Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-09T05:57:41.468Z Has data issue: false hasContentIssue false
  • English
  • Français

A Technique for Computing Sojourn Times in Large Networks of Interacting Queues

Published online by Cambridge University Press:  27 July 2009

V. Anantharam  and
M. Benchekroun
V. Anantharam
Affiliation:
School of Electrical Engineering, Cornell University, Ithaca, New York 14850
M. Benchekroun
Affiliation:
School of Electrical Engineering, Cornell University, Ithaca, New York 14850
Get access Rights & Permissions [Opens in a new window]

Abstract

Consider a large number of interacting queues with symmetric interactions. In the asymptotic limit, the interactions between any fixed finite subcollection become negligible, and the overall effect of interactions can be replaced by an empirical rate. The evolution of each queue is given by a time inhomogeneous Markov process. This may be considered a technique for writing dynamic Erlang fixed-point equations. We explore this as a tool to approximate sojourn time distributions.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 7 , Issue 4 , October 1993 , pp. 441 - 464
Copyright
Copyright © Cambridge University Press 1993

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

1.Anantharam, V. (1991). A mean field limit for a lattice caricature of dynamic routing in circuit switched networks. Annals of Applied Probability 1: 481–503.CrossRefGoogle Scholar
2.Billingsley, P. (1968). Convergence of probability measures. New York: Wiley.Google Scholar
3.Ethier, S.N. & Kurtz, T.G. (1986). Markov processes: Characterization and convergence. New York: Wiley.CrossRefGoogle Scholar
4.Gibbens, R.J., Hunt, P.J., & Kelly, F.P. (1990). Bistability in communication networks. In Grimmett, G.R. & Welsh, D.J.A. (eds.), Disorder in physical systems. Oxford University Press, pp. 113–128.Google Scholar
5.Hajek, B. & Krishna, T. (1990). Bounds on the accuracy of reduced-load blocking formula in some simple circuit-switched networks. Proceedings of the Bilkent University Conference on Communication, Control and Signal Processing. Ankara, Turkey: Bilkent University, pp. 1633–1642.Google Scholar
6.Hartman, T. (1964). Ordinary differential equations. New York: Wiley.Google Scholar
7.Hunt, P.J. (1990). Limit theorems for stochastic networks. Ph.D. thesis, University of Cambridge.Google Scholar
8.Kac, M. (1956). Foundations of kinetic theory. Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability. Berkeley, CA: University of California Press, pp. 154–161.Google Scholar
9.Kelly, F.P. (1986). Blocking probabilities in large circuit-switched networks. Advances in Applied Probability 18: 473–505.CrossRefGoogle Scholar
10.Kelly, F.P. (1988). Routing in circuit-switched networks: Optimization, shadow price and decentralization. Advances in Applied Probability 20: 112–144.CrossRefGoogle Scholar
11.Sznitman, A.-S. (1989). Propagation of chaos. Ecole d'Eté de Probabilités de Saint-Flour, Lecture Notes in Math. No. 1464, Berlin: Springer (preprint).Google Scholar
12.Uchiyama, K. (1986). Fluctuation in population dynamics. Lecture Notes in Biomathematics 70: 222–229.CrossRefGoogle Scholar
13.Whitt, W. (1985). Blocking when service is required from several facilities simultaneously. AT&T Technical Journal 64: 1807–1856.Google Scholar
14.Ziedins, I. & Kelly, F.P. (1989). Limit theorems for loss networks with diverse routing. Advances in Applied Probability 21: 804–830.CrossRefGoogle Scholar