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.

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.

DOI : 10.1051/ro:2007029
Classification : 60K20
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://archive.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://archive.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://archive.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://archive.numdam.org/articles/10.1051/ro:2007029/

[1] A. Boudi, Sur le contrôle adaptatif des populations de chaines de Markov finies. Thèse de 3ème cycle, Faculté des Sciences, Rabat, Morocco (1996).

[2] A. Huzurbazar, Flowgraph models for multistate Time-to-Event Data. Wiley, New York (2005). | MR | Zbl

[3] V.G. Kulkarni, V.F. Nicola and K.S. Trivedi, The completion time of a job on multimode systems. Adv. Appl. Probab. 19 (1987) 932-954. | Zbl

[4] N. Limnios and G. Oprişan, Semi-Markov Processes and Reliability. Birkhäuser, Boston (2001). | MR | Zbl

[5] N. Limnios, B. Ouhbi and A. Sadek, Empirical Estimator of Stationary Distribution For Semi-Markov Processes. Commun. Stat. Theory Methods 34 (2003) 987-995. | Zbl

[6] B. Ouhbi and N. Limnios, Non-parametric estimation for semi-Markov processes based on its hazard rate. Stat. Inference Stoch. Process. 2 (1999) 151-173. | Zbl

[7] B. Ouhbi and N. Limnios, Non-parametric estimation for semi-Markov kernels with application to reliability analysis. Appl. Stoch. Models Data Anal. 12 (1996) 209-220. | Zbl

[8] B. Ouhbi and N. Limnios, The Rate of Occurrence of Failures for Semi-Markov Processes and Estimation. Stat. Probab. Lett. 59 (2001) 245-255. | Zbl

[9] B. Ouhbi and N. Limnios, Non-parametric Reliability Estimation of Semi-Markov Processes. J. Stat. Plan. Inference 109 (2003) 155-165. | Zbl

[10] R. Pyke and R. Schaufele, The existence and uniqueness of stationary measures for Markov renewal processes. Ann. Math. Stat. 37 (1966) 1439-1462. | Zbl

[11] R. Pyke, Markov renewal processes: definitions and preliminary properties. Ann. Math. Stat. 32 (1961) 1231-1241. | Zbl

[12] R.M. Smith, K.S. Trivedi and A.V. Ramesh, Performability analysis : measures, an algorithm, and a case study. IEEE Trans. Comput. C-37 (1988) 406-417.

[13] A. Scenski, Cumulative operational time analysis of finite semi-Markov reliability models. Reliab. Eng. Syst. Saf. 44 (1994) 17-25.

[14] S. Ross, Applied probability models with optimization applications. Dover New York (1992). | MR

[15] G. Rubino and B. Sericola, Interval availability analysis using operational periods. Perform. Eval. 14 (1992) 257-272. | Zbl

[16] J. Janssen and N. Limnios Eds., Semi-Markov models and Applications. Kluwer Academic, Dordrecht (1999). | MR | Zbl

Cité par Sources :