Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-11T03:28:24.174Z Has data issue: false hasContentIssue false
  • English
  • Français

Dynamics of large uncontrolled loss networks

Part of: Operations research and management science Computer system organization Special processes Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Stan Zachary
Stan Zachary*
Affiliation:
Heriot-Watt University
*
∗Postal address: Department of Actuarial Mathematics and Statistics, Heriot-Watt University, Edinburgh EH14 4AS, Scotland, UK. Email address: s.zachary@ma.hw.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper studies the connection between the dynamical and equilibrium behaviour of large uncontrolled loss networks. We consider the behaviour of the number of calls of each type in the network, and show that, under the limiting regime of Kelly (1986), all trajectories of the limiting dynamics converge to a single fixed point, which is necessarily that on which the limiting stationary distribution is concentrated. The approach uses Lyapunov techniques and involves the evolution of the transition rates of a stationary Markov process in such a way that it tends to reversibility.

Keywords

Loss networkLyapunov function

MSC classification

Primary: 60K20: Applications of Markov renewal processes (reliability, queueing networks, etc.)
Secondary: 60G60: Random fields 68M20: Performance evaluation; queueing; scheduling 90B18: Communication networks
Type
Research Papers
Information
Journal of Applied Probability , Volume 37 , Issue 3 , September 2000 , pp. 685 - 695
Copyright
Copyright © Applied Probability Trust 2000 

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

Bean, N. G., Gibbens, R. J., and Zachary, S. (1995). Asymptotic analysis of large single resource loss systems under heavy traffic, with applications to integrated networks. Adv. Appl. Prob. 27, 273–292.Google Scholar
Bean, N. G., Gibbens, R. J., and Zachary, S. (1997). Dynamic and equilibrium behaviour of controlled loss networks. Ann. Appl. Prob. 7, 873–885.CrossRefGoogle Scholar
Fayolle, G., Malyshev, V. A., and Menshikov, M. V. (1995). Topics in the Constructive Theory of Countable Markov Chains. Cambridge University Press.Google Scholar
Hunt, P. J. (1995). Pathological behaviour in loss networks. J. Appl. Prob. 32, 519–533.Google Scholar
Hunt, P. J., and Kurtz, T. G. (1994). Large loss networks. Stoch. Proc. Appl. 53, 363–378.Google Scholar
Hunt, P. J., and Laws, C. N. (1993). Asymptotically optimal loss network control. Math. Operat. Res. 18, 880–900.Google Scholar
Kelly, F. P. (1979). Reversibility and Stochastic Networks. John Wiley, Chichester.Google Scholar
Kelly, F. P. (1986). Blocking probabilities in large circuit-switched networks. Adv. Appl. Prob. 18, 473–505.Google Scholar
Kelly, F. P. (1991). Loss networks. Ann. Appl. Prob. 1, 319–378.Google Scholar
Mitra, D. (1987). Asymptotic analysis and computational methods for a class of simple, circuit-switched networks with blocking. Adv. Appl. Prob. 19, 219–239.CrossRefGoogle Scholar
Moretta, B. (1995). Behaviour and control of single and two resource loss networks. Ph.D. Thesis, Heriot–Watt University.Google Scholar
Ross, K. W. (1995). Multiservice Loss Models for Broadband Telecommunication Networks. Springer, New York.Google Scholar
Whitt, W. (1985). Blocking when service is required from several facilities simultaneously. A.T.&T. Tech. J. 64, 1807–1856.Google Scholar
Zachary, S. (1996). The asymptotic behaviour of large loss networks. In Stochastic Networks: Theory and Applications. eds Kelly, F. P., Zachary, S. and Ziedins, I. R. Statist. Soc. Lecture Note Ser. 4, Oxford University Press, pp. 193-203.CrossRefGoogle Scholar
Zachary, S., and Ziedins, I. (2000). A refinement of the Hunt–Kurtz theory of large loss networks, with an application to virtual partitioning. Available at http://www.ma.hw.ac.uk/∼stan/papers/.Google Scholar
Ziedins, I. B., and Kelly, F. P. (1989). Limit theorems for loss networks with diverse routing. Adv. Appl. Prob. 21, 804–830.Google Scholar