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Two servers in series, studied in terms of a Markov renewal branching process

Published online by Cambridge University Press:  01 July 2016

Marcel F. Neuts
Marcel F. Neuts*
Affiliation:
Cornell University and Purdue University
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Abstract

This paper discusses the transient and limiting behavior of a system of queues, consisting of two service units in tandem in which the second unit has finite capacity. When the second unit reaches full capacity, a phenomenon termed “blocking” occurs. A wide class of rules to resolve blocking is defined and studied in a unified way.

The input to the first unit is assumed to be Poisson, the service times in the first unit are independent with a general, common distribution. When the system is not blocked, the second unit releases its customers according to a state-dependent, death process.

The analysis of the time-dependence relies heavily on several imbedded Markov renewal processes. In particular, the analog of the busy period for the M/G/1 queue is modeled here as a “Markov renewal branching process”. The study of this process requires the definition of a class of matrix functions which generalizes some classical definitions of matrix function. In terms of these “matrix functions” we are led to consider functional iterates and a matrix analog of Takács' functional equation for the transform of the distribution of the busy period in the M/G/1 model.

We further discuss the joint distribution of the queue lengths in units I and II and its marginal and limiting distributions.

A final section is devoted to an informal discussion on how the numerical analysis of this system of queues may be organized.

Type
Research Article
Information
Advances in Applied Probability , Volume 2 , Issue 1 , Spring 1970 , pp. 110 - 149
Copyright
Copyright © Applied Probability Trust 

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