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Comparing Markov Chains Simulated in Parallel

Published online by Cambridge University Press:  27 July 2009

Paul Glasserman  and
Pirooz Vakili
Paul Glasserman
Affiliation:
Columbia Business School, 403 Uris Hall, New York, New York 10027
Pirooz Vakili
Affiliation:
Manufacturing Engineering Department, Boston University, Boston, Massachusetts 02215
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Abstract

We investigate the dependence induced among multiple Markov chains when they are simulated in parallel using a shared Poisson stream of potential event occurrences. One expects this dependence to facilitate comparisons among systems; our results support this intuition. We give conditions on the transition structure of the individual chains implying that the coupled process is an associated Markov chain. Association implies that variance is reduced in comparing increasing functions of the chains, relative to independent simulations, through a routine argument. We also give an apparently new application of association to the problem of selecting the better of two systems from limited data. Under conditions, the probability of incorrect selection is asymptotically smaller when the systems compared are associated than when they are independent. This suggests a further advantage to linking multiple systems through parallel simulation.

Information

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 8 , Issue 3 , July 1994 , pp. 309 - 326
Copyright
Copyright © Cambridge University Press 1994

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References

1.Daley, D. (1968). Stochastically monotone Markov chains. Zeitschrift für Wahrscheinlichkeitstheorie und Verwunde Gebiete 10: 305–317.CrossRefGoogle Scholar
2.Esary, J.D., Proschan, F., & Walkup, D.W. (1967). Association of random variables. Annals of Mathematical Statistics 38: 1466–1474.CrossRefGoogle Scholar
3.Fox, B.L. & Glynn, P.W. (1990). Discrete-time conversion for simulating finite-horizon Markov processes. SIAM Journal on Applied Mathematics 50: 1457–1473.CrossRefGoogle Scholar
4.Glasserman, P. & Vakili, P. (1992). Correlation of uniformized Markov chains simulated in parallel. Proceedings of the Winter Simulation Conference. San Diego, CA: Society for Computer Simulation, pp. 475–482.CrossRefGoogle Scholar
5.Glasserman, P. & Yao, D.D. (1992). Some guidelines and guarantees for common random numbers. Management Science 38: 884–908.CrossRefGoogle Scholar
6.Goldsman, D., Nelson, B., & Schmeiser, B. (1991). Methods for selecting the best system. Proceedings of the 1991 Winter Simulation Conference. San Diego, CA: Society for Computer Simulation, pp. 177–186.CrossRefGoogle Scholar
7.Harris, T.E. (1977). A correlation inequality for Markov processes on partially ordered spaces. Annals of Probability 5: 451–454.CrossRefGoogle Scholar
8.Heidelberger, P. & Iglehart, D.L. (1979). Comparing stochastic systems using regenerative simulation and common random numbers. Advances in Applied Probability 11: 804–819.CrossRefGoogle Scholar
9.Heidelberger, P. & Nicol, D. (1991). Simultaneous parallel simulations of continuous time Markov chains at multiple parameter, settings. Proceedings of the 1991 Winter Simulation Conferences. San Diego, CA: Society for Computer Simulation, pp. 602–607.CrossRefGoogle Scholar
10.Ho, Y.C., Sreenivas, R., & Vakili, P. (1992). Ordinal optimization of discrete event dynamic systems. DEDSTA 2: 61–88.Google Scholar
11.Iscoe, I., Ney, P., & Nummelin, E. (1985). Large deviations of uniformly recurrent Markov additive processes. Advances in Applied Mathematics 6: 373–412.CrossRefGoogle Scholar
12.Karlin, S. & Taylor, H.M. (1981). A second course in stochastic processes. New York: Academic Press.Google Scholar
13.Keilson, J. & Kester, A. (1977). Monotone matrices and monotone Markov processes. Stochastic Processes and Their Applications 5: 231–241.CrossRefGoogle Scholar
14.Liggett, T.M. (1985). Interacting particle systems. New York: Springer.CrossRefGoogle Scholar
15.Lindqvist, B.H. (1988). Association of probability measures on partially ordered sets. Journal of Multivariate Analysis 26: 111–132.CrossRefGoogle Scholar
16.Massey, W.A. (1987). Stochastic orderings for Markov processes on partially ordered spaces. Mathematics of Operations Research 12: 350–367.CrossRefGoogle Scholar
17.Miller, H.D. (1961). A convexity property in the theory of random variables on a finite Markov chain. Annals of Mathematical Statistics 32: 1260–1270.CrossRefGoogle Scholar
18.Ney, P. & Nummelin, E. (1987). Markov additive processes, I. Eigenvalue properties and limit theorems, II. Large deviations. Annals of Probability 15: 561–609.Google Scholar
19.Vakili, P. (1992). Massively parallel and distributed simulation of a class of discrete event systems: A different perspective. TOMACS 2(3): 214–238.CrossRefGoogle Scholar