This paper is concerned with a certain property of the stationary distribution of a generalized semi-Markov process (GSMP) known as insensitivity. It is well-known that the so-called Matthes' conditions form a necessary and sufficient algebraic criterion for insensitivity. Most proofs of these conditions are basically algebraic. By interpreting a GSMP as a simple queueing network, we are able to show that Matthes' conditions are equivalent to the quasi-reversibility of the network, thus obtaining another simple proof of the sufficiency of these conditions. Furthermore, we apply our method to find a simple criterion for the insensitivity of GSMP's with generalized routing (in a sense that is introduced in the paper).

Our primary aim in this paper is to study a functional equation that arises in a problem of queueing. Consider a queue with compound Poisson arrivals and general service times with a gating mechanism. The gating mechanism takes in at most m(≤∞) customers at a time for service and serves these customers according to the processor-sharing discipline. In this paper, we examine various performance characteristics for this queue. The characteristics include waiting-time distribution, queue-length distribution, time spent with the server, and batch-size distribution for service.

Given a stochastic activity network in which the length of some or all of the arcs are random variables with known probability distributions. This paper concentrates on identifying the shortest path and the M shortest paths in the network and on using the M paths to identify surrogate stochastic networks which are amenable for deriving analytical solutions. First, it identifies the M shortest paths using a certain form of stochastic dominance. Second, it identifies the M shortest paths by applying the deterministic methods to the network resulting from replacing the random length of every arc by its mean value. The two sets of the M paths are compared with those obtained by Monte Carlo sampling. Finally, the paper investigates how the distributional properties of the shortest path in the surrogate network compare with those of the shortest path in the original stochastic network.