  We consider optimizing average queueing delay and average power consumption
in a nonpreemptive multi-class M/G/1 queue with dynamic power control that
affects instantaneous service rates. Four problems are studied: (1) satisfying
per-class average delay constraints; (2) minimizing a separable convex function
of average delays subject to per-class delay constraints; (3) minimizing
average power consumption subject to per-class delay constraints; (4)
minimizing a separable convex function of average delays subject to an average
power constraint. Combining an achievable region approach in queueing systems
and the Lyapunov optimization theory suitable for optimizing dynamic systems
with time average constraints, we propose a unified framework to solve the
above problems. The solutions are variants of dynamic $c\mu$ rules, and
implement weighted priority policies in every busy period, where weights are
determined by past queueing delays in all job classes. Our solutions require
limited statistical knowledge of arrivals and service times, and no statistical
knowledge is needed in the first problem. Overall, we provide a new set of
tools for stochastic optimization and control over multi-class queueing systems
with time average constraints.
