Hostname: page-component-8448b6f56d-c47g7 Total loading time: 0 Render date: 2024-04-24T09:30:46.784Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Get access Rights & Permissions [Opens in a new window]

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 

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

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.

References

[1] Aldous, D. J. (1983) Random walks on finite groups and rapidly mixing Markov chains. In Seminaire de Probabilités XVII, pp. 243–297. Lecture Notes in Mathematics 986, Springer-Verlag, Berlin.Google Scholar
[2] Anantharam, V. (1989) The optimal buffer allocation problem. IEEE Trans. Inf. Theory 35, 721–725.CrossRefGoogle Scholar
[3] Bollobás, B. (1979) Graph Theory: An Introductory Course. Graduate Texts in Math. 63, Springer-Verlag, New York.CrossRefGoogle Scholar
[4] Bhattacharya, R. N. and Lee, O. (1988) Ergodicity and central limit theorems for a class of Markov processes. J. Multivariate Anal. 27, 80–90.CrossRefGoogle Scholar
[5] Billingsley, P. (1968) Convergence of Probability Measures. Wiley, New York.Google Scholar
[6] Diaconis, P. (1988) Group Representations in Probability and Statistics. Institute of Mathematical Statistics, Lecture Notes-Monograph Series, Vol. 11.CrossRefGoogle Scholar
[7] Dynkin, E. B. and Yushkevich, A. A. (1969) Markov Processes: Theorems and Problems. Plenum Press, New York.Google Scholar
[8] Ethier, S. N. and Kurtz, T. G. (1986) Markov Processes: Characterization and Convergence. Wiley, New York.CrossRefGoogle Scholar
[9] Flatto, L., Odlyzko, A. M. and Wales, D. B. (1985) Random shuffles and group representations. Ann. Prob. 13, 154–178.CrossRefGoogle Scholar
[10] Freedman, D. (1983) Markov Chains. Springer-Verlag, New York.Google Scholar
[11] Good, I. J. (1951) Random walk on a finite Abelian group. Proc. Camb. Phil. Soc. 47, 756–762.CrossRefGoogle Scholar
[12] Jacod, J. and Shiryaev, A. N. (1987) Limit Theorems for Stochastic Processes. Springer-Verlag, Berlin.Google Scholar
[13] Kuelbs, J. (1973) The invariance principle for Banach space valued random variables. J. Multivariate Anal. 3, 161–172.CrossRefGoogle Scholar
[14] Maigret, N. (1978) Théorème de limite centrale fonctionnel pour une chaîne de Markov récurrente au sens de Harris et positive. Ann. Inst. H. Poincaré, B 14, 425–440.Google Scholar
[15] Pollard, D. (1984) Convergence of Stochastic Processes. Springer-Verlag, New York.Google Scholar
[16] Taylor, A. E. and Lay, D. C. (1980) Introduction to Functional Analysis, 2nd edn. Wiley, New York.Google Scholar
[17] Vakhania, N. N. (1981) Probability Distributions on Linear Spaces. North-Holland, New York.Google Scholar
[18] Walrand, J. (1988) An Introduction to Queueing Networks. Prentice-Hall, Englewood Cliffs, NJ.Google Scholar