Analysis of the discrete-time GI/Geo/1/ queue with single geometric vacation

Abstract

Queueing models are generally sorted into two types. They are continuous-time queue and discrete-time queue. Discrete models have a variety of assumptions. Because of the assumptions, the results look much different from continuous model results. GI/Geo/1/SGV queuing system is discrete system and the server takes exactly one geometric vacation each time the system empties. In this paper, Markov chain is defined by using EAS(Early Arrival System). Then from the transition probability matrix P associated with the imbedded Markov chain, the balance equations are obtained. From that, the probability generating functions of the stationary queue length and the stationary FCFS (First come First service) sojourn time are obtained. Then the results are compared with corresponding continuous-time counterparts. The results are also compared with GI/Geo/1 with multiple geometric vacation.