A new device in the optimization of queuing systems is introduced by using the method of phases. Non-exponential queues under control are considered with respect to the expected discounted reward criterion. For models with hyper-Erlang distributions equivalent phase-type systems are established. Approximation results for Markov decision models allow the extension to the case of general distribution functions. The approach is demonstrated by finding the form of an optimal policy for the GI/M/c queue with customer admission and batch arrival as well as for the GI/M/1 queue with interarrival time control.

A queueing model of M/G/1 type with controllable service time distribution is considered in this paper. Conditions on costs and service time distribution only are established such that there exists a monotone policy optimal relative to the average cost criterion. To prove this, it is shown that it suffices to search for a monotone optimal policy in a Markov decision model associated with the queueing model.

This paper employs the methods currently used to solve many queuing control models in order to investigate the behavior of a firm's optimal advertising and pricing strategies over time. Given that a firm's market position expands or deteriorates in a probabilistic way which depends upon the current position, the rate of advertising, and the price the firm charges, we present conditions which ensure that the optimal level of advertising is a monotonic function of the firm's market position, and we discuss the economic meaning of these conditions. Furthermore, although the primary focus is upon a non-competitive environment, we develop the above model as a non-zero sum, two-person stochastic game and show that an equilibrium strategy exists which is simple to compute.