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Realization probability in closed Jackson queueing networks and its application

Published online by Cambridge University Press:  01 July 2016

X. R. Cao
X. R. Cao*
Affiliation:
Harvard University
*
∗Present address: MRO1-1/L26, Digital Equipment Corporation, Marlboro, MA 01752, USA.
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Abstract

Perturbation analysis is a new technique which yields the sensitivities of system performance measures with respect to parameters based on one sample path of a system. This paper provides some theoretical analysis for this method. A new notion, the realization probability of a perturbation in a closed queueing network, is studied. The elasticity of the expected throughput in a closed Jackson network with respect to the mean service times can be expressed in terms of the steady-state probabilities and realization probabilities in a very simple way. The elasticity of the throughput with respect to the mean service times when the service distributions are perturbed to non-exponential distributions can also be obtained using these realization probabilities. It is proved that the sample elasticity of the throughput obtained by perturbation analysis converges to the elasticity of the expected throughput in steady-state both in mean and with probability 1 as the number of customers served goes to This justifies the existing algorithms based on perturbation analysis which efficiently provide the estimates of elasticities in practice.

Keywords

PERTURBATION ANALYSISCLOSED QUEUEING NETWORKSCONVERGENCE PROPERTIES
Type
Research Article
Information
Advances in Applied Probability , Volume 19 , Issue 3 , September 1987 , pp. 708 - 738
Copyright
Copyright © Applied Probability Trust 1987 

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Footnotes

This work was supported in part by the U.S. Office of Naval Contracts under N00014-79-C-0776 and N00014-84-K-0465, the National Science Foundation Grants ECS 82-13680 and CDR-85-001-08.

References

Billingsley, P. (1979) Probability and Measure. Wiley, New York.Google Scholar
Breiman, L. (1968) Probability. Addison-Wesley, Reading, Mass. Google Scholar
Buzen, J. P. (1973) Computational algorithms for closed queueing Networks with exponential servers. Comm. ACM 16, 527–531.CrossRefGoogle Scholar
Cao, X. R. (1985a) Convergence of parameter sensitivity estimates in a stochastic experiment. IEEE Trans. Autom. Control 30, 834–843.Google Scholar
Cao, X. R. (1985b) On the sample function of queueing networks with applications to perturbation analysis. Operat. Res. To appear.Google Scholar
Cao, X. R. and Ho, Y. C. (1985) Perturbation analysis of sojourn times in closed Jackson queueing networks. Submitted to Operat. Res. Google Scholar
Çinlar, E. (1975) Introduction to Stochastic Processes. Prentice-Hall, Englewood Cliffs, NJ.Google Scholar
Gordon, W. J. and Newell, G. F. (1967) Closed queueing systems with exponential servers. Operat. Res. 15, 254–265.CrossRefGoogle Scholar
Ho, Y. C. and Cao, X. R. (1983) Perturbation analysis and optimization of queueing networks. J. Optim. Theory Applic. 40, 559–582.CrossRefGoogle Scholar
Ho, Y. C. and Cao, X. R. (1985) Performance sensitivity to routing changes in queueing networks and flexible manufacturing systems using perturbation analysis. IEEE J. Robotics and Automation 1, 165–172.Google Scholar
Ho, Y. C., Suri, R., Cao, X. R., Diehl, G. W., Dille, J. W. and Zazanis, M. A. (1984) Optimization of large multiclass (non-product-form) queueing networks using perturbation analysis. Large Scale Systems 7, 165–180.Google Scholar
Kelly, F. P. (1979) Reversibility and Stochastic Networks. Wiley, New York.Google Scholar
Melamed, B. (1982) On Markov jump process imbedded at jump epochs and their queueing-theoretic applications. Math. Operat. Res. 7, 111–128.CrossRefGoogle Scholar
Pullman, N. J. (1976) Matrix Theory and its Applications. Dekker, New York.Google Scholar
Sevcik, K. C. and Mitrani, I. (1981) The distribution of queueing network states at input and output instants. J. ACM 28, 358–371.CrossRefGoogle Scholar
Suri, R. and Zazanis, M. A. (1985) Perturbation analysis gives strongly consistent estimates for the M/G/1 queue. Management Sci. Submitted. Google Scholar
Williams, A. C. and Bhandiwad, R. A. (1976) A generating function approach to queueing network analysis of multiprogrammed computers. Networks 6, 1–22.CrossRefGoogle Scholar