The Tail Behavior of a Longest-Queue-Served-First Queueing System: A Random Walk in the Half Plane

In this article, we consider the longest-queue-served-first queueing system, in which two types of customers arrive, independently and respectively according to two Poisson processes with possibly unequal arrival rates, to two separate queues served, with possibly unequal service rates, by a single exponential server. This system finds important applications and has been studied before. Our focus is on the tail asymptotic behavior for the stationary distribution when the system is stable. We model this system into a reflected random walk in the half plane. This random walk reveals two different properties, respectively, along the direction of the minimum of the two queues and the direction of the difference between the two queues. Specifically, the one-dimensional process for the minimum is reducible, which makes the tail analysis along the difference direction easier, but the process for the difference is irreducible, for which the analysis is more challenging (similar to the analysis for a non-singular genus one reflected random walk). We extend the kernel method for random walks in the quarter plane to this system to obtain exact tail asymptotic expressions along both directions, which are new contributions to the literature.