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Complementary generating functions for the MX/GI/1/k and GI/My/1/K queues and their application to the comparison of loss probabilities

Published online by Cambridge University Press:  14 July 2016

Masakiyo Miyazawa
Masakiyo Miyazawa*
Affiliation:
Science University of Tokyo
*
∗Postal address: Department of Information Sciences, Science University of Tokyo, Noda-city, Chiba, 278, Japan.
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Abstract

A direct proof is presented for the fact that the stationary system queue length distribution just after the service completion epochs in the Mx/GI/1/k queue is given by the truncation of a measure on Z+ = {0, 1, ·· ·}. The related truncation formulas are well known for the case of the traffic intensity ρ < 1 and for the virtual waiting time process in M/GI/1 with a limited waiting time (Cohen (1982) and Takács (1974)). By the duality of GI/MY/1/k to Mx/GI/1/k + 1, we get a similar result for the system queue length distribution just before the arrival of a customer in GI/MY/1/k. We apply those results to prove that the loss probabilities of Mx/GI/1/k and GI/MY/1/k are increasing for the convex order of the service time and interarrival time distributions, respectively, if their means are fixed.

Keywords

FINITE QUEUETRUNCATIONBATCH ARRIVALBATCH SERVICELOSS PROBABILITYCONVEX ORDERDUALITY
Type
Research Papers
Information
Journal of Applied Probability , Volume 27 , Issue 3 , September 1990 , pp. 684 - 692
Copyright
Copyright © Applied Probability Trust 1990 

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