Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-10T16:44:00.321Z Has data issue: false hasContentIssue false
  • English
  • Français

A versatile Markovian point process

Published online by Cambridge University Press:  14 July 2016

Marcel F. Neuts
Marcel F. Neuts*
Affiliation:
University of Delaware
*
∗Postal address: Department of Mathematical Sciences, University of Delaware, Newark, DE 19711, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Abstract

We introduce a versatile class of point processes on the real line, which are closely related to finite-state Markov processes. Many relevant probability distributions, moment and correlation formulas are given in forms which are computationally tractable. Several point processes, such as renewal processes of phase type, Markov-modulated Poisson processes and certain semi-Markov point processes appear as particular cases. The treatment of a substantial number of existing probability models can be generalized in a systematic manner to arrival processes of the type discussed in this paper.

Several qualitative features of point processes, such as certain types of fluctuations, grouping, interruptions and the inhibition of arrivals by bunch inputs can be modelled in a way which remains computationally tractable.

Keywords

POINT PROCESSESPROBABILITY DISTRIBUTIONS OF PHASE TYPECUMULATIVE PROCESSESCOMPUTATIONAL PROBABILITYQUEUEING THEORY
Type
Research Papers
Information
Journal of Applied Probability , Volume 16 , Issue 4 , December 1979 , pp. 764 - 779
Copyright
Copyright © Applied Probability Trust 1979 

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.)

Footnotes

Research sponsored by the Air Force Office of Scientific Research Air Force Systems Command USAF under Grant No. AFOSR–77–3236.

References

[1] Berman, M. (1978) Regenerative multivariate point processes. Adv. Appl. Prob. 10, 411–430.CrossRefGoogle Scholar
[2] Cox, D. R. (1962) Renewal Theory. Methuen, London.Google Scholar
[3] Grandell, J. (1976) Doubly Stochastic Poisson Processes. Springer–Verlag, Berlin.CrossRefGoogle Scholar
[4] Heffes, H. (1973) Analysis of first-come, first-served queueing systems with peaked inputs. Bell Syst. Tech. J. 7, 1215–1228.CrossRefGoogle Scholar
[5] Kosten, L. (1977) Parametrization of overflow traffic in telecommunication and data handling. Paper presented at the Seventh Conference on Stochastic Processes and their Applications, Enschede, The Netherlands. Abstract in Adv. Appl. Prob. 10 (1978), 337–338.CrossRefGoogle Scholar
[6] Kuczura, A. (1972) Queues with mixed renewal and Poisson inputs. Bell Syst. Tech. J. 51, 1305–1326.CrossRefGoogle Scholar
[7] Naor, P. and Yechiali, U. (1971) Queueing problems with heterogeneous arrivals and service. Opns Res. 19, 722–734.Google Scholar
[8] Neuts, M. F. (1971) A queue subject to extraneous phase changes. Adv. Appl. Prob. 3, 78–119.CrossRefGoogle Scholar
[9] Neuts, M. F. (1975) Probability distributions of phase type. In Liber Amicorum Professor Emeritus H. Florin , University of Louvain, 173–206.Google Scholar
[10] Neuts, M. F. (1975) Computational uses of the method of phases in the theory of queues. Computers Math. Appl. 1, 151–166.CrossRefGoogle Scholar
[11] Neuts, M. F. (1975) Computational problems related to the Galton–Watson process. Proceedings of the Actuarial Research Conference , Brown University.Google Scholar
[12] Neuts, M. F. (1977) Algorithms for the waiting time distributions under various queue disciplines in the M/G/1 queue with service time distributions of phase type. In Algorithmic Methods in Probability , North-Holland, Amsterdam, 177–197.Google Scholar
[13] Neuts, M. F. (1978) Renewal processes of phase type. Naval Res. Logist Quart. 25, 445–454.CrossRefGoogle Scholar
[14] Neuts, M. F. (1979) The M/M/1 queue with randomly varying arrival and service rates. Opsearch 15, 139–157.Google Scholar
[15] Purdue, P. (1974) The M/M/1 queue in a Markovian environment. Opns Res. 22, 562–569.CrossRefGoogle Scholar
[16] Ramaswami, V. (1980) The N/G/1 queue and its detailed analysis. Adv. Appl. Prob. 12 (2).CrossRefGoogle Scholar
[17] Ramaswami, V. and Neuts, M. F. (1980) Some explicit formulas and computational methods for infinite-server queues with phase type arrivals. J. Appl. Prob. 17 (2).CrossRefGoogle Scholar
[18] Rudemo, M. (1973) Point processes generated by transitions of Markov chains. Adv. Appl. Prob. 5, 262–286.CrossRefGoogle Scholar
[19] Yechiali, U. (1973) A queueing-type birth-and-death process defined on a continuous-time Markov chain. Opns Res. 21, 604–609.CrossRefGoogle Scholar