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Stochastic comparisons for queueing models via random sums and intervals

Part of: Distribution theory - Probability Special processes

Published online by Cambridge University Press:  01 July 2016

Alain Jean-Marie  and
Zhen Liu
Alain Jean-Marie
Affiliation:
INRIA
Zhen Liu*
Affiliation:
INRIA
*
∗Postal address for both authors: INRIA Sophia-Antipolis, 2004 Route des Lucioles, BP 109, 06565 Valbonne, France.
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Abstract

We consider the relationships among the stochastic ordering of random variables, of their random partial sums, and of the number of events of a point process in random intervals. Two types of result are obtained. Firstly, conditions are given under which a stochastic ordering between sequences of random variables is inherited by (vectors of) random partial sums of these variables. These results extend and generalize theorems known in the literature. Secondly, for the strong, (increasing) convex and (increasing) concave stochastic orderings, conditions are provided under which the numbers of events of a given point process in two ordered random intervals are also ordered.

These results are applied to some comparison problems in queueing systems. It is shown that if the service times in two M/GI/1 systems are compared in the sense of the strong stochastic ordering, or the (increasing) convex or (increasing) concave ordering, then the busy periods are compared for the same ordering. Stochastic bounds in the sense of increasing convex ordering on waiting times and on response times are provided for queues with bulk arrivals. The cyclic and Bernoulli policies for customer allocation to parallel queues are compared in the transient regime using the increasing convex ordering. Comparisons for the five above orderings are established for the cycle times in polling systems.

Keywords

STOCHASTIC ORDERINGPARTIAL SUMSPOINT PROCESSNUMBER OF EVENTSSTOCHASTIC BOUNDSBUSY PERIODBULK ARRIVALROUTINGPOLLING SYSTEMS

MSC classification

Primary: 60E05: Distributions
Secondary: 60K25: Queueing theory
Type
Research Article
Information
Advances in Applied Probability , Volume 24 , Issue 4 , December 1992 , pp. 960 - 985
Copyright
Copyright © Applied Probability Trust 1992 

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