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Exact Monte Carlo simulation for fork-join networks

Part of: Special processes Probabilistic methods, simulation and stochastic differential equations

Published online by Cambridge University Press:  01 July 2016

Hongsheng Dai
Hongsheng Dai*
Affiliation:
University of Brighton
*
∗ Postal address: Department of Mathematics, School of Computing, Engineering and Mathematics, University of Brighton, Lewes Road, Brighton BN2 4GJ, UK. Email address: h.dai@brighton.ac.uk
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Abstract

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In a fork-join network each incoming job is split into K tasks and the K tasks are simultaneously assigned to K parallel service stations for processing. For the distributions of response times and queue lengths of fork-join networks, no explicit formulae are available. Existing methods provide only analytic approximations for the response time and the queue length distributions. The accuracy of such approximations may be difficult to justify for some complicated fork-join networks. In this paper we propose a perfect simulation method based on coupling from the past to generate exact realisations from the equilibrium of fork-join networks. Using the simulated realisations, Monte Carlo estimates for the distributions of response times and queue lengths of fork-join networks are obtained. Comparisons of Monte Carlo estimates and theoretical approximations are also provided. The efficiency of the sampling algorithm is shown theoretically and via simulation.

Keywords

Coupling from the pastfork-join networkperfect samplingread-once coupling from the past

MSC classification

Primary: 65C05: Monte Carlo methods 65C50: Other computational problems in probability
Secondary: 60K20: Applications of Markov renewal processes (reliability, queueing networks, etc.) 60K25: Queueing theory
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 43 , Issue 2 , June 2011 , pp. 484 - 503
Copyright
Copyright © Applied Probability Trust 2011 

References

Ayhan, H. and Seo, D.-W. (2001). Laplace transform and moments of waiting times in Poisson driven (max,+) linear systems. Queueing Systems 37, 405–438.CrossRefGoogle Scholar
Baccelli, F., Massey, W. A. and Towsley, D. (1989). Acyclic fork-join queuing networks. J. Assoc. Comput. Math. 36, 615–642.CrossRefGoogle Scholar
Balsoma, S., Donatietllo, L. and van Dijk, N. M. (1998). Bound performance models of heterogeneous parallel processing systems. IEEE Trans. Parallel Distributed Systems 9, 1041–1056.CrossRefGoogle Scholar
Dai, H. (2008). Perfect sampling methods for random forests. Adv. Appl. Prob. 40, 897–917.CrossRefGoogle Scholar
Flatto, L. (1985). Two parallel queues created by arrivals with two demands. II. SIAM J. Appl. Math. 45, 861–878.CrossRefGoogle Scholar
Flatto, L. and Hahn, S. (1984). Two parallel queues created by arrivals with two demands. I. SIAM J. Appl. Math. 44, 1041–1053.CrossRefGoogle Scholar
Huber, M. (2004). Perfect sampling using bounding chains. Ann. Appl. Prob. 14, 734–753.CrossRefGoogle Scholar
Kendall, W. S. and Møller, J. (2000). Perfect simulation using dominating processes on ordered spaces, with application to locally stable point processes. Adv. Appl. Prob. 32, 844–865.CrossRefGoogle Scholar
Ko, S-S. and Serfozo, R. F. (2004). Response times in M/M/s fork-join networks. Adv. Appl. Prob. 36, 854–871.CrossRefGoogle Scholar
Nelson, R. and Tantawi, A. N. (1988). Approximate analysis of fork-join synchronization in parallel queues. IEEE Trans. Software Eng. 37, 739–743.Google Scholar
Propp, J. G. and Wilson, D. B. (1996). Exact sampling with coupled Markov chains and applications to statistical mechanics. Random Structures Algorithms 9, 223–252.3.0.CO;2-O>CrossRefGoogle Scholar
Raghavan, N. R. S. and Viswanadham, N. (2001). Generalized queueing network analysis of integrated supply chains. Internat. J. Production Res. 39, 205–224.CrossRefGoogle Scholar
Ross, S. M. (2007). Introduction to Probability Models, 8th edn. Academic Press, Burlington, MA.Google Scholar
Squillante, M. S., Zhang, Y., Sivasubramaniam, A. and Gautam, N. (2008) Generalized parallel-server fork-join queues with dynamic task scheduling. Ann. Operat. Res. 160, 227–255.CrossRefGoogle Scholar
Wilson, D. B. (2000). How to couple from the past using a read-once source of randomness. Random Structures Algorithms 16, 85–113.3.0.CO;2-H>CrossRefGoogle Scholar
Xia, C. H., Liu, Z., Towsley, D. and Lelarge, M. (2007). Scalability of fork/join queueing networks with blocking. In ACM SIGMETRICS Performance Evaluation Review, Association for Computing Machinery, New York, pp. 133–144.Google Scholar
Zhang, Z. (1990). Analytical results for waiting time and system size distributions in two parallel queueing systems. SIAM J. Appl. Math. 50, 1176–1193.CrossRefGoogle Scholar