Interests :
I'm interested in various topics of probability theory and its applications. In particular :Markov chains.
Stability of stochastic models.
Inventory control.
Queueing networks.
Production and manufacturing systems.
Thesis :
B. Rabta. Nouvelles conditions et nouvelles estimations de stabilité des chaines de Markov. Application aux modèles stochastiques de gestion des stocks. Thèse de Doctorat, (Directeur de thèse : Pr. Djamil Aissani) Université de Béjaia, 2 Mai 2006.Résumé / Abstract .pdf)In this thesis, new conditions for the stability and new perturbation bounds for Markov chains are obtained.
First, we have precised the conditions for the stability of Markov chains with general state space after perturbation of their transition kernels. Under these conditions, we have derived perturbation bounds for the deviation of the norm of the stationary distribution with respect to different quantities.
We have made the connection between the strong stability method and the absolute stability method. We have then, generalized several results from the case of finite state space to the case of denumerable state space. In particular, we have derived several perturbation bounds for the absolute and relative deviation of the individual components of the stationary vector of a discrete Markov chain with states in a finite or denumerable space.
We have shown that under some conditions, a geometrically ergodic Markov chain is strongly stable with respect to some norm. In particular, when the Lyapunov condition is satisfied. Then, we have obtained quantitative stability estimates for these chains.
We have applied some of the stability theory results to the study of the sensitivity of stochastic inventory models to the perturbations in their parameters. We have constructed a computer program to test numerically the performance of the results and to compare them.
Keywords : Markov chain, Perturbation, Stability, Quantitative estimates, Inventory control.
B. Rabta. Stabilité forte dans un modèle stochastique de gestion des stocks. Thèse de Magistère, (Directeur Djamil Aissani) Université de Béjaia, 23 Octobre 2002. ( Résumé / Abstract .doc)
In this work, we prove for the first time the applicability of the strong stability method to inventory systems. Inventory problems are often very complicated and we need to use approximations. Therefore, it is very important to justify these approximations and to estimate the resultant error.
First, we summarize some known results on inventory theory. Classical inventory models are analysed using a regenerative processus approach. Secondly, we study the strong stability of a periodic review inventory model with an (R,s,S) policy. After showing the strong stability of the imbedded Markov chain with respect to the disturbance of the demand distribution, we obtain quantitative estimates with exact computation of constants.
Keywords : Inventory control, (R,s,S) policy, Markov chain, Perturbation, Strong stability, Quantitative estimates.
