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Weak convergence for generalized semi-Markov processes

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  • Hordijk, A.
  • Schassberger, R.

Abstract

Generalized semi-Markov schemes were introduced by Matthes in 1962 under the designation 'Bedienungsschemata' (service schemes). They include a large variety of familiar stochastic models. It is shown in this paper that under appropriate regularity conditions the associated stochastic process describing the state at timet,t>=0, and the stationary distribution are continuous functions of the life-times of the active components. The supplementary-variable Markov process is shown to be the limit process of a sequence of discrete-state-process obtained through approximating the life-time distributions by mixtures of Erlang distributions and measuring ages and residual life-times in phases. This approach supplements the phase method.

Suggested Citation

  • Hordijk, A. & Schassberger, R., 1982. "Weak convergence for generalized semi-Markov processes," Stochastic Processes and their Applications, Elsevier, vol. 12(3), pages 271-291, May.
    • Handle: RePEc:eee:spapps:v:12:y:1982:i:3:p:271-291
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    Citations

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    Cited by:

    1. Dijk, J.N. & Akyildiz, I.F., 1988. "Networks with mixed processor sharing parallel queues and common pools," Serie Research Memoranda 0023, VU University Amsterdam, Faculty of Economics, Business Administration and Econometrics.
    2. Dijk, N.M. van, 1991. "Product forms for metropolitan area networks," Serie Research Memoranda 0007, VU University Amsterdam, Faculty of Economics, Business Administration and Econometrics.
    3. Dijk, N.M. van, 1988. "Error bounds for comparing open and closed queueing networks with an application to performability analysis," Serie Research Memoranda 0056, VU University Amsterdam, Faculty of Economics, Business Administration and Econometrics.
    4. Nico Dijk, 2005. "On product form tandem structures," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 62(3), pages 429-436, December.
    5. Dijk, N.M. van, 1987. "A LCFS finite buffer model with finite source batch input," Serie Research Memoranda 0049, VU University Amsterdam, Faculty of Economics, Business Administration and Econometrics.
    6. Dijk, N.M. van, 1989. "On 'stop=repeat' servicing for non-exponential queueing networks with blocking," Serie Research Memoranda 0023, VU University Amsterdam, Faculty of Economics, Business Administration and Econometrics.
    7. Richard J. Boucherie & Nico M. van Dijk, 2000. "On a Queueing Network Model for Cellular Mobile Telecommunications Networks," Operations Research, INFORMS, vol. 48(1), pages 38-49, February.
    8. Nico Dijk & Barteld Schilstra, 2022. "On two product form modifications for finite overflow systems," Annals of Operations Research, Springer, vol. 310(2), pages 519-549, March.
    9. Dijk, N.M. van, 1989. "A note on extended uniformization for non-exponential stochastic networks," Serie Research Memoranda 0026, VU University Amsterdam, Faculty of Economics, Business Administration and Econometrics.
    10. Dijk, N.M. van, 1988. "A LCFS finite buffer model with batch input and non-exponential services," Serie Research Memoranda 0007, VU University Amsterdam, Faculty of Economics, Business Administration and Econometrics.
    11. Niek Baer & Richard J. Boucherie & Jan-Kees C. W. van Ommeren, 2019. "Threshold Queueing to Describe the Fundamental Diagram of Uninterrupted Traffic," Transportation Science, INFORMS, vol. 53(2), pages 585-596, March.
    12. Dijk, N.M. van, 1989. "An equivalence of communication protocols for interconnection networks," Serie Research Memoranda 0034, VU University Amsterdam, Faculty of Economics, Business Administration and Econometrics.

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