 13 18 13951 14544 15802 03211 03173 0308860 7 9 13 11771 11963 12238 01652 01647 0163990 6 8 11 11149 11244 11358 01109 01108 01107

1

00 14 16 22 20525 20533 20481 06062 05479 0432530 8 11 15 13040 13342 13793 02650 02562 0242160 6 8 12 11656 11811 12011 01561 01545 0152490 6 7 11 11137 11229 11337 01099 01097 01094

100

00 5 6 10 11131 11215 11305 01097 01089 0108230 5 6 10 11099 11184 11273 01066 01061 0105660 5 6 10 11068 11153 11243 01037 01034 0103190 5 6 10 11040 11124 11214 01009 01008 01008

05

01

00 25 28 42 64099 65109 69481 09579 09412 0823605 21 25 41 44283 45761 52829 09062 08788 0745710 17 21 40 29028 31115 39708 07716 07451 0652015 15 20 38 21227 23112 30542 06121 05987 05579

1

00 17 19 38 27191 28952 36542 07863 07442 0627005 16 21 38 23956 25839 33851 07113 06784 0594910 16 21 38 21386 23326 31440 06331 06121 0561615 15 20 38 19441 21339 29316 05605 05505 05280

100

00 14 18 37 18203 20117 28092 05061 05049 0502705 14 20 37 18178 20080 28066 05046 05037 0502110 15 20 37 18146 20055 28041 05030 05025 0501415 15 20 37 18121 20030 28016 05015 05012 05007

09

01

00 74 99 246 112890 132254 285672 09893 09846 0950903 73 98 253 100678 119600 271965 09805 09735 0935006 71 97 260 87709 106206 257831 09593 09495 0913909 75 102 265 74716 92882 243478 09142 09057 08880

1

00 73 96 231 87160 108465 272102 09574 09479 0918703 72 96 233 84580 105880 269545 09445 09361 0913506 72 96 234 82028 103341 267009 09289 09228 0908009 71 95 235 79573 100868 264490 09111 09080 09022

100

00 81 110 265 78679 100268 268945 09013 09010 0900403 81 110 265 78651 100240 268905 09010 09007 0900306 81 110 265 78623 100213 268865 09006 09005 0900209 81 110 265 78596 100185 268825 09003 09002 09001








120588 120579 120583V119870119891

119871 119861119901


01

01

00 22 25 33 41401 41552 41155 04953 04909 0445210 10 13 18 18312 18449 18849 00376 00606 0106420 8 10 13 14721 14743 14805 00050 00133 0029430 7 8 11 13301 13307 13326 00013 00049 00117

1

00 9 12 16 16336 16535 16721 00203 00421 0073010 7 10 13 14723 14792 14875 00056 00163 0034320 7 9 12 13787 13811 13847 00022 00078 0018130 6 8 11 13165 13172 13189 00011 00044 00107

100

00 6 7 11 13037 13051 13062 00009 00039 0009610 6 7 11 13026 13039 13049 00009 00039 0009520 6 7 11 13015 13027 13036 00009 00038 0009330 6 7 11 13005 13016 13023 00009 00037 00092

05

01

00 36 41 58 61463 65540 81834 07544 07535 0709402 26 30 48 44310 47592 61809 05820 05952 0606204 20 24 43 33472 35687 46449 03792 04116 0487806 16 20 39 27521 28807 35933 02308 02694 03736

1

00 17 19 37 31041 32843 40155 03311 03663 0441402 16 19 37 29373 30968 38141 02832 03217 0411604 17 21 37 27875 29234 36269 02421 02821 0382506 16 21 37 26620 27814 34534 02074 02473 03544

100

00 16 19 37 26304 27513 34302 01989 02385 0346802 16 21 37 26286 27437 34280 01984 02380 0346304 16 21 37 26267 27418 34257 01978 02375 0345906 16 21 37 26249 27399 34234 01973 02370 03455

09

01

00 78 100 224 140668 163843 336740 09618 09582 0940301 75 98 224 127466 149913 319614 09424 09390 0925102 73 95 229 113801 135544 301618 09089 09077 0905203 71 93 234 99871 120651 282970 08541 08593 08796

1

00 76 101 239 97276 117783 283667 08612 08643 0880601 76 100 238 95526 116045 281630 08466 08523 0875802 75 100 237 93873 114217 279598 08312 08397 0870803 75 99 237 92075 112421 277421 08149 08265 08656

100

00 82 111 269 91219 111422 293701 08041 08176 0862201 82 111 269 91194 111398 293614 08038 08175 0862102 82 111 269 91169 111375 293527 08036 08173 0862103 82 111 269 91144 111351 293440 08033 08171 08620







120588 120579 120583V119870119891

119871 119861119901


01

01

00 20 23 32 32118 32976 34094 00679 00887 0140505 11 13 18 21893 22065 22152 00008 00034 0015910 9 11 14 18365 18407 18401 00000 00003 0002415 8 9 12 16458 16464 16462 00000 00001 00005

1

00 8 11 15 17766 17958 18184 00000 00005 0003505 8 10 14 17120 17219 17334 00000 00002 0001610 8 9 13 16587 16631 16680 00000 00001 0000815 8 9 12 16136 16145 16155 00000 00000 00004

100

00 8 8 12 16022 16050 16039 00000 00000 0000405 8 8 12 16016 16041 16028 00000 00000 0000410 8 8 12 16009 16032 16016 00000 00000 0000415 8 8 12 16002 16024 16005 00000 00000 00004

05

01

00 30 34 52 63770 68052 84877 04072 04526 0545601 24 30 46 53485 56091 69287 02546 03073 0441102 21 25 41 46431 48028 57143 01445 01924 0337603 18 22 39 41697 42487 48192 00797 01170 02476

1

00 18 20 39 45180 46593 51256 01208 01617 0288901 18 20 39 43618 44869 49560 01007 01399 0267902 19 23 38 42119 42803 48001 00837 01208 0247703 18 22 38 40862 41494 46449 00696 01042 02284

100

00 17 20 38 40659 41500 46956 00662 01001 0223301 17 20 38 40633 41474 46923 00659 00998 0223002 18 23 38 40514 41099 46889 00657 00995 0222603 18 23 38 40489 41074 46856 00655 00993 02223

09

01

000 78 101 229 151572 173703 354516 09045 09054 09082005 76 99 227 142268 163514 340044 08719 08759 08903010 75 98 225 131999 152452 324807 08265 08360 08684015 73 96 222 121491 140835 309017 07663 07840 08418

1

000 79 105 246 120928 139233 312097 07748 07881 08391005 79 104 245 119039 137614 309647 07595 07758 08343010 78 103 245 117512 135946 306805 07436 07629 08292015 78 103 244 115535 134025 304553 07270 07496 08241

100

000 84 114 272 115312 133305 356288 07160 07407 08207005 84 114 272 115273 133271 356049 07157 07405 08206010 84 114 272 115235 133237 355810 07155 07403 08205015 84 113 272 115196 133446 355572 07152 07401 08204






0 05 1 15 21

2

3

4

5

6

7M

ean

queu

e len

gth

Exp Erl Hyp

c = 1

Service rate 120583


0 05 1 15 2 25 31

2

3

4

5

6

7

8

9

Mea

n qu

eue l

engt

h

Exp Erl Hyp

c = 3

Service rate 120583




0 005 01 015 02 025 034

45

5

55

6

65

7

75

8

85

Mea

n qu

eue l

engt

h

Hyp Erl Exp

c = 6

Service rate 120583


0 02 04 06 08 15

10

15

20

25

30M

ean

queu

e len

gth c = 1

Service rate 120583

Exp Erl Hyp






0 005 01 015 02 025 038

10

12

14

16

18

20

22

24

26

28

Mea

n qu

eue l

engt

h

c = 3

Service rate 120583

Exp Erl Hyp


0 002 004 006 008 01 012 014 01610

15

20

25

30

35

Mea

n qu

eue l

engt

h c = 6

Service rate 120583

Exp Erl Hyp




0 2 4 6 8 1011

12

13

14

15

16

17

18

19

2

21

Mea

n qu

eue l

engt

h

c = 1

Service rate 120583

Exp Erl Hyp


0 05 1 15 2 25 3

14

16

18

2

22

24

26

28

3

Mea

n qu

eue l

engt

h

c = 3

Service rate 120583

Exp Erl Hyp





0 05 1 1516

18

2

22

24

26

28

3

32

Mea

n qu

eue l

engt

h

c = 6

Service rate 120583

Exp Erl Hyp


0 2 4 6 8 10118

12

122

124

126

128

13

Mea

n qu

eue l

engt

h c = 1 120579 = 01


Exp Erl Hyp





0 2 4 6 8 10146

1465

147

1475

148

1485

Mea

n qu

eue l

engt

h

c = 3 120579 = 01


Exp Erl Hyp


0 2 4 6 8 1026

265

27

275

Mea

n qu

eue l

engt

h

c = 6 120579 = 01


Exp Erl Hyp






0 02 04 06 08 1088

089

09

091

092

093

094

095

Bloc

king

pro

babi

lity

c = 1

Service rate 120583

Exp Erl Hyp


0 005 01 015 02 025 03081

082

083

084

085

086

087

088

089

Bloc

king

pro

babi

lity

c = 3

Service rate 120583

Exp Erl Hyp





0 002 004 006 008 01 012 014 016072

074

076

078

08

082

084

Bloc

king

pro

babi

lity

c = 6

Service rate 120583

Exp Erl Hyp


0 2 4 6 8 100

01

02

03

04

05

06M

ean

busy

serv

ers i

n no

nvac

atio

n


Exp Erl Hyp


119887= 10 and 120583V = 04



2 3 4 5 6 7 8 9 10005

01

015

02

025

03

035

04

045

05

Mea

n bu

sy se

rver

s in

nonv

acat

ion


Exp Erl Hyp


0 2 4 6 8 10078

08

082

084

086

088

09

092

094

096

098

Mea

n bu

sy se

rver

s in

nonv

acat

ion


Exp Erl Hyp




6 Conclusion


0 2 4 6 8 10064

065

066

067

068

069

07

071

072

073

Mea

n cu

stom

er lo

ss

Exp Erl Hyp



0 2 4 6 8 1001

015

02

025

03

035

04

045

Mea

n cu

stom

er lo

ss

Exp Erl Hyp




Disclosure



Competing Interests


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120588 120579 120583V119870119891

119871 119861119901


01

01

00 25 28 36 64229 64127 63354 09354 09143 0795530 10 13 18 13951 14544 15802 03211 03173 0308860 7 9 13 11771 11963 12238 01652 01647 0163990 6 8 11 11149 11244 11358 01109 01108 01107

1

00 14 16 22 20525 20533 20481 06062 05479 0432530 8 11 15 13040 13342 13793 02650 02562 0242160 6 8 12 11656 11811 12011 01561 01545 0152490 6 7 11 11137 11229 11337 01099 01097 01094

100

00 5 6 10 11131 11215 11305 01097 01089 0108230 5 6 10 11099 11184 11273 01066 01061 0105660 5 6 10 11068 11153 11243 01037 01034 0103190 5 6 10 11040 11124 11214 01009 01008 01008

05

01

00 25 28 42 64099 65109 69481 09579 09412 0823605 21 25 41 44283 45761 52829 09062 08788 0745710 17 21 40 29028 31115 39708 07716 07451 0652015 15 20 38 21227 23112 30542 06121 05987 05579

1

00 17 19 38 27191 28952 36542 07863 07442 0627005 16 21 38 23956 25839 33851 07113 06784 0594910 16 21 38 21386 23326 31440 06331 06121 0561615 15 20 38 19441 21339 29316 05605 05505 05280

100

00 14 18 37 18203 20117 28092 05061 05049 0502705 14 20 37 18178 20080 28066 05046 05037 0502110 15 20 37 18146 20055 28041 05030 05025 0501415 15 20 37 18121 20030 28016 05015 05012 05007

09

01

00 74 99 246 112890 132254 285672 09893 09846 0950903 73 98 253 100678 119600 271965 09805 09735 0935006 71 97 260 87709 106206 257831 09593 09495 0913909 75 102 265 74716 92882 243478 09142 09057 08880

1

00 73 96 231 87160 108465 272102 09574 09479 0918703 72 96 233 84580 105880 269545 09445 09361 0913506 72 96 234 82028 103341 267009 09289 09228 0908009 71 95 235 79573 100868 264490 09111 09080 09022

100

00 81 110 265 78679 100268 268945 09013 09010 0900403 81 110 265 78651 100240 268905 09010 09007 0900306 81 110 265 78623 100213 268865 09006 09005 0900209 81 110 265 78596 100185 268825 09003 09002 09001








120588 120579 120583V119870119891

119871 119861119901


01

01

00 22 25 33 41401 41552 41155 04953 04909 0445210 10 13 18 18312 18449 18849 00376 00606 0106420 8 10 13 14721 14743 14805 00050 00133 0029430 7 8 11 13301 13307 13326 00013 00049 00117

1

00 9 12 16 16336 16535 16721 00203 00421 0073010 7 10 13 14723 14792 14875 00056 00163 0034320 7 9 12 13787 13811 13847 00022 00078 0018130 6 8 11 13165 13172 13189 00011 00044 00107

100

00 6 7 11 13037 13051 13062 00009 00039 0009610 6 7 11 13026 13039 13049 00009 00039 0009520 6 7 11 13015 13027 13036 00009 00038 0009330 6 7 11 13005 13016 13023 00009 00037 00092

05

01

00 36 41 58 61463 65540 81834 07544 07535 0709402 26 30 48 44310 47592 61809 05820 05952 0606204 20 24 43 33472 35687 46449 03792 04116 0487806 16 20 39 27521 28807 35933 02308 02694 03736

1

00 17 19 37 31041 32843 40155 03311 03663 0441402 16 19 37 29373 30968 38141 02832 03217 0411604 17 21 37 27875 29234 36269 02421 02821 0382506 16 21 37 26620 27814 34534 02074 02473 03544

100

00 16 19 37 26304 27513 34302 01989 02385 0346802 16 21 37 26286 27437 34280 01984 02380 0346304 16 21 37 26267 27418 34257 01978 02375 0345906 16 21 37 26249 27399 34234 01973 02370 03455

09

01

00 78 100 224 140668 163843 336740 09618 09582 0940301 75 98 224 127466 149913 319614 09424 09390 0925102 73 95 229 113801 135544 301618 09089 09077 0905203 71 93 234 99871 120651 282970 08541 08593 08796

1

00 76 101 239 97276 117783 283667 08612 08643 0880601 76 100 238 95526 116045 281630 08466 08523 0875802 75 100 237 93873 114217 279598 08312 08397 0870803 75 99 237 92075 112421 277421 08149 08265 08656

100

00 82 111 269 91219 111422 293701 08041 08176 0862201 82 111 269 91194 111398 293614 08038 08175 0862102 82 111 269 91169 111375 293527 08036 08173 0862103 82 111 269 91144 111351 293440 08033 08171 08620







120588 120579 120583V119870119891

119871 119861119901


01

01

00 20 23 32 32118 32976 34094 00679 00887 0140505 11 13 18 21893 22065 22152 00008 00034 0015910 9 11 14 18365 18407 18401 00000 00003 0002415 8 9 12 16458 16464 16462 00000 00001 00005

1

00 8 11 15 17766 17958 18184 00000 00005 0003505 8 10 14 17120 17219 17334 00000 00002 0001610 8 9 13 16587 16631 16680 00000 00001 0000815 8 9 12 16136 16145 16155 00000 00000 00004

100

00 8 8 12 16022 16050 16039 00000 00000 0000405 8 8 12 16016 16041 16028 00000 00000 0000410 8 8 12 16009 16032 16016 00000 00000 0000415 8 8 12 16002 16024 16005 00000 00000 00004

05

01

00 30 34 52 63770 68052 84877 04072 04526 0545601 24 30 46 53485 56091 69287 02546 03073 0441102 21 25 41 46431 48028 57143 01445 01924 0337603 18 22 39 41697 42487 48192 00797 01170 02476

1

00 18 20 39 45180 46593 51256 01208 01617 0288901 18 20 39 43618 44869 49560 01007 01399 0267902 19 23 38 42119 42803 48001 00837 01208 0247703 18 22 38 40862 41494 46449 00696 01042 02284

100

00 17 20 38 40659 41500 46956 00662 01001 0223301 17 20 38 40633 41474 46923 00659 00998 0223002 18 23 38 40514 41099 46889 00657 00995 0222603 18 23 38 40489 41074 46856 00655 00993 02223

09

01

000 78 101 229 151572 173703 354516 09045 09054 09082005 76 99 227 142268 163514 340044 08719 08759 08903010 75 98 225 131999 152452 324807 08265 08360 08684015 73 96 222 121491 140835 309017 07663 07840 08418

1

000 79 105 246 120928 139233 312097 07748 07881 08391005 79 104 245 119039 137614 309647 07595 07758 08343010 78 103 245 117512 135946 306805 07436 07629 08292015 78 103 244 115535 134025 304553 07270 07496 08241

100

000 84 114 272 115312 133305 356288 07160 07407 08207005 84 114 272 115273 133271 356049 07157 07405 08206010 84 114 272 115235 133237 355810 07155 07403 08205015 84 113 272 115196 133446 355572 07152 07401 08204






0 05 1 15 21

2

3

4

5

6

7M

ean

queu

e len

gth

Exp Erl Hyp

c = 1

Service rate 120583


0 05 1 15 2 25 31

2

3

4

5

6

7

8

9

Mea

n qu

eue l

engt

h

Exp Erl Hyp

c = 3

Service rate 120583




0 005 01 015 02 025 034

45

5

55

6

65

7

75

8

85

Mea

n qu

eue l

engt

h

Hyp Erl Exp

c = 6

Service rate 120583


0 02 04 06 08 15

10

15

20

25

30M

ean

queu

e len

gth c = 1

Service rate 120583

Exp Erl Hyp






0 005 01 015 02 025 038

10

12

14

16

18

20

22

24

26

28

Mea

n qu

eue l

engt

h

c = 3

Service rate 120583

Exp Erl Hyp


0 002 004 006 008 01 012 014 01610

15

20

25

30

35

Mea

n qu

eue l

engt

h c = 6

Service rate 120583

Exp Erl Hyp




0 2 4 6 8 1011

12

13

14

15

16

17

18

19

2

21

Mea

n qu

eue l

engt

h

c = 1

Service rate 120583

Exp Erl Hyp


0 05 1 15 2 25 3

14

16

18

2

22

24

26

28

3

Mea

n qu

eue l

engt

h

c = 3

Service rate 120583

Exp Erl Hyp





0 05 1 1516

18

2

22

24

26

28

3

32

Mea

n qu

eue l

engt

h

c = 6

Service rate 120583

Exp Erl Hyp


0 2 4 6 8 10118

12

122

124

126

128

13

Mea

n qu

eue l

engt

h c = 1 120579 = 01


Exp Erl Hyp





0 2 4 6 8 10146

1465

147

1475

148

1485

Mea

n qu

eue l

engt

h

c = 3 120579 = 01


Exp Erl Hyp


0 2 4 6 8 1026

265

27

275

Mea

n qu

eue l

engt

h

c = 6 120579 = 01


Exp Erl Hyp






0 02 04 06 08 1088

089

09

091

092

093

094

095

Bloc

king

pro

babi

lity

c = 1

Service rate 120583

Exp Erl Hyp


0 005 01 015 02 025 03081

082

083

084

085

086

087

088

089

Bloc

king

pro

babi

lity

c = 3

Service rate 120583

Exp Erl Hyp





0 002 004 006 008 01 012 014 016072

074

076

078

08

082

084

Bloc

king

pro

babi

lity

c = 6

Service rate 120583

Exp Erl Hyp


0 2 4 6 8 100

01

02

03

04

05

06M

ean

busy

serv

ers i

n no

nvac

atio

n


Exp Erl Hyp


119887= 10 and 120583V = 04



2 3 4 5 6 7 8 9 10005

01

015

02

025

03

035

04

045

05

Mea

n bu

sy se

rver

s in

nonv

acat

ion


Exp Erl Hyp


0 2 4 6 8 10078

08

082

084

086

088

09

092

094

096

098

Mea

n bu

sy se

rver

s in

nonv

acat

ion


Exp Erl Hyp




6 Conclusion


0 2 4 6 8 10064

065

066

067

068

069

07

071

072

073

Mea

n cu

stom

er lo

ss

Exp Erl Hyp



0 2 4 6 8 1001

015

02

025

03

035

04

045

Mea

n cu

stom

er lo

ss

Exp Erl Hyp




Disclosure



Competing Interests


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120588 120579 120583V119870119891

119871 119861119901


01

01

00 22 25 33 41401 41552 41155 04953 04909 0445210 10 13 18 18312 18449 18849 00376 00606 0106420 8 10 13 14721 14743 14805 00050 00133 0029430 7 8 11 13301 13307 13326 00013 00049 00117

1

00 9 12 16 16336 16535 16721 00203 00421 0073010 7 10 13 14723 14792 14875 00056 00163 0034320 7 9 12 13787 13811 13847 00022 00078 0018130 6 8 11 13165 13172 13189 00011 00044 00107

100

00 6 7 11 13037 13051 13062 00009 00039 0009610 6 7 11 13026 13039 13049 00009 00039 0009520 6 7 11 13015 13027 13036 00009 00038 0009330 6 7 11 13005 13016 13023 00009 00037 00092

05

01

00 36 41 58 61463 65540 81834 07544 07535 0709402 26 30 48 44310 47592 61809 05820 05952 0606204 20 24 43 33472 35687 46449 03792 04116 0487806 16 20 39 27521 28807 35933 02308 02694 03736

1

00 17 19 37 31041 32843 40155 03311 03663 0441402 16 19 37 29373 30968 38141 02832 03217 0411604 17 21 37 27875 29234 36269 02421 02821 0382506 16 21 37 26620 27814 34534 02074 02473 03544

100

00 16 19 37 26304 27513 34302 01989 02385 0346802 16 21 37 26286 27437 34280 01984 02380 0346304 16 21 37 26267 27418 34257 01978 02375 0345906 16 21 37 26249 27399 34234 01973 02370 03455

09

01

00 78 100 224 140668 163843 336740 09618 09582 0940301 75 98 224 127466 149913 319614 09424 09390 0925102 73 95 229 113801 135544 301618 09089 09077 0905203 71 93 234 99871 120651 282970 08541 08593 08796

1

00 76 101 239 97276 117783 283667 08612 08643 0880601 76 100 238 95526 116045 281630 08466 08523 0875802 75 100 237 93873 114217 279598 08312 08397 0870803 75 99 237 92075 112421 277421 08149 08265 08656

100

00 82 111 269 91219 111422 293701 08041 08176 0862201 82 111 269 91194 111398 293614 08038 08175 0862102 82 111 269 91169 111375 293527 08036 08173 0862103 82 111 269 91144 111351 293440 08033 08171 08620







120588 120579 120583V119870119891

119871 119861119901


01

01

00 20 23 32 32118 32976 34094 00679 00887 0140505 11 13 18 21893 22065 22152 00008 00034 0015910 9 11 14 18365 18407 18401 00000 00003 0002415 8 9 12 16458 16464 16462 00000 00001 00005

1

00 8 11 15 17766 17958 18184 00000 00005 0003505 8 10 14 17120 17219 17334 00000 00002 0001610 8 9 13 16587 16631 16680 00000 00001 0000815 8 9 12 16136 16145 16155 00000 00000 00004

100

00 8 8 12 16022 16050 16039 00000 00000 0000405 8 8 12 16016 16041 16028 00000 00000 0000410 8 8 12 16009 16032 16016 00000 00000 0000415 8 8 12 16002 16024 16005 00000 00000 00004

05

01

00 30 34 52 63770 68052 84877 04072 04526 0545601 24 30 46 53485 56091 69287 02546 03073 0441102 21 25 41 46431 48028 57143 01445 01924 0337603 18 22 39 41697 42487 48192 00797 01170 02476

1

00 18 20 39 45180 46593 51256 01208 01617 0288901 18 20 39 43618 44869 49560 01007 01399 0267902 19 23 38 42119 42803 48001 00837 01208 0247703 18 22 38 40862 41494 46449 00696 01042 02284

100

00 17 20 38 40659 41500 46956 00662 01001 0223301 17 20 38 40633 41474 46923 00659 00998 0223002 18 23 38 40514 41099 46889 00657 00995 0222603 18 23 38 40489 41074 46856 00655 00993 02223

09

01

000 78 101 229 151572 173703 354516 09045 09054 09082005 76 99 227 142268 163514 340044 08719 08759 08903010 75 98 225 131999 152452 324807 08265 08360 08684015 73 96 222 121491 140835 309017 07663 07840 08418

1

000 79 105 246 120928 139233 312097 07748 07881 08391005 79 104 245 119039 137614 309647 07595 07758 08343010 78 103 245 117512 135946 306805 07436 07629 08292015 78 103 244 115535 134025 304553 07270 07496 08241

100

000 84 114 272 115312 133305 356288 07160 07407 08207005 84 114 272 115273 133271 356049 07157 07405 08206010 84 114 272 115235 133237 355810 07155 07403 08205015 84 113 272 115196 133446 355572 07152 07401 08204






0 05 1 15 21

2

3

4

5

6

7M

ean

queu

e len

gth

Exp Erl Hyp

c = 1

Service rate 120583


0 05 1 15 2 25 31

2

3

4

5

6

7

8

9

Mea

n qu

eue l

engt

h

Exp Erl Hyp

c = 3

Service rate 120583




0 005 01 015 02 025 034

45

5

55

6

65

7

75

8

85

Mea

n qu

eue l

engt

h

Hyp Erl Exp

c = 6

Service rate 120583


0 02 04 06 08 15

10

15

20

25

30M

ean

queu

e len

gth c = 1

Service rate 120583

Exp Erl Hyp






0 005 01 015 02 025 038

10

12

14

16

18

20

22

24

26

28

Mea

n qu

eue l

engt

h

c = 3

Service rate 120583

Exp Erl Hyp


0 002 004 006 008 01 012 014 01610

15

20

25

30

35

Mea

n qu

eue l

engt

h c = 6

Service rate 120583

Exp Erl Hyp




0 2 4 6 8 1011

12

13

14

15

16

17

18

19

2

21

Mea

n qu

eue l

engt

h

c = 1

Service rate 120583

Exp Erl Hyp


0 05 1 15 2 25 3

14

16

18

2

22

24

26

28

3

Mea

n qu

eue l

engt

h

c = 3

Service rate 120583

Exp Erl Hyp





0 05 1 1516

18

2

22

24

26

28

3

32

Mea

n qu

eue l

engt

h

c = 6

Service rate 120583

Exp Erl Hyp


0 2 4 6 8 10118

12

122

124

126

128

13

Mea

n qu

eue l

engt

h c = 1 120579 = 01


Exp Erl Hyp





0 2 4 6 8 10146

1465

147

1475

148

1485

Mea

n qu

eue l

engt

h

c = 3 120579 = 01


Exp Erl Hyp


0 2 4 6 8 1026

265

27

275

Mea

n qu

eue l

engt

h

c = 6 120579 = 01


Exp Erl Hyp






0 02 04 06 08 1088

089

09

091

092

093

094

095

Bloc

king

pro

babi

lity

c = 1

Service rate 120583

Exp Erl Hyp


0 005 01 015 02 025 03081

082

083

084

085

086

087

088

089

Bloc

king

pro

babi

lity

c = 3

Service rate 120583

Exp Erl Hyp





0 002 004 006 008 01 012 014 016072

074

076

078

08

082

084

Bloc

king

pro

babi

lity

c = 6

Service rate 120583

Exp Erl Hyp


0 2 4 6 8 100

01

02

03

04

05

06M

ean

busy

serv

ers i

n no

nvac

atio

n


Exp Erl Hyp


119887= 10 and 120583V = 04



2 3 4 5 6 7 8 9 10005

01

015

02

025

03

035

04

045

05

Mea

n bu

sy se

rver

s in

nonv

acat

ion


Exp Erl Hyp


0 2 4 6 8 10078

08

082

084

086

088

09

092

094

096

098

Mea

n bu

sy se

rver

s in

nonv

acat

ion


Exp Erl Hyp




6 Conclusion


0 2 4 6 8 10064

065

066

067

068

069

07

071

072

073

Mea

n cu

stom

er lo

ss

Exp Erl Hyp



0 2 4 6 8 1001

015

02

025

03

035

04

045

Mea

n cu

stom

er lo

ss

Exp Erl Hyp




Disclosure



Competing Interests


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120588 120579 120583V119870119891

119871 119861119901


01

01

00 20 23 32 32118 32976 34094 00679 00887 0140505 11 13 18 21893 22065 22152 00008 00034 0015910 9 11 14 18365 18407 18401 00000 00003 0002415 8 9 12 16458 16464 16462 00000 00001 00005

1

00 8 11 15 17766 17958 18184 00000 00005 0003505 8 10 14 17120 17219 17334 00000 00002 0001610 8 9 13 16587 16631 16680 00000 00001 0000815 8 9 12 16136 16145 16155 00000 00000 00004

100

00 8 8 12 16022 16050 16039 00000 00000 0000405 8 8 12 16016 16041 16028 00000 00000 0000410 8 8 12 16009 16032 16016 00000 00000 0000415 8 8 12 16002 16024 16005 00000 00000 00004

05

01

00 30 34 52 63770 68052 84877 04072 04526 0545601 24 30 46 53485 56091 69287 02546 03073 0441102 21 25 41 46431 48028 57143 01445 01924 0337603 18 22 39 41697 42487 48192 00797 01170 02476

1

00 18 20 39 45180 46593 51256 01208 01617 0288901 18 20 39 43618 44869 49560 01007 01399 0267902 19 23 38 42119 42803 48001 00837 01208 0247703 18 22 38 40862 41494 46449 00696 01042 02284

100

00 17 20 38 40659 41500 46956 00662 01001 0223301 17 20 38 40633 41474 46923 00659 00998 0223002 18 23 38 40514 41099 46889 00657 00995 0222603 18 23 38 40489 41074 46856 00655 00993 02223

09

01

000 78 101 229 151572 173703 354516 09045 09054 09082005 76 99 227 142268 163514 340044 08719 08759 08903010 75 98 225 131999 152452 324807 08265 08360 08684015 73 96 222 121491 140835 309017 07663 07840 08418

1

000 79 105 246 120928 139233 312097 07748 07881 08391005 79 104 245 119039 137614 309647 07595 07758 08343010 78 103 245 117512 135946 306805 07436 07629 08292015 78 103 244 115535 134025 304553 07270 07496 08241

100

000 84 114 272 115312 133305 356288 07160 07407 08207005 84 114 272 115273 133271 356049 07157 07405 08206010 84 114 272 115235 133237 355810 07155 07403 08205015 84 113 272 115196 133446 355572 07152 07401 08204






0 05 1 15 21

2

3

4

5

6

7M

ean

queu

e len

gth

Exp Erl Hyp

c = 1

Service rate 120583


0 05 1 15 2 25 31

2

3

4

5

6

7

8

9

Mea

n qu

eue l

engt

h

Exp Erl Hyp

c = 3

Service rate 120583




0 005 01 015 02 025 034

45

5

55

6

65

7

75

8

85

Mea

n qu

eue l

engt

h

Hyp Erl Exp

c = 6

Service rate 120583


0 02 04 06 08 15

10

15

20

25

30M

ean

queu

e len

gth c = 1

Service rate 120583

Exp Erl Hyp






0 005 01 015 02 025 038

10

12

14

16

18

20

22

24

26

28

Mea

n qu

eue l

engt

h

c = 3

Service rate 120583

Exp Erl Hyp


0 002 004 006 008 01 012 014 01610

15

20

25

30

35

Mea

n qu

eue l

engt

h c = 6

Service rate 120583

Exp Erl Hyp




0 2 4 6 8 1011

12

13

14

15

16

17

18

19

2

21

Mea

n qu

eue l

engt

h

c = 1

Service rate 120583

Exp Erl Hyp


0 05 1 15 2 25 3

14

16

18

2

22

24

26

28

3

Mea

n qu

eue l

engt

h

c = 3

Service rate 120583

Exp Erl Hyp





0 05 1 1516

18

2

22

24

26

28

3

32

Mea

n qu

eue l

engt

h

c = 6

Service rate 120583

Exp Erl Hyp


0 2 4 6 8 10118

12

122

124

126

128

13

Mea

n qu

eue l

engt

h c = 1 120579 = 01


Exp Erl Hyp





0 2 4 6 8 10146

1465

147

1475

148

1485

Mea

n qu

eue l

engt

h

c = 3 120579 = 01


Exp Erl Hyp


0 2 4 6 8 1026

265

27

275

Mea

n qu

eue l

engt

h

c = 6 120579 = 01


Exp Erl Hyp






0 02 04 06 08 1088

089

09

091

092

093

094

095

Bloc

king

pro

babi

lity

c = 1

Service rate 120583

Exp Erl Hyp


0 005 01 015 02 025 03081

082

083

084

085

086

087

088

089

Bloc

king

pro

babi

lity

c = 3

Service rate 120583

Exp Erl Hyp





0 002 004 006 008 01 012 014 016072

074

076

078

08

082

084

Bloc

king

pro

babi

lity

c = 6

Service rate 120583

Exp Erl Hyp


0 2 4 6 8 100

01

02

03

04

05

06M

ean

busy

serv

ers i

n no

nvac

atio

n


Exp Erl Hyp


119887= 10 and 120583V = 04



2 3 4 5 6 7 8 9 10005

01

015

02

025

03

035

04

045

05

Mea

n bu

sy se

rver

s in

nonv

acat

ion


Exp Erl Hyp


0 2 4 6 8 10078

08

082

084

086

088

09

092

094

096

098

Mea

n bu

sy se

rver

s in

nonv

acat

ion


Exp Erl Hyp




6 Conclusion


0 2 4 6 8 10064

065

066

067

068

069

07

071

072

073

Mea

n cu

stom

er lo

ss

Exp Erl Hyp



0 2 4 6 8 1001

015

02

025

03

035

04

045

Mea

n cu

stom

er lo

ss

Exp Erl Hyp




Disclosure



Competing Interests


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 05 1 15 21

2

3

4

5

6

7M

ean

queu

e len

gth

Exp Erl Hyp

c = 1

Service rate 120583


0 05 1 15 2 25 31

2

3

4

5

6

7

8

9

Mea

n qu

eue l

engt

h

Exp Erl Hyp

c = 3

Service rate 120583




0 005 01 015 02 025 034

45

5

55

6

65

7

75

8

85

Mea

n qu

eue l

engt

h

Hyp Erl Exp

c = 6

Service rate 120583


0 02 04 06 08 15

10

15

20

25

30M

ean

queu

e len

gth c = 1

Service rate 120583

Exp Erl Hyp






0 005 01 015 02 025 038

10

12

14

16

18

20

22

24

26

28

Mea

n qu

eue l

engt

h

c = 3

Service rate 120583

Exp Erl Hyp


0 002 004 006 008 01 012 014 01610

15

20

25

30

35

Mea

n qu

eue l

engt

h c = 6

Service rate 120583

Exp Erl Hyp




0 2 4 6 8 1011

12

13

14

15

16

17

18

19

2

21

Mea

n qu

eue l

engt

h

c = 1

Service rate 120583

Exp Erl Hyp


0 05 1 15 2 25 3

14

16

18

2

22

24

26

28

3

Mea

n qu

eue l

engt

h

c = 3

Service rate 120583

Exp Erl Hyp





0 05 1 1516

18

2

22

24

26

28

3

32

Mea

n qu

eue l

engt

h

c = 6

Service rate 120583

Exp Erl Hyp


0 2 4 6 8 10118

12

122

124

126

128

13

Mea

n qu

eue l

engt

h c = 1 120579 = 01


Exp Erl Hyp





0 2 4 6 8 10146

1465

147

1475

148

1485

Mea

n qu

eue l

engt

h

c = 3 120579 = 01


Exp Erl Hyp


0 2 4 6 8 1026

265

27

275

Mea

n qu

eue l

engt

h

c = 6 120579 = 01


Exp Erl Hyp






0 02 04 06 08 1088

089

09

091

092

093

094

095

Bloc

king

pro

babi

lity

c = 1

Service rate 120583

Exp Erl Hyp


0 005 01 015 02 025 03081

082

083

084

085

086

087

088

089

Bloc

king

pro

babi

lity

c = 3

Service rate 120583

Exp Erl Hyp





0 002 004 006 008 01 012 014 016072

074

076

078

08

082

084

Bloc

king

pro

babi

lity

c = 6

Service rate 120583

Exp Erl Hyp


0 2 4 6 8 100

01

02

03

04

05

06M

ean

busy

serv

ers i

n no

nvac

atio

n


Exp Erl Hyp


119887= 10 and 120583V = 04



2 3 4 5 6 7 8 9 10005

01

015

02

025

03

035

04

045

05

Mea

n bu

sy se

rver

s in

nonv

acat

ion


Exp Erl Hyp


0 2 4 6 8 10078

08

082

084

086

088

09

092

094

096

098

Mea

n bu

sy se

rver

s in

nonv

acat

ion


Exp Erl Hyp




6 Conclusion


0 2 4 6 8 10064

065

066

067

068

069

07

071

072

073

Mea

n cu

stom

er lo

ss

Exp Erl Hyp



0 2 4 6 8 1001

015

02

025

03

035

04

045

Mea

n cu

stom

er lo

ss

Exp Erl Hyp




Disclosure



Competing Interests


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 005 01 015 02 025 038

10

12

14

16

18

20

22

24

26

28

Mea

n qu

eue l

engt

h

c = 3

Service rate 120583

Exp Erl Hyp


0 002 004 006 008 01 012 014 01610

15

20

25

30

35

Mea

n qu

eue l

engt

h c = 6

Service rate 120583

Exp Erl Hyp




0 2 4 6 8 1011

12

13

14

15

16

17

18

19

2

21

Mea

n qu

eue l

engt

h

c = 1

Service rate 120583

Exp Erl Hyp


0 05 1 15 2 25 3

14

16

18

2

22

24

26

28

3

Mea

n qu

eue l

engt

h

c = 3

Service rate 120583

Exp Erl Hyp





0 05 1 1516

18

2

22

24

26

28

3

32

Mea

n qu

eue l

engt

h

c = 6

Service rate 120583

Exp Erl Hyp


0 2 4 6 8 10118

12

122

124

126

128

13

Mea

n qu

eue l

engt

h c = 1 120579 = 01


Exp Erl Hyp





0 2 4 6 8 10146

1465

147

1475

148

1485

Mea

n qu

eue l

engt

h

c = 3 120579 = 01


Exp Erl Hyp


0 2 4 6 8 1026

265

27

275

Mea

n qu

eue l

engt

h

c = 6 120579 = 01


Exp Erl Hyp






0 02 04 06 08 1088

089

09

091

092

093

094

095

Bloc

king

pro

babi

lity

c = 1

Service rate 120583

Exp Erl Hyp


0 005 01 015 02 025 03081

082

083

084

085

086

087

088

089

Bloc

king

pro

babi

lity

c = 3

Service rate 120583

Exp Erl Hyp





0 002 004 006 008 01 012 014 016072

074

076

078

08

082

084

Bloc

king

pro

babi

lity

c = 6

Service rate 120583

Exp Erl Hyp


0 2 4 6 8 100

01

02

03

04

05

06M

ean

busy

serv

ers i

n no

nvac

atio

n


Exp Erl Hyp


119887= 10 and 120583V = 04



2 3 4 5 6 7 8 9 10005

01

015

02

025

03

035

04

045

05

Mea

n bu

sy se

rver

s in

nonv

acat

ion


Exp Erl Hyp


0 2 4 6 8 10078

08

082

084

086

088

09

092

094

096

098

Mea

n bu

sy se

rver

s in

nonv

acat

ion


Exp Erl Hyp




6 Conclusion


0 2 4 6 8 10064

065

066

067

068

069

07

071

072

073

Mea

n cu

stom

er lo

ss

Exp Erl Hyp



0 2 4 6 8 1001

015

02

025

03

035

04

045

Mea

n cu

stom

er lo

ss

Exp Erl Hyp




Disclosure



Competing Interests


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 05 1 1516

18

2

22

24

26

28

3

32

Mea

n qu

eue l

engt

h

c = 6

Service rate 120583

Exp Erl Hyp


0 2 4 6 8 10118

12

122

124

126

128

13

Mea

n qu

eue l

engt

h c = 1 120579 = 01


Exp Erl Hyp





0 2 4 6 8 10146

1465

147

1475

148

1485

Mea

n qu

eue l

engt

h

c = 3 120579 = 01


Exp Erl Hyp


0 2 4 6 8 1026

265

27

275

Mea

n qu

eue l

engt

h

c = 6 120579 = 01


Exp Erl Hyp






0 02 04 06 08 1088

089

09

091

092

093

094

095

Bloc

king

pro

babi

lity

c = 1

Service rate 120583

Exp Erl Hyp


0 005 01 015 02 025 03081

082

083

084

085

086

087

088

089

Bloc

king

pro

babi

lity

c = 3

Service rate 120583

Exp Erl Hyp





0 002 004 006 008 01 012 014 016072

074

076

078

08

082

084

Bloc

king

pro

babi

lity

c = 6

Service rate 120583

Exp Erl Hyp


0 2 4 6 8 100

01

02

03

04

05

06M

ean

busy

serv

ers i

n no

nvac

atio

n


Exp Erl Hyp


119887= 10 and 120583V = 04



2 3 4 5 6 7 8 9 10005

01

015

02

025

03

035

04

045

05

Mea

n bu

sy se

rver

s in

nonv

acat

ion


Exp Erl Hyp


0 2 4 6 8 10078

08

082

084

086

088

09

092

094

096

098

Mea

n bu

sy se

rver

s in

nonv

acat

ion


Exp Erl Hyp




6 Conclusion


0 2 4 6 8 10064

065

066

067

068

069

07

071

072

073

Mea

n cu

stom

er lo

ss

Exp Erl Hyp



0 2 4 6 8 1001

015

02

025

03

035

04

045

Mea

n cu

stom

er lo

ss

Exp Erl Hyp




Disclosure



Competing Interests


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 02 04 06 08 1088

089

09

091

092

093

094

095

Bloc

king

pro

babi

lity

c = 1

Service rate 120583

Exp Erl Hyp


0 005 01 015 02 025 03081

082

083

084

085

086

087

088

089

Bloc

king

pro

babi

lity

c = 3

Service rate 120583

Exp Erl Hyp





0 002 004 006 008 01 012 014 016072

074

076

078

08

082

084

Bloc

king

pro

babi

lity

c = 6

Service rate 120583

Exp Erl Hyp


0 2 4 6 8 100

01

02

03

04

05

06M

ean

busy

serv

ers i

n no

nvac

atio

n


Exp Erl Hyp


119887= 10 and 120583V = 04



2 3 4 5 6 7 8 9 10005

01

015

02

025

03

035

04

045

05

Mea

n bu

sy se

rver

s in

nonv

acat

ion


Exp Erl Hyp


0 2 4 6 8 10078

08

082

084

086

088

09

092

094

096

098

Mea

n bu

sy se

rver

s in

nonv

acat

ion


Exp Erl Hyp




6 Conclusion


0 2 4 6 8 10064

065

066

067

068

069

07

071

072

073

Mea

n cu

stom

er lo

ss

Exp Erl Hyp



0 2 4 6 8 1001

015

02

025

03

035

04

045

Mea

n cu

stom

er lo

ss

Exp Erl Hyp




Disclosure



Competing Interests


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










2 3 4 5 6 7 8 9 10005

01

015

02

025

03

035

04

045

05

Mea

n bu

sy se

rver

s in

nonv

acat

ion


Exp Erl Hyp


0 2 4 6 8 10078

08

082

084

086

088

09

092

094

096

098

Mea

n bu

sy se

rver

s in

nonv

acat

ion


Exp Erl Hyp




6 Conclusion


0 2 4 6 8 10064

065

066

067

068

069

07

071

072

073

Mea

n cu

stom

er lo

ss

Exp Erl Hyp



0 2 4 6 8 1001

015

02

025

03

035

04

045

Mea

n cu

stom

er lo

ss

Exp Erl Hyp




Disclosure



Competing Interests


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Competing Interests


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article Phase-Type Arrivals and Impatient ...downloads.hindawi.com/journals/aor/2016/4024950.pdf · Chen et al. [] studied M/M/m/k queue with preemptive resume and impatience