Block-partitioning matrix solution of M/M/R/N queueing system with balking, reneging and server breakdowns

In this paper, we present analysis for an M/M/R/N queueing system with balking, reneging and server breakdowns. The server is subject to breakdowns with different Poisson breakdown rates $\alpha_0 $ and $\alpha$ for the empty period of the system and the nonempty period of the system, respectively. When the server breaks down, it will be repaired immediately by a repair facility attended by $R$ repairmen. The repair times of the servers are assumed to follow a negative exponential distribution with different repair rates $\beta_0$ and $\beta$ corresponding to whether the server breaks down in the empty period of the system and the nonempty period of the system. We study not only some queueing problems of the system, but also some reliability problems of the servers. By using the partitioned block matrix method, we solved the steady-state probability equations iteratively and derived the steady-state probabilities in a matrix form. Some performance measures of queueing and reliability are obtained. A cost model is developed to determine the optimum number of servers while the system availability is maintained at a certain level. The cost analysis is also investigated by numerical results.