Convexity and Optimal Load Distributions in Work Conserving */*/1 Queues

In this paper we develop fundamental convexity properties of unfinished work and packet waiting time in a work conserving */*/1 queue. The queue input consists of an uncontrollable background process and a rate-controllable input stream. We show that any moment of unfinished work is a convex function of the controllable input rate. The convexity properties are then extended to address the problem of optimal routing of arbitrary input streams over a collection of N queues in parallel with different (possibly time-varying) linespeeds (μ1(t),...,μN(t)). Our convexity results hold for stream-based routing (where individual packet streams must be routed to the same queue) as well as for packet-based routing where each packet is routed to a queue using some pre-determined splitting method, such as probabilistic splitting. Our analysis of these general systems is carried out by introducing a new function of the superposition of two input streams that we call the blocking function. Using this function facilitates analysis and provides much insight into the sample path dynamics of */*/1 queues.