Queueing Systems

, Volume 11, Issue 3, pp 255–272 | Cite as

Computation of the quasi-stationary distributions inM(n)/GI/1/K andGI/M(n)/1/K queues

  • Masaaki Kijima
  • Naoki Makimoto
Articles
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Abstract

In this paper, we provide numerical means to compute the quasi-stationary (QS) distributions inM/GI/1/K queues with state-dependent arrivals andGI/M/1/K queues with state-dependent services. These queues are described as finite quasi-birth-death processes by approximating the general distributions in terms of phase-type distributions. Then, we reduce the problem of obtaining the QS distribution to determining the Perron-Frobenius eigenvalue of some Hessenberg matrix. Based on these arguments, we develop a numerical algorithm to compute the QS distributions. The doubly-limiting conditional distribution is also obtained by following this approach. Since the results obtained are free of phase-type representations, they are applicable for general distributions. Finally, numerical examples are given to demonstrate the power of our method.

Keywords

Quasi-stationary distribution doubly-limiting conditional distribution finite quasi-birth-death process phase-type distribution queue length Perron-Frobenius eigenvalue 
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References

  1. [1]
    J.N. Darroch and E. Seneta, On quasi-stationary distributions in absorbing continuous-time finite Markov chains, J. Appl. Prob. 4(1967)192–196.Google Scholar
  2. [2]
    M. Kijima, Quasi-stationary distributions of phase-type queues, Math. Oper. Res. 18(1993), to appear.Google Scholar
  3. [3]
    M. Kijima and N. Makimoto, A unitified approach toGI/M(n)/1/K andM(n)/GI/1/K queues via finite quasi-birth-death processes, Commun. Statist. Stochastic Models 8(1992), to appear.Google Scholar
  4. [4]
    E.K. Kyprianou, On the quasi-stationary distribution of the virtual waiting time in queues with Poisson arrivals, J. Appl. Prob. 8(1971)494–507.Google Scholar
  5. [5]
    E.K. Kyprianou, On the quasi-stationary distributions of theGI/M/1 queues, J. Appl. Prob. 9(1972)117–128.Google Scholar
  6. [6]
    M.F. Neuts,Matrix-Geometric Solutions in Stochastic Models An Algorithmic Approach (Johns Hopkins University Press, Baltimore, 1981).Google Scholar
  7. [7]
    M.F. Neuts, The abscissa of convergence of the Laplace-Stieltjes transform of a PH-distribution, Commun. Statist. Comput. 13(1984)367–373.Google Scholar
  8. [8]
    P. Schweitzer, An iterative aggregation-disaggregation algorithm for computing the spectral radius of a non-negative matrix, Working Paper Series No. QM 8433, University of Rochester, Rochester, NY (1988).Google Scholar
  9. [9]
    E. Seneta,Non-Negative Matrices and Markov Chains, 2nd ed. (Springer, New York, 1981).Google Scholar
  10. [10]
    W. Whitt, Continuity of generalized semi-Markov processes, Math. Oper. Res. 5(1980)494–501.Google Scholar
  11. [11]
    D.V. Widder,The Laplace Transform (Princeton University Press, Princeton, 1946).Google Scholar
  12. [12]
    I. Ziedins, Quasi-stationary distributions and one-dimensional circuit-switched networks, J. Appl. Prob. 24(1987)965–977.Google Scholar

Copyright information

© J.C. Baltzer AG, Scientific Publishing Company 1992

Authors and Affiliations

  • Masaaki Kijima
    • 1
  • Naoki Makimoto
    • 2
  1. 1.Graduate School of Systems ManagementThe University of Tsukuba BunkyokuTokyoJapan
  2. 2.Department of Information SciencesTokyo Institute of TechnologyTokyoJapan

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