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On a Markovian queue with weakly correlated interarrival times

Published online by Cambridge University Press:  14 July 2016

Guy Latouche
Guy Latouche*
Affiliation:
Université Libre de Bruxelles
*
∗Postal address: Université Libre de Bruxelles, CP212, Faculté des Sciences, Boulevard du Triomphe, B-1050 Bruxelles, Belgium.
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Abstract

A queueing system with exponential service and correlated arrivals is analysed. Each interarrival time is exponentially distributed. The parameter of the interarrival time distribution depends on the parameter for the preceding arrival, according to a Markov chain. The parameters of the interarrival time distributions are chosen to be equal to a common value plus a factor ofε, where ε is a small number. Successive arrivals are then weakly correlated.

The stability condition is found and it is shown that the system has a stationary probability vector of matrix-geometric form. Furthermore, it is shown that the stationary probabilities for the number of customers in the system, are analytic functions ofε, for sufficiently smallε, and depend more on the variability in the interarrival time distribution, than on the correlations.

Keywords

QUEUEING SYSTEMCORRELATED ARRIVALSCOMPUTATIONAL PROBABILITY
Type
Research Papers
Information
Journal of Applied Probability , Volume 18 , Issue 1 , March 1981 , pp. 190 - 203
Copyright
Copyright © Applied Probability Trust 

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