Delay and rate-optimal control in a multi-class priority queue with adjustable service rates

We study two convex optimization problems in a multi-class M/G/1 queue with adjustable service rates: minimizing convex functions of the average delay vector, and minimizing average service cost, both subject to per-class delay constraints. Using virtual queue techniques, we solve the two problems with variants of dynamic cμ rules. These algorithms adaptively choose a strict priority policy, in response to past observed delays in all job classes, in every busy period. Our policies require limited or no statistics of the queue. Their optimal performance is proved by Lyapunov drift analysis and validated through simulations.