Asymptotic and monotonicity properties of some repairable systems

1998 ◽  
Vol 30 (04) ◽  
pp. 1089-1110 ◽  
Author(s):  
Günter Last ◽  
Ryszard Szekli
Keyword(s):  
Markov Chains ◽  
Total Variation ◽  
Repairable Systems ◽  
Lifetime Distributions ◽  
Failure Times ◽  
Imperfect Repair ◽  
Coupling Methods ◽  
Monotonicity Properties ◽  
Heavy Tailed

The paper studies a model of repairable systems which is flexible enough to incorporate the standard imperfect repair and many other models from the literature. Palm stationarity of virtual ages, inter-failure times and degrees of repair is studied. A Loynes-type scheme and Harris recurrent Markov chains combined with coupling methods are used. Results on the weak total variation and moment convergences are obtained and illustrated by examples with IFR, DFR, heavy-tailed and light-tailed lifetime distributions. Some convergences obtained are monotone and/or at a geometric rate.

Asymptotic and monotonicity properties of some repairable systems

1998 ◽  
Vol 30 (4) ◽  
pp. 1089-1110 ◽  
Author(s):  
Günter Last ◽  
Ryszard Szekli
Keyword(s):  
Markov Chains ◽  
Total Variation ◽  
Repairable Systems ◽  
Lifetime Distributions ◽  
Failure Times ◽  
Imperfect Repair ◽  
Coupling Methods ◽  
Monotonicity Properties ◽  
Heavy Tailed

The paper studies a model of repairable systems which is flexible enough to incorporate the standard imperfect repair and many other models from the literature. Palm stationarity of virtual ages, inter-failure times and degrees of repair is studied. A Loynes-type scheme and Harris recurrent Markov chains combined with coupling methods are used. Results on the weak total variation and moment convergences are obtained and illustrated by examples with IFR, DFR, heavy-tailed and light-tailed lifetime distributions. Some convergences obtained are monotone and/or at a geometric rate.

Stochastic comparison of repairable systems by coupling

1998 ◽  
Vol 35 (2) ◽  
pp. 348-370 ◽  
Author(s):  
Günter Last ◽  
Ryszard Szekli
Keyword(s):  
Point Processes ◽  
Stochastic Comparison ◽  
Repairable Systems ◽  
Imperfect Repair ◽  
Coupling Methods ◽  
Comparison Results ◽  
Special Cases ◽  
Repair Models

Stochastic comparison results for replacement policies are shown in this paper using the formalism of point processes theory. At each failure moment a repair is allowed. It is performed with a random degree of repair including (as special cases) perfect, minimal and imperfect repair models. Results for such repairable systems with schemes of planned replacements are also shown. The results are obtained by coupling methods for point processes.

Stochastic comparison of repairable systems by coupling

1998 ◽  
Vol 35 (02) ◽  
pp. 348-370 ◽  
Author(s):  
Günter Last ◽  
Ryszard Szekli
Keyword(s):  
Point Processes ◽  
Stochastic Comparison ◽  
Repairable Systems ◽  
Imperfect Repair ◽  
Coupling Methods ◽  
Comparison Results ◽  
Special Cases ◽  
Repair Models

Stochastic comparison results for replacement policies are shown in this paper using the formalism of point processes theory. At each failure moment a repair is allowed. It is performed with a random degree of repair including (as special cases) perfect, minimal and imperfect repair models. Results for such repairable systems with schemes of planned replacements are also shown. The results are obtained by coupling methods for point processes.

Optimizing Series Repairable Systems with Imperfect Repair

2013 ◽  
pp. 83-93
Author(s):  
Mohammed Hajeeh
Keyword(s):  
Mathematical Model ◽  
Closed Form ◽  
Optimal Solution ◽  
Closed Form Expression ◽  
Repairable Systems ◽  
Nonlinear Mathematical Model ◽  
Repair Rate ◽  
Imperfect Repair ◽  
Repair Models ◽  
Minimum Costs

Repairable systems are either repaired perfectly to a state of as good as new or imperfectly. In this work, a system which undergoes imperfect repair is investigated. A nonlinear mathematical model is formulated for a system with the objective of finding the optimum failure and repair rate with the minimum costs subject to attaining a pre-specified performance level. Two imperfect repair models are examined. In the first model, the system is replaced by a new one after several failures. In the second model, the system is either replaced with a specific probability (1-p) or is imperfectly repaired after each failure with probability p. The optimal solution is presented in a closed form expression.

Performance of Two-Component Systems with Imperfect Repair

2012 ◽  
pp. 292-304
Author(s):  
Mohammed Hajeeh
Keyword(s):  
Time To Failure ◽  
Repairable Systems ◽  
Two Component System ◽  
Imperfect Repair ◽  
Component Systems ◽  
Two Component ◽  
Operational Systems ◽  
Two Component Systems ◽  
Operational Probability ◽  
Over Time

Operational systems deteriorate over time and eventually fail by the failure of one or more of their components. Failed components are either replaced or repaired, and replacement is usually expensive. This article examines the behavior of repairable systems with imperfect repair, where a failed component is repaired once or more depending on factors such as repair cost, level of deterioration, and criticality of the component. When these systems are subjected to a customer use environment, their performance must endure different conditions. In imperfect repair, the performance of the system lessens after each failure. Three models of a two-component system studied are the series, parallel, and standby configurations, and the components are identical and independent. A closed form analytical expression for steady state operational probability is derived for different configurations under exponential distribution time to failure and repair time. Two examples are then discussed thoroughly.

Convergence of the number of failed components in a Markov system with nonidentical components

2001 ◽  
Vol 38 (4) ◽  
pp. 882-897 ◽  
Author(s):  
Jean-Louis Bon ◽  
Eugen Păltănea
Keyword(s):  
Total Variation ◽  
Random Function ◽  
Random Variable ◽  
Quality Parameter ◽  
Repair Time ◽  
Total Variation Distance ◽  
Repairable Systems ◽  
Markov System ◽  
Variation Distance

For most repairable systems, the number N(t) of failed components at time t appears to be a good quality parameter, so it is critical to study this random function. Here the components are assumed to be independent and both their lifetime and their repair time are exponentially distributed. Moreover, the system is considered new at time 0. Our aim is to compare the random variable N(t) with N(∞), especially in terms of total variation distance. This analysis is used to prove a cut-off phenomenon in the same way as Ycart (1999) but without the assumption of identical components.

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