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A functional central limit theorem for the jump counts of Markov processes with an application to Jackson networks

Part of: Markov processes

Published online by Cambridge University Press:  01 July 2016

Venkat Anantharam  and
Takis Konstantopoulos
Venkat Anantharam*
Affiliation:
Cornell University
Takis Konstantopoulos*
Affiliation:
University of Texas at Austin
*
* Present address: EECS Department, University of California, Berkeley, CA 94720, USA.
** Postal address: Department of Electrical and Computer Engineering, University of Texas, TX 78712, USA.
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Abstract

Each feasible transition between two distinct states i and j of a continuous-time, uniform, ergodic, countable-state Markov process gives a counting process counting the number of such transitions executed by the process. Traffic processes in Markovian queueing networks can, for instance, be represented as sums of such counting processes. We prove joint functional central limit theorems for the family of counting processes generated by all feasible transitions. We characterize which weighted sums of counts have zero covariance in the limit in terms of balance equations in the transition diagram of the process. Finally, we apply our results to traffic processes in a Jackson network. In particular, we derive simple formulas for the asymptotic covariances between the processes counting the number of customers moving between pairs of nodes in such a network.

Keywords

MARKOV CHAINCOUNTING PROCESSCOUPLINGINFINITE-DIMENSIONAL BROWNIAN MOTION

MSC classification

Primary: 60J25: Continuous-time Markov processes on general state spaces
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 27 , Issue 2 , June 1995 , pp. 476 - 509
Copyright
Copyright © Applied Probability Trust 1995 

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Footnotes

Research supported by NSF PYI award NCR 8857731, an IBM Faculty Development award, BellCore Inc. and the AT&T Foundation.

Research supported in part by NSF grant NCR-921143.

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