Whittle's index policy for a multi-class queueing system with convex holding costs

Multi-class service systems are of increasing importance in the practical modelling world but present a significant challenge for analysis. Most results to date concerning the optimal dynamic control of such systems have assumed holding cost rates to be linear in the number of customers present. In response to arguments that such an assumption is often inappropriate, we develop an index heuristic for a multi-class M/M/1 system with increasing convex holding cost rates. We use a prescription of Whittle to develop the required indices. A numerical study elucidates very strong performance of the index policy.