A Stochastic Fluid Model Approach to the Stationary Distribution of the Maximum Priority Process

In traditional priority queues, we assume that every customer upon arrival has a fixed, class-dependent priority, and that a customer may not commence service if a customer with a higher priority is present in the queue. However, in situations where a performance target in terms of the tails of the class-dependent waiting time distributions has to be met, such models of priority queueing may not be satisfactory. In fact, there could be situations where high priority classes easily meet their performance target for the maximum waiting time, while lower classes do not. Here, we are interested in the stationary distribution at the times of commencement of service of this maximum priority process. Until now, there has been no explicit expression for this distribution. We construct a mapping of the maximum priority process to a tandem fluid queue, which enables us to find expressions for this stationary distribution. We derive the results for the stationary distribution of the maximum priority process at the times of the commencement of service.Comment: The Eleventh International Conference on Matrix-Analytic Methods in Stochastic Models (MAM11), 2022, Seoul, Republic of Kore

A comparison of random walks in dependent random environments

We provide exact computations for the drift of random walks in dependent random environments, including $k$-dependent and moving average environments. We show how the drift can be characterized and evaluated using Perron–Frobenius theory. Comparing random walks in various dependent environments, we demonstrate that their drifts can exhibit interesting behavior that depends significantly on the dependency structure of the random environment