Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-10T15:06:44.407Z Has data issue: false hasContentIssue false
  • English
  • Français

State aggregation and discrete-state Markov chains embedded in a class of point processes

Part of: Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Xi-Ren Cao
Xi-Ren Cao*
Affiliation:
The Hong Kong University of Science and Technology
*
∗Present address: Department of EEE, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong.
Get access Rights & Permissions [Opens in a new window]

Abstract

One result that is of both theoretical and practical importance regarding point processes is the method of thinning. The basic idea of this method is that under some conditions, there exists an embedded Poisson process in any point process such that all its arrival points form a sub-sequence of the Poisson process. We extend this result by showing that on the embedded Poisson process of a uni- or multi-variable marked point process in which interarrival time distributions may depend on the marks, one can define a Markov chain with a discrete state that characterizes the stage of the interarrival times. This implies that one can construct embedded Markov chains with countable state spaces for the state processes of many practical systems that can be modeled by such point processes.

Keywords

THINNINGUNIFORMIZARONCOXIAN DISTRIBUTION

MSC classification

Secondary: 60G55: Point processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 32 , Issue 1 , March 1995 , pp. 39 - 51
Copyright
Copyright © Applied Probability Trust 1995 

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.)

References

Billingsley, P. (1979) Probability and Measure. Wiley, New York.Google Scholar
Brémaud, P. (1981) Point Processes and Queues: Martingale Dynamics. Springer-Verlag, New York.Google Scholar
Cao, X. R. (1993) Decomposition of random variables with bounded hazard rates. Operat. Res. Lett. 13, 113–120.Google Scholar
Kemeny, J. and Snell, J. L. (1960) Finite Markov Chains. Van Nostrand, Princeton, NJ.Google Scholar
Kleinrock, L. (1975) Queueing Networks, Vol. I: Theory. Wiley, New York.Google Scholar
Lewis, P. A. W. and Shedler, G. S. (1979) Simulation of nonhomogeneous Poisson processes by thinning. Naval Res. Logist. Quart. 26, 403–413.Google Scholar
Ogata, Y. (1981) On Lewis' simulation method for point processes. IEEE Trans. Inf. Theory 27, 23–31.Google Scholar
Shanthikumar, J. G. (1986) Uniformization and hybrid simulation/analytic methods of renewal processes. Operat. Res. 34, 573–580.Google Scholar
Van Dijk, N. M. (1990) On a simple proof of uniformization for continuous and discrete-state continuous-time Markov chains. Adv. Appl. Prob. 22, 749–750.Google Scholar
Walrand, J. (1988) An Introduction to Queueing Networks. Prentice-Hall, Englewood Cliffs, NJ.Google Scholar