Hostname: page-component-745bb68f8f-cphqk Total loading time: 0 Render date: 2025-01-12T00:57:46.913Z Has data issue: false hasContentIssue false
  • English
  • Français

Estimation de l'État d'une file d'attente et du temps de panne d'une machine par la méthode de semi-martingales

Published online by Cambridge University Press:  01 July 2016

P. Brémaud
P. Brémaud*
Affiliation:
CEREMADE, Université de Paris IX (Dauphine)
Get access Rights & Permissions [Opens in a new window]

Abstract

We establish the estimation equations corresponding to a system in which the state is a process with jumps and the observation is a point process. We then solve two problems: the estimation of the state of a queue when its output is observed, and the estimation of the time of disorder of a machine when the ‘complaints process’ is observed.

Keywords

POINT PROCESSQUEUEING PROCESSMARTINGALESTOCHASTIC INTEGRALPREDICTABLE PROCESS
Type
Research Article
Information
Advances in Applied Probability , Volume 7 , Issue 4 , December 1975 , pp. 845 - 863
Copyright
Copyright © Applied Probability Trust 1975 

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.)

References

[1] Brémaud, P. (1972) A martingale approach to point processes. Ph.D. Thesis, University of California at Berkeley. Memo ERL–M– 345, Electronics Research Laboratory.Google Scholar
[2] Brémaud, P. (1974) The martingale theory of point processes on the real half-line. Proceedings of the IRIA International Colloquium on Control Theory at Rocquencourt. Lecture Notes in Economics and Mathematical Systems 107, Springer Verlag, Berlin.Google Scholar
[3] Davis, M.H.A. (1973) Non-linear filtering with point process observations. Publication 73/10 of the Department of Computing and Control, Imperial College, London.Google Scholar
[4] Davis, M.H.A. (1973) Detection of signals with point process observations. Publication 73/8 of the Department of Computing and Control, Imperial College, London.Google Scholar
[5] Dellacherie, C. (1972) Capacités et Processus Stochastiques. Springer Verlag, Berlin.CrossRefGoogle Scholar
[6] Doleans, C. et Meyer, P.A. (1970) Intégrales stochastiques par rapport aux martingales locales. Dans Séminaire de Probabilités IV, Lecture Notes on Mathematics, Springer Verlag, Berlin.Google Scholar
[7] Fujisaki, M., Kallianpur, G. et Kunita, H. (1972) Stochastic differential equations for the non-linear filtering problem. Osaka J. Math 9, 19–40.Google Scholar
[8] Jacod, J. (1973) On the stochastic intensity of a random point process over the half line. Technical Report, Series 2, Department of Statistics, Princeton University.Google Scholar
[9] Jacod, J. (1974) Transformation of measures and Radon-Nikodym derivatives for point processes. Technical report, Series 2, Department of Statistics, Princeton University.Google Scholar
[10] Meyer, P.A. (1965) Probabilités et Potentiel. Hermann, Paris.Google Scholar
[11] Segall, A. (1973) A martingale approach to modelling, estimation and detection of jump processes. Ph.D. Thesis, Stanford University. Technical Report No. 7050–21, Center for Systems Research.Google Scholar
[12] Van Schuppen, J. (1973) Estimation theory for continuous time processes. Ph.D. Thesis, University of California at Berkeley. Memo M–405, Electronics Research Laboratory.Google Scholar
[13] Snyder, D. L. (1972) Filtering and detection for doubly stochastic Poisson processes. IEEE Trans. IT–18, 91–102.Google Scholar
[14] Watanabe, S. et Kunita, H. (1967) On square integrable martingales. Nagoya J. Math. 30, 209–245.Google Scholar
[15] Rubin, I. (1972) Regular point processes and their detection. IEEE Trans. IT–18, 547–557.Google Scholar
[16] Wong, E. (1973) Recent progress in stochastic processes. A review. IEEE Trans. IT–19, 262–275.Google Scholar
[17] Zakaï, M. (1969) On the optimal theory filtering of diffusion processes. Z. Wahrscheinlichkeitsth. 11, 230–243.Google Scholar
[18] Brémaud, P. (1975) La méthode des semi-martingales en filtrage lorsque l'observation est un processus ponctuel marqué. Séminaire de Probabilités 1974/75, Université de Strasbourg.Google Scholar
[19] Chou, C. S. et Meyer, P. A. (1974) Sur une représentation des martingales comme intégrales stochastiques dans les processus ponctuels. Dans Séminaire de Probabilités IX. Springer-Verlag, Berlin. (À paraître.) Google Scholar
[20] Brémaud, P. et Jacod, J. (1975) Revue des résultats recents sur la théorie des processus ponctuels et le point de vue des martingales. Rapport interne, Département de Mathématiques, Université de Rennes.Google Scholar