Ergodic conditions for a class of state dependent queueing systems

  • J. Izydorczyk
Stationarity And Ergodicity 2
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 60)


Queueing system stability criteria are frequently formulated in the form of testing the sign of queue length (or waiting time) mean increments for heavy load conditions. In this paper we deal with queueing systems which behaviour can be represented by Markov chains on RN. As stability of such systems is equivalent to ergodicity of proper Markov chains, it must be strongly noticed that criteria of this form are valid only together with some additional set of conditions imposed on the considered chain. We shall suggest four comparatively easy to verify conditions and prove that this conditions are sufficient for considered criteria to be valid. We shall also show that suggested conditions are essential in the sense that skipping one of them turns the criteria to be false and apply the obtained results to check ergodicity of some important class of queueing systems, namely queues with waiting time dependent service times.


Markov Chain Queue Length Ergodic Condition General State Space Heavy Load Condition 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [BORO76]
    Borovkov A.A. "Stochastic processes in queueing theory," Springer Verlag, 1976.Google Scholar
  2. [CALL73]
    Callahan J.R. "A queue with waiting time dependent service times," Naval Res. Logist. Quart. 20, p.321, 1973.Google Scholar
  3. [LASL78]
    Laslett G.M., Pollard D.B., Tweedie R.L. "Techniques for establishing ergodic and recurrence properties of continuous valued Markov chains," Naval Res. Logist. Quart. 25, p.455, 1978.Google Scholar
  4. [PUJO78]
    Pujolle G. "Applications of some Markov chains results to computer systems modelling," IRIA Rapport de Recherche 289, 1978.Google Scholar
  5. [SUGA65]
    Sugavara S., Takahashi M. "On some queues occurring in an integrated iron and steel works," J. of O.R. Japan 8, p.16, 1965.Google Scholar
  6. [TWEE75]
    Tweedie R.L. "Sufficient conditions for ergodicity and recurrence of Markov chains on a general state space," Stoch. Proc. Appl. 3, p.385, 1975.Google Scholar
  7. [TWEE76]
    Tweedie R.L. "Criteria for classifying general Markov chains" Adv. Appl. Prob. 8, p.737, 1976.Google Scholar

Copyright information

© Springer-Verlag 1984

Authors and Affiliations

  • J. Izydorczyk
    • 1
  1. 1.Depatment of Complex Control SystemsPolish Academy of SciencesGliwicePoland

Personalised recommendations