The scheduling of queues with non-linear holding costs
We consider multi-class, single server queueing systems and we seek to devise policies for server allocation which minimise some long-term cost function. In most of the work to date on the optimal dynamic control of such systems, holding cost rates are assumed to be linear in the number of customers present. Such assumptions have been argued to be unrealistic and thus inappropriate: see Van Meighem (1995). With pure priority policies, which often emerge from analyses based on linear holding cost assumptions, there is often the problem that service offered to lower priority traffic is unacceptably poor. Seeking to address such problems, we first investigate the performance of policies based on linear switching curves in an M/M/1 model with two customer types, imposing various constraints on the second moments of queue lengths. We then develop an index heuristic for a multi-class M/M/1 model with increasing convex holding cost rates. Following work by Whittle (1988), we develop the required indices and in a numerical study of two and three class systems, demonstrate the strong performance of these index policies. Performance of policies throughout the thesis, as measured by lowest costs achievable under a given policy class, (i. e. best linear switching, best threshold, or index policy) is compared with a lower bound on the minimum cost achievable under any policy. This lower bound is obtained by adopting the achievable region approach, see Bertsimas, Paschalidis & Tsitsiklis (1994) and Bertsimas & Nino-Mora (1996) in which we construct a set of constraints satisfied by the first and second moments of the queue lengths. These constraints define a relaxation of the set of achievable region performance vectors of the system. Optimisation over this relaxed region yields the lower bound. Numerical results indicate the strong performance of the index policy.