Reliability-based measures for a retrial system with mixed standby components

In this paper, we consider a retrial and repairable multi-component system with mixed warm and cold standby components. It is assumed that the failure times of primary (operating) and warm standby components follow exponential distributions. When a component fails, it is sent to a service station with a single server (repairman) and no waiting space. The failed component is repaired if the server is idle and it has to enter an orbit if the server is busy. The failed component in the orbit will try to get the repair service again after an exponentially distributed random time period. The repair time also has an exponential distribution. The mean time-to-failure, MTTF, and the steady-state availability, A_T(∞), are derived in this retrial and repairable system. Using a numerical example, we compare the systems with and without retrials in terms of the cost/benefit ratios. Sensitivity analysis for the mean time-to-failure and the steady-state availability are investigated as well.     

Cost and probabilistic analysis of series systems with mixed standby components

This paper deals with the reliability and availability characteristics of four different series system configurations with mixed standby (include cold standby and warm standby) components. The failure times of the primary and warm standby components are assumed to be exponentially distributed with parameters λ and , respectively. The repair time distribution of each server is also exponentially distributed with parameter μ. We derive the mean time-to-failure, MTTF, and the steady-state availability, A_T(∞), for four configurations and perform comparisons. For all four configurations, comparisons are done for specific values of distribution parameters and of the cost of the components. Finally, the configurations are ranked based on: MTTF, A_T(∞), and cost/benefit where benefit is either MTTF or A_T(∞).     

A multiple warm standby system with operational and repair times following phase-type distributions

We study a warm standby n-unit system. The system functions as long as there is one operative unit. When the unit online fails, a unit in standby becomes the new unit online, if any. When a unit fails it goes to repair. There is a repairman. The units are repaired following the arrival order. The unit online has a lifetime governed by a phase-time distribution. The repair times follow a phase-type distribution. The warm standby units have lifetimes exponentially distributed. All the other times are negligible. This system extends many others of frequent use in the literature. We show that this system is governed by a level-dependent quasi-birth-and-death process (LDQBD process). The availability, rate of occurrence of failures and other magnitudes of interest are calculated. The mathematical expressions are algorithmically and computationally implemented, using the Matlab programme.     

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 k-out-of-n: F balanced systems with multiple functional sectors

In this paper, new concepts of balanced systems are proposed based on real engineering problems. The system under study consists of l groups and each group has n functional sectors. The conception of balance difference is proposed for the first time. It is assumed that unbalanced systems are rebalanced by either forcing down some working units into standby or resuming some standby units into operation. In addition, a case that the forced-down units are subject to failure during standby is studied in this paper. Based on different balanced cases and system failure criteria, two reliability models for balanced systems are developed. The proposed systems have widespread applications in aerospace and military industries, such as wing systems of airplane and unmanned aerial vehicles with balanced engine systems. Markov process imbedding method is used to analyze the number of working units in each sector for each model. Finite Markov chain imbedding approach and universal generating function technique are used to obtain the system reliability for different models. Several case studies are finally presented to demonstrate the new models.     

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.     

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.     

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 ().     

Cost Analysis of the <Emphasis Type="Italic">M/M/R</Emphasis> Machine Repair Problem with Spares and Two Modes of Failure

In this paper we deal with the machine repair problem consisting of M operating machines with S spare machines, and R repairmen where machines have two failure modes under steady-state conditions. Spares are considered to be either cold-standby, warm-standby or hot-standby. The two failure modes have equal probability of repair. Failure time of the machines and repair time of the repairmen are assumed to follow a negative exponential distribution. A cost model is developed in order to determine the optimal values of the number of repairmen and the number of spares simultaneously, while maintaining a minimum specified level of system availability. Numerical results are presented in which several system characteristics are evaluated for three types of standby under optimal operating conditions.     

k-out-of-n-system with repair:T-policy

We consider a k-out-of-n system with repair underT-policy. Life time of each component is exponentially distributed with parameter λ. Server is called to the system after the elapse ofT time units since his departure after completion of repair of all failed units in the previous cycle or until accumulation ofn — k failed units, whichever occurs first. Service time is assumed to be exponential with rateμ.T is also exponentially distributed with parameter α. System state probabilities in finite time and long run are derived for (i) cold (ii) warm (iii) hot systems. Several characteristics of these systems are obtained. A control problem is also investigated and numerical illustrations are provided. It is proved that the expected profit to the system is concave in α and hence global maximum exists.     

An LDQBD process under degradation,inspection, and two types of repair

A warm standby n-system with operational and repair times following phase-type distributions is considered. The online unit goes through degradating levels, determined by inspections. Two types of repairs are performed, preventive and corrective, depending on the degradation level. The standby units undergo corrective repair. This systems is governed by a level-dependent-quasi-birth-and-death proces (LDQBD process), whose generator is constructed. The availability, rate of occurrence of failures, and other quantities of interest are calculated. A numerical example including an optimization problem and illustrating the calculations is presented. This system extend other previously studied in the literature.     

Estimation of the Failure Rate in a Partially Accelerated Life Test: A Sequential Approach

Abstract

Consider the situation in which a group of units are put on a partially accelerated life test. It is assumed that the lifelengths of the units are independent and exponentially distributed random variables with common failure rate θ, and that θ is the value of a random variable having a gamma distribution. A two‐stage sequential procedure for estimating θ under the squared error loss is proposed. In the first stage, the units are put on the test under normal stress up to time t, where t is determined as a stopping time that minimizes the expected loss plus cost of running the test. In the second stage, the stress is raised to a higher level for those units that did not fail by time t and held constant until they all fail. The accumulated data are then used to estimate θ with the Bayes estimator.     

Stability of hybrid stochastic delay systems whose discrete components have a large state space: A two-time-scale approach

This work is concerned with stability of stochastic differential delay equations with Markovian switching, where the modulating Markov chain has a large state space and is subject to both fast and slow movements. Under simple conditions, we demonstrate that if the limit systems are pth-moment exponentially stable, then the original systems are pth-moment exponentially stable in an appropriate sense. In addition, the exponential stability is also investigated. Moreover, stability in distribution is obtained for such hybrid systems.     

Preventive maintenance in a 1 out of n system: The uptime,downtime and costs

In repairable systems with redundancy, failed units can be replaced by spare units in order to reduce the system downtime. The failed units are sent to a repair shop or manufacturer for corrective maintenance and subsequently are returned for re-use. In this paper we consider a 1 out of n system with cold standby and we assume that repaired units are “as good as new”.When a unit has an increasing failure rate it can be advantageous to perform preventive maintenance in order to return it to its “as good as new” state, because preventive maintenance will take less time and tends to be cheaper. In the model we present we use age-replacement; a machine is taken out for preventive maintenance and replaced by a standby one if its age has reached a certain value, T_pm. In this paper we derive an approximation scheme to compute the expected uptime, the expected downtime and the expected costs per time unit of the system, given the total number of units and the age-replacement value, T_pm. Consequently the number of units and the value T_pm can be determined for maximum long-term economy.     

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.     

Analysis of s,S inventory systems with general lead time and demand distribution and adjustable reorder size

Stochastic Inventory systems of (s, S) type with general lead time distribution are studied when the time intervals between successive demands are independently and identically distributed. The demands are assumed to occur for one unit at a time and the quantity reordered is subject to review at the epoch of replenishment so as to level up the inventory to S. An explicit characterization of the inventory level is provided. The model is flexible enough to allow complete backlogging and or deal with shortages. A general method of dealing with cost over an arbitrary time interval is indicated. Special cases are discussed when either the lead time or the interval between successive demands is exponentially distributed.     

Robust L2–L∞ filtering for switched systems under asynchronous switching

This paper investigates the problem of robust L₂–L_∞ filtering for continuous-time switched systems under asynchronous switching. When there exists asynchronous switching between the filter and the system, based on the average dwell time approach, sufficient conditions for the existence of a linear filter that guarantee the filtering error system to be exponentially stable with a prescribed weighted L₂–L_∞ performance for switched systems are derived, and filter parameters can be obtained by solving a set of matrix inequalities. Finally, a numerical example is given to illustrate the effectiveness of the proposed method.     

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 the uniform exponential stability of a wide class of linear time-delay systems

This paper deals with the global uniform exponential stability independent of delay of time-delay linear and time-invariant systems subject to point and distributed delays for the initial conditions being continuous real functions except possibly on a set of zero measure of bounded discontinuities. It is assumed that the delay-free system as well as an auxiliary one are globally uniformly exponentially stable and globally uniform exponential stability independent of delay, respectively. The auxiliary system is typically a part of the overall dynamics of the delayed system but not necessarily the isolated undelayed dynamics as usually assumed in the literature. Since there is a great freedom in setting such an auxiliary system, the obtained stability conditions are very useful in a wide class of practical applications.