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A bivariate Poisson queueing process that is not infinitely divisible

Published online by Cambridge University Press:  24 October 2008

D. J. Daley
D. J. Daley
Affiliation:
Department of Statistics, Institute of Advanced Studies, The Australian National University
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Extract

We are using the term ‘bivariate Poisson process’ to describe a bivariate point process (N1(.), N2(.)) whose components (or, marginal processes) are Poisson processes. In this we are following Milne (2) who amongst his examples cites the case where N1(.) and N2(.) refer to the input and output processes respectively of the M/G/∈ queueing system. Such a bivariate point process is infinitely divisible. We shall now show that in a stationary M/M/1 queueing system (i.e. Poisson arrivals at rate λ, exponential service at rate µ > λ, single-server) a similar identification of (N1(.), N2(.)) yields a bivariate Poisson process that is not infinitely divisible.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 72 , Issue 3 , November 1972 , pp. 449 - 450
Copyright
Copyright © Cambridge Philosophical Society 1972

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References

REFERENCES

(1)Dwass, M. and Teicher, H.On infinitely divisible random vectors. Ann. Math. Statist. 28 (1957), 461–470.CrossRefGoogle Scholar
(2)Milne, R. K. Stochastic analysis of multivariate point processes. Ph.D. thesis, Australian National University (1971).Google Scholar
(3)Rejch, E.Waiting times when queues are in tandem. Ann. Math. Statist. 28 (1957), 768–773.CrossRefGoogle Scholar