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The continuity of queues

Published online by Cambridge University Press:  01 July 2016

Ward Whitt
Ward Whitt*
Affiliation:
Yale University
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Abstract

Kennedy (1972) showed that the standard single-server queueing model is continuous. These results are extended to the standard multi-server model here. Even when there is only one server, an additional condition is needed for the queue length process.

Keywords

CONTINUITY OF QUEUESAPPROXIMATIONS FOR QUEUESMULTI-SERVER QUEUESGI/G/s QUEUESWEAK CONVERGENCE ON FUNCTION SPACESTHE CONTINUOUS MAPPING THEOREM
Type
Research Article
Information
Advances in Applied Probability , Volume 6 , Issue 1 , March 1974 , pp. 175 - 183
Copyright
Copyright © Applied Probability Trust 1974 

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References

Billingsley, P. (1968) Convergence of Probability Measures. John Wiley, New York.Google Scholar
Brumelle, S. L. (1971) Some inequalities for parallel-server queues. Operat. Res. 19, 402–413.Google Scholar
Brumelle, S. L. (1972) Bounds on the wait in a GI/M/k queue. Management Sci. 19, 773–777.CrossRefGoogle Scholar
Cox, D. R. and Smith, W. L. (1961) Queues. Methuen, London.Google Scholar
Jacobs, D. R. Jr. and Schach, S. (1972) Stochastic order relationships between GI/G/k systems. Ann. Math. Statist. 43, 1623–1633.CrossRefGoogle Scholar
Keifer, J. and Wolfowitz, J. (1955) On the theory of queues with many servers. Trans. Amer. Math. Soc. 78, 1–18.Google Scholar
Kennedy, D. P. (1972) The continuity of the single server queue. J. Appl. Prob. 9, 370–381.Google Scholar
Kingman, J. F. C. (1963) Poisson counts for random sequences of events. Ann. Math. Statist. 34, 1217–1232.CrossRefGoogle Scholar
Luchak, G. (1958) The continuous time solution of the equations of the single channel queue with a general class of service time distributions by the method of generating functions. J. Roy. Statist. Soc. B 20, 176–181.Google Scholar
Prabhu, N. U. and Lalchandani, A. P. (1966) Markov process analysis of Luchak's queueing model. Technical Report No. 16, Department of Industrial Engineering and Operations Research, Cornell University.Google Scholar
Ross, S. M. (1973) Bounds on the delay distribution in GI/G/1 queues. Operations Research Center, University of California, Berkeley.CrossRefGoogle Scholar
Schassberger, R. (1970) On the waiting time in the queueing system GI/G/1. Ann. Math. Statist. 41, 182–187.Google Scholar
Schassberger, R. (1972) On the work load process in a general preemptive resume priority queue. J. Appl. Prob. 9, 588–603.Google Scholar
Stoyan, D. (1972) Monotonicity of stochastic models. ZAMM 52, 23–30 (in German).Google Scholar
Whitt, W. (1968) Weak convergence theorems for queues in heavy traffic , Department of Operations Research, Cornell University. (Technical Report No. 2, Department of Operations Research, Stanford University.) Google Scholar
Whitt, W. (1974a) Continuity of several functions on the function space D. Ann. Probability, 2. To appear.Google Scholar
Whitt, W. (1974b) Representation and convergence of point processes on the line. Ann. Probability, 2. To appear.Google Scholar
Yu, O. S. (1973) Stochastic bounding relations for a heterogeneous-server queue with Erlang service times. Stanford Research Institute, Menlo Park, California.Google Scholar