Hostname: page-component-788cddb947-rnj55 Total loading time: 0 Render date: 2024-10-12T23:02:37.744Z Has data issue: false hasContentIssue false

Poisson functionals of Markov processes and queueing networks

Published online by Cambridge University Press:  01 July 2016

Richard F. Serfozo*
Affiliation:
Georgia Institute of Technology

Abstract

We present conditions under which a point process of certain jump times of a Markov process is a Poisson process. The central idea is that if the Markov process is stationary and the compensator of the point process in reverse time has a constant intensity a, then the point process is Poisson with rate a. A known example is that the output flow from an M/M/1 queueing system is Poisson. We present similar Poisson characterizations of more general marked point process functionals of a Markov process. These results yield easy-to-use criteria for a collection of such processes to be multivariate Poisson, compound Poisson, or marked Poisson with a specified dependence or independence. We discuss several applications for queueing systems with batch arrivals and services and for networks of queues. We also indicate how our results extend to functionals of non-Markovian processes.

Type
Research Article
Copyright
Copyright © Applied Probability Trust 1989 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

This research was sponsored in part by Air Force Office of Scientific Research contracts 84–0367 and F49620 85 C 0144; it was partly carried out at the University of North Carolina at Chapel Hill.

References

Benveniste, A. and Jacod, J. (1973) Systèmes de Lévy des processus de Markov. Invent. Math. 21, 183–98.CrossRefGoogle Scholar
Bremaud, P. (1975) An extension of Watanabe's theorem on characterization of Poisson processes. J. Appl. Prob. 12, 396399.CrossRefGoogle Scholar
Bremaud, P. (1981) Point Processes and Queues: Martingale Dynamics. Springer-Verlag, New York.CrossRefGoogle Scholar
Burke, P. J. (1956) The output of a queueing system. Operat. Res. 4, 699704.CrossRefGoogle Scholar
Çinlar, E. (1975) Introduction to Stochastic Processes. Prentice-Hall, Englewood Cliffs, NJ.Google Scholar
Disney, R. L. and Kiessler, P. C. (1987) Traffic Processes in Queueing Networks: A Markov Renewal Approach. Johns Hopkins University Press, Baltimore.Google Scholar
Disney, R. L. and König, D. (1985) Queueing networks: a survey of their random processes. SIAM Rev. 27, 335403.CrossRefGoogle Scholar
Gross, D. and Harris, C. M. (1985) Fundamentals of Queueing Theory. Wiley, New York.Google Scholar
Kallenberg, O. (1983) Random Measures, 3rd edn. Akademie-Verlag, Berlin; Academic Press, New York.CrossRefGoogle Scholar
Karr, A. F. (1986) Point Processes and Their Statistical Inference. Dekker, New York.Google Scholar
Kelly, F. P. (1979) Reversibility and Stochastic Networks. Wiley, New York.Google Scholar
Kook, K. (1989) Ph.D. Dissertation, Industrial and Systems Engineering, Georgia Institute of Technology.Google Scholar
Liptser, R. Sh. and Shiryayev, A. N. (1978) Statistics of Random Processes, Vol II. Springer-Verlag, New York.CrossRefGoogle Scholar
Melamed, B. (1979) On Poisson traffic processes in discrete-state Markovian systems with applications to queueing theory. Adv. Appl. Prob. 11, 218239.CrossRefGoogle Scholar
Reich, E. (1957) Waiting times when queues are in tandem. Ann. Math. Statist. 28, 768773.CrossRefGoogle Scholar
Varaiya, P. and Walrand, J. (1981) Flows in queueing networks: a martingale approach. Math. Operat. Res. 6, 387404.Google Scholar
Watanabe, S. (1964) On discontinuous additive functionals and Levy measures of a Markov process. Japanese J. Math. 34, 5370.CrossRefGoogle Scholar
Whittle, P. (1986) Systems in Stochastic Equilibrium. Wiley, New York.Google Scholar