Time-Dependent Palm Probabilities and Queueing ... - Eurandom

Time-Dependent Palm Probabilities and Queueing Applications Brian H. Fralix, Germ´an Ria˜ no, Richard F. Serfozo July 17, 2007 Abstract This study shows that time-dependent Palm probabilities of a nonstationary process are expressible as integrals of a certain stochastic intensity process. A consequence is a characterization of a Poisson process in terms of time-dependent Palm probabilities. These two results are analogous to results of Papangelou and Mecke, respectively, for stationary point processes. Included is a new proof of Watanabe’s characterization of a Poisson process. Next, using stochastic intensities of time-dependent Palm probabilities, we present conditions under which the distribution of a stochastic process (e.g., a queueing process) at a fixed time is equal to its Palm probability distribution conditioned on a jump at that time. This result is a time-dependent analogue of an ASTA property (arrivals see time averages) for stationary processes. Another result is that, for an asymptotically stationary process, a Palm probability Pt conditioned on a point at a time t converges weakly to a Palm probability for a stationary process as t → ∞. We also present formulas for time-dependent Palm probabilities of Markov processes, and Little laws for queueing systems that relate queue-length processes to time-dependent Palm probabilities of sojourn times of the items in the system.

1

Introduction

Consider a point process N on < that is stationary (N (B) denotes the number of points in a set B, and the joint distribution of its increments is invariant under shifts in time). Let P 0 denote the Palm probability of N conditioned that N has a point at 0. Papangelou [19] proved that the absolute continuity P 0