In this paper we face the problem of estimating the elements that define a homogeneous semi-Marko... more In this paper we face the problem of estimating the elements that define a homogeneous semi-Markov process. Given an observed sample path of the process in a finite interval , non-parametric estimators of the entries of the semi-Markov kernel and its derivatives can be constructed. These estimators exhibit good properties as consistency and asymptotic normality. On the other hand, in our approach we consider that we have some prior information about the underlying process. This information concerns the mean sojourn times in the different states of the process and has to be taken into account in the estimation procedure. So we start from the smoothed estimators and construct minimum divergence estimators (maximum entropy) under some constraints for the associated sojourn moments.
... For example: (1) Syntax error Gate type input 1 input 2 input 3 GXY OR GAB XYZABC XYZAB A for... more ... For example: (1) Syntax error Gate type input 1 input 2 input 3 GXY OR GAB XYZABC XYZAB A formal definition of fault tree graph models 269 Here, if the length of the character chain for events description is limited to 5, input 2 and 3 will be the same event XYZAB, which in ...
Markov and Semi-Markov Processes Stochastic Systems with Switching Stochastic Systems in the Seri... more Markov and Semi-Markov Processes Stochastic Systems with Switching Stochastic Systems in the Series Scheme Stochastic Systems with Split and Merging Phase Merging Principles Weak Convergence Poisson Approximation Applications Appendices: Weak Convergence of Probability Measures Some Limit Theorems for Stochastic Processes Some Auxiliary Results
In this paper, we introduce a Markov decision model with absorbing states and a constraint on the... more In this paper, we introduce a Markov decision model with absorbing states and a constraint on the asymptotic failure rate. The objective is to find a stationary policy which minimizes the infinite horizon expected average cost, given that the system never fails. Using Perron-Frobenius theory of non-negative matrices and spectral analysis, we show that the problem can be reduced to a linear programming problem. Finally, we apply this method to a real problem for an aeronautical system.
In this chapter we present basic results for the countable case. By “countable case” we mean that... more In this chapter we present basic results for the countable case. By “countable case” we mean that the state space E is finite or countable and that e = P(E). We give two different kinds of results: the first one concerns some specialization of the general case; the second one concerns results which can be stated only in the countable case, as the recurrence time to a fixed state.
The study of the semi-Markov process is closely related to the theory of Markov renewal processes... more The study of the semi-Markov process is closely related to the theory of Markov renewal processes (MRP) which can be considered as an extension of the classical renewal theory (see, e.g., Feller [32], Cox [23]).
In this paper, we introduce the notion of “failure delay systems”. A classical assumption in reli... more In this paper, we introduce the notion of “failure delay systems”. A classical assumption in reliability studies of systems, is that time between occurrence of a cut set and system failure is small enough to be neglected: we call this the simultaneity assumption. By considering non negligible time (delay) a large family of systems that we call “failure delay systems” are introduced. For these systems there are two conditions for failure: first, the occurrence of a particular failure, i.e. a cut set, and second, its continuous presence for a given time, fixed or random. We formally define failure delay systems and give the lifetime structure for reliability measures evaluation. We describe next some classes of failure delay systems. A single component system without and with repair are studied and closed-form solutions for lifetime distribution are derived. Finally, as an example of failure delay systems we study the reservoir.
Let Q(x,A,t), x ∈ E, A ∈ e,t ∈ IR+, be a semi-Markov kernel on (E,e) and let (J n ,S n )n∈N and (... more Let Q(x,A,t), x ∈ E, A ∈ e,t ∈ IR+, be a semi-Markov kernel on (E,e) and let (J n ,S n )n∈N and (J n ,X n )n∈N be, respectively, the associated MRP and the (J-X)-process (see Section 2.2).
ABSTRACT In this article, the weak convergence of impulsive recurrent process with semi-Markov sw... more ABSTRACT In this article, the weak convergence of impulsive recurrent process with semi-Markov switching in the scheme of Poisson approximation is proved. Singular perturbation problem for the compensating operator of the extended Markov renewal process is used to prove the relative compactness.
Explicit formulations for throughput availability, demand availability, on-stream availability, a... more Explicit formulations for throughput availability, demand availability, on-stream availability, availability at a particular level and time, limiting throughput availability, and mean throughput to first failure are developed in terms of discrete and continuous parameter Markov chains. These measures are particularly adapted to continuous production or transportation systems. A numerical example illustrates the results
This paper treats a particular type of Markov system: time nonhomogeneous Markov system in discre... more This paper treats a particular type of Markov system: time nonhomogeneous Markov system in discrete time. In order to adapt measures to this kind of system, performability measures are formulated, such as new indicators more dedicated to electrical power systems like `instantaneous mean load curtailed' and `mean energy not supplied on a time interval'. This method considers hazard-rate time variation
Communications in Statistics - Theory and Methods, 2004
ABSTRACT In this paper, we define a discrete-time semi-Markov model and propose a computation pro... more ABSTRACT In this paper, we define a discrete-time semi-Markov model and propose a computation procedure for solving the corresponding Markov renewal equation, necessary for all our reliability measurements. Then, we compute the reliability and its related measures, and we apply the results to a three-state system.
In this paper we face the problem of estimating the elements that define a homogeneous semi-Marko... more In this paper we face the problem of estimating the elements that define a homogeneous semi-Markov process. Given an observed sample path of the process in a finite interval , non-parametric estimators of the entries of the semi-Markov kernel and its derivatives can be constructed. These estimators exhibit good properties as consistency and asymptotic normality. On the other hand, in our approach we consider that we have some prior information about the underlying process. This information concerns the mean sojourn times in the different states of the process and has to be taken into account in the estimation procedure. So we start from the smoothed estimators and construct minimum divergence estimators (maximum entropy) under some constraints for the associated sojourn moments.
... For example: (1) Syntax error Gate type input 1 input 2 input 3 GXY OR GAB XYZABC XYZAB A for... more ... For example: (1) Syntax error Gate type input 1 input 2 input 3 GXY OR GAB XYZABC XYZAB A formal definition of fault tree graph models 269 Here, if the length of the character chain for events description is limited to 5, input 2 and 3 will be the same event XYZAB, which in ...
Markov and Semi-Markov Processes Stochastic Systems with Switching Stochastic Systems in the Seri... more Markov and Semi-Markov Processes Stochastic Systems with Switching Stochastic Systems in the Series Scheme Stochastic Systems with Split and Merging Phase Merging Principles Weak Convergence Poisson Approximation Applications Appendices: Weak Convergence of Probability Measures Some Limit Theorems for Stochastic Processes Some Auxiliary Results
In this paper, we introduce a Markov decision model with absorbing states and a constraint on the... more In this paper, we introduce a Markov decision model with absorbing states and a constraint on the asymptotic failure rate. The objective is to find a stationary policy which minimizes the infinite horizon expected average cost, given that the system never fails. Using Perron-Frobenius theory of non-negative matrices and spectral analysis, we show that the problem can be reduced to a linear programming problem. Finally, we apply this method to a real problem for an aeronautical system.
In this chapter we present basic results for the countable case. By “countable case” we mean that... more In this chapter we present basic results for the countable case. By “countable case” we mean that the state space E is finite or countable and that e = P(E). We give two different kinds of results: the first one concerns some specialization of the general case; the second one concerns results which can be stated only in the countable case, as the recurrence time to a fixed state.
The study of the semi-Markov process is closely related to the theory of Markov renewal processes... more The study of the semi-Markov process is closely related to the theory of Markov renewal processes (MRP) which can be considered as an extension of the classical renewal theory (see, e.g., Feller [32], Cox [23]).
In this paper, we introduce the notion of “failure delay systems”. A classical assumption in reli... more In this paper, we introduce the notion of “failure delay systems”. A classical assumption in reliability studies of systems, is that time between occurrence of a cut set and system failure is small enough to be neglected: we call this the simultaneity assumption. By considering non negligible time (delay) a large family of systems that we call “failure delay systems” are introduced. For these systems there are two conditions for failure: first, the occurrence of a particular failure, i.e. a cut set, and second, its continuous presence for a given time, fixed or random. We formally define failure delay systems and give the lifetime structure for reliability measures evaluation. We describe next some classes of failure delay systems. A single component system without and with repair are studied and closed-form solutions for lifetime distribution are derived. Finally, as an example of failure delay systems we study the reservoir.
Let Q(x,A,t), x ∈ E, A ∈ e,t ∈ IR+, be a semi-Markov kernel on (E,e) and let (J n ,S n )n∈N and (... more Let Q(x,A,t), x ∈ E, A ∈ e,t ∈ IR+, be a semi-Markov kernel on (E,e) and let (J n ,S n )n∈N and (J n ,X n )n∈N be, respectively, the associated MRP and the (J-X)-process (see Section 2.2).
ABSTRACT In this article, the weak convergence of impulsive recurrent process with semi-Markov sw... more ABSTRACT In this article, the weak convergence of impulsive recurrent process with semi-Markov switching in the scheme of Poisson approximation is proved. Singular perturbation problem for the compensating operator of the extended Markov renewal process is used to prove the relative compactness.
Explicit formulations for throughput availability, demand availability, on-stream availability, a... more Explicit formulations for throughput availability, demand availability, on-stream availability, availability at a particular level and time, limiting throughput availability, and mean throughput to first failure are developed in terms of discrete and continuous parameter Markov chains. These measures are particularly adapted to continuous production or transportation systems. A numerical example illustrates the results
This paper treats a particular type of Markov system: time nonhomogeneous Markov system in discre... more This paper treats a particular type of Markov system: time nonhomogeneous Markov system in discrete time. In order to adapt measures to this kind of system, performability measures are formulated, such as new indicators more dedicated to electrical power systems like `instantaneous mean load curtailed' and `mean energy not supplied on a time interval'. This method considers hazard-rate time variation
Communications in Statistics - Theory and Methods, 2004
ABSTRACT In this paper, we define a discrete-time semi-Markov model and propose a computation pro... more ABSTRACT In this paper, we define a discrete-time semi-Markov model and propose a computation procedure for solving the corresponding Markov renewal equation, necessary for all our reliability measurements. Then, we compute the reliability and its related measures, and we apply the results to a three-state system.
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