Queueing models for a single machine subject to multiple types of interruptions

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1 IIE Transactions (011 43, Copyright C IIE ISSN: X print / online DOI: / X Queueing models for a single machine subject to multiple types of interruptions KAN WU 1,, LEON MCGINNIS and BERT ZWART 1 School of Mechanical and Aerospace Engineering, Nanyang Technological University, Singapore kan66@gmail.com School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA 3033, USA Received June 009 and accepted October 010 Queueing models are commonly applied to quantify the performance of production systems. Prior research has usually focused on deriving queueing models for a specific type of interruptions. However, machines generally suffer multiple types of interruptions in practical manufacturing systems. To satisfy this need, an integrated model is proposed, in which multiple types of interruptions commonly seen on the shop floor are considered. Keywords: Queuing theory, manufacturing, interruptions 1. Introduction Consider a single machine, where the problem is to estimate the cycle time. Observing this machine reveals: (i variation in the inter-arrival times of jobs; (ii variations in the processing times of jobs; and (iii a variety of interruptions, all of which have the effect of preventing the machine from fulfilling the function of processing jobs. In the absence of the interruptions, it would be straightforward to estimate the cycle time using standard queuing models from the literature, because service time is well-defined it is simply the time during which a job claims the machine for processing. The interruptions, however, create problems. In the literature on this problem a common strategy is to incorporate the interruption time with the actual processing time to create a modified processing time. Using some variant of modified processing time, either an exact M/G/1 model or an approximate G/G/1 model is then formulated. No single model in the literature addresses all possible types of interruptions, so there is no unified comprehensive approach to integrating interruptions in queuing models. In this article, we provide a unified treatment of the problem of estimating cycle times in the presence of interruptions. We first carefully define a comprehensive interruption classification scheme. Based on this scheme, we provide a unifying definition of a generalized service time that incorporates interruptions from our classification scheme in a unified way. We then derive a unified M/G/1 model with interruptions. The extension to the corresponding G/G/1 Corresponding author model is then addressed. The article concludes with observations about the problem and suggestions for future research.. Classifying interruption events Classifying phenomena requires a point of view. For example, inspired by Buzacott and Hanifin (1978, Wu and Hui (008 classified interruptions as either run-based or time-based events. Run-based interruptions are associated with job processing, whose time-based interruptions are associated with time, whether or not a job is present at the station. For example, interruptions caused by power outages, preventive maintenance (PM, or process experiments are time dependent and should be classified as time-based events, while interruptions due to differences in recipes (setups or out-of-specification inputs constitute run-based events. We adopt the basic run-based versus time-based distinction and elaborate upon it. Both run-based and time-based events can be further classified as preemptive or non-preemptive. A preemptive event can occur anytime during processing, but a nonpreemptive event can only occur before or after processing. Therefore, a run-based non-preemptive event can only occur before job processing starts or after job processing ends, since it cannot preempt processing yet is associated with the processing of jobs. Setups are typical examples of this type of interruption. Out-of-spec inputs are runbased preemptive events, power outages are time-based preemptive events, and PM is a time-based non-preemptive event X C 011 IIE

2 754 Wu et al. 1. Natural variability (S 1 ; λ 1 Arrival rates: Arrival interval fluctuation Service rates: Natural fluctuation, product mix. Interruptions -1. Run-based events -. Time-based events Preemptive events (D r ; η r : Out-of-spec input a. State-induced events (T -1-. Non-preemptive events: s Setup b. Product-induced events (T p ; N p --1. Preemptive events (D t ; η t : Power outage --. Non-preemptive events (S ; λ : PM Fig. 1. Classification of factors that affect queue times. Run-based non-preemptive events can be further classified as state-induced or product-induced events. Stateinduced events correspond to a machine changing state either from busy to idle or idle to busy. As an example, a machine that goes into a sleep mode when it is idle, and requires a warm-up time when it returns to production mode, experiences a run-based, non-preemptive, stateinduced interruption. Product-induced events correspond to changing machine settings for different products; i.e., product-based setup. There is a fundamental difference between these two types of events: state-induced setups do not occur when a machine is fully utilized, since no state change occurs if a machine is always busy. However, product-induced setups are determined by external customer demands. Thus, they cannot be completely avoided. We may alter the frequencies of product-induced setups by changing scheduling rules, but we simply cannot run one product all the time if customers demand more than one product. The complete classificationschemeissummarizedinfig.1. Since capacity for a single machine system is the maximum throughput rate of the machine, the previously described difference between product-induced and stateinduced events implies different impacts on capacity: product-induced events (-1-b have impacts on both cycle time and capacity; state-induced events (-1-a have impacts on cycle time but not capacity. In the treatment of non-preemptive time-based events (- - such as PM, there are alternatives in practice. At one extreme, the PM has a high priority and must be performed as soon as the machine becomes available (current job is completed. At the other extreme, the PM has a low priority and is performed at the next machine idle time (no job waiting for processing. For an intermediate priority, the PM might wait for some number of jobs to complete and then claim the machine, even though it is not idle. Thus, one approach to dealing with non-preemptive time-based events in queuing models is to treat them as a different class of jobs, with a priority reflecting how quickly they must be handled..1. Defining service time To derive the integrated model, we have to first define a Generalized Service Time (GST in accordance with the previous classification. The GST is defined as the difference between the job departure time and the time epoch at which the job first claims the capacity of the machine,wherethejob departure time is the time that a job releases the machine capacity. A job claims capacity of a machine if: (i the job is present at the machine; (ii the preceding job has released the machine capacity; and (iii the machine is ready to process this job, or is ready to be engaged in a productinduced event. Therefore, GST is the summation of the product-induced setup time (if any, the service time, and the downtimes of all preemptive events occurring during that service time: G = S + N(S i=1 D i + T p, (1 where S is the service time, N(S is the number of preemptive events (e.g., breakdowns during S, D i is the ith downtime, and T p is the duration of a run-based non-preemptive product-induced event experienced by a job. Here we assume there is no preemptive interruption during setups or, if there is, those interruptions can be modeled by T p... M/G/1 models of each interruption category For each interruption category, Wu (009 has introduced its corresponding M/G/1 model. Table 1 gives a nonexhaustive set of examples, also connecting with various particular models in the literature. The form of the Pollaczek Khintchine (P K formula in (.1.1 and (.1.b reveals the possibility to extend M/G/1 models to M/G/m approximations based on the approach by Sakasegawa (1977. It also suggests a way to extend M/G/1 models to Kingman s G/G/1 approximations (Kingman, 1965; Hegman, 1975 by considering the variance of interarrival times. This observation plays an

3 Queueing models for a single machine 755 Table 1. M/G/1 model for each interruption category Category E(QT Source ρg E(RG (1 ρ G = ( 1+c G ( ρg 1 ρ G E(G Gaver (196 and Hopp and Spearman ( E(QT =.1.a E(QT = ρ E(RS + 1/λ (1 ρ 1/λ+E(T E(T s s + E(Ts 1/λ+E(T E(R s T Adan and Resing (001 ( ρg E(RG 1+c.1.b E(QT = (1 ρ G = G ρg E(RG..1 E(QT = (1 ρ G + E(Dη 1+E(Dη E(R D Gaver (196.. E(QT 1 = (ρ 1 E(R 1 + ρ E(R / (1 ρ 1 Adan and Resing (001 E(QT = E(QT = ρ1 E(R1+ρ E(R (1 ρ 1 ρ (1 ρ 1 λ1 λ 1+λ E(QT 1 + λ λ 1+λ E(QT ( ρg 1 ρ G E(G Hopp and Spearman (1996 and Wu (009 important role in Section 3.1 where we extend the integrated M/G/1 model to its G/G/1 counterpart. By considering the variance of inter-arrival times, we could also apply the model to queueing networks under the assumption of stochastic independence. Each of the queueing models presented in Table 1 deals with a single interruption type, which can be used to predict system performance under the influence of a specific interruption type. The definition and derivation of each model can be found in the referred sources. However, multiple interruptions may occur at the same workstation in practical systems. An integrated queueing model considering multiple types of interruptions is necessary and will be derived in the next section. 3. Integrated models of service interruptions We surveyed several types of interruptions and models in the literature that deal with some of those interruptions. The main technical contribution of this article is to propose integrated models where all such interruptions are taken into account simultaneously. Although the resulting formulae look laborious, they are explicit, and they can be specialized to specific practical situations where some types of interruptions may not arise. Thus, in this section, we first develop the integrated models and then provide a table with several specific examples, connecting with the existing literature. The notation is summarized in Table. We will conduct mean value analysis for the queue time of the integrated model. Before proceeding to the detailed mean value analysis, we have some observations on the concept of the GST. Based on Equation (1, the productinduced setup time and all of the preemptive interruptions that occur during processing have been considered in GST. The preemptive interruptions during processing can be either run based or time based. While the run-based interruptions are fully covered by GST, GST only covers part of the time-based interruptions, since some of the timebased interruptions may occur during non-processing time. Therefore, the GST contains the original service time, runbased preemptive events (-1-1, run-based non-preemptive product-induced events (-1-b, and some of the timebased preemptive events (--1; that is, G 1 = S 1 + N r (S 1 i=1 D r i + N t (S 1 i=1 D t i + T p, ( where G 1 stands for the GST, S 1 stands for the service time, N r (S 1 is the number of run-based preemptive events during S 1, D r is the length of the run-based preemptive Table. Summary of notation S 1 : Service time of a job λ 1 : Arrival rate of a job G 1 : Generalized service time of a job ρ 1 = λ 1 E(G 1 QT 1 : Queue time of a job CT 1 : Cycle time of a job S : Duration of a time-based non-preemptive event λ : Arrival rate of a time-based non-preemptive event G : Generalized recover time of time-based non-preemptive events ρ = λ E(G D r : Duration of a run-based preemptive event η r : Arrival rate of a run-based preemptive event D t : Duration of a time-based preemptive event η t : Arrival rate of a time-based preemptive event T s : Duration of a run-based non-preemptive state-induced event T p : Duration of a run-based non-preemptive product-induced event experienced by a job P: Duration of a run-based non-preemptive product-induced event N p : Average number of jobs processed by a machine between two consecutive setups

4 756 Wu et al. events, N t (S 1 is the number of time-based preemptive events during S 1, D t is the length of the time-based preemptive events, and T p stands for the duration of a run-based, non-preemptive, product-induced event experienced by a job. The first and second moments of generalized service time in the integrated models are given as follows. E(G 1 = E(S 1 [1 + η r E(D r + η t E(D t ] + E(T p, (3 E(G 1 = E(S 1 ([1 + η re(d r ] + [1 + η t E(D t ] [ + E(T p 1 + E(S 1 η r E(D r + E(T p E(D r ] [ + E(S 1 η t E(D t + E(T p E(D t ] + E(Tp. (4 We treat time-based non-preemptive interruptions as a second class of jobs, with its own arrival rate and service time and with a priority to determine whether the interruption claims the machine at the next job completion or waits until there are no jobs to process. Therefore, there are two possible cases for the time-based non-preemptive events (-- the interruption has a low priority (relative to the waiting jobs in Case 1 and high priority in Case. Analysis for Case 1 In Case 1, we will postpone a PM to the completion of all jobs in queue. The time-based non-preemptive event has low priority, while the job has high priority. By PASTA (Wolff, 198, an arriving job finds on average E(L q 1 jobs in the queue and sees the machine occupied by a job (with probability ρ 1, occupied by a time-based nonpreemptive event (with probability ρ, or none of the above (with probability (1 ρ 1 ρ. If an arriving job finds the machine is not occupied by a job or a time-based nonpreemptive event, it must find the machine in one of the following three states: up, during a state-induced setup, or during a time-based preemptive event. If the machine is up, the arriving job will induce a stateinduced setup before processing starts. If the job arrives when the machine is in a state-induced setup, it must wait until the remaining part of the setup process is complete. If the job arrives when the machine is in a time-based preemptive event, it must wait until the machine is recovered plus a state-induced setup time. Therefore, E(QT 1 = E(L q 1 E(G 1 + ρ 1 E(R G1 + ρ E(R G [ ] (1 ρ E(Ts /(λ 1 + η t + E(T s E(R Ts +, +E(D t η t (E(R Dt + E(T s /(λ 1 + η t (5 Combining Little s law (Little, 1961 and Equation (5 we get: E(QT 1 = ρ 1 1 ρ 1 E(R G1 + ρ (1 ρ (1 ρ 1 E(R G + 1 ρ 1 E(T s /(λ 1 + η t + E(T s E(R Ts + E(D t η t (E(R Dt + E(T s /(λ 1 + η t, (6 and E(CT 1 = E(QT 1 + E(G 1, where = 1/(λ 1 + η t + E(D t η t /(λ 1 + η t + E(T s,ρ i = λ i E(G i fori = 1,, and ρ = ρ 1 + ρ ; E(R Gi = E(Gi / E(G i, for i = 1, ; E(G = E(S, E(G = E(S, and E(G 1 is given by Equation (3, E(G 1 isgivenbyequation (4; E(T p = E(P/N p, E(R Ts = E(Ts /E(T s; and E(R Dt = E(Dt /E(D t, and S is the downtime of timebased non-preemptive events as described in Fig. 1. Analysis for Case In Case, we will postpone a PM to the completion of the current job. The time-based non-preemptive event has a high priority, while the job has low priority. Since the jobs have a low priority, in order to get the average queue time of a job, we have to calculate the queue time of a PM first. Using similar arguments as before (PASTA, Little, 1961 we obtain: E(QT = ρ 1 E(R G1 + ρ (1 ρ E(R G + 1 ρ 1 ρ (1 ρ [ E(Ts E(R Ts + E(D ] tη t /(λ 1 + η t E(R Dt. (7 The queue time of a job can be derived accordingly. By PASTA, an arriving low-priority job finds on average E(L q 1 jobs and E(L q time-based non-preemptive events in the queues and sees the machine occupied by a job (with probability ρ 1, occupied by a time-based non-preemptive event Table 3. Parameters (unit: minutes in the first experiment with Poisson arrivals Run-based Time-based Non-preemptive Preemptive Non-preemptive Service time (1/λ 1, S 1 Preemptive (l/η r, D r state-induced (, T s (l/η t, D t (1/ λ, S MTTF or IAT SCV Mean or MTTR SCV

5 Queueing models for a single machine 757 (with probability ρ ornoneoftheabove(withprobability (1 ρ 1 ρ. If an arriving job finds the machine not occupied by a job or a time-based non-preemptive event, it must find the machine in one of the following three states: up, during a state-induced setup, or during a time-based preemptive event. Therefore, E(QT 1 = E(L q 1 E(G 1 + E(L q E(G + ρ 1 E(R G1 + ρ E(R G [ ] (1 ρ E(Ts /(λ 1 + η t + E(T s E(R Ts +. +E(D t η t (E(R Dt + E(T s /(λ 1 + η t (8 Applying Little s (1961 law to Equation (8 we obtain: E(QT 1 = ρ 1 E(R G1 + ρ [E(QT + E(R G ] 1 ρ 1 1 ρ 1 E(T s /(λ 1 + η t + E(T s E(R Ts (1 ρ + +E(D (1 ρ 1 t η t (E(R Dt + E(T s /, (λ 1 + η t E(CT 1 = E(QT 1 + E(G 1, (9 where = 1/(λ 1 + η t + E(D t η t /(λ 1 + η t + E(T s,ρ i = λ i E(G i fori = 1,, and ρ = ρ 1 + ρ ; E(R Gi = E(Gi / E(G i for i = 1,, E(G = E(S, E(G = E(S, and E(G 1 is given by Equation (3, E(G 1 isgivenbyequation (4; E(T p = E(P/N p ; E(R Ts = E(Ts /E(T s, and E(R Dt = E(Dt /E(D t. As with the conclusions given in the analysis of the timebased non-preemptive events (--, depending on the PM scheduling policy, the queue time of a job in practice can be the values of Equation (6 or (9 or any value in between. Equation (6 gives a lower bound, and Equation (9 gives an upper bound of the true queue time. In the above two models, we assume that a preemptive event only preempts the processing of a job instead of another interruption. For example, a power outage (--1 will not occur during an out-of-spec input (-1-1, setup (-1-, PM (--, or another time-based preemptive event (-- 1. We need to note that Equation (9 would become an approximation if the preemptive events could preempt interruptions. After all, in reality we cannot prevent a power outage from occurring during a setup or PM Extension to non-poisson arrivals Equations (6 and (9 give a lower bound and upper bound of the true queue time. However, they are limited to Poisson arrivals. The P K formula in the (.1.1 M/G/1 models in Table 1 suggests that Equations (6 and (9 can be extended to general arrivals as follows: E(QT 1 = a1 + cg1 + ρ1 (1 ρ (1 ρ 1 ρ E(G 1 + a + cg E(G 1 ρ 1 1 ρ 1 E(T s /(λ 1 + η t + E(T s E(R (( Ts c adt + c D + E(D t η t t /,(10 E(D t + E(T s (λ 1 + η t E(QT 1 = a1 + cg1 ( ρ1 c E(G 1 + a + c G ρ E(G 1 ρ 1 1 ρ 1 + ρ E(QT 1 ρ 1 E(T s /(λ 1 + η t + E(T s E(R (( Ts (1 ρ c adt + c D + + E(D t η t t (1 ρ 1, (11 E(D t + E(T s /(λ 1 + η t E(QT = a1 + cg1 + ρ1 (1 ρ (1 ρ ρ E(G 1 + a + cg E(G 1 ρ 1 ρ adt + c D + E(T s E(R TS + t E(D t η t,(1 (λ 1 + η t Table 4. Queue time comparison for the first experiment Utilization (% SQT AQT Percentage difference SST GST ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±

6 758 Wu et al. Table 5. Parameters (unit: minutes in the second experiment with non-poisson arrivals Run-based Time-based Non-preemptive Preemptive Non-preemptive U Service time (1/λ 1, S 1 Preemptive (l/η r, D r state-induced (, T s (l/η t, D t (1/λ, S MTTF or IAT SCV Mean or MTTR SCV where c a1 is the coefficient of variation (CV of job arrival intervals, c a is the CV of the interarrival times of timebased non-preemptive events, c adt is the CV of the meantime-to-failure of time-based preemptive events, c G1 is the CV of G 1, c G is the CV of G,andc Dt is the CV of D t. When arrival processes are generally and independently distributed, Equations (10, (11, and (1 can be used to approximate the lower bound and upper bound queue times respectively. From the above equations, we can also see that the interarrival time squared coefficient of variation of run-based interruptions has no impact on the mean queue time, but the inter-arrival time squared coefficient of variation of time-based interruptions does impact the mean queue time. 4. Simulation validation To validate the models presented in Section 3, we design two simulation experiments that are inspired by the situations in practical production lines. In the first experiment, Poisson arrivals (for both jobs and interruptions are assumed. In the second experiment, arrival processes can be generally and independently distributed. In the first experiment, we assume that the service time has a small natural variability due to the motion of a human (and/or robot. Due to occasional out-of-spec input, the machine suffers run-based preemptive interruptions. Whenever the machine is idle, a state-induced setup is needed before the machine can process jobs again. Timebased preemptive interruptions, such as facility-related events (e.g., power outages and natural disasters (e.g., earthquakes, occur a few times a year. Time-based nonpreemptive events, such as experiments or PM, occur several times a month but only when the machine is idle and no production job is in a queue (i.e., the interruption has a lower priority. The parameters for these scenarios are summarized in Table 3, where MTTF is mean-time-to-failure, IAT is the mean inter-arrival time (for time-based nonpreemptive events, MTTR is mean-time-to-repair, and SCV is the squared coefficient of variation. The service time and all downtimes follow gamma distributions. All arrivals (except for the state-induced events follow Poisson distribution with ten different mean arrival rates resulting in utilizations from 10 to 95%. For each input rate, the mean and variance are computed based on 100 replications. For utilizations less than 80%, each replication is composed of jobs with the first jobs being discarded. For utilizations higher than 90%, each replication is composed of jobs with the first half being discarded as the warm-up. The queue times from simulations and Equation (6 are given in Table 4. In Table 4, SQT is the mean simulated queue time, AQT is the mean approximated queue time, and SST is the mean simulated service time. The half-width of the 90% confidence intervals for SQT and SST are given immediately after its mean. Equation (6 gives reliable prediction especially in heavy traffic. Because we assume no interruption Table 6. Queue time comparison for the second experiment Utilization(% SQT AQT Percentage difference (% SST GST ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±

7 Queueing models for a single machine 759 could occur during another type of interruption, AQT from Equation (6 tends to slightly underestimate the true queue time. In the second experiment, all of the conditions are identical to the first one, except that the arrival processes follow gamma distributions. The parameters are listed in Table 5. The queue times from simulations and Equation (10 are given in Table 6. The simulation results show that Equation (10 gives reliable prediction for the considered scenario. The largest error is 6.4%, which occurs at around a 40% utilization. 5. Conclusions In this article, a comprehensive classification of interruptions in practical manufacturing systems has been proposed. Using the classification, we can model the impact of different types of interruptions on manufacturing process performance with queueing theory. One key to this classification and the associated queuing models is the clear definition of capacity and its inverse, service time. In Section 3, we used mean value analysis to derive integrated modelsforthecycle time of an M/G/1 system under the impacts of multiple types of events. In Section 4, the integrated model is validated with simulation experiments and gives reliable predictions to system performance. Although most shop floor activities fit into the new classification we propose, there is still a possibility that some events have not been considered. For example, model C, breakdowns occur homogeneous in time, repair process initiated if customers are present at station, introduced by Avi-Itzhak and Naor (1963, can be viewed as a variation of the time-based preemptive downtime events. However, the chance of its occurrence in a water fabrication plant will be small, since machines are highly automated and the majority of the time-based breakdowns can be recognized (by alarms, etc. at the beginning of the failures. The classifications presented in this article and the corresponding integrated models are the most complete today. No doubt, future research will enhance both the classification scheme and the integrated model. References Adan, I. and Resing J. (001 Lecture notes on queueing theory, available at iadan/, accessed, December 008. Avi-Itzhak, B. and Naor. P. (1963 Some queueing problems with the service station subject to breakdown. Operations Research, 11, Buzacott, J.A. and Hanifin, L.E. (1978 Models of automatic transfer lines with inventory banks a review and comparison. AIIE Transaction, 10(, Gaver, D.P. (196 A waiting line with interrupted service, including priorities. Journal of the Royal Statistical Society, 4, Gross, D. and Harris, C.M. (1998 Queueing Theory, Wiley, New York. Heyman, D.P. (1975 A diffusion model approximation for the GI/G/1 queue in heavy traffic. Bell System Technical Journal, 54(9, Hopp, W.J. and Spearman, M.L. (1996 Factory Physics, Irwin, Chicago, IL. Kingman, J.F.C. (1965 The heavy traffic approximation in the theory of queues, in Proceedings of the Symposium on Congestion Theory, pp Little, J.D.C. (1961 A proof of the queueing formula: L = λw. Operations Research, 9, Sakasegawa, H. (1977 An approximation formula L q = αρ β /(1 ρ. Annual of the Institute for Statistical Mathematics, 9, Wolff, R.W. (198 Poisson arrivals see time averages. Operations Research, 30, Wu, K. (009 New results in factory physics, Ph.D. dissertation, Georgia Institute of Technology, Atlanta, GA. Wu, K. and Hui, K. (008 The determination and indetermination of service times in manufacturing systems. IEEE Transactions on Semiconductor Manufacturing, 1(1, 7 8. Biographies Kan Wu is an Assistant Professor at Nanyang Technological University. He received an M.S. degree in Industrial Engineering and Operations Research, an M.E. degree in Nuclear Engineering from the University of California at Berkeley, and a Ph.D. degree in Industrial and Systems Engineering from Georgia Institute of Technology. He has been a senior engineer with Tefen, Ltd., and Taiwan Semiconductor Manufacturing Company, an IE manager at Inotera Memories Inc., and the CTO and founding team member of Sensor Analytics, Inc. His Ph.D. dissertation was awarded the third place in the IIE Pritsker Doctoral Dissertation Award in 010. His research interests focus on quantifying and optimizing the performance of manufacturing systems. Leon McGinnis is the Gwaltney Professor of Manufacturing Systems at Georgia Tech. He is internationally known for his leadership in the material handling research community and his research in the area of discrete event logistics systems. He has received several awards for his innovative research, including the David F. Baker Award from the IIE, the Reed- Apple Award from the Material Handling Education Foundation, and the Material Handling Innovation Pioneer award from Material Handling Management Magazine. He is author or editor of seven books and more than 110 technical publications. At Georgia Tech, he has held leadership positions in a number of industry-focused centers and programs, including the Material Handling Research Center, the Computer Integrated Manufacturing Systems Program, the Manufacturing Research Center, and the newly formed Product/Systems Lifecycle Management Center. His current research explores the application of PLM technologies to the design and management of highly capitalized factories. Bert Zwart is senior researcher at the Center for Mathematics and Computer Science in Amsterdam and is a Professor at VU University Amsterdam, while on leave from his tenure at Georgia Tech. He holds an M.A. in Econometrics from VU University, Amsterdam, and a Ph.D. in Applied Mathematics from Eindhoven University of Technology. He serves on several conference program committees in applied probability and performance analysis and serves on the editorial boards of six international journals. His research is concerned with modeling, analysis, and simulation of stochastic systems arising in actuarial and financial mathematics, computer and communication systems, manufacturing systems, and customer contact centers. His research is partly supported by NSF grants and , an IBM faculty award, and a VIDI grant from NWO.