The paper examines multivariate delayed marked renewal processes, of which one component is formed by a delayed compound Poisson process observed at epochs of some point process. In addition, the values of these observations (and other components) are watched when crossing their respective thresholds and the value of the original Poisson process at any moment of time, past the first passage time, is the objective of this investigation. The results (which are imperative for classes of semiregenerative processes) are given in closed analytical forms and illustrated on various stochastic models.

This paper studies the queueing process in a class of D-policy models with Poisson bulk input, general service time, and four different vacation scenarios, among them a multiple vacation, single vacation and idle server. The D-policy specifies a busy period discipline, which requires an idle or vacationing server to resume his service when the workload process crosses some fixed level D. The analysis of the queueing process is based on the theory of fluctuations for three-dimensional marked counting processes presented in the paper. For all models, we derive the stationary distributions for the embedded and continuous time parameter queueing processes in closed analytic forms and illustrate the results by a number of examples and applications.