The Characterization of the Stationary Distribution of the Supplemented Self-Clocking Jump Process

We introduce a Self-Clocking Jump Process (SCJP), which is a natural generalization of a generalized semi-Markov process and applies to a large class of queueing models. As a stochastic process, a SCJP is a special case of a process with an embedded marked point process. We characterize the stationary distribution of the supplemented SCJP and discuss some properties of it. We also give a mathematical base to the usual approach of Kolmogorov differential equations. We use an operator theoretic argument but most of the proofs are elementary.