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A note on networks of infinite-server queues

Published online by Cambridge University Press:  14 July 2016

J. Michael Harrison  and
Austin J. Lemoine
J. Michael Harrison*
Affiliation:
Stanford University
Austin J. Lemoine*
Affiliation:
Systems Control, Inc.
*
∗Postal address: Graduate School of Business, Stanford University, Stanford, CA 94305, U.S.A.
∗∗Postal address: Systems Control, Inc., 1801 Page Mill Road, Palo Alto, CA 94304, U.S.A.
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Abstract

The subject of this paper is networks of queues with an infinite number of servers at each node in the system. Our purpose is to point out that independent motions of customers in the system, which are characteristic of infinite-server networks, lead in a simple way to time-dependent distributions of state, and thence to steady-state distributions; moreover, these steady-state distributions often exhibit an invariance with regard to distributions of service in the network. We consider closed systems in which a fixed and finite number of customers circulate through the network and no external arrivals or departures are permitted, and open systems in which customers originate from an external source according to a Poisson process, possibly non-homogeneous, and each customer eventually leaves the system.

Keywords

NETWORKS OF QUEUESINFINITE-SERVER STATIONSCLOSED SYSTEMOPEN SYSTEMMIGRATION PROCESSESCOLONIESREGENERATIVE ROUTINGPOISSON PROCESSORDER-STATISTIC PROPERTYINVARIANCE PROPERTY OF STEADY-STATE DISTRIBUTION
Type
Short Communications
Information
Journal of Applied Probability , Volume 18 , Issue 2 , June 1981 , pp. 561 - 567
Copyright
Copyright © Applied Probability Trust 

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Footnotes

This work was sponsored in part by the National Science Foundation under Grant No. ENG-7824568.

References

[1] Barbour, A. D. (1976) Networks of queues and the method of stages. Adv. Appl. Prob. 8, 584–591.Google Scholar
[2] Baskett, F., Chandy, K. M., Muntz, R. R. and Palacios, F. G. (1975) Open, closed and mixed networks of queues with different classes of customers. J. Assoc. Comput. Mach. 22, 248–260.Google Scholar
[3] Brown, M. and Ross, S. M. (1969) Some results for infinite server Poisson queues. J. Appl. Prob. 6, 604–611.Google Scholar
[4] Burman, D. Y. (1982) Insensitivity in queueing systems. Adv. Appl. Prob. 14 (1). To appear.Google Scholar
[5] Cox, D. R. and Lewis, P. A. W. (1966) The Statistical Analysis of Series of Events. Methuen, London.Google Scholar
[6] Daley, D. J. (1976) Queueing output processes. Adv. Appl. Prob. 8, 395–415.Google Scholar
[7] Feller, W. (1971) An Introduction to Probability Theory and its Applications, Vol. II. Wiley, New York.Google Scholar
[8] Gaver, D. P. and Lehoczky, J. P. (1977) A diffusion approximation solution for a repairman problem with two types of failure. Management Sci. 24, 71–81.Google Scholar
[9] Harrison, J. M. and Lemoine, A. J. (1977) Limit theorems for periodic queues. J. Appl. Prob. 14, 566–576.Google Scholar
[10] Kelly, F. P. (1975) Networks of queues with customers of different classes. J. Appl. Prob. 12, 542–554.Google Scholar
[11] Kelly, F. P. (1976) Networks of queues. Adv. Appl. Prob. 8, 416–432.Google Scholar
[12] Kelly, F. P. (1979) Reversibility and Stochastic Networks. Wiley, New York.Google Scholar
[13] Miller, D. R. (1972) Existence of limits in regenerative processes. Ann. Math. Statist. 43, 1275–1282.Google Scholar
[14] Puri, P. S. (1978) A limit theorem for point processes with applications. J. Appl. Prob. 15, 726–747.Google Scholar
[15] Schassberger, R. (1978) The insensitivity of stationary probabilities in networks of queues.Google Scholar