A geometric process repair model for a repairable cold standby system with priority in use and repair

In this paper, a deteriorating cold standby repairable system consisting of two dissimilar components and one repairman is studied. For each component, assume that the successive working times form a decreasing geometric process while the consecutive repair times constitute an increasing geometric process, and component 1 has priority in use and repair. Under these assumptions, we consider a replacement policy N based on the number of repairs of component 1 under which the system is replaced when the number of repairs of component 1 reaches N. Our problem is to determine an optimal policy N^* such that the average cost rate (i.e. the long-run average cost per unit time) of the system is minimized. The explicit equation of the average cost rate of the system is derived and the corresponding optimal replacement policy N^* can be determined analytically or numerically. Finally, a numerical example with Weibull distribution is given to illustrate some theoretical results in this paper.     

An optimal geometric process model for a cold standby repairable system

In this paper a cold standby repairable system consisting of two identical components and one repairman is studied. Assume that each component after repair is not ‘as good as new'. Under this assumption, by using a geometric process, we consider a replacement policy N based on the number of repairs of component 1. Our problem is to determine an optimal replacement policy N* such that the long-run expected reward per unit time is maximized. The explicit expression of the long-run expected reward per unit time is derived and the corresponding optimal repair replacement policy can be determined analytically or numerically. Finally, a numerical example is given.     

Reward optimization of a repairable system

This paper analyzes a system subject to repairable and non-repairable failures. Non-repairable failures lead to replacement of the system. Repairable failures, first lead to repair but they lead to replacement after a fixed number of repairs. Operating and repair times follow phase type distributions (PH-distributions) and the pattern of the operating times is modelled by a geometric process. In this context, the problem is to find the optimal number of repairs, which maximizes the long-run average reward per unit time. To this end, the optimal number is determined and it is obtained by efficient numerical procedures.     

An optimal replacement policy for a repairable system based on its repairman having vacations

This paper studies a cold standby repairable system with two different components and one repairman who can take multiple vacations. If there is a component which fails and the repairman is on vacation, the failed component will wait for repair until the repairman is available. In the system, assume that component 1 has priority in use. After repair, component 1 follows a geometric process repair, while component 2 can be repaired as good as new after failures. Under these assumptions, a replacement policy N based on the failed times of component 1 is studied. The system will be replaced if the failure times of component 1 reach N. The explicit expression of the expected cost rate is given, so that the optimal replacement time N^? is determined. Finally, a numerical example is given to illustrate the theoretical results of the model.     

考虑使用与维修优先权的冷贮备系统可靠性模型分析

针对装备系统中部件关键重要程度不同的情况,以具有使用与维修优先权的多部件冷贮备系统为研究对象,利用适用性较强的phase-type（PH）分布代替以往在可靠性建模中采取的指数分布、Weibull分布等典型分布,统一描述了系统各部件的寿命和维修时间,并建立了通用性更好的系统可靠性模型。在模型解析过程中,采用矩阵解析方法,获得了系统的可靠度函数、稳态可用度以及系统平均开工时间、平均停工时间等工程实践中常用的系统可靠性指标解析表达式,并利用算例验证了模型的适用性,讨论了贮备部件数量对系统可靠性指标的影响。研究结果表明,利用PH分布对具有使用和维修优先权的冷贮备系统进行可靠性解析建模,不仅具有良好的解析特性,而且能够有效地保证模型的适用性,在工程实践中具有良好的应用价值。     

An extended extreme shock maintenance model for a deteriorating system

In this paper, we study a deteriorating system which is suffering random shocks from its environment. Assume that in the system's operating stage, whenever a shock arrives, it will do some damage to the system, but shocks with a “small” level of damage are harmless for the system, while shocks with a “large” level of damage may result in the system's failure. The system's deterioration is caused by both the external shocks and the internal load. In the external, the magnitude of the shock damage the system can bear is decreasing with respect to the number of repairs taken. In the internal, the consecutive repair time is increasing in the number of repairs taken. A replacement policy N, by which the system is replaced at the time of the Nth failure, is adopted. An explicit expression of the long run average cost per unit time is derived, and an optimal policy N^* for minimizing the long run average cost per unit time is determined analytically. A numerical example is also given.     

Lifetime replacement policy in discrete time for a single unit system

We consider, in discrete time, a single unit system which operates for a period of time represented by a general distribution. This unit is subjected to failures during operations. Some of these failures are repairable and the unit is repaired in the repair facility. When the unit experiences a non-repairable failure then it has to be replaced with a new one. We consider a replacement policy based on the lifetime of the unit. This policy can be studied from two different approaches. The first approach, named Model I, is to replace the unit by a new one when the unit attains a predetermined lifetime. The other approach, named Model II, is to close repair facility when the lifetime of the unit attains a predetermined quantity. For each model, we obtain the stationary distribution and some performance measures of interest.     

Two models for a repairable two-system with phase-type sojourn time distributions

This paper investigates a general repairable two-system. The operational and repair times are general, but for applicability, are approached by phase-time distributions, given that this class is dense in the set of distribution functions on the positive real line. Two models are studied, depending on the remembering of the failure phase when the unit is repaired. The versatility of this class of functions is shown. For these models, the availability and the rate of occurrence of failures are calculated. These performance measures are presented in a well-structured form, and are computationally implemented. The method and results are illustrated by a numerical example. The present work generalizes others in the specialized literature, and completes the study of two-systems under the Markov system.     

Failure estimates with replacement/repair

The expected number of circuit pack failures is an important input to the accurate planning of the inventory level of spares. The failure rate of a circuit pack depends on its age and is generally higher during the infant mortality period (frequently one year) as compared to the steady-state period. In some of the existing models, it is assumed that all of the defective circuit packs are replaced with new circuit packs. On the other extreme, some models assume that each of the defective circuit packs is repaired and shipped back to the field. Depending on the modelling assumption, the estimates of expected number of failures can differ by as much as 15 per cent. The Markov process model described in this paper includes a parameter (fraction of non-repairable circuit packs) which permits us to investigate cases where only a portion of the circuit packs are non-repairable (i.e. need to be replaced with new circuit packs). The model estimates the expected number of failures based on the infant mortality rate, the steady-state failure rate, age distribution of the installed circuit packs, growth potential and fraction of non-repairable circuit packs.     

Reliability analysis for a circular consecutive-2-out-of-n:F repairable system with priority in repair

A circular consecutive-2-out-of-n:F repairable system with one repairman is studied in this paper. When there are more than one failed component, priorities are assigned to the failed components. Both the working time and the repair time of each component is assumed to be exponentially distributed. Every component after repair is as good as new. By using the definition of generalized transition probability and the concept of critical component, we derive the state transition probability matrix of the system. Methodologies are then presented for the derivation of system reliability indexes such as availability, rate of occurrence of failure, mean time between failures, reliability, and mean time to first failure.     

Optimal maintenance policy for a Markovian system under periodic inspection

This study proposes a state-dependent maintenance policy R_i,j(T,N,α) for a multi-state continuous-time Markovian deteriorating system subject to aging and fatal shocks and with states 0 (new state) <1<2<…<L (failed-state). Under R_i,j(T,N,α), the system is inspected at each kT for k=1,2,3… to identify the current state as, say a, and then do-nothing, repair and replacement are taken immediately according to 0≤a≤i−1, i≤a≤j−1 and j≤a≤L−1, respectively in case i<j. Additionally, the replacement is carried out whenever L occurs due to fatal shocks. This policy includes numerous maintenance policies in the literature as special cases and can be applied quite generally. We then try to determine the optimal i^*, j^* and T^* such that the expected long-run cost rate is minimized. A numerical example is given to evaluate the performance of the policy.     

A bivariate optimal replacement policy for a multistate repairable system

In this paper, a deteriorating simple repairable system with k+1 states, including k failure states and one working state, is studied. It is assumed that the system after repair is not “as good as new” and the deterioration of the system is stochastic. We consider a bivariate replacement policy, denoted by (T,N), in which the system is replaced when its working age has reached T or the number of failures it has experienced has reached N, whichever occurs first. The objective is to determine the optimal replacement policy (T,N)^* such that the long-run expected profit per unit time is maximized. The explicit expression of the long-run expected profit per unit time is derived and the corresponding optimal replacement policy can be determined analytically or numerically. We prove that the optimal policy (T,N)^* is better than the optimal policy N^* for a multistate simple repairable system. We also show that a general monotone process model for a multistate simple repairable system is equivalent to a geometric process model for a two-state simple repairable system in the sense that they have the same structure for the long-run expected profit (or cost) per unit time and the same optimal policy. Finally, a numerical example is given to illustrate the theoretical results.     

Quantitative analysis of a fault tree with priority AND gates

A method for calculating the exact top event probability of a fault tree with priority AND gates and repeated basic events is proposed when the minimal cut sets are given. A priority AND gate is an AND gate where the input events must occur in a prescribed order for the occurrence of the output event. It is known that the top event probability of such a dynamic fault tree is obtained by converting the tree into an equivalent Markov model. However, this method is not realistic for a complex system model because the number of states which should be considered in the Markov analysis increases explosively as the number of basic events increases. To overcome the shortcomings of the Markov model, we propose an alternative method to obtain the top event probability in this paper. We assume that the basic events occur independently, exponentially distributed, and the component whose failure corresponds to the occurrence of the basic event is non-repairable. First, we obtain the probability of occurrence of the output event of a single priority AND gate by Markov analysis. Then, the top event probability is given by a cut set approach and the inclusion–exclusion formula. An efficient procedure to obtain the probabilities corresponding to logical products in the inclusion–exclusion formula is proposed. The logical product which is composed of two or more priority AND gates having at least one common basic event as their inputs is transformed into the sum of disjoint events which are equivalent to a priority AND gate in the procedure. Numerical examples show that our method works well for complex systems.     

A quantification algorithm for a repairable system in the GO methodology

The GO methodology is an effective method of system reliability analysis. It has been applied to non-repairable systems. This paper discusses the application of the GO method to a repairable system which is described by a Markov model and presents the quantification algorithm of the steady characteristics of the repairable system. The calculation formulas of the ordinary operators and the logical gates are derived and the steady reliability parameters of the system such as average operation probability and average failure frequency can be directly computed by the GO method. The result of an example shows that the algorithm is correct. The algorithm will be useful for the safety analysis of most engineering repairable systems.     

Redundancy analysis for repairable multi-state system by using combined stochastic processes methods and universal generating function technique

This paper discusses a type of redundancy that is typical in a multi-state system. It considers two interconnected multi-state systems where one multi-state system can satisfy its own stochastic demand and also can provide abundant resource (performance) to another system in order to improve the assisted system reliability. Traditional methods are usually not effective enough for reliability analysis for such multi-state systems because of the “dimensional curse” problem. This paper presents a new method for reliability evaluation for the repairable multi-state system considering such kind of redundancy. The proposed method is based on the combination of the universal generating function technique and random processes methods. The numerical example is presented to illustrate the proposed method.     

Simultaneous determination of preventive maintenance and replacement policies in a queue-like production system with minimal repair

This paper addresses the joint effects of preventive maintenance and replacement policies on a queue-like production system with minimal repair at failures. We consider a policy which calls for a preventive maintenance operation whenever N parts have been processed. If a failure occurs and at least K preventive maintenance operations have been carried out, the system is replaced by a new one. Otherwise, a failure is handled by minimal repair. An analytical model is developed and the argument of renewal–reward theory is used to provide long-run expected profit per unit time for a given maintenance and replacement policy. Numerical examples are given to provide some managerial insights.     

Bracketing of failure path probability in a system with ageing repair times

In practical terms, the distributions of repair times are not known; we have only mean values. We show here that by assuming these distributions to be ageing, it is possible in many cases to bracket availability and reliability using only Mean Times to Repair. Furthermore, this computation can be carried out using Markov models.     

Optimal restarting distribution after repair for a Markov deteriorating system

We consider a repairable system such that different completeness degrees are possible for the repair (or corrective maintenance) that go from a ‘minimal’ up to a ‘complete’ repair. Our question is: to what extent must the system be repaired in case of failure for the long-run availability to be optimal? The system evolves in time according to a Markov process as long as it is running, whereas the duration of repairs follows general distributions. After repair, the system starts again in the up-state i with probability d(i). We observe from numerical examples that the optimal restarting distribution d^opt (such that the long-run availability is optimal) is generally random and does not correspond to a new start in a fixed up-state. Sufficient conditions under which the optimal restarting distribution is non-random are given. Also, the optimal restarting distribution is provided for two classical structures in reliability.     

Dependability and performance stochastic modelling of a two-unit repairable production system with preventive maintenance

In this paper, a two-unit multistate repairable production system is considered in which preventive maintenance (PM) is implemented in order to improve its dependability and performance. A general model is provided for the production system using a semi-Markov process, for examining system’s limiting behaviour. Apart from combining redundancy with PM, we introduce scenarios like imperfect and failed maintenance which are usually met in real life production systems. For the proposed model, we calculate the availability, the mean time to failure and the total operational cost and we formulate optimisation problems settled with respect to the system’s inspection times. The main aim of our work is to determine the optimal inspection times and consequently the optimal PM policies to be adopted in order to optimise system’s dependability and performance.