SOME RESULTS ON INTERARRIVAL TIMES OF NONHOMOGENEOUS POISSON PROCESSES

It is well known that in the case of a Poisson process with constant i ntensity function the interarrival times are independent and identical ly distributed, each having exponential distribution. We study this pr oblem 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 (decreasin g) in the hazard rate ordering sense and they are also jointly likelih ood ratio ordered (cf. Shanthikumar and Yao, 1991, Bivariate character ization of some stochastic order relations, Advances in Applied Probab ility 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 success ive number of events from one process occurring between two consecutiv e occurrences from the second process.