Modelling a general standby system and evaluation of its performance

Redundancy or standby is a technique that has been widely applied to improving system reliability and availability in system design. In this paper, a general method for modelling standby system is proposed and system performance measures are derived. It is shown that the proposed general standby system includes the cases of cold, hot and warm standby systems with units of exponential distribution, which were studied in the literature, as special cases. An optimal allocation problem for a standby system is also discussed. Copyright © 2007 John Wiley & Sons, Ltd.     

On allocation of redundancies in two-component series systems

Allocation of active [standby] redundancies in a system is a topic of great interest in reliability engineering and system safety because optimal configurations can significantly increase the reliability of a system. In this paper, we study the problem of allocating two exponentially distributed active [standby] redundancies in a two-component series system using the tools of stochastic ordering. We establish two interesting results on likelihood ratio ordering which have no restriction on the parameters.     

Optimal allocation of redundancies in series systems

It is of great interest for the problem of how to allocate redundancies in a system so as to optimize the system performance in reliability engineering and system security. In this paper, we consider the problems of optimal allocation of both active and standby redundancies in series systems in the sense of various stochastic orderings. For the case of allocating one redundancy to a series system with two exponential components, we establish two likelihood ratio order results for active redundancy case and standby redundancy case, respectively. We also discuss the case of allocating K active redundancies to a series system and establish some new results. The results developed here strengthen and generalize some of the existing results in the literature. Specifically, we give an answer to an open problem mentioned in Hu and Wang [T. Hu, Y. Wang, Optimal allocation of active redundancies in r-out-of-n systems, Journal of Statistical Planning and Inference 139 (2009) 3733–3737]. Numerical examples are provided to illustrate the theoretic results established here.     

Optimal maintenance policy in a multistate deteriorating standby system

An optimal maintenance policy for a multistate deteriorating standby system is proposed in this study. Traditionally, a system could only presume two operational states: success or failure, and the maintenance policy is to determine the optimal number of standby components, subject to factors such as maintenance capability, cost of the standby items, etc., so as to minimize the operational cost. This study considers a more general production system in which progressive deterioration is incurred during the operating time, hence resulting in degrading performance. By modeling the system as a multistate deteriorating system, an optimal maintenance policy is obtained by determining the optimal number of standby components required in the system and the optimal state in which the replacement of deteriorating components shall be made.     

A redundant repairable system with imperfect coverage and fuzzy parameters

This paper proposes a procedure to construct the membership functions of the system characteristics of a redundant repairable system with two primary units and one standby in which the coverage factor is the same for an operating unit failure as that for a standby unit failure. Times to failure and times to repair of the operating and standby units are assumed to follow fuzzified exponential distributions. The α-cut approach is used to extract from the fuzzy repairable system a family of conventional crisp intervals for the desired system characteristics, determined with a set of parametric nonlinear programs using their membership functions. A numerical example is solved successfully to illustrate the practicality of the proposed approach. Because the system characteristics are governed by the membership functions, more information is provided for use by management, and because the redundant system is extended to the fuzzy environment, general repairable systems are represented more accurately and the analytic results are more useful for designers and practitioners.     

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.     

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.     

Cold vs. hot standby mission operation cost minimization for 1-out-of-N systems

It is well recognized that using the hot standby redundancy provides fast restoration in the case of failures. However the redundant elements are exposed to working stresses before they are used, which reduces the overall system reliability. Moreover, the cost of maintaining the hot redundant elements in the operational state is usually much greater than the cost of keeping them in the cold standby mode. Therefore, there exists a tradeoff between the cost of losses associated with the restoration delays and the operation cost of standby elements. Such a trade-off can be obtained by designing both hot and cold redundancy types into the same system. Thus a new optimization problem arises for the standby system design. The problem, referred to in this work as optimal standby element distributing and sequencing problem (SE-DSP) is to distribute a fixed set of elements between cold and hot standby groups and select the element initiation sequence so as to minimize the expected mission operation cost of the system while providing a desired level of system reliability. This paper first formulates and solves the SE-DSP problem for 1-out-of-N: G heterogeneous non-repairable standby systems. A numerical method is proposed for evaluating the system reliability and expected mission cost simultaneously. This method is based on discrete approximation of time-to-failure distributions of the system elements. A genetic algorithm is used as an optimization tool for solving the formulated optimization problem. Examples are given to illustrate the considered problem and the proposed solution methodology.     

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.     

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.