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

Abstract

We study the functionals of a Poisson marked processΠobserved by a renewal process. A sequence of observations continues untilΠcrosses some fixed level at one of the observation epochs (the first passage time). In various stochastic models applications (such as queueing withN-policy combined with multiple vacations), it is necessary to operate with the value ofΠprior 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.