A new look at transient versions of Little's law, and M/G/1 preemptive last-come-first-served queues

Abstract

We take a new look at transient, or time-dependent Little laws for queueing systems. Through the use of Palm measures, we show that previous laws (see Bertsimas and Mourtzinou (1997)) can be generalized. Furthermore, within this framework, a new law can be derived as well, which gives higher-moment expressions for very general types of queueing system; in particular, the laws hold for systems that allow customers to overtake one another. What is especially novel about our approach is the use of Palm measures that are induced by nonstationary point processes, as these measures are not commonly found in the queueing literature. This new higher-moment law is then used to provide expressions for all moments of the number of customers in the system in an M/G/1 preemptive last-come-first-served queue at a time t > 0, for any initial condition and any of the more famous preemptive disciplines (i.e. preemptive-resume, and preemptive-repeat with and without resampling) that are analogous to the special cases found in Abate and Whitt (1987c), (1988). These expressions are then used to derive a nice structural form for all of the time-dependent moments of a regulated Brownian motion (see Abate and Whitt (1987a), (1987b)).