Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-13T00:52:21.712Z Has data issue: false hasContentIssue false
  • English
  • Français

Event and time averages: a review

Part of: Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Pierre Brémaud ,
Raghavan Kannurpatti  and
Ravi Mazumdar
Pierre Brémaud*
Affiliation:
CNRS and École Polytechnique
Raghavan Kannurpatti*
Affiliation:
Columbia University
Ravi Mazumdar*
Affiliation:
Université du Québec
*
∗ Postal address: Laboratoire des Signaux et Systèmes, CNRS-ESE, Plateau du Moulon, 91190 Gif-sur-Yvette Cédex, France.
∗∗ Postal address: Department of Electrical Engineering and Center for Telecommunications Research, Columbia University, New York, NY 10027, USA.
∗∗∗ Postal address: INRS-Télécommunications, Université du Québec, Ile-des-Soeurs, PQ, H3E 1H6, Canada.
Get access Rights & Permissions [Opens in a new window]

Abstract

This article reviews results related to event and time averages (EATA) for point process models, including PASTA, ASTA and ANTIPASTA under general hypotheses. In particular, the results for the stationary case relating the Palm and martingale approach are reviewed. The non-stationary case is discussed in the martingale framework where minimal conditions for ASTA generalizing earlier work are presented in a unified framework for the discrete- and continuous-time cases. In addition, necessary and sufficient conditions for ASTA to hold in the stationary case are discussed in the case even when stochastic intensities may not exist and a short proof of the ANTIPASTA results known to date are given.

Keywords

STATIONARY POINT PROCESSESPALM PROBABILITIESSTOCHASTIC INTENSITYNON-STATIONARY POINT PROCESSESMARTINGALESRADON-NIKODYM DERIVATIVESQUEUEING PROCESSES

MSC classification

Primary: 60G55: Point processes
Type
Research Article
Information
Advances in Applied Probability , Volume 24 , Issue 2 , June 1992 , pp. 377 - 411
Copyright
Copyright © Applied Probability Trust 1992 

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

This paper was presented at the TIMS/ORSA Conference, Monterey, California, January 1991.

References

[1] Baccelli, F. and Bremaud, P. (1987) Palm Probabilities and Stationary Queueing Systems. Lecture Notes in Statistics 41, Springer-Verlag, Berlin.CrossRefGoogle Scholar
[2] Bremaud, P. (1981) Point Processes and Queues: Martingale Dynamics. Springer-Verlag, Berlin.CrossRefGoogle Scholar
[3] Bremaud, P. (1989) Characteristics of queueing systems observed at events and the connection between stochastic intensity and Palm probability. QUESTA 5, 99–111.Google Scholar
[4] Bremaud, P. (1989) Palm-martingale calculus, event and time averages: the stationary and non-stationary cases. Presented at the Vth Vilnius International Conference on Probability Theory and Mathematical Statistics, 26 June-1 July 1989. To appear in the proceedings of the conference.Google Scholar
[5] Dacunha-Castelle, D. and Duflo, M. (1982) Probabilités et statistiques, Vol. 2: Problèmes à temps mobile. Masson, Paris.Google Scholar
[6] Dellacherie, C. and Meyer, P. A. (1980) Probabilités et potentiel. Hermann, Paris.Google Scholar
[7] Georgiadis, L. (1987) Relations between arrival and time averages of a process in discrete-time systems with conditional lack of anticipation. Preprint, Electrical Engineering Department, University of Virginia, Charlottesville.Google Scholar
[8] Green, L. and Melamed, B. (1990) An ANTIPASTA result for Markovian systems. Operat. Res. 38, 173–175.CrossRefGoogle Scholar
[9] Hsiao, M. T. and Lazar, A. A. (1989) An extension to Norton's equivalent. QUESTA 5, 401–412.Google Scholar
[10] Jansen, U., König, D. and Nawrotzki, K. (1979) A criterion of insensitivity for a class of queueing systems with random marked point processes. Math. Operationforsch. Statist., Series Optimization 10, 379–403.CrossRefGoogle Scholar
[11] Kabanov, J. M., Liptser, R. S. and Shiryayev, A. N. (1979) Absolute continuity and singularity of locally absolutely continuous probability distributions: I. Mat. Sb. 35, 631–679 (English translation).CrossRefGoogle Scholar
[12] Kendall, D. G. (1951) Some problems in the theory of queues (with discussion). J. R. Statist. Soc. B 13, 151–185.Google Scholar
[13] König, D. and Schmidt, V. (1980) Imbedded and non-imbedded stationary characteristics of queueing systems with varying service rate and point processes. J. Appl. Prob. 17, 753–767.CrossRefGoogle Scholar
[14] König, D. and Schmidt, V. (1989) EPSTA: the coincidence of time stationary and customer stationary distributions. QUESTA 5, 247–264.Google Scholar
[15] König, D. and Schmidt, V. (1990) Extended and conditional versions of the PASTA property. Adv. Appl. Prob. 22, 510–512.CrossRefGoogle Scholar
[16] König, D., Miyazawa, M. and Schmidt, V. (1980) On identification of Poisson Arrivals in queues coinciding with time stationary and customer stationary distributions. J. Appl. Prob. 20, 860–871.CrossRefGoogle Scholar
[17] Liptser, R. S. and Shiryayev, A. N. (1978) Statistics of Random Processes II (English edition). Springer-Verlag, Berlin.CrossRefGoogle Scholar
[18] Liptser, R. S. and Shiryayev, A. N. (1990) Theory of Martingales. Kluwer, Dordrecht.Google Scholar
[19] Makowski, A., Melamed, B. and Whitt, W. (1989) On averages seen by arrivals in discrete time. Proc. 28th IEEE Conf. Decision and Control, Tampa, 1084–1086.Google Scholar
[20] Matthes, K., Kerstan, J. and Mecke, J. (1978) Infinitely Divisible Point Processes (English edition) Wiley, New York.Google Scholar
[21] Mecke, J. (1967) Stationäre Zuffalige auf Lokalkompakten Abelschen Gruppen. Z. Wahrscheinlich keitsth. 8, 39–56.Google Scholar
[22] Melamed, B. (1979) On Poisson traffic processes in discrete space Markovian systems with applications to queueing theory. Adv. Appl. Prob. 11, 218–239.CrossRefGoogle Scholar
[23] Melamed, B. and Whitt, W. (1990) On arrivals that see time averages. Operat. Res. 37, 156–172.CrossRefGoogle Scholar
[24] Melamed, B. and Whitt, W. (1990) On arrivals that see time averages: a martingale approach. J. Appl. Prob. 27, 376–384.CrossRefGoogle Scholar
[25] Miyazawa, M. and Wolff, R. W. (1990) Further results on ASTA for general stationary processes and related problems. J. Appl. Prob. 27, 792–804.CrossRefGoogle Scholar
[26] Miyazawa, M. and Yamazaki, G. (1988) The basic equations for a supplemented GSMP and its application to queues. J. Appl. Prob. 25, 563–578.CrossRefGoogle Scholar
[27] Papangelou, F. (1972) Integrability of expected increments of point processes and a related change of time. Trans. Amer. Math. Soc. 165, 483–506.CrossRefGoogle Scholar
[28] Regterschot, G. J. K. and Van Doorn, E. A. (1988) Conditional PASTA. Operat. Res. Lett. 7, 229–232.Google Scholar
[29] Rosenkranz, W. A. and Simha, R. (1989) Poisson arrivals see time averages: a generalization. Preprint, Department of Mathematics, University of Massachusetts, Amherst.Google Scholar
[30] Serfozo, R. (1989) Poisson functionals of Markov processes and queueing networks. Adv. Appl Prob. 21, 595–611.CrossRefGoogle Scholar
[31] Shalmon, M. (1988) Analysis of the GI/GI/1 queue and its variations via the LCFS preemptive resume discipline and its random walk interpretation. Prob. Eng. Inf. Sci. 2, 215–230.CrossRefGoogle Scholar
[32] Shanthikumar, J. G. and Sumita, U. (1986) On G/G/1 queue with LIFO-PP service discipline. J. Operat. Res. Soc. Japan 29, 220–231.Google Scholar
[33] Stidham, S. (1972) Regenerative processes in the theory of queues, with applications to the alternating-priority queues. Adv. Appl. Prob. 4, 542–577.CrossRefGoogle Scholar
[34] Stidham, S. (1982) Sample-path analysis. In Applied Probability-Computer Science: The Interface, Vol. II, ed. Disney, R. L. and Ott, T., Birkhauser, Boston, 41–70.CrossRefGoogle Scholar
[35] Stidham, S. and El-Taha, M. (1989) Sample path analysis of processes with imbedded point processes. QUESTA 5, 131–165.Google Scholar
[36] Stidham, S. and El-Taha, M. (1990) An extension to ASTA. Preprint.Google Scholar
[37] Strauch, R. E. (1970) When a queue looks the same to an arriving customer as to an observer. Management Sci. 17, 140–141.CrossRefGoogle Scholar
[38] Tijms, H. C. (1986) Stochastic Modelling and Analysis. Wiley, New York.Google Scholar
[39] Walrand, J. (1988) An Introduction to Queueing Networks. Prentice-Hall, Englewood Cliffs, N.J. Google Scholar
[40] Watanabe, S. (1964) On discontinuous additive functionals and Levy measures of a Markov process. Japan J. Math. 34, 53–70.CrossRefGoogle Scholar
[41] Wolff, R. W. (1982) Poisson arrivals see time averages. Operat. Res. 30, 223–231.CrossRefGoogle Scholar