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Comparing counting processes and queues

Published online by Cambridge University Press:  01 July 2016

Ward Whitt
Ward Whitt*
Affiliation:
Bell Laboratories
*
∗Postal address: Bell Laboratories, Holmdel, NJ 07733, U.S.A.
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Abstract

Several partial orderings of counting processes are introduced and applied to compare stochastic processes in queueing models. The conditions for the queueing comparisons involve the counting processes associated with the interarrival and service times. The two queueing processes being compared are constructed on the same probability space so that each sample path of one process lies below the corresponding sample path of the other process. Stochastic comparisons between the processes and monotone functionals of the processes follow immediately from this construction. The stochastic comparisons are useful to obtain bounds for intractable systems. For example, the approach here yields bounds for queues with time-dependent arrival rates.

Keywords

QUEUEINGSTOCHASTIC ORDERSTOCHASTIC COMPARISONSCOUNTING PROCESSPOINT PROCESS ON THE LINEBOUNDS
Type
Research Article
Information
Advances in Applied Probability , Volume 13 , Issue 1 , March 1981 , pp. 207 - 220
Copyright
Copyright © Applied Probability Trust 1981 

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References

Arjas, E. and Lehtonen, T. (1978) Approximating many server queues by means of single server queues. Math. Operat. Res. 3, 205–223.CrossRefGoogle Scholar
Barlow, R. E. and Proschan, F. (1975) Statistical Theory of Reliability and Life Testing. Holt, Rinehart and Winston, New York.Google Scholar
Breiman, L. (1968) Probability. Addison Wesley, Reading, MA.Google Scholar
Iglehart, D. L. and Whitt, W. (1970) Multiple channel queues in heavy traffic, I. Adv. Appl. Prob. 2, 150–177.Google Scholar
Jacobs, D. R. and Schach, S. (1972) Stochastic order relationships between GI/G/k systems. Ann. Math. Statist. 43, 1623–1633.Google Scholar
Jensen, A. (1953) Markov chains as an aid in the study of Markov processes. Skand. Akt. 36, 87–91.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
Lindvall, T. (1973) Weak convergence of probability measures and random functions in the function space D[0, ∞). J. Appl. Prob. 10, 109–121.Google Scholar
Miller, D. R. (1979) Almost sure comparisons of renewal processes and Poisson processes, with applications to reliability theory. Math. Operat. Res. 4, 406–413.Google Scholar
Niu, S. (1977) Bounds and comparisons for some queueing systems. Operations Research Center Report 77–32, University of California, Berkeley.Google Scholar
O'Brien, G. L. (1975) Inequalities for queues with dependent interarrival and service times. J. Appl. Prob. 12, 653–656.CrossRefGoogle Scholar
Parthasarathy, K. R. (1967) Probability Measures on Metric Spaces. Academic Press, New York.Google Scholar
Schassberger, R. (1976) On the equilibrium distribution of a class of finite-state generalized semi-Markov processes. Math. Operat. Res. 1, 395–406.CrossRefGoogle Scholar
Schmidt, V. (1976) On the non-equivalence of two criteria of comparability of stationary point processes. Zast. Mat. 15, 33–37.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 multi-server queues with finite waiting rooms, I: same number of servers. Adv. Appl. Prob. 11, 439–447.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. (1980) Comparing semi-Markov processes. Math. Operat. Res. 5, 110–119.Google Scholar
Stidham, S. (1970) On the optimality of single-server queueing systems. Operat. Res. 18, 708–732.Google Scholar
Stoyan, D. (1977) Bounds and approximations in queueing through monotonicity and continuity. Operat. Res. 25, 851–863.CrossRefGoogle Scholar
Stoyan, D. and Stoyan, H. (1976) Some qualitative properties of single server queues. Math. Nachr. 70, 29–35.Google Scholar
Yu, O. S. (1974) Stochastic bounds for heterogeneous-server queues with Erlang service times. J. Appl. Prob. 11, 785–796.Google Scholar