Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-24T08:57:09.498Z Has data issue: false hasContentIssue false
  • English
  • Français

Comparing multi-server queues with finite waiting rooms, I: Same number 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.
Get access Rights & Permissions [Opens in a new window]

Abstract

We compare two queueing systems with the same number of servers that differ by having stochastically ordered service times and/or interarrival times as well as different waiting room capacities. We establish comparisons for the sequences of actual-arrival and departure epochs, and demonstrate by counterexample that many stochastic comparisons possible with infinite waiting rooms no longer hold with finite waiting rooms.

Keywords

QUEUESBOUNDSSTOCHASTIC COMPARISONSFINITE WAITING ROOMSMULTI-SERVER QUEUESPATH ORDER
Type
Research Article
Information
Advances in Applied Probability , Volume 11 , Issue 2 , June 1979 , pp. 439 - 447
Copyright
Copyright © Applied Probability Trust 1979 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Chung, K. L. (1974) A Course in Probability Theory. Academic Press, New York.Google Scholar
Daley, D. J. and Moran, P. A. P. (1968) Two-sided inequalities for waiting time and queue size in GI/G/1. Theory Prob. Appl. 13, 356–358.CrossRefGoogle Scholar
Jacobs, D. R. and Schach, S. (1972) Stochastic order relationships between GI/G/k systems. Ann. Math. Statist. 43, 1623–1632.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
Kiefer, J. and Wolfowitz, J. (1955) On the theory of queues with many servers. Trans. Amer. Math. Soc. 78, 1–18.Google 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.Google Scholar
Sonderman, D. (1978) Comparison Results for Stochastic Processes Arising in Queueing Systems. Ph.D. Dissertation. Yale University.Google Scholar
Sonderman, D. (1979a) Comparing uniformizable semi-Markov processes. Maths. Opns. Res. To appear.Google Scholar
Sonderman, D. (1979b) Comparing multi-server queues with finite waiting rooms, II: Different numbers of servers. Adv. Appl. Prob. 11, 448–455.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.Google 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.Google 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.Google Scholar