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The N/G/1 queue and its detailed analysis

Published online by Cambridge University Press:  01 July 2016

V. Ramaswami
V. Ramaswami*
Affiliation:
Drexel University
*
∗Postal address: Department of Mathematical Sciences, Drexel University, Philadelphia, PA 19104, U.S.A.
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Abstract

We discuss a single-server queue whose input is the versatile Markovian point process recently introduced by Neuts [22] herein to be called the N-process. Special cases of the N-process discussed earlier in the literature include a number of complex models such as the Markov-modulated Poisson process, the superposition of a Poisson process and a phase-type renewal process, etc. This queueing model has great appeal in its applicability to real world situations especially such as those involving inhibition or stimulation of arrivals by certain renewals. The paper presents formulas in forms which are computationally tractable and provides a unified treatment of many models which were discussed earlier by several authors and which turn out to be special cases. Among the topics discussed are busy-period characteristics, queue-length distributions, moments of the queue length and virtual waiting time. We draw particular attention to our generalization of the Pollaczek–Khinchin formula for the Laplace–Stieltjes transform of the virtual waiting time of the M/G/1 queue to the present model and the resulting Volterra system of integral equations. The analysis presented here serves as an example of the power of Markov renewal theory.

Keywords

COMPUTATIONAL PROBABILITYSINGLE-SERVER QUEUEPHASE-TYPE DISTRIBUTIONSN-PROCESSQUEUE LENGTHWAITING TIME
Type
Research Article
Information
Advances in Applied Probability , Volume 12 , Issue 1 , March 1980 , pp. 222 - 261
Copyright
Copyright © Applied Probability Trust 1980 

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Footnotes

This paper is based on Part I of a Ph.D. dissertation submitted to Purdue University. This research was supported by AFOSR-72-2350C at the Department of Statistics, Purdue University and AFOSR-77-3236 at the Department of Statistics and Computer Science, University of Delaware. A detailed version of the paper in technical report form may be obtained by writing to the author.

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