NEW DEVELOPMENT OF OPTIMAL COMPUTING BUDGET ALLOCATION FOR DISCRETE EVENT SIMULATION

NEW DEVELOPMENT OF OPTIMAL COMPUTING BUDGET ALLOCATION FOR DISCRETE EVENT SIMULATION Hsiao-Chang Chen Dept. of Systems Engineering University of Pennsyvania Phiadephia, PA 904-635, U.S.A. Chun-Hung Chen Dept. of Systems Engineering University of Pennsyvania Phiadephia, PA 904-635, U.S.A. Liyi Dai Enver Y ucesan Dept. of Systems Science and Mathematics Washington University St. Louis, MO 6330, U.S.A. ABSTRACT This paper deas with ranking and seection probem via simuation. We present an optima computing budget aocation technique which can seect the best of k simuated designs. This approach can inteigenty determine the best simuation engths for a simuation experiments and significanty reduce the tota computation cost to obtain the same confidence eve. Numerica testing resuts are incuded. Aso we provide the resuts of anaysis for some parameters which affect the performance of our approach. Besides, we compare our method with traditiona two-stage procedures. Numerica resuts show that our approach is much faster than the traditiona two-stage procedures. INTRODUCTION In order to design and efficienty manage arge man-made systems such as communication networks, traffic systems, and automated manufacturing faciities, it is often necessary to appy extensive simuation to study their performance since no cosed-form anaytica soutions exist for such probems. Unfortunatey, simuation can be both expensive and time consuming. Suppose we want to compare k different system designs. We conduct N simuation repications for each of the k designs. Therefore, we need kn simuation repications. The simuation resuts become more accurate as ncreases. If the accuracy requirement is high (s not sma), and if the tota number of designs in a decision probem is not sma (k is arge), then kn can be very arge, which may easiy make tota simuation cost extremey high and precude the feasibiity of simuation approach. The effective reduction of computation costs whie obtaining a good decision is crucia in simuation. INSEAD Technoogy Management Area Fontainebeau, FRANCE Dudewicz and Daa (975) deveop a two-stage procedure for seecting the best design or a design which is very cose to the best design. In the first stage, a designs are simuated with n 0 repications. Based on the resuts obtained from the first stage, the number of additiona simuation repications is determined for each design in the second stage to achieve the desired confidence eve. Rinott (978) presents aternative way to estimate the number of required simuation repications in the second stage. Many researchers have extended this idea to more genera ranking and seection probem in conjunction with new deveopments (Charnes, 99, Matejcik and Neson, 993, Bechhofer, Santner, and Godsman, 995, and Hsu, 996). To further reduce the overa computation cost, Chen (995) formuates the procedures of seecting the best design as another optimization probem. The ideas are as foows. Intuitivey, some inferior designs can be discarded at an eary stage of simuation. As the simuation proceeds, some designs can be further ignored when higher simuation accuracy for the remaining designs is obtained. This procedure is repeated unti a desired confidence eve is achieved. Proceeding in this fashion, itte effort is wasted on simuating inferior designs; the overa simuation time is hence reduced. Then the question is how to systematicay do this? When? And which designs? Ideay, we want to optimay choose the number of simuation repications for a designs to minimize the tota simuation cost, whie obtaining the desired confidence eve. In fact, this question is equivaent to optimay decide which designs wi receive additiona computing budget for continuing simuation or to find an optima way to reach an optima design. Chen et a. (996) provide an approach to sove such an optimization probem. They use Chernoff bounds to estimate the gradient information and then appy steepest

Optima Computing Budget Aocation for Discrete Event Simuation 335 descent method to sove this optimization probem. Whie this approach can effectivey sove the optima computing budget aocation probem, the compexity of In this paper, we present another simpe way of estimating gradient information. From numerica resuts we observe that the performance of this new approach is even better than that in Chen et a. (996). Additionay, we compare our approach with the traditiona two-stage procedures by conducting a numerica experiment. Numerica resuts show that our approach is more than ten times faster than the two-stage procedures. Since our approach is based on the Bayesian mode presented in Chen (995), we wi give an overview of that mode for seecting the best design in the next Section. In Section 3, we define the optima computing budget aocation probem and propose a sequentia approach. We aso demonstrate the numerica testing in a simpe exampe. Section 4 discusses two crucia factors, initia simuation ength and one-time incrementa computing budget. Section 5 gives a brief review of traditiona two-stage procedure and compares our method with the two-stage procedure. Section 6 concudes this paper. PROBLEM FORMULATION AND CON- FIDENCE LEVEL QUANTIFICATION Suppose that our goa is to seect a design associated with the smaest mean performance measure among k designs with unknown variances that are not necessariy equa. Further assume that the computing budget is imited and the number of designs is arge. Denote k : the tota number of designs, X ij : the j th i.i.d. sampe of the performance measure from design i, : the number of simuation repications for design i, µ i : the sampe mean for design i, µ i = X ij, N j= i µ i : the mean performance measure; µ i = E(X ij ), σ : the variance for design i. When N s are arge, µ i can be a good approximation for µ i, since, according to the aw of arge numbers, P{µ i µ i }, as. Batch means method (Schmeiser 98) can be used if the simuation sampes from any design are not independent. Given the fact that we can conduct ony a finite number of simuation repications, µ i is simpy an approximation to µ i. Using the approximation resuts to seect the best design we have to access the probabiity of correct seection (without oss of generaity, we consider minimization probems in this paper; thus, the best design means the design with the smaest µ i ). Correct seection can be Chernoff bounds becomes a major obstace to incude some second-order optimization techniques, such as Newton s method, to achieve a faster convergence rate. defined as a design with the smaest sampe performance measure which is actuay the best design. In the remainder of this paper, et CS denote correct seection There exists a arge iterature on seecting the best design. Godsman and Neson (994) provide an exceent survey on current approaches (e.g., Godsman, Neson, and Schmeiser, 99, Gupta and Panchapakesan, 979, and Law and Keton, 99) to estimating simuation confidence eve. In addition, Bechhofer, Santner, and Godsman (995) give a systematic and more detaied discussion on this issue. These approaches are mainy suitabe for probems having a sma number of designs (e.g. Godsman and Neson (994) suggest to 0 designs). However, for rea-ife probems, the number of designs can easiy grow extremey arge. Chen (996) provides an effective way to quantify confidence eve when the number of designs is arge. From Chen (996) we have P{CS} =P{ a design with the smaest sampe mean performance is reay the best design} k = P{ ˆµ b < ˆµ i }= APCS () i= i b where index b designates the design having the smaest sampe mean performance and µ i is the posterior distribution which consists of information from both prior distribution and the sampes {X ij, j =,,..., }. Under the assumption of normaity, ˆµ i ~N( X ij, σ i ) for i =,,...,k. j= We refer to this ower bound of the correct seection probabiity as the Approximate Probabiity of Correct Seection (APCS) Whie P{CS} is very difficut to obtain, APCS can be computed easiy. We wi use APCS to approximate P{CS}. Numerica testing in Chen (996) shows that it can provide reasonaby good approximation. Furthermore, the sensitivity information of the confidence eve with respect to simuation repications can be easiy obtained when the approach in Chen (996) is appied, which provide the basis to determine the aocation of the computing budget in this paper.

336 Chen, Chen, Dai, and Yucesan 3 GRADIENT APPROACH FOR OPTIMAL COMPUTING BUDGET ALLOCATION We wish to minimize the tota computation cost whie obtaining a desired confidence eve in seecting the best system out of k competing designs. If simuations are performed on a sequentia computer, the computation cost can be approximated by N +N +L+N k. Ideay we want min( N +N +L+N k ) () s.t. APCS P* where P* is a user-defined confidence eve requirement. Note that N, N,,..., N k are integers and APCS(N, N,..., N k ) can be computed ony after exhausting the tota simuation budget, N +N +L+N k. Soving this probem can be difficut, especiay when k is extremey arge. Since the purpose of soving () is to reduce computation cost with a desired confidence eve, we shoud not spend too much effort soving () during simuation. Otherwise, the additiona cost of soving () wi overwhem the benefits of the computing budget aocation. Hence, we need to sove () very efficienty, even if this means obtaining a sub-optima soution. Efficiency is more crucia than optimaity in this setting. We now present a sequentia approach to determine the number of simuation repications. Before conducting the simuation, there is neither knowedge about APCS nor an idea about how to aocate budget, therefore a designs are simuated with n 0 repications, and the posterior distribution for design i is n 0 N( X ij, σ i ) n j= 0 n 0 We use this statistica information to decide on further aocation. In other words, after running n 0 repications for each design, we have a basic idea about each design and can decide which designs are worthy of being aocated more of the computing budget. Furthermore, et i be the additiona computing budget aocated to design i in each step ( i is a non-negative integer). In order to effectivey aocate the computing budget for further simuation, it is necessary to know how APCS woud be affected if further repication i is added independenty to each design i. Under a Bayesian mode, it is convenient to use the statistica information at to estimate APCS at + i by using an approximated posterior distribution σ i N( X ij, ) for design i N j= i + i We refer this approximation to EAPCS (Estimated Approximate Probabiity of Correct Seection). We assume that i is not arge and is cose to + i ; otherwise EAPCS is not a good estimator for APCS. Note that can be repaced by n 0 for each design when posterior distribution is approximated at n 0 + i. We hope that APCS becomes arger as simuation proceeds; we sequentiay add computing budget by each time unti that APCS achieves a satisfactory eve P *. In order to minimize the tota computation cost, this budget shoud be aocated among designs so as to maximize the EAPCS. Thus, at step, =,,..., max,, k k s.t. i= EAPCS(N i = and +,N + + + N k i + k ) 0 for a i. (3) We assume the system parameter are continuous so that the gradient method (Luenberger 984) is appied to approximatey sove (3). A major issue for using the gradient method is the estimation of gradient information. Chen et a. (996) appy Chernoff bounds to estimate the gradient information and provide an effective soution to (3). However, the compexity of Chernoff bounds becomes a major obstace for appying some second-order optimization techniques, such as Newton s method, to achieve a faster convergence rate. In this paper, finite differencing, we approximate the gradient using a straightforward formua: EAPCS = APCS(N,N,L, + τ,l N k ) APCS(N,N,L, L,N k ) τ where τ is a sma number. To avoid spending too much time in iterativey finding the soution for (3), we ony execute a sma number of iterations (e.g. times) when appying the gradient method. In summary, we have the foowing agorithm: A Sequentia Agorithm for Optima Computing Budget Aocation (OCBA) Step 0. PERFORM SIMULATION with n 0 repication for a designs, 0, N =N =L =N k = n 0. Step. If APCS( N,N,K,N k ) P, stop, otherwise, go to Step.

Optima Computing Budget Aocation for Discrete Event Simuation 337 Step. Sove (3), + = + i, for i =,...,k. +. Step 3. PERFORM SIMULATION unti ( N,N,K,N k ); go to Step. Numerica Testing To compare the method provided in this paper with our previous approach (Chen et a. 996), we test a simpe G/G/ queue (k = 0). There is one server with uniformy distributed service times and customer interarriva times are aso distributed uniformy. In this singe-node exampe, a designs have the same arriva time uniformy distributed over [0.,.9], and service time in design i is uniform[0.,.3 + 0.05i], i =,,..., 0. We want to find a design with minimum average system time for customers served in the first 0 time units (terminating simuation). Obviousy, higher service rate resuts in shorter system time in this exampe, therefore, design is the true best design. In the numerica experiment, we compare the computation costs and the actua convergence probabiities P{CS} for different approaches. We set = and n 0 = 0 in this exampe. To avoid spending too much time in soving (3), we ony do two iterations in the gradient method. 0,000 independent experiments are performed so that the average computation cost and P{CS} can be estimated. Different confidence eve requirements are aso tested. Tabe contains the test resuts using a Chernoff bound approach (Chen et a. 996). Tabe shows the resuts using the approach presented in this paper. Tabe. Average tota number of simuation repications and P{CS} by using Chernoff bounds (n 0 = 0 and = ). P* With Chernoff P(CS) Bounds 60% 90.5 0.7 80% 35.73 0.89 90% 54.37 0.96 95% 764.58 0.98 Tabe. Average tota number of simuation repications and P{CS} for OCBA appication (n 0 = 0 and = ). P* With OCBA P(CS) 60% 96.47 0.7 80% 344.3 0.866 90% 53.64 0.963 95% 735.43 0.98 From Tabe and Tabe, we observe that the performances of the two approaches are not much different for sma P * (e.g., 80%). whie the new approach is faster when the confidence eve requirement P * is high (e.g., > 90%). 4 SELECTION FOR n 0 AND To appy our approach, we need to seect the initia simuation repication n 0 and one-time increment of computing budget. These two parameters may affect the performance of our approach. In this section we give recommendations for seecting these two parameters, athough a good choice can be probem-specific. Initia Simuation Repication Number n 0 We test the exampe presented in Section 3 for different vaues of n 0. Figure shows the numerica resuts with respect to different confidence eve requirements P. It is we understood that n 0 can not be too sma; otherwise, the estimates of mean and variance may be very poor, resuting in poor computing budget aocationtion. On the other hand, if n 0 is too arge, we may waste our computation time to reach a confidence eve which is much higher than the desired eve. Intuitivey such an effect is ess significant when P is arge. The computation cost vs. n 0 for different confidence eve requirements is shown in Figure. From Figure, we observe that the computation cost is not sensitive to n 0 when P is greater than 90%. For the cases that P is ess than 80%, which is not common in rea-ife appications, a good choice for n 0 is a number between 0 and 0 based on our empirica experiences. One-time Incrementa Computing Budget Again, the seection of is probem-specific. Generay speaking, arge may resut in wasting computation time to obtain an unnecessariy high confidence eve. On the other hand, if is too sma, we need to sove the budget aocation probem many times, diuting the benefit of this approach. We test the same exampe for different seections of. To have a fair comparison, instead of comparing the simuation repication numbers, we compare tota CPU time, which consists of simuation time (the CPU time for simuation ony) and optimization time (the CPU time to sove budget aocation by appying the gradient method). Figure shows the numerica resuts with respect to P = 80%. The optimization time for arge is much smaer than that for sma. Conversey, the simuation time for arge is arger than that for sma. We observe that a good seection of for a 0-designs probem is a number between 5 and 30 based on our empirica experiences.

338 Chen, Chen, Dai, and Yucesan 5 COMPARISON WITH OTHER METHODS In this section we compare our approach with two-stage procedures given by Dudewicz and Daa (975) and Rinott (977) by using the exampe presented in Section 3. Unike our Bayesian approach, these two-stage procedures are deveoped based on cassic statistica mode. It is convenient to et (),...,(k) denote the unknown indices such that µ () µ () L µ (k) The goa in two-stage procedures is the same as ours: to seect a design with the smaest mean, µ (). Based on the indifference-zone idea we may be wiing to choose design () if µ () and µ () are very cose (e.g. µ () µ () d, where d is the indifference-zone). By doing so, the procedure can avoid making a arge number of simuation repications to separate sma difference. More specificay, we intend to determine the number of simuation repications to ensure P{µ () < µ (i), i µ (i) µ () d } P The procedure is as foows. Before performing the simuation we specify three parameters: the confidence eve requirement P, the simuation repication in the first stage n 0, and indifference zone d. Let h sove Rinott's or Dudewicz s integra given P, n 0, and k (h can aso be found from the tabes in Wicox (984) for Rinott s procedure, and those in Gibbons (977) for Dudewicz s procedure). In the first stage, a designs are simuated with n 0 repications. Based on the resuting variance estimate obtained from the first stage, we determine how many additiona simuation repications for each design shoud be conducted in the second stage in order to reach the desired confidence eve: =max(n 0, (hs i /d * ) ), for i =,,, k. where is the integer round-up function. Bigger d means arger indifference zone, resuting in ower simuation cost. In our previous testing, we are interested in finding the design with smaest mean µ (). To have a fair comparison, d can not be greater than µ () µ () which is around 0.059, estimated by using simuation. Two confidence eves P = 0.9, 0.95 and initia sampe repication n 0 = 0 and n 0 = 0 have been tested. In these tests, we have iterations in each sequentia optimization step when using OCBA to sove (3) soved. For the case of OCBA we appy one-time incrementa computing budget =. When appying Rinott s and Dudewicz s procedures, we set the indifference zone d = 0.059 ( µ () µ () 0.059). In both cases 0,000 independent experiments are run to evauate the computationa efficiency and to estimate the actua convergence probabiities P{CS}. The computation cost and P{CS} are given in Tabe 3 and Tabe 4 corresponding to n 0 = 0 and n 0 = 0, respectivey. From Tabe 3, significant speedup is observed for our method over both two-stage procedures, whie the actua convergence probabiities for a approaches are no ess than the desired eves. The average speedup is approximatey 3.74 times and.48 times over the Rinott s and Dudewicz s procedures, respectivey, when P = 95% and n 0 = 0. Figures 3 and 4 show the average for a i over the 0,000 experiments using our approach and Rinott s procedure respectivey (P = 90% and n 0 = 0). From these two figures, we observe that the budget aocation using our approach is quite different from that using Rinott s procedure. Two-stage procedures determine the number of simuation repications for a designs using ony the information of sampe variances. The information on their sampe means is not utiized. On the other hand, our approach expoits the information of both sampe means and variances, achieving much faster performance. Tabe 3. Average tota number of simuation runs and the probabiity of correct seection as n 0 = 0. n 0 = 0 P*=90% P(CS) P*=95% P(CS) OCBA 53.64 0.96 755.3 0.989 Dudewicz's 8059.97 0.984 03. 0.995 Speedup 5.39 4.6 Rinott s 9479.37 0.986 646. 0.995 Speedup 8. 6.5 Tabe 4. Average tota number of simuation runs and the probabiity of correct seection as n 0 = 0. n 0 = 0 P*=90% P(CS) P*=95% P (CS) OCBA 557.64 0.974 738.6 0.989 Dudewicz's 683.0 0.975 95.33 0.989 Speedup.5.48 Rinott s 7978.6 0.99 050.56 0.995 Speedup 4.3 3.74

Optima Computing Budget Aocation for Discrete Event Simuation 339 6 CONCLUDING REMARKS In this paper we present an optima computing budget aocation technique that can seect the best of k simuated designs. We aso compare our approach with traditiona two-stage procedures by conducting a numerica experiment. Preiminary numerica testing shows that our approach is more than ten times faster than two-stage procedures. Further testing and anaysis of our approach is under way. Aso the deveopment of a more efficient budget aocation technique which utiizes methods with higher convergence rate, ike Newton s method, is one of our research topics. Average tota simuation repications 800 700 600 500 400 300 00 00 0 0 3 6 9 5 8 4 7 30 The initia simuation repication number 60% 80% 90% 95% Figure. Average tota simuation repications vs. n 0 for different P as OCBA appied ( iterations). The resuts are based on an average over 0,000 repicates ( =). Seconds 7.5 7 6.5 6 5.5 5 4.5 4 3.5 3.5.5 0.5 0 0 3 6 9 5 8 4 7 30 33 36 39 4 45 48 5 The one-time incrementa computing budget Tota cpu time Simuation Optimization Figure. The tota CPU time vs. for P =80% and n 0 = 0 when OCBA is appied ( iterations) based on an average over 0,000 repicates (tota CPU time = simuation time + optimization time). The resuts are

340 Chen, Chen, Dai, and Yucesan Designs 0 9 8 7 6 5 4 3 0 0 0 30 40 50 60 70 80 90 0000304050 Average simuation repications Figure 3. Computing budget aocation are determined by OCBA when P = 0.90, and n 0 =0 (system is the best design). 9 Designs 7 5 3 0 00 400 600 800 000 00 400 600 800 Average simuaion repications Figure 4. Computing budget aocation determined by in Rinott s procedure when P =0.90, and n 0 =0 (design is the best design). ACKNOWLEDGMENTS This work was supported in part by the Nationa Science Foundation under grant No. ECS-96479. REFERENCES Bechofer R. E., Santner, T. J., and Godsman, D. M. 995. Design and Anaysis of Experiments for Statistica Seection, Screening, and Mutipe Comparisons. John Wiey & Sons, Inc.

Optima Computing Budget Aocation for Discrete Event Simuation 34 Charnes, J. M. 99. Mutivariate Simuation Output Anaysis. Proceedings of the 99 Winter Simuation Confidence, 87-93. Chen, C. H. 995. An Effective Approach to Smarty Aocate Computing Budget for Discrete Event Simuation. Proceedings of the 34th IEEE Conference on Decision and Contro, 598-605. Chen, C. H. 996. A Lower Bound for the Correct Subset-Seection Probabiity and Its Appication to Discrete Event System Simuations. IEEE Transactions on Automatic Contro., Vo. 4, No. 8, 7-3. Chen, C. H., Chen, H. C., and Dai, L. 996. A Gradient Approach for Smarty Aocating Computing Budget for Discrete Event Simuation. Proceedings of the 996 Winter Simuation Conference, 398-405. Dudewicz, E. J. and Daa, S. R. 975. Aocation of Observations in Ranking and Seection with Unequa Variances. Sankhya, B37:8-78. Gibbons, J. D., Okin, I., and Sobe, M. 977. Seecting and Ordering Popuations: A New Statistica Methodoogy. John Wiey & Sons, Inc. Godsman, G., B. L. Neson, and B. Schmeiser. 99. Methods for Seecting the Best System. Proceedings of the 99 Winter Simuation Conference, 77-86. Godsman, G., and B. L. Neson. 994. Ranking, Seection, and Mutipe Comparison in Computer Simuation. Proceedings of the 994 Winter Simuation Conference, 9-99. Hsu, J. C. 996. Mutipe Comparisons Theory and Methods. Chapman & Ha. Gupta, S. S. and S. Panchapakesan. 979. Mutipe Decision Procedures: Theory and Methodoogy of Seecting and Ranking Popuations. John Wiey. Law, A. M. and W. D. Keton. 99. Simuation Modeing & Anaysis. McGraw-Hi, Inc. Luenberger, D. G., 984. Linear and Noninear Programming. Addison-Wesey. Matejcik, F. J. and Neson, B. L. 99. Simutaneous Ranking, Seection and Mutipe Comparisons for Simuation. Proceeding of the 993 Winter Simuation Confidence, 386-39. Rinott, Y. 978. On Two-stage Seection Procedures and Reated Probabiity Inequaities. Communications in Statistics A7, 799-8. Schmeiser, B. 98. Batch Size Effects in the Anaysis of Simuation Output. Operations Research, Vo. 30, No. 3,, 556-567. Wicox, R. R. 984. A Tabe For Rinott s Seection Procedure. Journa of Quaity Technoogy, Vo. 6, No., 97-00. AUTHOR BIOGRAPHIES HSIAO-CHANG CHEs a Ph.D. candidate at the Systems Engineering Department, University of Pennsyvania. He received a doube B.S. degree in Mathematics and Computer Science from the Eastern Michigan University in 99, and he received an M.S. degree in Systems Science and Mathematics from Washington University, St. Louis in 994. He is working on deveoping efficient approaches for discrete event simuation. CHUN-HUNG CHEs an Assistant Professor of Systems Engineering at the University of Pennsyvania, Phiadephia, PA. He received his Ph.D. degree in Simuation and Decision from Harvard University in 994. His research interests cover a wide range of areas in Monte Caro simuation, optima contro, stochastic decision processes, ordina optimization, perturbation anaysis, and their appications to manufacturing systems. Dr. Chen won the 994 Harvard University Eiahu I. Jury Award for the best thesis in the fied of contro. He is aso one of the recipients of the 99 MasPar Parae Computer Chaenge Award. LIYI DAI is an assistant professor in the Department of Systems Science and Mathematics at Washington University, MO. He received the M.S. degree from the Institute of Systems Science, Academia Sinica, Beijing, China, in 986, and the Ph.D. degree from Harvard University in 993. His research interests incude discrete event dynamic systems, simuation, stochastic optimization, communication systems, and singuar systems. He has coauthored over 30 papers in various journas and is the author of Singuar Contro Systems (Berin: Springer-Verag, 989). Dr. Dai is isted in Who's Who among Asian Americans and is a recipient of the NSF CAREER award. ENVER Y ucesas an associate Professor of Operations Research at INSEAD in Fontainebeau, FRANCE. He hods a BSIE degree from Pudue University, and an MS and a Ph.D. both in OR, from Corne University. He is currenty visiting the Department of Systems Engineering at the University of Pennsyvania.