Hostname: page-component-7dd5485656-wlg5v Total loading time: 0 Render date: 2025-10-31T22:58:43.702Z Has data issue: false hasContentIssue false
  • English
  • Français

Marked point processes as limits of Markovian arrival streams

Part of: Stochastic processes Special processes

Published online by Cambridge University Press:  14 July 2016

Søren Asmussen  and
Ger Koole
Søren Asmussen*
Affiliation:
Aalborg University
Ger Koole*
Affiliation:
University of Leiden
*
∗ Postal address: Institute of Electronic Systems, Aalborg University, Fr. Bajersv. 7, DK-9220 Aalborg, Denmark.
∗∗ Postal address: Department of Mathematics and Computer Science, University of Leiden, P.O. Box 9512, 2300 RA Leiden, The Netherlands.
Get access Rights & Permissions [Opens in a new window]

Abstract

A Markovian arrival stream is a marked point process generated by the state transitions of a given Markovian environmental process and Poisson arrival rates depending on the environment. It is shown that to a given marked point process there is a sequence of such Markovian arrival streams with the property that as m →∞. Various related corollaries (involving stationarity, convergence of moments and ergodicity) and counterexamples are discussed as well.

Keywords

MARKOVIAN POINT PROCESSMARKOV-MODULATED POISSON PROCESSWEAK CONVERGENCE OF POINT PROCESSESCONTINUITY OF QUEUES

MSC classification

Primary: 60G55: Point processes
Secondary: 60K25: Queueing theory

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 30 , Issue 2 , June 1993 , pp. 365 - 372
Copyright
Copyright © Applied Probability Trust 1993 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

[1] Asmussen, S. and Johansen, H. (1986) Über eine Stetigkeitsfrage betreffend des Bedienungs-systems GI/GI/k. Elektron. Inf. Kyb. – EIK 22, 565–570.Google Scholar
[2] Asmussen, S. and Perry, D. (1991) On cycle maxima, first passage problems and extreme value theory for queues. Stoch. Models 8, 421–458.Google Scholar
[3] Borovkov, A. A. (1976) Stochastic Processes in Queueing Theory. Springer-Verlag, New York.Google Scholar
[4] Burman, D. Y. and Smith, D. R. (1986) An asymptotic analysis of a queueing system with Markov-modulated arrivals. Operat. Res. 34, 105–119.CrossRefGoogle Scholar
[5] Daley, D. J. and Vere-Jones, D. (1988) An Introduction to the Theory of Point Processes. Springer-Verlag, New York.Google Scholar
[6] Franken, P., König, D., Arndt, U. and Schmidt, V. (1981) Queues and Point Processes. Wiley, Chichester.Google Scholar
[7] Herrmann, U. (1965) Ein Approximationssatz für Verteilungen stationärer zufälliger Punktfolgen. Math. Nachrichten 30, 377–381.Google Scholar
[8] Hordijk, A. and Koole, G. (1990) On the optimality of the generalized shortest queue policy. Prob. Eng. Inf. Sci. 4, 477–487.CrossRefGoogle Scholar
[9] Hordijk, A. and Koole, G. (1992) On the shortest queue policy for the tandem parallel queue. Prob. Eng. Inf. Sci. 6, 63–79.Google Scholar
[10] Hordijk, A. and Koole, G. (1993) On the optimality of LEPT and µc rules for parallel processors and dependent arrival processes. Adv. Appl. Prob. 25(4).Google Scholar
[11] Hordijk, A. and Schassberger, R. (1982) Weak convergence for generalized semi-Markov processes. Stoch. Proc. Appl. 12, 271–291.Google Scholar
[12] Kallenberg, O. (1983) Random Measures. Academic Press, New York.CrossRefGoogle Scholar
[13] Lucantoni, D. (1991) New results on the single server queue with a batch Markovian arrival process. Stoch. Models 7, 1–46.CrossRefGoogle Scholar
[14] Lucantoni, D., Meier-Hellstern, K. S. and Neuts, M. F. (1990) A single server queue with server vacations and a class of non-renewal arrival processes. Adv. Appl. Prob. 22, 676–705.CrossRefGoogle Scholar
[15] Neuts, M. F. (1979) A versatile Markovian point process. J. Appl. Prob. 16, 764–779.Google Scholar
[16] Ramaswami, V. (1980) The N/G/Ì queue and its detailed analysis. Adv. Appl. Prob. 12, 222–261.CrossRefGoogle Scholar
[17] Regterschot, G. J. K. and De Smit, J. H. A. (1986) The queue M/G/1 with Markov-modulated arrivals and services. Math. Operat. Res. 11, 456–483.Google Scholar
[18] Rolski, T. (1981) Stationary Random Processes Associated with Point Processes. Lecture Notes in Statistics 5, Springer-Verlag, New York.Google Scholar
[19] Rudemo, M. (1973) Point processes generated by transitions of Markov chains. Adv. Appl. Prob. 5, 262–286.Google Scholar
[20] Schassberger, R. (1973) Warteschlangen. Springer, Wien.Google Scholar
[21] Sengupta, B. (1990) The semi-Markov queue: theory and applications. Stoch. Models 6, 383–413.Google Scholar
[22] Serfozo, R. F. (1990) Point processes. In Handbooks in Operations Research and Management Science. Vol. 2: Stochastic Models, ed. Heyman, D. P. and Sobel, M. J., pp. 1–93. North-Holland, Amsterdam.Google Scholar
[23] Stoyan, D. (1983) Comparison Methods for Queues and Other Stochastic Models. Wiley, Chichester.Google Scholar