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Reversibility in Queueing Models

  1. Masakiyo Miyazawa

Published Online: 19 JUL 2013

DOI: 10.1002/9780470400531.eorms1079

Wiley Encyclopedia of Operations Research and Management Science

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.

Author Information

  1. Tokyo University of Science, Noda, Japan

Publication History

  1. Published Online: 19 JUL 2013

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.

Keywords:

  • reversibility in structure;
  • queueing network;
  • quasi-reversibility;
  • symmetric service;
  • balanced service;
  • batch movement