A server backup model with Markovian arrivals and phase type services

We consider a finite capacity queueing system with one main server who is supported by a backup server. We assume Markovian arrivals, phase type services, and a threshold-type server backup policy with two pre-determined lower and upper thresholds. A request for a backup server is made whenever the buffer size (number of customers in the queue) hits the upper threshold and the backup server is released from the system when the buffer size drops to the lower threshold or fewer at a service completion of the backup server. The request time for the backup server is assumed to be exponentially distributed. For this queuing model we perform the steady state analysis and derive a number of performance measures. We show that the busy periods of the main and backup servers, the waiting times in the queue and in the system, are of phase type. We develop a cost model to obtain the optimal threshold values and study the impact of fixed and variable costs for the backup server on the optimal server backup decisions. We show that the impact of standard deviations of the interarrival and service time distributions on the server backup decisions is quite different for small and large values of the arrival rates. In addition, the pattern of use of the backup server is very different when the arrivals are positively correlated compared to mutually independent arrivals.