Cost analysis of a finite M/M/R queueing system with balking, reneging, and server breakdowns

We study an M/M/R queueing system with finite capacity plus balking, reneging, and server breakdowns. Arriving customers balk (do not enter) with a probability (1 - b(n)) and renege (leave the queue after entering) according to a negative exponential distribution. The server can break down at any time even if no customers are in the system. Arrival and service times of the customers, and breakdown times and repair times of the servers are assumed to follow a negative exponential distribution. We use a matrix geometric method to derive the steady-state probabilities, using which various system performance measures that can be obtained. A cost model is developed to determine the optimum number of servers. Under the optimal operating conditions, numerical results are presented in which several system performance measures are evaluated based on assumed numerical values given to the system parameters. Sensitivity analysis is also investigated.