Steady state analysis on the batch arrival Geo/G/1 queue with multiple adaptive vacations

Abstract

This paper concentrates on the steady state analysis on the discrete-time batch arrival Geo/G/1 queue with vacations. In this model, after serving all customers in the system, the server will take a random maximum number of vacations before returning to the service mode. The distributions of busy period, vacation period and vacation cycle have been given. Meanwhile, the PGFs for the off-line period and on-line period are expressed explicitly. The PGFs are also derived for the steady-state queue length, the unfinished work as well as the waiting time of an arbitrary customer in the FCFS and LCFS systems. According to the relationship of the discrete-time and continuous-time systems, we also have obtained the mean queue length, the mean unfinished work and the mean waiting time in continuous-time system. Several vacation models are special cases of the vacation model presented in this study. Moreover, we give some numerical examples to observe the effect of the vacation policy on the queue model.