First Passage Analysis in a Queue with State Dependent Vacations

Abstract

This paper deals with a single-server queue where the server goes on maintenance when the queue is exhausted. Initially, the maintenance time is fixed by deterministic or random number T. However, during server’s absence, customers are screened by a dispatcher who estimates his service times based on his needs. According to these estimates, the dispatcher shortens server’s maintenance time and as the result the server returns earlier than planned. Upon server’s return, if there are not enough customers waiting (under the N-Policy), the server rests and then resumes his service. At first, the input and service are general. We then prove a necessary and sufficient condition for a simple linear dependence between server’s absence time (including his rest) and the number of waiting customers. It turns out that the input must be (marked) Poisson. We use fluctuation and semi-regenerative analyses (previously established and embellished in our past work) to obtain explicit formulas for server’s return time and the queue length, both with discrete and continuous time parameter. We then dedicate an entire section to related control problems including the determination of the optimal T-value. We also support our tractable formulas with many numerical examples and validate our results by simulation.