A deteriorating cold standby repairable system with priority in use

In this paper, a cold standby repairable system consisting of two dissimilar components and one repairman is studied. In this system, it is assumed that the working time distributions and the repair time distributions of the two components are both exponential and component 1 is given priority in use. After repair, component 2 is “as good as new” while component 1 follows a geometric process repair. Under these assumptions, using the geometric process and a supplementary variable technique, some important reliability indices such as the system availability, reliability, mean time to first failure (MTTFF), rate of occurrence of failure (ROCOF) and the idle probability of the repairman are derived. A numerical example for the system reliability R(t) is given. And it is considered that a repair-replacement policy based on the working age T of component 1 under which the system is replaced when the working age of component 1 reaches T. Our problem is to determine an optimal policy T^∗ such that the long-run average cost per unit time of the system is minimized. The explicit expression for the long-run average cost per unit time of the system is evaluated, and the corresponding optimal replacement policy T^∗ can be found analytically or numerically. Another numerical example for replacement model is also given.     

Reliability analysis of a three unit warm standby redundant system with repair

Two-unit warm standby redundant systems have been investigated extensively in the past. The most general model is the one in which both the lifetime and repair time distributions of the units are arbitrary. However the study of standby systems with more than two units, though very important, has received much less attention, possibly because of the built-in difficulties in analyzing them. Such systems have been studied only when either the lifetime or the repair time is exponentially distributed. When both these distributions are general, the problem appears to be intractable even in the case of cold standby systems. The present contribution is an improvement in the state of art in the sense that a three unit warm standby system is shown to be capable of comprehensive analysis. In particular we show that there are imbedded renewal points that render the analysis possible. Using these imbedded renewal points we obtain the reliability and availability functions. Emeritus Deceased 23rd December 2003.     

Exponential asymptotic property of a parallel repairable system with warm standby under common-cause failure

We discuss exponential asymptotic property of the solution of a parallel repairable system with warm standby under common-cause failure. This system can be described by a group of partial differential equations with integral boundary. First we show that the positive contraction C₀-semigroup T(t) [Weiwei Hu, Asymptotic stability analysis of a parallel repairable system with warm standby under common-cause failure, Acta Anal. Funct. Appl. 8 (1) (2006) 5-20] which is generated by the operator corresponding to these equations is a quasi-compact operator. Then by using [Weiwei Hu, Asymptotic stability analysis of a parallel repairable system with warm standby under common-cause failure, Acta Anal. Funct. Appl. 8 (1) (2006) 5-20] that 0 is an eigenvalue of the operator with algebraic index one and the C₀-semigroup T(t) is contraction, we conclude that the spectral bound of the operator is zero. By using the above results the exponential asymptotical stability of the time-dependent solution of the system follows easily.     

Reliability analysis of two-unit cold standby repairable systems under Poisson shocks

This paper analyses the reliability of a cold standby system consisting of two repairable units, a switch and a repairman. At any time, one of the two units is operating while the other is on cold standby. The repairman may not always at the job site, or take vacation. We assume that shocks can attack the operating unit. The arrival times of the shocks follow a homogeneous Poisson process and their magnitude is a random variable following a known distribution. Time on repairing a failed unit and the length of repairman’s vacation follow general continuous probability distributions, respectively. The paper derives a number of reliability indices: system reliability, mean time to first failure, steady-state availability, and steady-state failure frequency.     

Long-run availability of a warm standby system

We analyze the long-run availability of a duplex system characterized by warm standby and attended by two repairmen. In order to describe the random behavior of the system, we employ a stochastic process endowed with stationary measures satisfying coupled steady-state differential equations. The solution procedure is based on the theory of sectionally holomorphic functions. As an example, we consider the particular case of deterministic repair (replacement).     

APPROXIMATING ANALYSIS OF A KIND OF REPAIRABLE STANDBY SYSTEMS

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RELIABILITY ANALYSIS FOR A REPAIRABLE PARALLEL SYSTEM WITH TIME-VARYING FAILURE RATES

To solve a real problem :how to calculate the reliability of a system with time-varying failure rates in industry systems,this paper studies a model for the load-sharing parallel system with time-varying failure rates,and obtains calculating formulas of reliability and availability of the system by solving differential equations. In this paper, the failure rates are expressed in polynomial configuration. The constant,linear and Weibull failure rate are in their special form. The polynomial failure rates provide flexibility in modeling the practical time-varying failure rates.     

Reliability analysis of a renewable multiple cold standby system

We present a general reliability analysis of the basic multiple cold standby system attended by a single repair facility. The particular case of deterministic repair provides some explicit results for the survival function illustrated by computer-plotted graphs.     

Evaluating a warm standby system with components having proportional hazard rates

This paper investigates the heterogeneity of components with proportional hazard rates in a redundant system. The total number of those standbys surviving the failure time of some active component is derived, and the algorithm to determine the optimal number of standbys is also discussed.     

The behavior of warm standby components with respect to a coherent system

This paper is concerned with a coherent system consisting of active components and equipped with warm standby components. In particular, we study the random quantity which denotes the number of surviving warm standby components at the time of system failure. We represent the distribution of the corresponding random variable in terms of system signature and discuss its potential utilization with a certain optimization problem.     

A complex discrete warm standby system with loss of units

A redundant complex discrete system is modelled through phase type distributions. The system is composed of a finite number of units, one online and the others in a warm standby arrangement. The units may undergo internal wear and/or accidental external failures. The latter may be repairable or non-repairable for the online unit, while the failures of the standby units are always repairable. The repairability of accidental failures for the online unit may be independent or not of the time elapsed up to their occurrence. The times up to failure of the online unit, the time up to accidental failure of the warm standby ones and the time needed for repair are assumed to be phase-type distributed. When a non-repairable failure occurs, the corresponding unit is removed. If all units are removed, the system is then reinitialized. The model is built and the transient and stationary distributions determined. Some measures of interest associated with the system, such as transition probabilities, availability and the conditional probability of failure are achieved in transient and stationary regimes. All measures are obtained in a matrix algebraic algorithmic form under which the model can be applied. The results in algorithmic form have been implemented computationally with Matlab. An optimization is performed when costs and rewards are present in the system. A numerical example illustrates the results and the CPU (Central Processing Unit) times for the computation are determined, showing the utility of the algorithms.     

Reliability analysis of non‐repairable systems modeled by dynamic fault trees with priority AND gates

Many real‐life systems are typically involved in sequence‐dependent failure behaviors. Such systems can be modeled by dynamic fault trees (DFTs) with priority AND gates, in which the occurrence of the top events depends on not only combinations of basic events but also their failure sequences. To the author's knowledge, the existing methods for reliability assessment of DFTs with priority AND gates are mainly Markov‐state‐space‐based, inclusion–exclusion‐based, Monte Carlo simulation‐based, or sequential binary decision diagram‐based approaches. Unfortunately, all these methods have their shortcomings. They either suffer the problem of state space explosion or are restricted to exponential components time‐to‐failure distributions or need a long computation time to obtain a solution with a high accuracy. In this article, a novel method based on dynamic binary decision tree (DBDT) is first proposed. To build the DBDT model of a given DFT, we present an adapted format of the traditional Shannon's decomposition theorem. Considering that the chosen variable index has a great effect on the final scale of disjoint calculable cut sequences generated from a built DBDT, which to some extent determines the computational efficiency of the proposed method, some heuristic branching rules are presented. To validate our proposed method, a case study is analyzed. The results indicate that the proposed method is reasonable and efficient. Copyright © 2015 John Wiley & Sons, Ltd.     

A bivariate optimal replacement policy for a cold standby repairable system with preventive repair

In this paper, the repair-replacement problem for a deteriorating cold standby repairable system is investigated. The system consists of two dissimilar components, in which component 1 is the main component with use priority and component 2 is a supplementary component. In order to extend the working time and economize the running cost of the system, preventive repair for component 1 is performed every time interval T, and the preventive repair is “as good as new”. As a supplementary component, component 2 is only used at the time that component 1 is under preventive repair or failure repair. Assumed that the failure repair of component 1 follows geometric process repair while the repair of component 2 is “as good as new”. A bivariate repair-replacement policy (T, N) is adopted for the system, where T is the interval length between preventive repairs, and N is the number of failures of component 1. The aim is to determine an optimal bivariate policy (T, N)^∗ such that the average cost rate of the system is minimized. The explicit expression of the average cost rate is derived and the corresponding optimal bivariate policy can be determined analytically or numerically. Finally, a Gamma distributed example is given to illustrate the theoretical results for the proposed model.     

Cost benefit analysis of series systems with warm standby components

This paper deals with the cost benefit analysis of series systems with warm standby components. The time-to-repair and the time-to-failure for each of the primary and warm standby components is assumed to have the negative exponential distribution. We develop the explicit expressions for the mean time-to-failure, MTTF, and the steady-state availability, A _T () for three configurations and perform a comparative analysis. Under the cost/benefit (C/B) criterion, comparisons are made based on assumed numerical values given to the distribution parameters, and to the cost of the components. The configurations are ranked based on: MTTF, A _T (), and C/B where B is either MTTF or A _T ().     

The well-posedness and stability of a repairable standby human–machine system

In this paper, the solution of a standby human–machine system is investigated. By using the method of functional analysis, especially, the linear operator theory and the C₀ semigroup theory on Banach space, we prove the well-posedness and the existence of a positive solution of the system. And under some appropriate hypotheses, we study the asymptotic stability of solution of the system.     

On the availability of a warm standby system: a numerical approach

We consider a basic duplex system characterized by warm standby and attended by two general heterogeneous repairmen. In order to derive computational results for the point availability of the engineering system, we first employ a stochastic process endowed with time-dependent transition measures satisfying coupled partial differential equations. However, an explicit evaluation of the (exact) solution is in general excluded. Therefore, we also propose a numerical solution of the equations. Our methodology is based on new modification of the first-order upwind scheme applied to a semiinfinite region. As an application, we consider the important case of Weibull–Gnedenko repair and provide an in-depth analysis of some key features of the duplex system.     

Reliability analysis of a single warm-standby system subject to repairable and nonrepairable failures

An n-unit system provisioned with a single warm standby is investigated. The individual units are subject to repairable failures, while the entire system is subject to a nonrepairable failure at some finite but random time in the future. System performance measures for systems observed over a time interval of random duration are introduced. Two models to compute these system performance measures, one employing a policy of block replacement, and the other without a block replacement policy, are developed. Distributional assumptions involving distributions of phase type introduce matrix Laplace transformations into the calculations of the performance measures. It is shown that these measures are easily carried out on a laptop computer using Microsoft Excel. A simple economic model is used to illustrate how the performance measures may be used to determine optimal economic design specifications for the warm standby.     

Well-posedness and stability of the repairable system with N failure modes and one standby unit

The well-posedness and stability of the repairable system with N failure modes and one standby unit were discussed by applying the c₀ semigroups theory of function analysis. Firstly, the integro-differential equations described the system were transformed into some abstract Cauchy problem of Banach space. Secondly, the system operator generates positive contractive c₀ semigroups T(t) and so the well-posedness of the system was obtained. Finally, the spectral distribution of the system operator was analyzed. It was proven that 0 is strictly dominant eigenvalue of the system operator and the dynamic solution of the system converges to the steady-state solution. The steady-state solution was shown to be the eigenvector of the system operator corresponding to the eigenvalue 0. At the same time the dynamic solution exponentially converges to the steady-state solution.     

Stochastic analysis of a n-unit repairable system with slow switch

This paper deals with the costn–benefit analysis of a cold standby system composed of n identical repairable units, subject to slow switch. Two models of system functioning are studied in this paper. In model 1, the repair time of a unit is assumed to follow exponential distribution and the other time distributions as arbitrary, while in model 2, the repair time of a unit is assumed to be arbitrarily distributed and the other time distributions follow exponential law. For both the models, the system characteristics, namely

(i) the expected upn–time of the system during the period (O,t]

(ii) the expected busyn–period of the repair facility during the period (0,t] and

(iii) the expected time the units spend in the switchover/installation state during the period (O,t]

are studied by identifying the system a t suitable regeneration epochs. The cost-benefit analysis is carried out using these characteristics     

Reliability analysis of a k-out-of-n:G repairable system with single vacation

This paper analyzes a k-out-of-n:G   repairable system with one repairman who takes a single vacation, the duration of which follows a general distribution. The working time of each component is an exponentially distributed random variable and the repair time of each failed component is governed by an arbitrary distribution. Moreover, we assume that every component is “as good as new” after being repaired. Under these assumptions, several important reliability measures such as the availability, the rate of occurrence of failures, and the mean time to first failure of the system are derived by employing the supplementary variable technique and the Laplace transform. Meanwhile, their recursive expressions are obtained. Furthermore, through numerical examples, we study the influence of various parameters on the system reliability measures. Finally, the Monte Carlo simulation and two special cases of the system which are (n-1)$(n-1)$-out-of-n:G repairable system and 1-out-of-n:G repairable system are presented to illustrate the correctness of the analytical results.