Analysis of a queueing model with contingent additional server

Abstract

We consider a two-server queueing system with a finite buffer. Customers arrive to the system according to the Markovian arrival process. Normally, only one server is active. The service time of a customer has a phase-type distribution. An additional server is activated only if the queue length exceeds some fixed preassigned threshold. The service time by the additional server also has a phase-type distribution with the same state space. While the underlying Markov chains of service at two servers have non-coinciding states, service in two servers is provided independently. But if it occurs that the underlying Markov chain for one server, say, server 1, needs transition to the state, at which the underlying Markov chain for server 2 is currently staying, service in the server 1 is postponed until the Markov chain for server 2 transits to another state. Dynamics of the system is described by the multidimensional Markov chain. The generator of this Markov chain is written down. Expressions for computation of performance measures are derived. Problem of numerical determination of the optimal threshold is solved.