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Simplification of DES models of M/M/1 tandem queues by approximating WIP-dependent inter-departure times Daniel Huber, John Fowler and Dieter Armbruster SIMULATION published online 28 August 2014 DOI: 10.1177/0037549714546665

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Simulation

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

Simulation: Transactions of the Society for Modeling and Simulation International 1–9 ! 2014 The Society for Modeling and Simulation International DOI: 10.1177/0037549714546665 sim.sagepub.com

Daniel Huber1, John Fowler1 and Dieter Armbruster2

Abstract This paper presents two algorithms to analytically approximate work in process (WIP)-dependent inter-departure times for tandem queues composed of a series of M/M/1 systems. The first algorithm is used for homogeneous tandem queues, the second for such with bottlenecks. Both algorithms are based on the possible combinations of distributing the WIP on the queues. For each combination the time to the next departure is estimated. A weighted average of all estimated times of each WIP level is calculated to get the expected mean inter-departure time. The generated interdeparture times are used in a simple model of the tandem queue. The inter-departure times, the average WIP and average 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 in many applications.

Keywords discrete event simulation, simplification, WIP-dependent inter-departure times

1. Introduction The use of work in process (WIP)-dependent interdeparture times for building aggregated or simple discrete event simulation models of complex production systems was introduced and refined by Veeger et al.1–2 In their approach Veeger et al. use real-world data of a production system to generate the inter-departure times. This is done by calculating the inter-departure time and actual WIP at each departure event. Since the WIP can only take integer numbers, the inter-departure times are averaged for every WIP value. The quality of the results of the simple model presented was very good, such that it is appealing to use this method not only for building simple models but also for the simplification of existing complex simulation models. When simplifying existing simulation models, running simulation experiments could generate data similar to data from the real-world production system. To have greater impact on the simplification process in terms of simulation runtime reduction, it would be much better if the inter-departure times could be calculated analytically, without simulating the original complex model. After all, not having to simulate the original complex model is the intention of model simplification.

One well-known and often used method for getting steady-state WIP-dependent output, or inter-departure times, 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 and reaches an upper limit defined by the modeled production system. When translating this clearing function definition from output to inter-departure times, it becomes a convex function with a lower limit for high WIP. Clearing functions are mainly used to improve production planning and scheduling methods, e.g. linear programming, by giving them non-linear WIP-dependent behavior.4–6 They can be derived analytically using steady-state queueing models. As using WIP-dependent behavior in model simplification is very promising in achieving simple models with 1

WPC Supply Chain Management, Arizona State University, Tempe, AZ, USA 2 School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ, USA Corresponding author: Daniel Huber, Fraunhofer Institute for Intelligent Analysis and Information Systems IAIS, Department ART, Schloss Birlinghoven, 53757 Sankt Augustin, Germany. Email: [email protected]


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Simulation: Transactions of the Society for Modeling and Simulation International

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.

little approximation error, it is not the solution in every application frame.7 In his approach Rose used a server with WIP-dependent processing times to improve a delay element in a simple model consisting of a bottleneck server taken from an original model in combination with three simple delay elements but could not achieve satisfactory 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 depicted in Figure 1. Jobs enter the system at queue Q1 according to a Markovian arrival process with mean inter-arrival time ta. Jobs are waiting in the first in first out (FIFO) queues Qi to be processed by server Si and leave the system after being processed at station k. The departure process has a WIP-dependent mean inter-departure time td(w). In Figure 2 a clearing function of a M/M/1 tandem queue of length 3 is presented and the inter-departure times of the same system calculated by using the method of Veeger et al.1 The clearing function was calculated by using 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. This clearly shows that the clearing function, defined as the steady-state average inter-departure time over the steadystate average WIP, is different from the WIP-dependent inter-departure time of Veeger et al., which is defined as the average td over the actual WIP. Thus, clearing functions do not give an adequate approximation for our purpose. Veeger et al.9 did work on getting analytical interdeparture times of queuing systems, but they were focusing on a single M/G/n queue in heavy traffic, i.e. high WIP, such that all n servers are busy all of the time. No other publications regarding the analytical approximation of WIP-dependent inter-departure times could be found.

2. Basics of the approximation of WIP-dependent inter-departure times In a M/M/1 tandem queue of length k there are k queues and servers. Each server i = 1.k has a mean service time


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

Q1

w

Q2

Q3

c

td (c)

¯td (C)

td (w)

cd (w)

6/3

2.000

0.818

1 1 1

1 0 0

0 1 0

0 0 1

3 2 1

2 2 2 2 2 2 .. .

2 1 0 1 0 0

0 1 2 0 1 0

0 0 0 1 1 2

3 2 2 1 1 1

Figure 3. Structure of the simple model.

10/6

1.667

0.897

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. The highest utilization in the tandem queue shall be used as the description of the utilization of the whole system. WIP-dependent inter-departure times are defined as follows. One job departs the system and leaves w jobs behind. The next job departs in td(w) time units. Here td(w) is the mean of a random distribution with a coefficient of variation of cd(w). The key idea of our method to calculate td(w) is to look at all possible combinations of the job allocation. One joballocation combination Pc is an array of the number of jobs ji per queue i, with ki= 1 ji = w. For each combination c 2 C(w) the jobs on the queue Qi=1.k with the highest index i are defining the inter-departure time. The interdeparture time is the sum of all service times of the servers Sj=i.k. To get the expected inter-departure time for w, the weighted average of the inter-departure times of all combinations c is used. In Table 1 the C, td and cd are shown for WIP-Levels w = 1,2 in a M/M/1 tandem queue of length k = 3, where all servers have the mean service time of te = 1. The data for td(w) and cd(w) was generated by analyzing a discrete event 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 process is not Markovian. When analyzing a bigger interval of WIP-levels (like in Figure 5), it shows that limw!N(td) = te and limw!N(cd) = 1. This behavior can be explained when looking at the allocation combinations. As can be seen in Table 1, there are 3 possible combinations for w = 1 and 6 for w = 2. Since all service processes are Markovian and thus memoryless, we do not have to be concerned about how long a server is already processing a job when looking at the current state of the system. As soon as there is one or

more jobs in a queue, the corresponding server is busy and can be handled as if it just has started processing. At w = 1 in a third of the combinations the last server is busy, at w = 2 in half the combinations the last server is busy. In these combinations the inter-departure time is the service time of the last server, i.e. exponential distributed time. In the other combinations it is a sum of service times, i.e. not exponentially distributed time. The ratio of combinations with the last server occupied is growing with w, resulting in the limits above. The possible allocation combinations at two consecutive departures are not independent of each other since some combinations become impossible following certain other combinations. In general, combinations with jobs close to the beginning of the tandem queue have a higher probability of being impossible, thus an increase in td(w) is expected when looking at specific departure events and assuming independence, i.e. always considering all combinations. But when taking the long time average WIP, cycle time and throughput as key performance indicators, we expect no significant influence of assuming independence. The inter-departure times can be used as input data in a simple discrete event model which then can be used as a simplification of the original model. In the next section we will present this simple model. Then, we will present two algorithms to approximate the inter-departure times analytically for homogenous tandem queues and such with bottlenecks. We will present results showing the quality of the td-approximation as well as the quality of the simple model when compared with the original one and conclude with some limitations of our approach and ideas for further research.

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


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This setup of call functions is necessary, because td(w = 0) is not defined the same way as for w . 0, since td would also be dependent on the arrival process. In the case of an arrival at w = 0,P the release gate is opened after the raw processing time i = 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 systems and at all times the first two moments are known, we use a gamma-distribution to sample random values.

4. Approximation of inter-departure times The approximation of td(w) in a tandem queue with bottlenecks, i.e. all servers can have different te, is much more complicated than in the homogenous case. This is because in the latter case all allocation combinations c 2 C at a WIP level have the same probability of occurrence. If there is a bottleneck, combinations with a high ratio of jobs in the queue of the bottleneck are much more likely than those with a high ratio of jobs in other queues. In the following two subsections the two algorithms are explained 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 combination c is occupied, the next departure will happen in average te time units. If the server Sk2‘ is occupied and all downstream servers Sk2j, 0 4 j 4 ‘, 0 4 ‘ 4 k21 are idle, the next departure will happen on average at time b(‘) = (‘ + 1) ! te :

ð2Þ

The number of combinations with ‘ or more consecutive idle servers, starting from Sk is m%Cl (‘) =

(k $ ‘ $ 1 + w)! : (k $ ‘ $ 1)! ! w!

ð3Þ

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

k $1 ! X

‘=0

" mCl (‘) b(‘) ! : mC

cd (w) =

" k $1 ! X StD(gamma(‘ + 1, te )) mCl (‘) ! : b(‘) mC ‘=0

ð4Þ

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

ð5Þ

4.2. Tandem queues with bottlenecks In tandem queues with bottlenecks, at least one server has a different service time than the other servers in the tandem queue. To get to an ‘‘allocation-probability’’ for each queue in the tandem queue, we first define a bottleneck indicator z = max(te ( ! )), (! ) " te (i) 2 G= ,i=1...k : z

ð6Þ ð7Þ

In the vector G = {g1.g k} the element of the bottleneck server g bn has a value of 1 and all other g i6¼bn are put into squared relation and are smaller than 1. For evaluating the probability of occurrence r(c) of each combination c 2 C, we first calculate an intermediate e(c):

ð1Þ

The number of allocation combinations for a tandem queue of length k and WIP level w is given by (standard stochastic problem to put w not distinguishable balls into k distinguishable urns) (k + w $ 1)! : mC = (k $ 1)! ! w!

departure is expected to occur in te, with cd = 1. In all other cases the next expected departure is expected to occur in (‘ + 1) te. The convolution of (‘ + 1) exponential distributions with equal mean te can be expressed by an Erlang or Gamma distribution with shape (‘ + 1) and scale te. By calculating the standard deviation (SD) of such a Gamma distribution, the coefficient of variation is

e(c) =

k X

i=1

(ci ! gi )

ð8Þ

eu!e(c) r(c) = P u!e(c) : Ce

ð9Þ

Experiments showed that the deviations between probabilities had to be increased to get a good approximation for r(c), thus in Equation (9) an exponential function is used, with u a constant. Using an exponential function showed to be a better approximation than other methods we tested, like exponentiation, to distinguish between likely 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) ,

ð10Þ

where for a given combination c 2 C(w) ‘max(c) is the index 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 by Equations (5) and (10) which in this case is not exact, since the te in the tandem queues are not the same. An exact approach would involve convolution of ‘ + 1


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Table 2. Test models. te Model I II III IV V VI VII VIII IX X

S1

S2

S3

1 1 1 0.8 0.8 0.5 0.5 0.975 0.95 0.9

1 1 1 1 1 1 1 0.975 0.95 0.9

1 1 1 0.8 0.7 0.6 0.5 1 1 1

S4

S5

S6 . . . S10

1 1

1 1

1

0.8

0.9

0.5 0.975 0.95 0.9

0.9 0.975 0.95 0.9

exponential distributions, each having a different mean. The calculations are quite involved and as we will show, the approximation is quite good making the exact approach unnecessary. In this algorithm each combination c must be handled separately, which becomes a computational problem when k increases and the utilization is high, since then the maximum WIP level for which td have to be calculated is higher. Here mC grows extremely fast with w and k as can be seen in Equation (2).

5. Results To test the algorithms, 10 different original models were used (see Table 2), each with a utilization u of 0.3, 0.5 and 0.8. When using simulation, all analyses were done with the average of four simulation replications (efficient parallelization on the quad-core test platform), each covering 106 time units. Models I–III are homogeneous queues, models IV and V and VI and VII have the same asymmetric bottleneck characteristic, but of different strength and models VIII–X have the same symmetric bottleneck characteristic of increasing strength. These models were chosen to test the algorithms in a wide range of queue configurations and to discover trends in the approximation quality. The original models and the simple model were implemented with MatLab 2012b’s SimEvents extension. All algorithms were implemented in MatLab 2012b. The test platform was a 2012 Mac mini with an i7 processor and 8 GB of RAM. In Figure 4 the td(w) obtained from simulation of the original model are shown for model I. With growing utilization, the maximum WIP level is growing. For each of the curves representing different utilizations, as WIP increases there is a region where the number of samples gets small, resulting in noisy data. In the low-noise area the covering of the data points is very good, meaning that

td is independent of the arrival process. It is easy to see that limw!N(td) = 1. In Figure 5 the td(w) and cd(w) for models I, II and III at u = 0.8 are shown (obtained from simulations of the original models). For low WIP-levels td grows with k. Independent of the tandem length, the limit for high WIP is 1 for all models. For model III there is some noise in td(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 good u for Equation (9) are shown. The relative errors of td(w) were calculated for the models VIII, IX and X, then the absolute values of these errors were averaged over all w and u. As can be seen for u = 0.6 there is a minimum in model X, the model with the strongest bottleneck. The differences in the other models are less significant, only in model VIII the error at u = 0.6 is not the lowest one. Not shown here are all of the other values 0.5 . u . 0.65 tested, but the errors were larger in all tested configurations. Here u = 0.6 was chosen to be a good choice for a wide range of model configurations and was used in all following experiments, but without further evidence for it being the best choice. Figure 7 shows how the errors in WIP develop when the strength of one single bottleneck in the tandem queue is increased. The errors are presented for utilizations 0.5 and 0.8. When the simple model is run with td(w) calculated for homogenous tandem queues (legend: h.Alg), the error increases quickly and surpasses 5% at utilization 0.8 even in the model of the least inhomogeneous tandem queue. This clearly shows the necessity of the algorithm for tandem queues with bottlenecks, with which the error is kept significantly smaller (legend: bn.Alg). In Figure 8 the relative error between the empirical td(w) and those calculated with the presented algorithms are shown for models II, V and VII (all tandem length 5, model VII should act as a worst case for maximum length and heterogeneity). For the homogenous model II the error is generally smaller than 61%. For the heterogeneous models V and VII the error is generally smaller than 63%. Comparing models V and VII, the strength of the bottlenecks does not seem to have much of an influence on the over all quality of the result. For the homogenous model the error fluctuates around zero, where as for models V and VII the errors seem to be on an arbitrary trajectory. Starting with 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 models and the simple model using the generated td(w) are shown. Negative values represent a lower average WIP in the simple models than in the original ones. The error in average cycle time was also calculated: values were very similar to these, as expected by Little’s law. Comparing


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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 for homogeneous models I, II and III.

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

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


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Figure 6. Error indicator for models VIII, IX and X to find good θ. Error indicator is averaged over all utilizations and WIP levels of the 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).

tendencies are visible. Looking at all models and utilizations, the errors are clearly smaller than 6 5%. In Table 4, the errors of the coefficient of variation of the average WIP are shown. Looking at each model separately, larger errors in WIP correlate to larger errors

in coefficient in variation. Also the variability of the errors is larger than in Table 3. All values are smaller than 6 8.5%, with models IV–VII performing significantly better. Reviewing the results obtained, it can be stated that homogenous tandem lines can be approximated very well,


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

Table 3. Average WIP analysis.

Table 4. Average coefficient of variation analysis.

Utilization

Utilization

Model I II III IV V VI VII

Error in average WIP 0.3

0.5

0.8

1.57% − 2.39% − 4.49% 0.44% − 0.63% − 0.97% − 0.68%

− 0.53% − 2.78% − 3.09% − 0.11% 0.15% − 2.34% 0.20%

− 1.63% − 1.37% − 0.82% 0.20% 3.22% − 4.96% − 1.86%

especially for high utilizations. Tandem queues with bottlenecks can be approximated with similar error levels. The algorithm created can be used for a variety of queue configurations. The small error levels in average WIP of the simple model should render it applicable in many realistic applications. The higher errors in the coefficient of variation should have smaller impact on the applicability, since when using a simple model, the average of the key values is generally more important than their second moments.

6. Conclusion and future research We have developed methods to simplify models of M/M/1 tandem queues. Results show small relative errors in a wide range of tandem queue parameterization.

Error in cv of average WIP 0.3

Model I II III IV V VI VII

− 0.08% − 3.42% − 8.46% 0.33% − 0.46% − 1.83% − 1.42%

0.5

0.8

− 2.42% − 4.39% − 6.32% 0.0% − 0.03% − 1.95% − 0.58%

− 0.71% − 1.53% − 1.39% 0.74% 0.41% 0.27% − 1.65%

In addition to the M/M/1 tandem queues, we tried to extend the approach to M/M/s tandem queues. They are much harder to approximate, since jobs can overtake each other and the most progressed job in the tandem queue is not necessarily the next job to exit. Furthermore, even in a tandem queue without bottlenecks, every combination has its own probability of occurrence, strongly avoiding unbalanced combinations, also due to the overtaking process. This behavior increases with the number of parallel servers s. When looking at single combinations it was not possible to develop an algorithm to universally approximate td. Since jobs can overtake each other, the td for low WIP levels are also dependent on the arrival process. We were able to get decent results for homogenous tandem queues with s = 2 and 3 using an adapted bottleneck algorithm, but for higher values of s the error was unacceptable.


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We also looked at M/M/1/g tandem queues, where the queues have a limiting capacity of g. In these systems the inter-departure times are also harder to approximate, because even 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. Even with perfect inter-departure times (for testing, td(w) was obtained from simulation data) the simple model of such a system has a high error. Let every queue in the original model has a capacity of g, such that the whole model has a maximum WIP of wmax = gk + k. If wmax is set as the capacity of the queue in the simple model (Figure 3), which intuitively needs to be done, it will block incoming jobs when its capacity restriction is reached. Thus, the simple model blocks significantly less often than the original model, since the original model blocks at wmax and when the first queue has an allocation of g. The average WIP in the simple model is significantly higher than in the original model. The number of combinations that need to be evaluated for tandem queues with bottlenecks increase so rapidly that with the test system used and applying MatLab’s parallelization tools, systems with k' 5 could not be approximated in a practical runtime. Because of the parallelization on a single 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 a user-defined maximum, but using interpolation between calculated values, etc. However, it seems that a significant reduction in runtime can only be achieved by not calculating all possible combinations for a WIP level.

5. Kacar NB and Uzsoy R. Estimating clearing functions from simulation data. In Johansson B, Jain S, Montoya-Torres J et al. (eds.), Proceedings of the 2010 Winter Simulation Conference. Piscataway, NJ: Institute of Electrical and Electronics 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 semiconductor 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 interdeparture times from the M/G/s queue. Queueing Systems 2011; 68: 353–360.

Funding

John Fowler is Chair and Professor of the WP Carey Supply Chain Management Department at Arizona State University. His research interests include modeling, analysis, and control of manufacturing and service systems. He is a Fellow of the Institute of Industrial Engineers and is the SCS representative on the Board of Directors of the Winter Simulation Conference. He is an Area Editor of Transactions 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.

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

References 1. Veeger C, Etman P, Rooda J, et al. Cycle time distributions of semiconductor workstations using aggregate modeling. In Rossetti MD, Hill RR, Johansson B, et al (eds.), Proceedings of the 2009 Winter Simulation Conference. Piscataway, NJ: Institute of Electrical and Electronics Engineers, Inc., pp. 1610–1621. 2. Etman P, Veeger C, Lefeber E, et al. Aggregate modeling of semiconductor equipment using effective process times. In Jain S, Creasey RR, Himmelspach J, et al. (eds.), Proceedings of the 2011 Winter Simulation Conference. Piscataway, NJ: Institute of Electrical and Electronics Engineers, Inc., pp. 1795–1807. 3. Karmarkar US. Capacity loading and release planning with work-in-progress (WIP) and lead-times. J Manufact Operat Management 1989; 2: 105–123. 4. Missbauer H and Uzsoy R. Optimization models for production planning. In Kempf KG, Keskinocak P, Uszoy R (eds.), Planning Production and Inventories in the Extended Enterprise. New York: Springer, 2011, pp. 437–507.

Author biographies Daniel Huber is a visiting scholar at the Arizona State University with a scholarship of the Deutsche Forschungsgemeinschaft. He studied industrial engineering at the Universita¨t Paderborn, Germany. In 2009 he was awarded a doctorate in Business Computing at the Heinz Nixdorf Institut (HNI) at the Universita¨t Paderborn. Following this, he was working as a postdoc at the HNI and as a simulation specialist in the automotive industry. His main research interests are material flow simulation, modeling methodology and automatic model simplification.

Dieter Armbruster received a PhD in physics from the Universita¨t Tu¨bingen, Germany, in 1984. He was a postdoc at Cornell University and a part-time Professor for Systems Engineering at Eindhoven University of Technology until 2011. He is currently Professor in the School of Mathematical and Statistical Sciences, Arizona State University, Tempe. His research interests are broad based and focus on applied mathematics for real world problems. His projects range from dynamical systems theory and chaos to the dynamics of complex networks and production systems.
