http://sim.sagepub.com/SIMULATION

http://sim.sagepub.com/content/early/2014/08/27/0037549714546665The online version of this article can be found at:

DOI: 10.1177/0037549714546665

published online 28 August 2014SIMULATIONDaniel Huber, John Fowler and Dieter Armbruster

timesSimplification of DES models of M/M/1 tandem queues by approximating WIP-dependent inter-departure

Published by:

http://www.sagepublications.com

On behalf of:

Society for Modeling and Simulation International (SCS)

can be found at:SIMULATIONAdditional services and information for

http://sim.sagepub.com/cgi/alertsEmail Alerts:

http://sim.sagepub.com/subscriptionsSubscriptions:

http://www.sagepub.com/journalsReprints.navReprints:

http://www.sagepub.com/journalsPermissions.navPermissions:

http://sim.sagepub.com/content/early/2014/08/27/0037549714546665.refs.htmlCitations:

What is This?

- Aug 28, 2014OnlineFirst Version of Record >>

at ARIZONA STATE UNIV on September 9, 2014sim.sagepub.comDownloaded from at ARIZONA STATE UNIV on September 9, 2014sim.sagepub.comDownloaded from

Simulation

Simulation: Transactions of the Society forModeling and Simulation International1–9! 2014 The Society for Modeling andSimulation InternationalDOI: 10.1177/0037549714546665sim.sagepub.com

Simplification of DES models of M/M/1tandem queues by approximatingWIP-dependent inter-departure times

Daniel Huber1, John Fowler1 and Dieter Armbruster2

AbstractThis paper presents two algorithms to analytically approximate work in process (WIP)-dependent inter-departure timesfor tandem queues composed of a series of M/M/1 systems. The first algorithm is used for homogeneous tandemqueues, the second for such with bottlenecks. Both algorithms are based on the possible combinations of distributingthe WIP on the queues. For each combination the time to the next departure is estimated. A weighted average of allestimated times of each WIP level is calculated to get the expected mean inter-departure time. The generated inter-departure times are used in a simple model of the tandem queue. The inter-departure times, the average WIP and aver-age cycle time of the tandem queue and the simple model are compared in several tandem queue parameterizations.Results show only a small error between the simple model and the tandem queue, rendering this approach applicable inmany applications.

Keywordsdiscrete event simulation, simplification, WIP-dependent inter-departure times

1. Introduction

The use of work in process (WIP)-dependent inter-departure times for building aggregated or simple discreteevent simulation models of complex production systemswas introduced and refined by Veeger et al.1–2 In theirapproach Veeger et al. use real-world data of a productionsystem to generate the inter-departure times. This is doneby calculating the inter-departure time and actual WIP ateach departure event. Since the WIP can only take integernumbers, the inter-departure times are averaged for everyWIP value.

The quality of the results of the simple model presentedwas very good, such that it is appealing to use this methodnot only for building simple models but also for the sim-plification of existing complex simulation models. Whensimplifying existing simulation models, running simula-tion experiments could generate data similar to data fromthe real-world production system.

To have greater impact on the simplification process interms of simulation runtime reduction, it would be muchbetter if the inter-departure times could be calculated ana-lytically, without simulating the original complex model.After all, not having to simulate the original complexmodel is the intention of model simplification.

One well-known and often used method for gettingsteady-state WIP-dependent output, or inter-departuretimes, of a productions system is using a clearing function.Karmarkar3 proposed a non-linear clearing function,where output increases as a concave function of WIP andreaches an upper limit defined by the modeled productionsystem. When translating this clearing function definitionfrom output to inter-departure times, it becomes a convexfunction with a lower limit for high WIP. Clearing func-tions are mainly used to improve production planning andscheduling methods, e.g. linear programming, by givingthem non-linear WIP-dependent behavior.4–6 They can bederived analytically using steady-state queueing models.

As using WIP-dependent behavior in model simplifica-tion is very promising in achieving simple models with

1WPC Supply Chain Management, Arizona State University, Tempe, AZ,USA2School of Mathematical and Statistical Sciences, Arizona State University,Tempe, AZ, USA

Corresponding author:Daniel Huber, Fraunhofer Institute for Intelligent Analysis and InformationSystems IAIS, Department ART, Schloss Birlinghoven, 53757 SanktAugustin, Germany.Email: d@nielhuber.de

at ARIZONA STATE UNIV on September 9, 2014sim.sagepub.comDownloaded from

little approximation error, it is not the solution in everyapplication frame.7 In his approach Rose used a serverwith WIP-dependent processing times to improve a delayelement in a simple model consisting of a bottleneck ser-ver taken from an original model in combination withthree simple delay elements but could not achieve satisfac-tory result quality.

In our approach we want to focus on flow shop models,which in an abstract form can be modeled as a series of M/M/1 queues of length k (also known as tandem queue).Such a system consist of k queues and servers, as depictedin Figure 1. Jobs enter the system at queue Q1 according toa Markovian arrival process with mean inter-arrival timeta. Jobs are waiting in the first in first out (FIFO) queuesQi to be processed by server Si and leave the system afterbeing processed at station k. The departure process has aWIP-dependent mean inter-departure time td(w).

In Figure 2 a clearing function of a M/M/1 tandemqueue of length 3 is presented and the inter-departuretimes of the same system calculated by using the methodof Veeger et al.1 The clearing function was calculated byusing the approximation for cycle time and WIP of factory

physics.8 Both curves have the same limit for high WIP,but show a significant deviation for small WIP. Thisclearly shows that the clearing function, defined as thesteady-state average inter-departure time over the steady-state average WIP, is different from the WIP-dependentinter-departure time of Veeger et al., which is defined asthe average td over the actual WIP. Thus, clearing func-tions do not give an adequate approximation for ourpurpose.

Veeger et al.9 did work on getting analytical inter-departure times of queuing systems, but they were focus-ing on a single M/G/n queue in heavy traffic, i.e. highWIP, such that all n servers are busy all of the time. Noother publications regarding the analytical approximationof WIP-dependent inter-departure times could be found.

2. Basics of the approximation ofWIP-dependent inter-departure times

In a M/M/1 tandem queue of length k there are k queuesand servers. Each server i = 1.k has a mean service time

Figure 1. M/M/1 tandem queue of length k.

Figure 2. Comparison of clearing function and WIP-dependent inter-departure times td(w) for a 3 M/M/1 queue with te( · ) = 1.

2 Simulation: Transactions of the Society for Modeling and Simulation International

at ARIZONA STATE UNIV on September 9, 2014sim.sagepub.comDownloaded from

te(i). The expected utilization of each server is u(i) = te(i)/ta. To have a stable system it must hold u(i)\ 1, 8i. Thehighest utilization in the tandem queue shall be used as thedescription of the utilization of the whole system.

WIP-dependent inter-departure times are defined as fol-lows. One job departs the system and leaves w jobs behind.The next job departs in td(w) time units. Here td(w) is themean of a random distribution with a coefficient of varia-tion of cd(w).

The key idea of our method to calculate td(w) is to lookat all possible combinations of the job allocation. One job-allocation combination c is an array of the number of jobsji per queue i, with

Pki= 1 ji =w. For each combination

c 2 C(w) the jobs on the queue Qi=1.k with the highestindex i are defining the inter-departure time. The inter-departure time is the sum of all service times of the serversSj=i.k. To get the expected inter-departure time for w, theweighted average of the inter-departure times of all combi-nations c is used.

In Table 1 the C, td and cd are shown for WIP-Levelsw = 1,2 in a M/M/1 tandem queue of length k = 3, whereall servers have the mean service time of te = 1. The datafor td(w) and cd(w) was generated by analyzing a discreteevent simulation running 106 time units in 4 replications.As can be seen, the mean of td(c) for WIP-Level w, !td(C),is equal to td(w).

The cd in Table 1 are not 1, meaning the departure pro-cess is not Markovian. When analyzing a bigger interval ofWIP-levels (like in Figure 5), it shows that limw!N(td) = teand limw!N(cd) = 1. This behavior can be explained whenlooking at the allocation combinations. As can be seen inTable 1, there are 3 possible combinations for w = 1 and 6for w = 2. Since all service processes are Markovian andthus memoryless, we do not have to be concerned abouthow long a server is already processing a job when lookingat the current state of the system. As soon as there is one or

more jobs in a queue, the corresponding server is busy andcan be handled as if it just has started processing. At w = 1in a third of the combinations the last server is busy, atw = 2 in half the combinations the last server is busy. Inthese combinations the inter-departure time is the servicetime of the last server, i.e. exponential distributed time. Inthe other combinations it is a sum of service times, i.e. notexponentially distributed time. The ratio of combinationswith the last server occupied is growing with w, resultingin the limits above.

The possible allocation combinations at two consecu-tive departures are not independent of each other sincesome combinations become impossible following certainother combinations. In general, combinations with jobsclose to the beginning of the tandem queue have a higherprobability of being impossible, thus an increase in td(w)is expected when looking at specific departure events andassuming independence, i.e. always considering all combi-nations. But when taking the long time average WIP, cycletime and throughput as key performance indicators, weexpect no significant influence of assuming independence.

The inter-departure times can be used as input data in asimple discrete event model which then can be used as a sim-plification of the original model. In the next section we willpresent this simple model. Then, we will present two algo-rithms to approximate the inter-departure times analyticallyfor homogenous tandem queues and such with bottlenecks.We will present results showing the quality of the td-approxi-mation as well as the quality of the simple model when com-pared with the original one and conclude with somelimitations of our approach and ideas for further research.

3. The simple model

The structure of the simple model is given in Figure 3. Themodel contains a source and a sink, which are equal tothose in the original model (equal parameterization andgeneration of the same job-objects). Between these two isa FIFO queue and a release gate (circle with cross symbol).The diamond shaped object between source and queuecalls the release timer whenever a job passes and the queueis empty. The diamond shaped object between release gateand sink calls the release timer whenever a job passes.

Table 1. Allocation combinations and WIP-dependent td and cdof 3 M/M/1 queue with te( · ) = 1.

Q1 Q2 Q3

w c td(c) td(C) td(w) cd(w)

1 1 0 0 31 0 1 0 2 6/3 2.000 0.8181 0 0 1 1

10/6 1.667 0.897

222222

210100

012010

000112

322111

...

Figure 3. Structure of the simple model.

Huber et al. 3

at ARIZONA STATE UNIV on September 9, 2014sim.sagepub.comDownloaded from

This setup of call functions is necessary, because td(w = 0)is not defined the same way as for w . 0, since td wouldalso be dependent on the arrival process. In the case of anarrival at w = 0, the release gate is opened after the rawprocessing time

Pi= 1...k te(i) of the tandem queue. In all

other cases, the release gate is opened in td(w) time units.Since td is exponentially distributed in heavily loaded sys-tems and at all times the first two moments are known, weuse a gamma-distribution to sample random values.

4. Approximation of inter-departure times

The approximation of td(w) in a tandem queue with bottle-necks, i.e. all servers can have different te, is much morecomplicated than in the homogenous case. This is becausein the latter case all allocation combinations c 2 C at aWIP level have the same probability of occurrence. Ifthere is a bottleneck, combinations with a high ratio ofjobs in the queue of the bottleneck are much more likelythan those with a high ratio of jobs in other queues.

In the following two subsections the two algorithms areexplained in detail.

4.1. Homogenous tandem queues

In homogenous tandem queues, te(1) = te(2) = ! ! ! = te(k).When the last server in the tandem queue (Sk) in combina-tion c is occupied, the next departure will happen in aver-age te time units. If the server Sk2‘ is occupied and alldownstream servers Sk2j, 0 4 j 4 ‘, 0 4 ‘ 4 k21are idle, the next departure will happen on average at time

b(‘)= (‘+ 1) ! te: "1#

The number of allocation combinations for a tandemqueue of length k and WIP level w is given by (standardstochastic problem to put w not distinguishable balls into kdistinguishable urns)

mC =(k +w$ 1)!

(k $ 1)! ! w!: "2#

The number of combinations with ‘ or more consecutiveidle servers, starting from Sk is

m%Cl(‘)=(k $ ‘$ 1+w)!

(k $ ‘$ 1)! ! w!: "3#

The number of combinations with exactly ‘ consecutiveidle servers mCl, can be calculated by mCl(‘) = m*Cl(‘)2 m*Cl (‘+ 1), giving the inter-departure time

td(w)=Xk$1

‘= 0

b(‘) ! mCl(‘)

mC

! ": "4#

For each ‘ the time to next departure b(‘) is weighted byits probability of occurrence and all of these terms aresummed up. For combinations with Sk occupied, the next

departure is expected to occur in te, with cd = 1. In allother cases the next expected departure is expected tooccur in (‘+ 1) te. The convolution of (‘+ 1) exponentialdistributions with equal mean te can be expressed by anErlang or Gamma distribution with shape (‘+ 1) andscale te. By calculating the standard deviation (SD) of sucha Gamma distribution, the coefficient of variation is

cd(w)=Xk$1

‘= 0

StD(gamma(‘+ 1, te))

b(‘)! mCl(‘)

mC

! ": "5#

4.2. Tandem queues with bottlenecks

In tandem queues with bottlenecks, at least one server hasa different service time than the other servers in the tan-dem queue. To get to an ‘‘allocation-probability’’ for eachqueue in the tandem queue, we first define a bottleneckindicator

z=max(te( ! )), "6#

G=te(i)

z

! "2

, i= 1 . . . k

( )

: "7#

In the vector G = {g1.gk} the element of the bottle-neck server gbn has a value of 1 and all other gi6&bn are putinto squared relation and are smaller than 1. For evaluatingthe probability of occurrence r(c) of each combinationc 2 C, we first calculate an intermediate e(c):

e(c)=Xk

i= 1

(ci ! gi) "8#

r(c)=eu!e(c)

PC eu!e(c) : "9#

Experiments showed that the deviations between prob-abilities had to be increased to get a good approximationfor r(c), thus in Equation (9) an exponential function isused, with u a constant. Using an exponential functionshowed to be a better approximation than other methodswe tested, like exponentiation, to distinguish betweenlikely and unlikely combinations.

The inter-departure time can now be calculated as

td(w)=X

C(w)

X‘max(c)

‘= 0

te(k $ ‘)

!

! r(c)

" #

, "10#

where for a given combination c 2 C(w) ‘max(c) is theindex of the last busy server Sk2‘ in the tandem queue, i.e.queues k,k 21.k 2 ‘+ 1 are all zero.

The coefficient of variation is approximated byEquations (5) and (10) which in this case is not exact,since the te in the tandem queues are not the same. Anexact approach would involve convolution of ‘+ 1

4 Simulation: Transactions of the Society for Modeling and Simulation International

at ARIZONA STATE UNIV on September 9, 2014sim.sagepub.comDownloaded from

exponential distributions, each having a different mean.The calculations are quite involved and as we will show,the approximation is quite good making the exactapproach unnecessary.

In this algorithm each combination c must be handledseparately, which becomes a computational problem whenk increases and the utilization is high, since then the maxi-mum WIP level for which td have to be calculated ishigher. Here mC grows extremely fast with w and k as canbe seen in Equation (2).

5. Results

To test the algorithms, 10 different original models wereused (see Table 2), each with a utilization u of 0.3, 0.5 and0.8. When using simulation, all analyses were done withthe average of four simulation replications (efficient paral-lelization on the quad-core test platform), each covering106 time units. Models I–III are homogeneous queues,models IV and V and VI and VII have the same asym-metric bottleneck characteristic, but of different strengthand models VIII–X have the same symmetric bottleneckcharacteristic of increasing strength. These models werechosen to test the algorithms in a wide range of queue con-figurations and to discover trends in the approximationquality.

The original models and the simple model were imple-mented with MatLab 2012b’s SimEvents extension. Allalgorithms were implemented in MatLab 2012b. The testplatform was a 2012 Mac mini with an i7 processor and8 GB of RAM.

In Figure 4 the td(w) obtained from simulation of theoriginal model are shown for model I. With growing utili-zation, the maximum WIP level is growing. For each ofthe curves representing different utilizations, as WIPincreases there is a region where the number of samplesgets small, resulting in noisy data. In the low-noise areathe covering of the data points is very good, meaning that

td is independent of the arrival process. It is easy to seethat limw!N(td) = 1.

In Figure 5 the td(w) and cd(w) for models I, II and IIIat u = 0.8 are shown (obtained from simulations of theoriginal models). For low WIP-levels td grows with k.Independent of the tandem length, the limit for high WIPis 1 for all models. For model III there is some noise intd(w \ 10), since only a few samples were available.For low WIP levels, cd(w) is significantly lower than 1,but it converges much faster than td(w).

In Figure 6 the results of the experiments to find a goodu for Equation (9) are shown. The relative errors of td(w)were calculated for the models VIII, IX and X, then theabsolute values of these errors were averaged over all wand u. As can be seen for u = 0.6 there is a minimum inmodel X, the model with the strongest bottleneck. The dif-ferences in the other models are less significant, only inmodel VIII the error at u = 0.6 is not the lowest one. Notshown here are all of the other values 0.5 . u . 0.65tested, but the errors were larger in all tested configura-tions. Here u = 0.6 was chosen to be a good choice for awide range of model configurations and was used in all fol-lowing experiments, but without further evidence for itbeing the best choice.

Figure 7 shows how the errors in WIP develop whenthe strength of one single bottleneck in the tandem queueis increased. The errors are presented for utilizations 0.5and 0.8. When the simple model is run with td(w) calcu-lated for homogenous tandem queues (legend: h.Alg), theerror increases quickly and surpasses 5% at utilization 0.8even in the model of the least inhomogeneous tandemqueue. This clearly shows the necessity of the algorithmfor tandem queues with bottlenecks, with which the erroris kept significantly smaller (legend: bn.Alg).

In Figure 8 the relative error between the empiricaltd(w) and those calculated with the presented algorithmsare shown for models II, V and VII (all tandem length 5,model VII should act as a worst case for maximum lengthand heterogeneity). For the homogenous model II the erroris generally smaller than 61%. For the heterogeneousmodels V and VII the error is generally smaller than 63%.Comparing models V and VII, the strength of the bottle-necks does not seem to have much of an influence on theover all quality of the result. For the homogenous modelthe error fluctuates around zero, where as for models V andVII the errors seem to be on an arbitrary trajectory. Startingwith w = 25 there are bigger jumps in the error trajectory.This is due to noise in the simulation data, since only 3%of the samples of a simulation run have w . 24.

In Table 3 the relative errors between the original mod-els and the simple model using the generated td(w) areshown. Negative values represent a lower average WIP inthe simple models than in the original ones. The error inaverage cycle time was also calculated: values were verysimilar to these, as expected by Little’s law. Comparing

Table 2. Test models.

teModel S1 S2 S3 S4 S5 S6 . . . S10

I 1 1 1II 1 1 1 1 1III 1 1 1 1 1 1IV 0.8 1 0.8V 0.8 1 0.7 0.8 0.9VI 0.5 1 0.6VII 0.5 1 0.5 0.5 0.9VIII 0.975 0.975 1 0.975 0.975IX 0.95 0.95 1 0.95 0.95X 0.9 0.9 1 0.9 0.9

Huber et al. 5

at ARIZONA STATE UNIV on September 9, 2014sim.sagepub.comDownloaded from

models I, II and III, the error seems to increase with k forlow utilizations and decrease for high utilizations. Also theerror is decreasing for higher utilizations. A reason for thiscould be that the simple model has a problem with

operating in an area of td(w), where there is a high gradientin the values. For high utilization the error is smaller,because all models operate most of the times in a regionclose to the limit. For models IV–VII no general

Figure 4. Average WIP-dependent inter-departure time td for utilizations u = 0.3, 0.5 and 0.8 for model I.

Figure 5. Average WIP-dependent inter-departure time td(w) and its coefficient of variation cd(w) in dependence of WIP forhomogeneous models I, II and III.

6 Simulation: Transactions of the Society for Modeling and Simulation International

at ARIZONA STATE UNIV on September 9, 2014sim.sagepub.comDownloaded from

tendencies are visible. Looking at all models and utiliza-tions, the errors are clearly smaller than 6 5%.

In Table 4, the errors of the coefficient of variationof the average WIP are shown. Looking at each modelseparately, larger errors in WIP correlate to larger errors

in coefficient in variation. Also the variability of the errorsis larger than in Table 3. All values are smaller than 68.5%, with models IV–VII performing significantly better.

Reviewing the results obtained, it can be stated thathomogenous tandem lines can be approximated very well,

Figure 6. Error indicator for models VIII, IX and X to find good !. Error indicator is averaged over all utilizations and WIP levels ofthe absolute relative errors in td(w).

Figure 7. Error in average WIP of the simple model when using both algorithms for calculating td(w) on the same models(increasing bottleneck strength: II, VIII, IX and X).

Huber et al. 7

at ARIZONA STATE UNIV on September 9, 2014sim.sagepub.comDownloaded from

especially for high utilizations. Tandem queues with bot-tlenecks can be approximated with similar error levels.The algorithm created can be used for a variety of queueconfigurations. The small error levels in average WIP ofthe simple model should render it applicable in many rea-listic applications. The higher errors in the coefficient ofvariation should have smaller impact on the applicability,since when using a simple model, the average of thekey values is generally more important than their secondmoments.

6. Conclusion and future research

We have developed methods to simplify models of M/M/1tandem queues. Results show small relative errors in awide range of tandem queue parameterization.

In addition to the M/M/1 tandem queues, we tried toextend the approach to M/M/s tandem queues. They aremuch harder to approximate, since jobs can overtake eachother and the most progressed job in the tandem queue isnot necessarily the next job to exit. Furthermore, even in atandem queue without bottlenecks, every combination hasits own probability of occurrence, strongly avoiding unba-lanced combinations, also due to the overtaking process.This behavior increases with the number of parallelservers s. When looking at single combinations it was notpossible to develop an algorithm to universally approximatetd. Since jobs can overtake each other, the td for low WIPlevels are also dependent on the arrival process. We wereable to get decent results for homogenous tandem queueswith s = 2 and 3 using an adapted bottleneck algorithm, butfor higher values of s the error was unacceptable.

Figure 8. Error of average WIP-dependent inter-departure time td(w) for models II, IV and VII.

Table 3. Average WIP analysis.

Utilization Error in average WIP

0.3 0.5 0.8

ModelI 1.57% ! 0.53% ! 1.63%II ! 2.39% ! 2.78% ! 1.37%III ! 4.49% ! 3.09% ! 0.82%IV 0.44% ! 0.11% 0.20%V ! 0.63% 0.15% 3.22%VI ! 0.97% ! 2.34% ! 4.96%VII ! 0.68% 0.20% ! 1.86%

Table 4. Average coefficient of variation analysis.

Utilization Error in cv of average WIP

0.3 0.5 0.8

ModelI ! 0.08% ! 2.42% ! 0.71%II ! 3.42% ! 4.39% ! 1.53%III ! 8.46% ! 6.32% ! 1.39%IV 0.33% 0.0% 0.74%V ! 0.46% ! 0.03% 0.41%VI ! 1.83% ! 1.95% 0.27%VII ! 1.42% ! 0.58% ! 1.65%

8 Simulation: Transactions of the Society for Modeling and Simulation International

at ARIZONA STATE UNIV on September 9, 2014sim.sagepub.comDownloaded from

We also looked at M/M/1/g tandem queues, where thequeues have a limiting capacity of g. In these systems theinter-departure times are also harder to approximate, becauseeven in homogenous systems (same as with M/M/ s systems)the probability of occurrence of combinations is varying,preferring combinations with jobs in the front queues. Evenwith perfect inter-departure times (for testing, td(w) wasobtained from simulation data) the simple model of such asystem has a high error. Let every queue in the originalmodel has a capacity of g, such that the whole model has amaximumWIP of wmax = gk+ k. If wmax is set as the capac-ity of the queue in the simple model (Figure 3), which intui-tively needs to be done, it will block incoming jobs when itscapacity restriction is reached. Thus, the simple modelblocks significantly less often than the original model, sincethe original model blocks at wmax and when the first queuehas an allocation of g. The average WIP in the simple modelis significantly higher than in the original model.

The number of combinations that need to be evaluatedfor tandem queues with bottlenecks increase so rapidly thatwith the test system used and applying MatLab’s paralleli-zation tools, systems with k' 5 could not be approximatedin a practical runtime. Because of the parallelization on asingle computer, memory size was the limiting factor.There are easy ways to reduce the computational costs, e.g.by not calculating td for every w beginning from 1 to auser-defined maximum, but using interpolation betweencalculated values, etc. However, it seems that a significantreduction in runtime can only be achieved by not calculat-ing all possible combinations for a WIP level.

Funding

This research was financially supported by the DeutscheForschungsgemeinschaft (grant number HU1912/2-1). DA wassupported by a grant from the Volkswagen Foundation under theprogram on Complex Networks.

References

1. Veeger C, Etman P, Rooda J, et al. Cycle time distributionsof semiconductor workstations using aggregate modeling. InRossetti MD, Hill RR, Johansson B, et al (eds.), Proceedings of

the 2009 Winter Simulation Conference. Piscataway, NJ: Instituteof Electrical and Electronics Engineers, Inc., pp. 1610–1621.

2. Etman P, Veeger C, Lefeber E, et al. Aggregate modeling ofsemiconductor equipment using effective process times. In JainS, Creasey RR, Himmelspach J, et al. (eds.), Proceedings of the

2011 Winter Simulation Conference. Piscataway, NJ: Institute ofElectrical and Electronics Engineers, Inc., pp. 1795–1807.

3. Karmarkar US. Capacity loading and release planning withwork-in-progress (WIP) and lead-times. J Manufact Operat

Management 1989; 2: 105–123.4. Missbauer H and Uzsoy R. Optimization models for produc-

tion planning. In Kempf KG, Keskinocak P, Uszoy R (eds.),Planning Production and Inventories in the Extended

Enterprise. New York: Springer, 2011, pp. 437–507.

5. Kacar NB and Uzsoy R. Estimating clearing functions fromsimulation data. In Johansson B, Jain S, Montoya-Torres J

et al. (eds.), Proceedings of the 2010 Winter Simulation

Conference. Piscataway, NJ: Institute of Electrical andElectronics Engineers, Inc., pp. 1699–1710.

6. Armbruster D and Uzsoy R. Continuous dynamic models,

clearing functions, and discrete-event simulation in aggregate

production planning. In Tutorials in Operations Research,

INFORMS 2012.7. Rose O. Improved simple simulation models for semiconduc-

tor wafer factories. In Henderson SG, Biller B, Hsieh MH,

et al. (eds.) Proceedings of the 2007 Winter Simulation

Conference. Piscataway, NJ: Institute of Electrical and

Electronics Engineers, Inc., pp. 1709–1712.8. Hopp WJ and Spearman ML. Factory Physics. New York:

McGraw-Hill, 2001.9. Veeger C, Kerner Y, Etman P, et al. Conditional inter-

departure times from the M/G/s queue. Queueing Systems

2011; 68: 353–360.

Author biographies

Daniel Huber is a visiting scholar at the Arizona StateUniversity with a scholarship of the DeutscheForschungsgemeinschaft. He studied industrial engineer-ing at the Universitat Paderborn, Germany. In 2009 hewas awarded a doctorate in Business Computing at theHeinz Nixdorf Institut (HNI) at the Universitat Paderborn.Following this, he was working as a postdoc at the HNIand as a simulation specialist in the automotive industry. Hismain research interests are material flow simulation, model-ing methodology and automatic model simplification.

John Fowler is Chair and Professor of the WP CareySupply Chain Management Department at Arizona StateUniversity. His research interests include modeling, analy-sis, and control of manufacturing and service systems. Heis a Fellow of the Institute of Industrial Engineers and isthe SCS representative on the Board of Directors of theWinter Simulation Conference. He is an Area Editor ofTransactions of the Society for Computer Simulation

International, an Associate Editor of IEEE Transactions

on Semiconductor Manufacturing, and Editor of IIE

Transactions on Healthcare Systems Engineering.

Dieter Armbruster received a PhD in physics from theUniversitat Tubingen, Germany, in 1984. He was a post-doc at Cornell University and a part-time Professor forSystems Engineering at Eindhoven University ofTechnology until 2011. He is currently Professor in theSchool of Mathematical and Statistical Sciences, ArizonaState University, Tempe. His research interests are broadbased and focus on applied mathematics for real worldproblems. His projects range from dynamical systems the-ory and chaos to the dynamics of complex networks andproduction systems.

Huber et al. 9

at ARIZONA STATE UNIV on September 9, 2014sim.sagepub.comDownloaded from