M/M/1 Queue with Controllable Service Rate

Abstract

In this paper we consider an M/M/1 queue, in which the customer service rate is allowed to be increased and decreased by a fixed value at each customer service completion. These changes in service rate are controlled by probabilities depending on the actual number of customers and the actual service rate. The dependency on the number of customers follows a specific power form, while the dependency on actual service rate is general and independent of the dependency on the number of customers. We describe how to compute the partial stationary distribution of the service rate values when the system is empty. Based on it we provide a computational procedure for computing the stationary probability vectors of the number of customers in the system. We derive also the vector probability generating function and the vector mean of the stationary number of customers. We establish a methodology which utilizes the specific structure of the model. This methodology inherits some element from the stationary analysis of the standard QBD model and provides a first order, forward algorithm for computing the stationary probability vectors of the number of customers in the system.