Performance Analysis of an M/E k /1 Queue with Balking and Two Service Rates Based on a Single Vacation Policy

Abstract

In this chapter, we present an analysis for an M/E k /1 queue with balking and two service rates based on a single vacation policy. Customers are served at two different rates depending on the number of customers in the system. If customers on arrival find other customers in the system, they either decide to enter the queue or balk with a constant probability. The server takes a single vacation when the system becomes empty. We first formulate a quasi birth-and-death process for the queueing system. Then, we obtain the equilibrium condition of the system. By using the matrix-geometric solution method, we obtain the matrix-geometric form solution for the steady-state probability vectors. The computation of the boundary steady-state probability vectors is also discussed. Then, we derive explicitly performance measures of the system. Based on this performance analysis, we develop a cost model to determine numerically the system's optimal cost and optimal critical value. Finally, we perform a sensitivity analysis through numerical experiments.