Solving convex optimization with side constraints in a multi-class queue by adaptive c\mu rule

We study convex optimization problems with side constraints in a multi-class \(M/G/1\) queue with controllable service rates. In the simplest problem of optimizing linear costs with fixed service rate, the \(c\mu \) rule is known to be optimal. A natural question to ask is whether such simple policies exist for more complex control objectives. In this paper, combining the achievable region approach in queueing systems and the Lyapunov drift theory suitable to optimize renewal systems with time-average constraints, we show that convex optimization problems can be solved by variants of adaptive \(c\mu \) rules. These policies greedily re-prioritize job classes at the end of busy periods in response to past observed delays in each job class. Our method transforms the original problems into a new set of queue stability problems, and the adaptive \(c\mu \) rules are queue stable policies. An attractive feature of the adaptive \(c\mu \) rules is that they use limited statistics of the queue, where no statistics are required for the problem of satisfying average queueing delay in each job class.