Relationships in stationary jump processes with countable state space and their applications to queues

The authors consider a stationary continuous time process with a finite or countable space, which is usually called a stationary pure jump process because it changes its state by jumps. Relations between the stationary distributions at an arbitrary time, just before and after jump instants are obtained by using conditional sojourn times in states and in sets of states. For a skip free process, these distributions have product form expressions in terms of the conditional mean sojourn times. The results are applied to queueing models. The authors extend some known relationships between queueing characteristics to batch arrival and batch serve queues with statinary inputs. The results also give a unified approach for truncation expressions for finite queues.