Queueing Systems

, Volume 59, Issue 1, pp 63–86

Continuity theorems for the M/M/1/n queueing system

Article

Abstract

In this paper continuity theorems are established for the number of losses during a busy period of the M/M/1/n queue. We consider an M/GI/1/n queueing system where the service time probability distribution, slightly different in a certain sense from the exponential distribution, is approximated by that exponential distribution. Continuity theorems are obtained in the form of one or two-sided stochastic inequalities. The paper shows how the bounds of these inequalities are changed if further assumptions, associated with specific properties of the service time distribution (precisely described in the paper), are made. Specifically, some parametric families of service time distributions are discussed, and the paper establishes uniform estimates (given for all possible values of the parameter) and local estimates (where the parameter is fixed and takes only the given value). The analysis of the paper is based on the level crossing approach and some characterization properties of the exponential distribution.

Keywords

Continuity theorems Loss systems M/GI/1/n and M/M/1/n queues Busy period Branching process Number of level crossings Kolmogorov (uniform) metric Stochastic ordering Stochastic inequalities

Mathematics Subject Classification (2000)

60K25 60B05 60E15 62E17

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