On the Exiting Patterns of Multivariate Renewal-Reward Processes with an Application to Stochastic Networks

This article is a study of vector-valued renewal-reward processes on Rd. The jumps of the process are assumed to be independent and identically distributed nonnegative random vectors with mutually dependent components, each of which may be either discrete or continuous (or a mixture of discrete and continuous components). Each component of the process has a fixed threshold. Operational calculus techniques and symmetries with respect to permutations are used to find a general result for the probability of an arbitrary weak ordering of threshold crossings. The analytic and numerical tractability of the result are demonstrated by an application to the reliability of stochastic networks and some other special cases. Results are shown to agree with empirical probabilities generated through simulation of the process.