Optimal control of retrial queues with finite population and state-dependent service rate

Abstract

The research of wide class of retrial queuing systems faces the problem of calculating characteristics of the system in stationary regime. Markov chain that describes service process in such system is multidimensional and its transaction matrix usually does not have special properties that would streamline the explicit solution of Kolmogorov set of equations. In addition, the probabilities of transition between states of the controlled system depend on its current state that complicates their obtaining even more. Therefore, only the simplest models are explicitly researched on this moment. In this paper we consider a finitesource retrial queue with c servers and controlled parameters. The primary calls arrive from n customers. Each customer after some random period of time which is exponential distributed random variable tries to get service and is served immediately if there is any free server. Service times are also exponentially distributed. The customer who finds all servers busy leaves the system and returns after an exponential time. Two- and three-dimensional Markov models that describe threshold and hysteresis control policies are taken into account. Explicit vector-matrix representations of stationary distributions are main results in both cases. Also we state and give an algorithm for solving a multi-criteria problem of maximization of total income from the system.