International Journal of Mathematics and Mathematical Sciences
Volume 26 (2001), Issue 7, Pages 427-436

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

Jewgeni H. Dshalalow1 and Jean-Baptiste Bacot2

1Applied Mathematics Program, Florida Institute of Technology, Melbourne 32901, FL, USA
2Operations Research Program, Florida Institute of Technology, Melbourne 3201, FL, USA

Received 22 May 2000

Copyright © 2001 Jewgeni H. Dshalalow and Jean-Baptiste Bacot. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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.