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A model for interaction of a Poisson and a renewal process and its relation with queuing theory

Published online by Cambridge University Press:  14 July 2016

Lennart Råde
Lennart Råde*
Affiliation:
Chalmers University of Technology, Gothenburg, Sweden
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Abstract

This paper discusses the response process when a Poisson process interacts with a renewal process in such a way that one or more points of the Poisson process eliminate a random number of consecutive points of the renewal process. A queuing situation is devised such that the c.d.f. of the length of the busy period is the same as the c.d.f. of the length of time intervals of the renewal response process. The Laplace-Stieltjes transform is obtained and from this the expectation of the time intervals of the response process is derived. For a special case necessary and sufficient conditions for the response process to be a Poisson process are found.

Keywords

RENEWAL PROCESSPOISSON PROCESSINTERACTIONQUEUING THEORYFINITE WAITING ROOM
Type
Short Communications
Information
Journal of Applied Probability , Volume 9 , Issue 2 , June 1972 , pp. 451 - 456
Copyright
Copyright © Applied Probability Trust 1972 

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References

[1] Coleman, R. and Gastwirth, J. L. (1969) Some models for interaction of renewal processes related to neuron firing. J. Appl. Prob. 6, 38–58.CrossRefGoogle Scholar
[2] Ten Hoopen, M. and Reuver, H. (1965) Selective interaction of two independent recurrent processes. J. Appl. Prob. 2, 286–292.CrossRefGoogle Scholar