Reversibility in Queueing Models

Abstract

In stochastic models for queues and their networks, random events evolve in time. A process for their backward evolution is referred to as a time‐reversed process . If some property is unchanged under time reversal, we may better understand the model. A concept of reversibility is invented for this invariance. Local balance for a stationary Markov chain has been used, but it is still too strong for queueing applications. We are concerned with a continuous‐time Markov chain, but do not assume it has a stationary distribution. We define reversibility in structure as an invariant property of a family of the set of models under a certain operation. Any member of this set is a pair of transition rate functions and its supporting measure, and each set represents dynamics of queueing systems such as arrivals and departures. We use a permutation Γ of the family members to describe the change of the dynamics under time reversal. This reversibility is called Γ ‐reversibility in structure . To apply these definitions, we introduce new classes of models, called reacting systems and self‐reacting systems . Using them, we give a unified view for queues and their networks. They include symmetric service, batch movements, and state‐dependent routing.

Structure-Reversibility of a Two-Dimensional Reflecting Random Walk and Its Application to Queueing Network

“…A Markov chain is said to be reversible if its time-reversed process is stochastically identical with the original Markov chain. However, this condition is too strong (see, e.g., [9] for queueing models). Instead of this reversibility, we consider weaker concept of reversibility for a reflecting random walk.…”

“…However, this condition is too strong (see, e.g., [9] for queueing models). Instead of this reversibility, we consider weaker concept of reversibility for a reflecting random walk.…”

“…Namely, we take twodimensional reflecting random walks for this class motivated by Miyazawa [9]. They may be interpreted as queueing models, but our intention is to consider the time-reversed processes within a class of two-dimensional reflecting random walks.…”