A rate balance principle and its application to queueing models

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Abstract

We introduce a rate balance principle for general (not necessarily Markovian) stochastic processes. Special attention is given to processes with birth-and-death-like transitions, for which it is shown that for any state n, the rate of two consecutive transitions from \(n-1\) to \(n+1\) coincides with the corresponding rate from \(n+1\) to \(n-1\). We demonstrate how useful this observation is by deriving well-known, as well as new, results for non-memoryless queues with state-dependent arrival and service processes. We also use the rate balance principle to derive new results for a state-dependent queue with batch arrivals, which is a model with non-birth-and-death-like transitions.

Keywords

Rate balance G/M/1 M/G/1 Birth–death process Batch arrivals Conditional distribution Residual lifetime 

Mathematics Subject Classification

60G17 60K25 

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Department of StatisticsUniversity of AucklandAucklandNew Zealand
  2. 2.Department of Industrial EngineeringTechnische Universiteit EindhovenEindhovenThe Netherlands
  3. 3.Department of Statistics and Federmann Center for the Study of RationalityThe Hebrew University of JerusalemJerusalemIsrael

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