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Comparing multi-server queues with finite waiting rooms, II: Different numbers of servers

Published online by Cambridge University Press:  01 July 2016

David Sonderman
David Sonderman*
Affiliation:
Yale University
*
∗Postal address: Box 1070, New Haven, CT 06504, U.S.A.
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Abstract

We compare two queueing systems with identical general arrival streams, but different numbers of servers, different waiting room capacities, and stochastically ordered service time distributions. Under appropriate conditions, it is possible to construct two new systems on the same probability space so that the new systems are probabilistically equivalent to the original systems and each sample path of the stochastic process representing system size in one system lies entirely below the corresponding sample path in the other system. This construction implies stochastic order for these processes and many associated quantities of interest, such as a busy period, the number of customers lost in any interval, and the virtual waiting time.

Keywords

QUEUESBOUNDSSTOCHASTIC COMPARISONSFINITE WAITING ROOMSMULTI-SERVER QUEUESPATH ORDER
Type
Research Article
Information
Advances in Applied Probability , Volume 11 , Issue 2 , June 1979 , pp. 448 - 455
Copyright
Copyright © Applied Probability Trust 1979 

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References

Chung, K. L. (1974) A Course in Probability Theory. Academic Press, New York.Google Scholar
Kamae, T., Krengel, U. and O'Brien, G. L. (1977) Stochastic inequalities on partially ordered spaces. Ann. Prob. 5, 899–912.CrossRefGoogle Scholar
Kirstein, B. M. (1976) Monotonicity and comparability of time-homogeneous Markov processes with discrete state space. Math. Operationsforsch. Statist. 7, 151–168.Google Scholar
Lehmann, E. L. (1955) Ordered families of distributions. Ann. Math. Statist. 26, 399–419.Google Scholar
O'Brien, G. L. (1975) The comparison method for stochastic processes. Ann. Prob. 3, 80–88.CrossRefGoogle Scholar
Sonderman, D. (1978) Comparison Results for Stochastic Processes Arising in Queueing Systems. Ph.D Dissertation. Yale University.Google Scholar
Sonderman, D. (1979a) Comparing multi-server queues with finite waiting rooms, I: Same number of servers. Adv. Appl. Prob. 11, 439–447.CrossRefGoogle Scholar
Sonderman, D. (1979b) Comparing uniformizable semi-Markov processes. Maths. Opns. Res. To appear.Google Scholar
Sonderman, D. and Whitt, W. (1979) Comparing multi-server queues with finite waiting rooms, III; embedded sequences. To appear.CrossRefGoogle Scholar
Stidham, S. (1970) On the optimality of single-server queueing systems. Opns. Res. 18, 708–732.CrossRefGoogle Scholar
Stoyan, D. (1973) Monotonieeigenschaften einliniger Bedienungssysteme mit exponentiellen Bedienungszeiten. Apl. Mat. 18, 268–279.Google Scholar
Stoyan, D. (1977a) Qualitative Eigenschaften und Abschatzungen stochastischer Modelle. Akademie-Verlag, Berlin.CrossRefGoogle Scholar
Stoyan, D. (1977b) Bounds and approximations in queueing through monotonicity and continuity. Opns Res. 25, 851–863.Google Scholar
Yu, O. S. (1974) Stochastic bounds for heterogeneous-server queues with Erlang service-times. J. Appl. Prob. 11, 785–796.CrossRefGoogle Scholar