On finite single-server queue subject to non-preemptive breakdowns

The paper deals with modelling and simulation of a finite single-server queueing system with a server subject to breakdowns. We consider that customers come to the queueing system in the Poisson stream. Customers incoming to the system are served according to the FCFS discipline, service times are considered to follow the Erlang distribution defined by the shape parameter and the scale parameter. Customers can wait for the service in the queue which length is limited by (m-1) places, that means the total capacity of the queueing system is equal to m places. Further, we assume that the server can break down. Breakdowns of the server are considered to be non-preemptive that means when a breakdown occurs during customer servicing it is possible to finish it before server repair is started. Times between breakdowns and repair times are assumed to follow the exponential distribution. We model the queue as a quasi-birth death process for which we present steady-state diagram and equation system describing the system behaviour in the steady-state. Solving the equation system in Matlab we get stationary probabilities which are used for computing basic performance measures. The mathematical model is supported by a simulation model in order to validate the outcomes of the mathematical model.