Equilibrium joining strategies in M/M/1 Queues with working vacation and vacation interruptions^∗

Department of Mathematics, Beijing Jiaotong University, Beijing 100044, P.R. China.

^† Corresponding author (Jinting Wang). Email: jtwang@bjtu.edu.cn

Received: 17 September 2014
Accepted: 1 July 2015

Abstract

We study the equilibrium joining strategies for customers in an M/M/1 queue with working vacations and vacation interruptions. The service rate switches between a low and a high value depending on system dynamics. The server will take a multiple working vacation when the system is empty, during which a low service rate is provided to the arriving customers if any. Upon completion of the first customer’s service, given that the system is not empty, the working vacation will be terminated which means the server comes back and serves the following customers with a higher service rate. Otherwise, if the system is found empty upon completion of the first service, the server will continue his working vacation. Arriving customers may or may not know the state of the server and/or the number of the customers upon arrival, but they have to decide whether to enter the system or balk based on a linear reward-cost structure. We investigate customer behavior according to different levels of information regarding the system state. The equilibrium strategies for the customers are derived and the stationary behavior of the system under these strategies are analyzed. Finally, the effect of different levels of information on equilibrium thresholds and equilibrium entrance probabilities is illustrated by several numerical examples.