Random Walk Processes in a Bilevel (M-N)-Policy Queue with Multiple Vacations

Abstract

In this article we introduce a bulk queue with a multiple vacations policy. When the queue is exhausted, the server goes on multiple vacations until the queue replenishes to M or more customers. On his return, if the queue crosses an N as well, the server resumes his service. Otherwise, he waits until the queue to reach or exceed N. We use fluctuation analysis and game-theoretic approach to obtain a closed form functional of the queue length at the beginning of a busy period followed by a Kendall-like formula for the queueing process upon departures. The formation of the queueing process during vacations and waiting periods can be modeled as a multistage game of two players, with the use of past results from stochastic games to arrive at explicit functionals of the queueing process. Then we continue investigating the queueing process, now with continuous time parameter, using time sensitive analysis and semi-regenerative techniques. This enables us to work on potentials of related processes and obtain various performance measures, including the mean buffer load, switchovers rate, and mean stationary service cycle. Special cases demonstrate analytical tractability of the results.