Reversibility in Queueing Models
Published Online: 19 JUL 2013
Copyright © 2010 John Wiley & Sons, Inc. All rights reserved.
Wiley Encyclopedia of Operations Research and Management Science
How to Cite
Miyazawa, M. 2013. Reversibility in Queueing Models. Wiley Encyclopedia of Operations Research and Management Science. 1–19.
- Published Online: 19 JUL 2013
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.
- reversibility in structure;
- queueing network;
- symmetric service;
- balanced service;
- batch movement