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The infinite server queue with semi-Markovian arrivals and negative exponential services

Published online by Cambridge University Press:  14 July 2016

Marcel F. Neuts  and
Shun-Zer Chen
Marcel F. Neuts
Affiliation:
Purdue University, Lafayette, Indiana
Shun-Zer Chen
Affiliation:
Purdue University, Lafayette, Indiana
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Abstract

The queue with an infinite number of servers with a semi-Markovian arrival process and with negative exponential service times is studied. The queue length process and the type of the last customer to join the queue before time t are studied jointly, both in continuous and in discrete time. Limiting stationary probabilities are also obtained.

Keywords

INFINITE SERVER QUEUESEMI-MARKOV ARRIVALSNEGATIVE EXPONENTIAL SERVICE TIMESTRANSIENT BEHAVIORLIMITING BEHAVIORMARKOV RENEWAL THEORY
Type
Short Communications
Information
Journal of Applied Probability , Volume 9 , Issue 1 , March 1972 , pp. 178 - 184
Copyright
Copyright © Applied Probability Trust 1972 

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References

[1] Pyke, R. (1961) Markov renewal processes: Definitions and preliminary properties. Ann. Math. Statist. 32, 1231–1242.Google Scholar
[2] Pyke, R. (1961) Markov renewal processes with finitely many states. Ann. Math. Statist. 32, 1243–1259.Google Scholar
[3] Takács, L. (1962) Introduction to the Theory of Queues. Oxford University Press New York.Google Scholar