Pseudodynamic cost limit replacement model under minimal repair

This paper presents a mathematical model to evaluate pseudodynamic cost limit replacement policies for a system that follows a general time-to-failure distribution. When the failed system requires repair, it is first inspected and the repair cost is estimated. Minimal repair is only then undertaken if the estimated cost is less than the exponentially declining repair cost limit. A negative exponential distribution of estimated repair cost is assumed for analytic tractability. An example with a Weibull time-to-failure distribution is given to illustrate the computational results.     

Cost limit replacement policy under minimal repair

This paper presents a policy for either repairing or replacing a system that has failed. When a system requires repair, it is first inspected and the repair cost is estimated. Repair is only then undertaken if the estimated cost is less than the “repair cost limit”. However, the repair cannot return the system to “as new” condition but instead returns it to the average condition for a working system of its age. Examples include complex systems where the repair or replacement of one component does not materially affect the condition of the whole system. A Weibull distribution of time to failure and a negative exponential distribution of estimated repair cost are assumed for analytic amenability. An optimal “repair cost limit” policy is developed that minimizes the average cost per unit time for repairs and replacement. It is shown that the optimal policy is finite and unique.     

Reliability analysis and optimization applications of a two-unit standby redundant system with spare units

Consider a two-unit standby redundant system with two main units, one repair facility, and n spare units. When the main unit has failed and the other is under repair, a spare unit takes over the operation and if it fails, it is replaced by a new one until the repair of the failed unit is completed. The system fails when the last spare unit fails while one main unit is under repair and the other has failed. In this paper, we derive expressions for 1) the distribution function of the first time to system failure, 2) the probability that the total number of failed spare units during the time interval (0,t] is n and 3) the mean of the total number of failed spare units in (0,t] and its asymptotic behaviour. Introducing costs incurred for each failed main unit and each failed spare unit, the expected cost per unit of time of the system was also derived. Finally an optinmization problem is discussed in order to compare the expected cost of the system with both main units and spare units with that of spare units only, and particular cases are considered.     

Optimal replacement policies for systems with multiple standbycomponents

The authors consider two new preventive replacement policies for a multiple-component cold-standby system. The failure rate of the component in operation is constant. The system is inspected at random points over time to determine whether it is to be replaced. The replacement decision is based on the number of failed components at the time of inspection. There are two replacement options if the complete system fails during operation: (i) replace the system if an inspection reveals that it has failed (system failure is not self-announcing), and (ii) replace the system the instant it fails (system failure is self-announcing). There is a threshold value on the number of failed components (at the time of inspection) which minimizes the mean total cost. The authors develop a simple efficient procedure to find the optimal threshold value. They compare the cost of operating a system that is inspected at random points in time, with the cost of operating a system that is monitored continuously through an attached monitoring device, and discuss cost tradeoffs     

Cost analysis of a two dissimilar-unit cold standby redundant system subject to inspection and two types of repair

This paper deals with the cost analysis of a two dissimilar-unit cold standby redundant system subject to inspection and two types of repair where each unit of the system has two modes, normal and failed. It is assumed that the failure, repair, replacement and inspection times are stochastically independent random variables each having an arbitrary distribution. The cold standby unit replaces the failed operative unit after a random amount of time. An inspection is required to decide whether it needs type I (minor repair) or type 2 (major repair). In this system the repairman is not always available with the system, but is called whenever the operative unit fails. The system is analysed by the semi-Markov process technique. Some reliability measures of interest to system designers as well as operations managers have been obtained. Pointwise availability, steady-state availability, busy period by a server and the expected cost per unit time of the system are obtained. Certain important results have been derived as particular cases.     

Evaluation of Generating System Reliability Indices Under Limited Repair Facility

A generating system having n units, where each unit is represented by a 2-state device, will have more than 2n possible states when repair facility of the system is limited, i.e. number of repair crews is less than the number of units. When the number of units failed is more than the number of repair crews available, the most recent failures queue for repairs. This paper gives a general equation to determine the number of system states under limited repair facility and an approximate technique to determine the probabilities of system states and reliability indices of the system.     

An Optimal Decision Rule for Repair vs Replacement

This paper presents a policy for either repairing or replacing a system that has failed. The policy applies to systems whose mean residual life function is decreasing. An optimal policy is developed that minimizes the cost per unit time for repair and replacement. Results are shown graphically for a particular distribution of time to failure and are motivated in terms of an automobile replacement problem.     

Periodic replacement when minimal repair costs depend on the age and the number of minimal repair for a multi-unit system

A policy of periodic replacement with minimal repair at failure is considered for the multi-unit system which have the specific multivariate distribution. Under such a policy the system is replaced at multiples of some period T while minimal repair is performed at any intervening component failures. The cost of a minimal repair to the component is assumed to be a function of its age and the number of minimal repair. A simple expression is derived for the expected minimal repair cost in an interval in terms of the cost function and the failure rate of the component. Necessary and sufficient conditions for the existence of an optimal replacement interval are exhibited.     

Determining Optimum Burn-In and Replacement Times Using Bayesian Decision Theory

An important problem facing a manufacturer is the determination of the amount of time to burn-in items (in order to eliminate early failures) and the age at which to replace items (to avoid failures due to wearout). This problem becomes difficult to solve if the time-to-failure distribution of an item is unknown and must be estimated from test and operational data. This paper describes a method of statistical data analysis which is readily applied to the solution of this decision problem under a realistic but general loss (or gain) function. The method is a multiparameter Bayesian analysis which requires multiple integration of the (multivariate) posterior of the parameters of the time-to-failure distribution to obtain the expected loss (or gain) resulting from a particular choice of burn-in time and item replacement age. This integration is performed by a Monte Carlo Procedure using importance sampling. An example demonstrates the flexibility of this method of analysis. The data are a mixture of ``point' and truncated data, which often create difficulties when using conventional methods of decision analysis. In addition, since the method permits up to ten parameters for the family of time-to-failure distributions, a ``bathtub' hazard rate function is used to generate the data for the example. The results are presented in the form of Bayesian confidence intervals for the true hazard rate function and a presentation of the expected loss as a function of burn-in time and age at replacement.     

Periodic replacement with minimal repair at failure and general random repair cost for a multi-unit system

A policy of periodic replacement with minimal repair at failure is considered for the multi-unit system which have the specific multivariate distribution. Under such a policy an operating system is completely replaced whenever it reaches age T (T > 0) at a cost c₀ while minimal repair is performed at any intervening component failures. The cost of the j-th minimal repair to the component which fails at age y is g(C(y),c_j(y)), where C(y) is the age-dependent random part, c_j(y) is the deterministic part which depends on the age and the number of the minimal repair to the component, and g is an positive nondecreasing continuous function. A simple expression is derived for the expected minimal repair cost in an interval in terms of the cost function and the failure rate of the component. Necessary and sufficient conditions for the existence of an optimal replacement interval are exhibited.     

Cost analysis of a two unit repairable system subject to on-line preventive maintenance and/or repair

The purpose of this paper is to carry out the cost analysis of a two-unit repairable system subject to on-line preventive maintenance(PM) and repair. The policy adopted here is that the on-line PM work of the operating unit is undertaken followed by repair work of the failed unit(if any). All the random variables that. arise in the analysis are assumed to be independently and arbitrarily distributed. A mathematical model is developed using semi-regenerative phenomena and system of integral equations satisfied by various state probabilities corresponding to various initial conditions are obtained. An iterative numerical method is used to solve the system of integral equations. A cost function is built based on the expected number of various jobs completed by the server in [0, t].     

Optimal monitoring strategies for slowly deteriorating repairablesystems

A simple model for determination of an optimal limit on taking corrective action in a slowly deteriorating repairable system is presented. The performance of such a system is assumed to be characterized by a single parameter which is continuously being monitored. The underlying deterioration process is assumed to be governed by a Brownian motion process with a positive drift. When the measured value of the parameter reaches the action limit, the repair/replacement procedure is initiated. The optimal action limit is derived so that the expected long run average total cost is minimized. Some simple numerical examples illustrate the model and the optimization     

Optimal periodic preventive repair and replacement policy assuming geometric process repair

In this paper, a simple deteriorating system with repair is studied. When failure occurs, the system is replaced at high cost. To extend the operating life, the system can be repaired preventively. However, preventive repair does not return the system to a "good as new" condition. Rather, the successive operating times of the system after preventive repair form a stochastically decreasing geometric process, while the consecutive preventive repair times of the system form a stochastically increasing geometric process. We consider a bivariate preventive repair policy to solve the efficiency for a deteriorating & valuable system. Thus, the objective of this paper is to determine an optimal bivariate replacement policy such that the average cost rate (i.e., the long-run average cost per unit time) is minimized. The explicit expression of the average cost rate is derived, and the corresponding optimal replacement policy can be determined numerically. An example is given where the operating time of the system is given by a Weibull distribution.     

Periodic replacement when minimal repair costs depend on the age and the number of minimal repairs for a multi-unit system

A policy of periodic replacement with minimal repair at failure is considered for a multi-unit system which has a specific multivariate distribution. Under such a policy the system is replaced at multiples of some period T while minimal repair is performed for any intervening component failure. The cost of a minimal repair to the component is assumed to be a function of its age and the number of minimal repairs. A simple expression is derived for the expected minimal repair cost in an interval in terms of the cost function and the failure rate of the component. The necessary and sufficient conditions for the existence of an optimal replacement interval are found.     

Optimal redundancies for reliability and availability of series systems

Five different series system configurations with standby units are compared based on their overall reliability and availability. The time-to-failure of a component and its repair time are assumed to have the negative exponential distribution. The mean time-to-failure, MTTF, and the steady-state availability, A_T(∞), are obtained for the first three simple configurations and comparisons are performed. For all five configurations, comparisons are made based on assumed numerical values given to the distribution parameters, as well as to the cost of the components. The configurations are ranked based on: MTTF, A_T(∞), cost and cost-benefit where benefit is either MTTF or A_T(∞).     

A maintenance model for two-unit redundant system

In this paper, a maintenance model for two-unit redundant system with one repairman is studied. At the beginning, unit 1 is operating, unit 2 is the standby unit. The costs include the operating reward, repair cost and replacement cost, besides, a penalty cost is incurred if the system breaks down. Two kinds of replacement policy, based on the number of failures for two units and the working age, respectively are used. The long-run average cost per unit time for each kind of replacement policy is derived. Also, a particular model in which the system is deteriorative, two units are identical and the penalty cost rate is high, is thoroughly studied.     

A geometric-process repair-model with good-as-new preventive repair

This paper studies a deteriorating simple repairable system. In order to improve the availability or economize the operating costs of the system, the preventive repair is adopted before the system fails. Assume that the preventive repair of the system is as good as new, while the failure repair of the system is not, so that the successive working times form a stochastic decreasing geometric process while the consecutive failure repair times form a stochastic increasing geometric process. Under this assumption and others, by using geometric process we consider a replacement policy N based on the failure number of the system. Our problem is to determine an optimal replacement policy N such that the average cost rate (i.e., the long-run average cost per unit time) is minimized. The explicit expression of the average cost rate is derived, and the corresponding optimal replacement policy can be determined analytically or numerically. And the fixed-length interval time of the preventive repair in the system is also discussed. Finally, an appropriate numerical example is given. It is seen from that both the optimal policies N** and N* are unique. However, the optimal policy N** with preventive repair is better than the optimal policy N* without preventive repair     

Common-cause failures and critical human errors in repairable and non-repairable systems

This paper presents a reliability analysis of a k-out-of-N:G redundant system with common-cause failures, critical human errors and r repair facilities. The system is in a failed state when common-cause failures or critical human errors occurred or k units failed. When less than k units failed, the failed units are to be repaired. If the whole system is in a failed state, it cannot be repaired. Laplace transorms of state probabilities and system reliability are derived. Various versions of mean time to failure of a system are also reported.     

Design of exact regenerating hierarchical code for distributed storage system

Erasure code is widely used as the redundancy scheme in distributed storage system. When a storage node fails, the repair process often requires to transfer a large amount of data. Regenerating code and hierarchical code are two classes of codes proposed to reduce the repair bandwidth cost. Regenerating codes reduce the amount of data transferred by each helping node, while hierarchical codes reduce the number of nodes participating in the repair process. In this paper, we propose a "sub-code nesting framework" to combine them together. The resulting regenerating hierarchical code has low repair degree as hierarchical code and lower repair cost than hierarchical code. Our code can achieve exact regeneration of the failed node, and has the additional property of low updating complexity.     

An age replacement policy with increasing minimal repair cost

A replacement policy for a system in which minimal repair cost increases in system age is considered. If a system fails before age T, it is minimally repaired. Otherwise, the system is replaced when if fails for the first time after age T. The mean cost rate is used as a criterion for optimization. It is shown that the optimal T minimizing the mean cost rate is finite and unique.