On a reliability problem by stochastic control methods (original) (raw)
Stochastic Processes: Use and Limitations in Reliability Theory
Lecture Notes in Economics and Mathematical Systems, 1991
Stochastic processes are powerful tools for the investigation of the reliability and availability of repairable equipment and systems. Because of the involved models and in order to be mathematically tractable, these processes are generally confined to the class of regenerative stochastic processes with a finite state space, to which belong renewal processes, Markov processes, semi-Markov processes, and more general regenerative processes with only few (in the limit case only one) regeneration states. This contribution introduce briefly these processes and uses them to solve some reliability problems encountered in pratical applications. Investigations deal with different kinds of reliabilities and availabilities for one item, series, parallel, and series/ parallel structures. For series/parallel structures useful approximate expressions are developed. M. J. Beckmann et al. (eds.), Stochastic Processes and their Applications © Springer-Verlag Berlin Heidelberg 1991 and more difficult, even if constant failure and repair rates are assumed. Useful approximating expressions for practical applications can be obtained by assuming that each element of the reliability block diagram at system level has its own repair crew or by using macro structures.
Multi-state reliability systems under discrete time semi-Markovian hypothesis
2011
We consider repairable reliability systems with m components, the lifetimes and repair times of which are independent. The l-th component can be either in the failure state 0 or in the perfect state dl or in one of the degradation states {1, 2,..., dl − 1}. The time of staying in any of these states is a random variable following a discrete distribution not geometric. Thus, the state of every component and consequently of the whole system is described by a discrete-time semi-Markov chain together with the backward recurrence chain. Using recently obtained results concerning the discrete-time semi-Markov chains, we derive basic reliability measures in a general form. Finally, we present some numerical results of our proposed approach in specific reliability systems. 1 The General Model Let us consider a multi-state system (MSS) of order m, which means that it consists of m components which in their run are multi-state. As usually, the states of components determine the state of the s...
Optimal control limit for degradation process of a unit modelled as a Markov chain
Kopnov V.A. Residual Life, Linear Fatigue Damage Accumulation and Optimal Stopping. Reliability Engineering & System Safety, 1994, 43(1), pp. 29-35, 1994
An optimal maintenance problem of a plain bearing is discussed where the bearing is a critical unit of a piece of metallurgical equipment. The bearing shell wear process is investigated as a degradation one; maximum bearing degradation results in the whole unit and system in general break-down. The restoration cost of the failed system is definitely higher than the inspection and preventive replacement cost. To describe the degradation process a Markov chain model is used on the basis of statistical inference methods. To estimate the system operation cost a mathematical expectation of losses per unit time in a steady-state regime is evaluated. Linear algebraic equations are developed to assess the losses depending on a control limit value for the degradation process; for the given operation condition the optimal limit of degradation has been found.
Asymptotic failure rate of a Markov deteriorating system with preventive maintenance
Journal of Applied Probability, 2003
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A study on stochastic modelling of the repairable system
Journal of Computational Mathematica, 2021
All reliability models consisting of random time factors form stochastic processes. In this paper we recall the definitions of the most common point processes which are used for modelling of repairable systems. Particularly this paper presents stochastic processes as examples of reliability systems for the support of the maintenance related decisions. We consider the simplest one-unit system with a negligible repair or replacement time, i.e., the unit is operating and is repaired or replaced at failure, where the time required for repair and replacement is negligible.When the repair or replacement is completed, the unit becomes as good as new and resumes operation. The stochastic modelling of recoverable systems constitutes an excellent method of supporting maintenance related decision-making processes and enables their more rational use.
Optimal Maintenance Problems for Markovian Deteriorating Systems
Stochastic Models in Reliability and Maintenance, 2002
This chapter deals with optimal maintenance problems for Markovian deteriorating systems. The function of the system deteriorates with time, and the grade of deterioration is classified as one of s+2 discrete states, O, 1, ... ,s, s+ 1, in the order of increasing deterioration. State O is a good state, i. e., the system is like new, the states 1, ... ,s are deteriorat ion states and the state s + 1 is a failure state. In a normal operation, these states are assumed to constitute a discrete or continuous time Markovian process with an absorbing state s+ 1. In Section 8.1, we first introduce a basic replacement problem for a discrete time Markovian deteriorat ing system. In Section 8.2, we discuss an optimal inspection and replacement problem for the system in Section 8.1. In Section 8.3, we consider an optimal inspection and replacement problem under incomplete system observation. In Section 8.4, we treat a continuous time Markovian deteriorating system and discuss an optimal inspection and replacement problem. In Section 8.5, we deal with a maintenance problem in queueing system and discuss an optimal maintenance policy based on both the queue length and the server state.
In this chapter, the weak convergence of additive functionals of processes with locally independent increments and with semi-Markov switching in the scheme of Poisson approximation is investigated. Singular perturbation problem for the compensating operator of the extended Markov renewal process is used to prove the relative compactness. This approach can be used in applications and especially in shock and degradation in random environment arising in reliability. Keywords and phrases Poisson approximation - semimartingale - semi-Markov process - locally independent increments process - piecewise deterministic Markov process - weak convergence - singular perturbation - degradation - reliability
Reliability analysis of a system using semi Markov process
International Journal of Statistics and Applied Mathematics 2024; 9(4): 145-151 , 2024
This study present we describe reliability analysis of a system using semi Markov process. The system consists one main unit A and two associate units B and C. System will be operable when main unit and at least one associate unit is in operative mode and the system will be in semi operable state state when the main unit is failed but both associate are in operative mode. As soon as job arrives, all the unit works with load. It is assumed that only one job is taken for processing at a time. There is a single server who visits the system immediately when preventive maintenance and repair required. The unit works as new after preventive maintenance and repair. The failures of the unit are distributed exponentially while the distribution of PM and repair time are taken is arbitrary semi-Markov and regenerating point technique are used in this model.