Performance analysis of an M/M/c/N queueing system with balking, reneging and synchronous vacations of partial servers

In this paper, we present an analysis for an M/M/c/N queueing system with balking, reneging and synchronous vacations of partial servers together. It is assumed that arriving customers balk with a probability and renege according to an exponential distribution. The vacation policy prescribes that any d (0 < d < c) servers take a vacation together when these d servers find that no customers are waiting in the line at a service completion instant and c-d servers are always available, either serving the customers or remaining idle. At a vacation completion instant, if the number of customers in the system is not more than c-d, these d servers take another vacation together; otherwise, these d servers return to serving the queue. The service times and the vacation times are assumed to follow independent exponential distributions. By using the Markov process theory, we first develop the equations of the steady state probabilities and derive a matrix form solution of the steady-state probabilities. Then we give some performance measures of the system such as the expected number of waiting customers, the expected number of the customers in the system, the average rate of customer loss due to impatience, etc. Based on the performance analysis, we formulate a cost model to determine the optimal number of servers on vacation. Finally, we perform sensitivity analysis through numerical experiments.