In this paper we, firstly, present a recursive formula of the empirical estimator of the semi-Markov kernel. Then a non-parametric estimator of the expected cumulative operational time for semi-Markov systems is proposed. The asymptotic properties of this estimator, as the uniform strongly consistency and normality are given. As an illustration example, we give a numerical application.
Mots clés : expected cumulative operational time, semi-Markov process, non-parametric estimation
@article{RO_2007__41_4_399_0,
author = {Ouhbi, Brahim and Boudi, Ali and Tkiouat, Mohamed},
title = {The expected cumulative operational time for finite {semi-Markov} systems and estimation},
journal = {RAIRO - Operations Research - Recherche Op\'erationnelle},
pages = {399--410},
publisher = {EDP-Sciences},
volume = {41},
number = {4},
year = {2007},
doi = {10.1051/ro:2007029},
mrnumber = {2361293},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ro:2007029/}
}
TY - JOUR AU - Ouhbi, Brahim AU - Boudi, Ali AU - Tkiouat, Mohamed TI - The expected cumulative operational time for finite semi-Markov systems and estimation JO - RAIRO - Operations Research - Recherche Opérationnelle PY - 2007 SP - 399 EP - 410 VL - 41 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ro:2007029/ DO - 10.1051/ro:2007029 LA - en ID - RO_2007__41_4_399_0 ER -
%0 Journal Article %A Ouhbi, Brahim %A Boudi, Ali %A Tkiouat, Mohamed %T The expected cumulative operational time for finite semi-Markov systems and estimation %J RAIRO - Operations Research - Recherche Opérationnelle %D 2007 %P 399-410 %V 41 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/ro:2007029/ %R 10.1051/ro:2007029 %G en %F RO_2007__41_4_399_0
Ouhbi, Brahim; Boudi, Ali; Tkiouat, Mohamed. The expected cumulative operational time for finite semi-Markov systems and estimation. RAIRO - Operations Research - Recherche Opérationnelle, Tome 41 (2007) no. 4, pp. 399-410. doi : 10.1051/ro:2007029. http://www.numdam.org/articles/10.1051/ro:2007029/
[1] , Sur le contrôle adaptatif des populations de chaines de Markov finies. Thèse de 3ème cycle, Faculté des Sciences, Rabat, Morocco (1996).
[2] , Flowgraph models for multistate Time-to-Event Data. Wiley, New York (2005). | MR | Zbl
[3] , and, The completion time of a job on multimode systems. Adv. Appl. Probab. 19 (1987) 932-954. | Zbl
[4] and, Semi-Markov Processes and Reliability. Birkhäuser, Boston (2001). | MR | Zbl
[5] , and, Empirical Estimator of Stationary Distribution For Semi-Markov Processes. Commun. Stat. Theory Methods 34 (2003) 987-995. | Zbl
[6] and, Non-parametric estimation for semi-Markov processes based on its hazard rate. Stat. Inference Stoch. Process. 2 (1999) 151-173. | Zbl
[7] and, Non-parametric estimation for semi-Markov kernels with application to reliability analysis. Appl. Stoch. Models Data Anal. 12 (1996) 209-220. | Zbl
[8] and, The Rate of Occurrence of Failures for Semi-Markov Processes and Estimation. Stat. Probab. Lett. 59 (2001) 245-255. | Zbl
[9] and, Non-parametric Reliability Estimation of Semi-Markov Processes. J. Stat. Plan. Inference 109 (2003) 155-165. | Zbl
[10] and, The existence and uniqueness of stationary measures for Markov renewal processes. Ann. Math. Stat. 37 (1966) 1439-1462. | Zbl
[11] , Markov renewal processes: definitions and preliminary properties. Ann. Math. Stat. 32 (1961) 1231-1241. | Zbl
[12] , and, Performability analysis : measures, an algorithm, and a case study. IEEE Trans. Comput. C-37 (1988) 406-417.
[13] , Cumulative operational time analysis of finite semi-Markov reliability models. Reliab. Eng. Syst. Saf. 44 (1994) 17-25.
[14] , Applied probability models with optimization applications. Dover New York (1992). | MR
[15] and, Interval availability analysis using operational periods. Perform. Eval. 14 (1992) 257-272. | Zbl
[16] and Eds., Semi-Markov models and Applications. Kluwer Academic, Dordrecht (1999). | MR | Zbl
Cité par Sources :
