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Loss networks and Markov random fields

Part of:Stochastic processesComputer system organizationSpecial processes

Published online by Cambridge University Press:  14 July 2016

Stan Zachary and
Ilze Ziedins
Stan Zachary*
Affiliation:
Heriot-Watt University
Ilze Ziedins*
Affiliation:
The University of Auckland
*
∗ Postal address: Department of Actuarial Mathematics and Statistics, Heriot-Watt University, Edinburgh EH14 4AS, UK. Email address: s.zachary@ma.hw.ac.uk.
∗∗ Postal address: Department of Statistics, The University of Auckland, Private Bag 92019, Auckland, New Zealand.
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Abstract

This paper examines the connection between loss networks without controls and Markov random field theory. The approach taken yields insight into the structure and computation of network equilibrium distributions, and into the nature of spatial dependence in networks. In addition, it provides further insight into some commonly used approximations, enables the development of more refined approximations, and permits the derivation of some asymptotically exact results.

Keywords

Loss networksMarkov random fieldsreduced load approximationsErlang fixed point

MSC classification

Primary:60K20: Applications of Markov renewal processes (reliability, queueing networks, etc.)
Secondary:60G60: Random fields68M20: Performance evaluation; queueing; scheduling

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 36 , Issue 2 , June 1999, pp. 403 - 414
Copyright
Copyright © Applied Probability Trust 1999 

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References

Bean, N. G., and Stewart, M. A. (1999). A new perspective on the normalisation of invariant measures for loss networks and other product form systems. To appear in Math. Comput. Modelling.Google Scholar
Bebbington, M. S., Pollett, P. K., and Ziedins, I. (1997). Improved fixed point methods for loss networks with linear structure. Proc. 4th Internat. Conf. Telecommun., Melbourne, Australia 3, 1411–1416.Google Scholar
Burman, D. Y., Lehoczky, J. P., and Lim, Y. (1984). Insensitivity of blocking probabilities in a circuit-switched network. J. Appl. Prob. 21, 850–859.Google Scholar
Buzen, J. P. (1973). Computational algorithms for closed queueing networks with exponential servers. Commun. ACM 16, 527–531.CrossRefGoogle Scholar
Choudhury, K. K., Leung, G. L., and Whitt, W. (1995). An algorithm to compute blocking probabilities in multi-rate multi-class multi-resource loss models. Adv. Appl. Prob. 27, 1104–1143.Google Scholar
Chung, S. P., and Ross, K. W. (1991). Reduced load approximations for multirate loss networks. IEEE Trans. Commun. 41, 726–736.Google Scholar
Dziong, Z., and Roberts, J. W. (1987). Congestion probabilities in a circuit-switched integrated services network. Perf. Eval. 7, 267–284.CrossRefGoogle Scholar
Hunt, P. J., and Kelly, F. P. (1989). On critically loaded loss networks. Adv. Appl. Prob. 21, 831–841.Google Scholar
Hunt, P. J., and Laws, C. N. (1991). Asymptotically optimal loss network control. Math. Operat. Res. 18, 880–900.Google Scholar
Kelly, F. P. (1985). Stochastic models of computer communication systems. J. R. Statist. Soc. B 47, 379–395.Google Scholar
Kelly, F. P. (1986). Blocking probabilities in large circuit-switched networks. Adv. Appl. Prob. 18, 473–505.CrossRefGoogle Scholar
Kelly, F. P. (1991). Loss networks. Ann. Appl. Prob. 1, 319–378.Google Scholar
MacPhee, I. M., and Ziedins, I. (1996). Admission controls for loss networks with diverse routing. In Stochastic Networks: Theory and Applications, eds Kelly, F. P., Zachary, S. and Ziedins, I. OUP, Oxford, pp. 205–214.CrossRefGoogle Scholar
Ross, K. W. (1995). Multiservice loss models for broadband telecommunication networks. Springer, New York.Google Scholar
Seneta, E. (1981). Non-negative matrices and Markov chains, 2nd edn. Springer.CrossRefGoogle Scholar
Whitt, W. (1985). Blocking when service is required from several facilities simultaneously. AT&T Technical Journal 64, 1807–1856.Google Scholar
Zachary, S. (1983). Countable state space Markov random fields and Markov chains on trees. Ann. Prob. 11, 894–903.CrossRefGoogle Scholar
Zachary, S. (1985). Bounded, attractive and repulsive Markov specifications on trees and on the one-dimensional lattice. Stoch. Proc. Appl. 20, 247–256.Google Scholar
Zachary, S. (1991). On blocking in loss networks. Adv. Appl. Prob. 23, 355–372.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