On the two-moment approximation of the discrete-time \(\mathrm{GI}/\mathrm{G}/1\) queue with a single vacation (Q2398526)

Summary: We consider a discrete-time \(\mathrm{GI}/\mathrm{G}/1\) queue in which the server takes exactly one vacation each time the system becomes empty. The interarrival times of arriving customers, the service times, and the vacation times are all generic discrete random variables. Under our study, we derive an exact transform-free expression for the stationary system size distribution through the modified supplementary variable technique. Utilizing obtained results, we introduce a simple two-moment approximation for the system size distribution. From this, approximations for the mean system size along with the system size distribution could be obtained. Finally, some numerical examples are given to validate the proposed approximation method.