A Filtered ASTA property

Muhammad El-Taha, Department of Mathematics and Statistics, University of Southern Maine

Shaler Stidham, Jr., Department of Operations Research, University of North Carolina

Abstract

Recently the PASTA (Poisson arrivals see time averages) property has been extended to ASTA (arrivals see time averages) by eliminating the need for Poisson arrivals and weakening the LAA (lack of anticipation) assumption. This paper presents a strengthening of ASTA under the original LAA assumption of Wolff. We consider a stochastic process, X, with an associated point process, N, that admits a stochastic intensity and satisfies LAA. Various authors have noted in various contexts that ASTA holds if and only if the arrival intensity is state independent. For a class of point processes that includes doubly stochastic as well as ordinary Poisson processes, we prove that the point process obtained by restricting the process X to any given set of states not only has the same intensity but the same probabilistic structure as the original point process. In particular if the original point process is Poisson, the new point process is still Poisson with the same parameter as the original point process. For a discrete-time version, of interest in its own right, we provide a simple proof of a strengthened version of ASTA in discrete time. Unlike other discrete-time versions of ASTA, ours is valid for point processes with stationary but not necessarily independent increments. The continuous-time results are obtained using martingale theory. A corollary is a simple proof of PASTA under conditions that require only that the relevant limits exist. Our results may also provide some insight into characterizing Poisson flows in queueing systems.