Analysis of an M/M/1/N Queue with Balking, Reneging and Server Vacations

This paper presents an analysis for an M/M/1/N queueing system with balking, reneging and server vacations. Arriving customers balk (do not enter) with a probability and renege (leave the queue after entering) according to a negative exponential distribution. It is assumed that the server has a multiple vacation. By using the Markov process method, we first develop the equations of the steady state probabilities. Then, we derive the matrix form solution of the steady-state probabilities. Next, we give some performance measures of the system. Based on the performance analysis, we formulate a cost model to determine the optimal service rate. Finally, we present some numerical examples to demonstrate how the various parameters of the model influence the behavior of the system.