Stochastic Comparisons in Vacation Models.

Abstract

We consider single-server queueing systems, known as vacation models, where at every service completion the server might either serve the next customer from the queue (if any) or take a vacation (i.e., become unavailable to the customers for a random period of time) depending on the service schedule of the model. Using coupling arguments, we make stochastic comparisons between quantities of interest of one vacation model to those of another vacation model under a different service schedule. We first establish a stochastic ordering for sequences of service completion epochs, from which stochastic comparisons for waiting time sequences and queue size processes can be deduced. These comparisons are then used to obtain some monotonicity results for vacation models with limited and Bernoulli service schedules.