The Run Probabilities of Tes Processes

The run statistics of a discrete-time, real-valued stochastic process are the statistics of process excursions above a given level. As such they are a special case of first passage times (hitting times) in discrete time. The study of run probabilities is motivated by applications such as compressed video, where a random and autocorrelated sequence of compressed frames arrives deterministically at a finite buffer, and the loss probability of consecutive frames (runs) constitutes a better measure of service quality than simple loss probabilities. This paper studies the run probabilities of a subclass of TES processes. A uniform TES process is a modulo-1 autoregressive stochastic process, uniform on [0; 1); general TES processes are obtained by transforming a basic TES process to ones with general marginals. The paper develops an integral equation in the generating function of the run probabilities of TES processes. An exact matrix solution of theoretical interest is obtained, but the solution requires expensive computations including inverting and margining. We, therefore, derive additionally a Sokolov-type method to obtain numerical approximations. We also identify a class of TES processes for which a closed-form solution can be exhibited. Finally, we give some numerical examples that support the efficacy of our approach by comparing the approximate run probabilities to Monte Carlo simulation statistics, as well as the exact solution in the transform domain.