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Some Results on Interarrival Times of Nonhomogeneous Poisson Processes

Published online by Cambridge University Press:  27 July 2009

Subhash C. Kochar
Subhash C. Kochar
Affiliation:
Indian Statistical Institute, 7, SJS Sansanwal Marg, New Delhi 110016, India
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Abstract

It is well known that in the case of a Poisson process with constant intensity function the interarrival times are independent and identically distributed, each having exponential distribution. We study this problem when the intensity function is monotone. In particular, we show that in the case of a nonhomogeneous Poisson process with decreasing (increasing) intensity the interarrival times are increasing (decreasing) in the hazard rate ordering sense and they are also jointly likelihood ratio ordered (cf. Shanthikumar and Yao, 1991, Bivariate characterization of some stochastic order relations, Advances in Applied Probability 23: 642–659). This result is stronger than the usual stochastic ordering between the successive interarrival times. Also in this case, the interarrival times are conditionally increasing in sequence and, as a consequence, they are associated. We also consider the problem of comparing two nonhomogeneous Poisson processes in terms of the ratio of their intensity functions and establish some results on the successive number of events from one process occurring between two consecutive occurrences from the second process.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 10 , Issue 1 , January 1996 , pp. 75 - 85
Copyright
Copyright © Cambridge University Press 1996

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