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Networks of queues

Published online by Cambridge University Press:  01 July 2016

F. P. Kelly
F. P. Kelly*
Affiliation:
University of Cambridge
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Abstract

The behaviour in equilibrium of networks of queues is studied. Equilibrium distributions are obtained and in certain cases it is shown that the state of an individual queue is independent of the state of the rest of the network. The processes considered in this paper are irreversible; however, the method used to establish equilibrium distributions is one which has previously only been used when dealing with reversible processes. Results are obtained for models of communication networks, machine interference and birth-illness-death processes.

Keywords

NETWORK OF QUEUESMARKOV PROCESSEQUILIBRIUM DISTRIBUTIONTIME REVERSALCOMMUNICATION NETWORKMACHINE INTERFERENCEBIRTH-ILLNESS-DEATH PROCESS
Type
Research Article
Information
Advances in Applied Probability , Volume 8 , Issue 2 , June 1976 , pp. 416 - 432
Copyright
Copyright © Applied Probability Trust 1976 

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References

Barbour, A. D. (1976) Networks of queues and the method of stages. Adv. Appl. Prob 8, No. 3.Google Scholar
Bartholomew, D. J. (1967) Stochastic Models for Social Processes. Wiley, London.Google Scholar
Bartlett, M. S. (1949) Some evolutionary stochastic processes. J. R. Statist. Soc. B 11, 211–229.Google Scholar
Benson, F. and Cox, D. R. (1951) The productivity of machines requiring attention at random intervals. J. R. Statist. Soc. B 13, 65–82.Google Scholar
Billingsley, P. (1968) Convergence of Probability Measures. Wiley, New York.Google Scholar
Chiang, C. L. (1968) Introduction to Stochastic Processes in Biostatistics. Wiley, New York.Google Scholar
Cox, D. R. and Smith, W. L. (1961) Queues. Methuen, London.Google Scholar
Jackson, J. R. (1957) Networks of waiting lines. Operat. Res. 5, 518–521.CrossRefGoogle Scholar
Jackson, J. R. (1963) Jobshop-like queueing systems. Management Sci. 10, 131–142.Google Scholar
Jackson, R. R. P. (1954) Queueing systems with phase-type service. Operat. Res. Q. 5, 109–120.CrossRefGoogle Scholar
Jackson, R. R. P. (1956) Random queueing processes with phase-type service. J. R. Statist. Soc. B18, 129–132.Google Scholar
Karlin, S. (1966) A First Course in Stochastic Processes. Academic Press, New York.Google Scholar
Kelly, F. P. (1975) Networks of queues with customers of different types. J. Appl. Prob. 12, 542–554.Google Scholar
Kendall, D. G. (1964) Some recent work and further problems in the theory of queues. Theor. Prob. Appl. 9, 1–13.CrossRefGoogle Scholar
Kendall, D. G. (1975) Some problems in mathematical genealogy. In Perspectives in Probability and Statistics: Papers in Honour of M. S. Bartlett, ed. Gani, J. Applied Probability Trust, Sheffield, Distributed by Academic Press, London, 325–345.Google Scholar
Kingman, J. F. C. (1969) Markov population processes. J. Appl. Prob. 6, 1–18.CrossRefGoogle Scholar
Kleinrock, L. (1964) Communication Nets. McGraw-Hill, New York.Google Scholar
Koenigsberg, E. (1958) Cyclic queues. Operat. Res. Q. 9, 22–35.Google Scholar
Mirasol, N. M. (1963) The output of an M/G/∞ queueing system is Poisson. Operat. Res. 11, 282–284.Google Scholar
Moran, P. A. P. (1968) An Introduction to Probability Theory. Clarendon Press, Oxford.Google Scholar
Reich, E. (1957) Waiting times when queues are in tandem. Ann. Math. Statist. 28, 768–773.CrossRefGoogle Scholar
Reich, E. (1965) Departure processes. In Proceedings of the Symposium on Congestion Theory. University of North Carolina Press, Chapel Hill, 439–457.Google Scholar
Sevastyanov, B. A. (1957) An ergodic theorem for Markov processes and its application to telephone systems with refusals. Theor. Prob. Appl. 2, 104–112.CrossRefGoogle Scholar
Shanbhag, D. N. (1972) Letter to the editor. J. Appl. Prob. 9, 470.Google Scholar
Shanbhag, D. N. and Tambouratzis, D. G. (1973) Erlang's formula and some results on the departure process for a loss system. J. Appl. Prob. 10, 233–240.CrossRefGoogle Scholar
Spitzer, F. (1970) Interaction of Markov processes. Adv. Math. 5, 246–290.CrossRefGoogle Scholar
Takács, L. (1969) On Erlang's formula. Ann. Math. Statist. 40, 71–78.CrossRefGoogle Scholar
Taylor, J and Jackson, R. R. P. (1954) An application of the birth and death process to the provision of spare machines. Operat. Res. Q. 5, 95–108.CrossRefGoogle Scholar
Whitt, W. (1974) The continuity of queues. Adv. Appl. Prob. 6, 175–183.CrossRefGoogle Scholar
Whittle, W. (1967) Nonlinear migration processes. In Proceedings of the 36th Session of the International Statistical Institute, 642–647.Google Scholar
Whittle, P. (1968) Equilibrium distributions for an open migration process. J. Appl. Prob. 5, 567–571.Google Scholar
Whittle, P. (1975) Reversibility and acyclicity. In Perspectives in Probability and Statistics: Papers in Honour of M. S. Bartlett, ed. Gani, J. Applied Probability Trust, Sheffield. Distributed by Academic Press, London, 217–224.Google Scholar