Stochastic scheduling in a multiclass G/G/1 queue

The problem of scheduling customers in a multiclass G/G/1 queue is addressed so as to minimize a weighted sum of the work-loads of the different classes. It is established that the nonidling preemptive fixed priority policy that schedules customers belonging to the class having the maximum weight, minimizes the cost function pathwise at any point in time. This result is based on the application of elementary forward induction arguments and is shown to hold for a very general class of policies. A proof for the optimality of the μc-rule in the multiclass G/M/1 queue is then obtained as an easy corollary of the first result