International Journal of Mathematics and Mathematical Sciences
Volume 26 (2001), Issue 7, Pages 427-436
doi:10.1155/S0161171201005221

On functionals of a marked Poisson process observed by a renewal process

Abstract

We study the functionals of a Poisson marked process Π observed by a renewal process. A sequence of observations continues until Π crosses some fixed level at one of the observation epochs (the first passage time). In various stochastic models applications (such as queueing with N-policy combined with multiple vacations), it is necessary to operate with the value of Π prior to the first passage time, or prior to the first passage time plus some random time. We obtain a time-dependent solution to this problem in a closed form, in terms of its Laplace transform. Many results are directly applicable to the time-dependent analysis of queues and other stochastic models via semi-regenerative techniques.