Optimal Control of State-Dependent Service Rates in a MAP/M/1 Queue

Abstract

In this paper, we study the optimal control of service rates in a queueing system with a Markovian arrival process (MAP) and exponential service times. The service rate is allowed to be state dependent, i.e., we can adjust the service rate according to the queue length and the phase of the MAP. The cost function consists of holding cost and operating cost. The goal is to find the optimal service rates that minimize the long-run average total cost. To achieve that, we use the matrix-analytic methods (MAM) together with the sensitivity-based optimization (SBO) theory. A performance difference formula is derived, which can quantify the difference of the long-run average total cost under any two different settings of service rates. Based on the difference formula, we show that the long-run average total cost is monotone in the service rate and the optimal control is a bang–bang control. We also show that, under some mild conditions, the optimal control policy of service rates is of a quasi-threshold-type. By utilizing the MAM theory, we propose a recursive algorithm to compute the value function related quantities. An iterative algorithm to efficiently find the optimal policy, which is similar to policy iteration, is proposed based on the SBO theory. We further study some special cases of the problem, such as the optimality of the threshold-type policy for the M/M/1 queue. Finally, a number of numerical examples are presented to demonstrate the main results and explore the impact of the phase of the MAP on the optimization in the MAP/M/1 queue.