A general model for consecutive-k-out-of-n: F repairable system with exponential distribution and (k−1)-step Markov dependence

In this paper, a general model for consecutive-k-out-of-n: F repairable system with exponential distribution and (k−1)-step Markov dependence is introduced. The lifetime of a component is an exponential random variable, its parameter depends on the number of consecutive failed components that precede the component. The repair time is also an exponential random variable. A priority repair rule on the basis of the system failure risk is adopted. Then the transition density matrix of the system is determined. Some reliability indices, including the system availability, rate of occurrence of failures and reliability are evaluated accordingly. For the demonstration of the model and methodology, a linear system example and a circular system example are investigated.     

A number-dependent replacement policy for a system with continuous preventive maintenance and random lead times

This paper considers a number-dependent replacement policy for a system with two failure types that is replaced at the nth type I (minor) failure or the first type II (catastrophic) failure, whichever occurs first. Repair or replacement times are instantaneous but spare/replacement unit delivery lead times are random. Type I failures are repaired at zero cost since preventive maintenance is performed continuously. Type II failures, however, require costly system replacement. A model is developed for the average cost per unit time based on the stochastic behavior of the system and replacement, storage, and downtime costs. The cost-minimizing policy is derived and discussed. We show that the optimal number of type I failures triggering replacement is unique under certain conditions. A numerical example is presented and a sensitivity analysis is performed.     

The influence of delivery times on repairable k-out-of-N systems with spares

The k-out-of-N structure is a popular type of redundancy in fault-tolerant systems with wide applications in computer and communication systems, and power transmission and distribution systems, among others, during the past several decades. In this paper, our interest is in such a reliability system with identical, repairable components having exponential life times, in which at least k out of N components are needed for the system to perform its functions. There is a single repairman who attends to failed components on a first-come-first-served basis. The repair times are assumed to be of phase type. The system has K spares which can be tapped to extend the lifetime of the system using a probabilistic rule. We assume that the delivery time of a spare is exponentially distributed and there could be multiple requests for spares at any given time. Our main goal is to study the influence of delivery times on the performance measures of the k-out-of-N reliability system. To that end, the system is analyzed using a finite quasi-birth-and-death process and some interesting results are obtained.     

Analysis of R out of N systems with several repairmen,exponential life times and phase type repair times: An algorithmic approach

An R out of N repairable system consisting of N independent components is operating if at least R components are functioning. The system fails whenever the number of good components decreases from R to R − 1. A failed component is sent to a repair facility having several repairmen. Life times of working components are i.i.d random variables having an exponential distribution. Repair times are i.i.d random variables having a phase type distribution. Both cold and warm stand-by systems are considered. We present an algorithm deriving recursively in the number of repairmen the generator of the Markov process that governs the process. Then we derive formulas for the point availability, the limiting availability, the distribution of the down time and the up time. Numerical examples are given for various repair time distributions. The numerical examples show that the availability is not very sensitive to the repair time distribution while the mean up time and the mean down time might be very sensitive to the repair time distributions.     

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.     

Optimal order-replacement policy for a phase-type geometric process model with extreme shocks

A system is subject to random shocks that arrive according to a phase-type (PH) renewal process. As soon as an individual shock exceeds some given level the system will break down. The failed system can be repaired immediately. With the increasing number of repairs, the maximum shock level that the system can withstand will be decreasing, while the consecutive repair times after failure will become longer and longer. Undergoing a specified number of repairs, the existing system will be replaced by a new and identical one. The spare system for the replacement is available only by sending a purchase order to a supplier, and the duration of spare system procurement lead time also follows a PH distribution. Based on the number of system failures, a new order-replacement policy (also called $(K\text{,}N)$ policy) is proposed in this paper. Using the closure property of the PH distribution, the long-run average cost rate for the system is given by the renewal reward theorem. Finally, through numerical calculation, it is determined an optimal order-replacement policy such that the long-run expected cost rate is minimum.     

Vacation policies for machine repair problem with two type spares

This paper studies the machine repair problem consisting of M operating machines with two types of spare machines (S = S₁ + S₂), and R servers (repairmen) who leave for a vacation of random length when there are no failed machines queuing up for repair in the repair facility. At the end of the vacation the servers return and operate two vacation policies. First, the servers take vacations repeatedly until they find the repair facility has at least one waiting failed machine in the queue. Second, the servers do not take a vacation again and remain idle until the first arriving failed machine arrives, which starts a busy period in the repair facility. For both policies, the servers have two service rates for repair-slow and fast. The matrix geometric theory is used to find the steady-state probabilities of the number of failed machines in the system as well as the performance measures. Some special cases are given. A direct search algorithm is used to simultaneously determine the optimal values of the number of two types of spares and the number of servers while maintaining a minimum specified level of system availability.     

Reliability Analysis of a One-Unit System with Unrepairable Spare Units and its Optimization Applications

It is assumed that a unit is either in operation or is in repair. When the main unit is under repair, spare units which cannot be repaired are used. In this system the following quantities are of interest: (i) The time distribution and the mean time to first-system failure, given that the n spare units are provided at time 0. (ii) The probability that the number of the failed spare units are equal to exactly n during the interval (0, t], and its expected number during the interval (0, t]. These quantities are derived by solving the renewal-type equations.Two optimization problems are discussed using the results obtained, viz.: (i) The expected cost of two systems, one with both a main unit and spare units and the other with only spare units is considered. (ii) A preventive maintenance policy of the main unit is considered in order to minimize the expected cost rate. Some policies of the two problems are discussed under suitable conditions. Numerical examples are also presented.     

A study of repairable parts inventory system operating under performance-based contract

Performance-Based Logistics (PBL) is becoming a dominant logistics support strategy, especially in the defense industry. PBL contracts are designed to serve the customer’s key performance measures, while the traditional contracts for after-sales services, such as Fixed-price (FP) and Cost-plus (C+), only provide insurance or incentive. In this research, we develop an inventory model for a repairable parts system operating under a PBL contract. We model the closed-loop inventory system as an M/M/m queue in which component failures are Poisson distributed and the repair times at the service facility are exponential. Our model provides the supplier and the customer increased flexibility in achieving target availability. Analysis of key parameters suggests that to improve the availability of the system with repairable spare parts, the supplier should work to improve the components reliability and efficiency of repair facility, rather than the base stock level, which has minimal impact on system availability.     

Fisher information in window censored renewal process data and its applications

Suppose we have a renewal process observed over a fixed length of time starting from a random time point and only the times of renewals that occur within the observation window are recorded. Assuming a parametric model for the renewal time distribution with parameter θ, we obtain the likelihood of the observed data and describe the exact and asymptotic behavior of the Fisher information (FI) on θ contained in this window censored renewal process. We illustrate our results with exponential, gamma, and Weibull models for the renewal distribution. We use the FI matrix to determine optimal window length for designing experiments with recurring events when the total time of observation is fixed. Our results are useful in estimating the standard errors of the maximum likelihood estimators and in determining the sample size and duration of clinical trials that involve recurring events associated with diseases such as lupus.     

修如新模型中备件量的计算及其置信上限的Fiducial推断

本文研究了在修如新模型下, 对预定贮存期为$T$同时开始贮存的$N$个系统, 给出在$P_0$可修复率下所需备件数的计算公式; 针对贮存寿命服从威布尔分布的系统, 利用枢轴量, 在$P_0$可修复率和预定贮存期为$T$的条件下, 给出$N$个系统所需备件数的置信上限的定义; 并基于系统寿命试验的完全样本, 利用Fiducial方法得出备件数置信上限的计算方法.     

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.     

Mean First-passage Times for a <Emphasis Type="Italic">k</Emphasis>-out-of-<Emphasis Type="Italic">n</Emphasis> Sytem with Repair

Consider a k-out-of-n system where the lifetimes of the components are independent and identically distributed exponential (λ) random variables. Each component has its own repair facility, the repair times being independent and identically distributed exponential (μ) random variables, independent of the failure times. The mean operating time and mean repair time during the cycle between two successive breakdowns are found using renewal theory and the expression for the system availability. Using these, the mean first-passage times from any of the operating states of the system to the down state, and the mean first-passage times from any of the down states to the operating state are found recursively.     

A disaster queue with Markovian arrivals and impatient customers

We consider a single server queueing system in which arrivals occur according to a Markovian arrival process. The system is subject to disastrous failures at which times all customers in the system are lost. Arrivals occurring during the time the system undergoes repair are stored in a buffer of finite capacity. These customers can become impatient after waiting a random amount of time and leave the system. However, these customers do not become impatient once the system becomes operable. When the system is operable, there is no limit on the number of customers who can be admitted. The structure of this queueing model is of GI/M/1-type that has been extensively studied by Neuts and others. The model is analyzed in steady state by exploiting the special nature of this type queueing model. A number of useful performance measures along with some illustrative examples are reported.     

Constraint Generation for Network Reliability Problems

This paper presents a constraint generation approach to the network reliability problem of adding spare capacity at minimum cost that allows the traffic on a failed link to be rerouted to its destination. Any number of non-simultaneous link failures can be part of the requirements on the spare capacity. The key result is a necessary and sufficient condition for a multicommodity flow to exist, which is derived in the appendix. Computational results on large numbers of random networks are presented.     

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 standby system with two types of repair persons

A continuously monitored one‐unit system, backed by an identical standby unit, is perfectly repaired by an in‐house repair person, if achievable within a random or deterministic patience time (DPT), or else by a visiting expert, who repairs one or all failed units before leaving. We study four models in terms of the limiting availability and limiting profit per unit time, using semi‐Markov processes, when all distributions are exponential. We show that a DPT is preferable to a random patience time, and we characterize conditions under which the expert should repair multiple failed units (rather than only one failed unit) during each visit. We also extend the method when life‐ and repair times are non‐exponential. Copyright © 2009 John Wiley & Sons, Ltd.     

Optimal age-replacement model with age-dependent type of failure and random lead time based on a cumulative repair-cost limit policy

In this paper, we consider an age-replacement model with minimal repair based on a cumulative repair cost limit and random lead time for replacement delivery. A cumulative repair cost limit policy uses information about a system’s entire repair cost history to decide whether the system is repaired or replaced; a random lead time models delay in delivery of a replacement once it is ordered. A general cost model is developed for the average cost per unit time based on the stochastic behavior of the assumed system, reflecting the costs of both storing a spare and of system downtime. The optimal age for preventive replacement minimizing that cost rate is derived, its existence and uniqueness is shown, and structural properties are presented. Various special cases are included, and a numerical example is given for illustration. Because the framework and analysis are general, the proposed model extends several existing results.     

On the interaction between maintenance,spare part inventories and repair capacity for a k-out-of-N system with wear-out

In this paper we consider a k-out-of-N system with identical, repairable components under a condition-based maintenance policy. Maintenance consists of replacing all failed and/or aged components. Next, the replaced components have to be repaired. The system availability can be controlled by the maintenance policy, the spare part inventory level, the repair capacity and repair job priority setting. We present two approximate methods to analyse the relation between these control variables and the system availability. Comparison with simulation results shows that we can generate accurate approximations using one of these models, depending on the system size.     

Comparative analysis of availability between three systems with general repair times, reboot delay and switching failures

This paper compares the availability characteristics between three different systems with reboot delay and standby switching failures. Three systems are studied under the assumption that the time-to-failure and the time-to-repair of the primary and standby units are exponentially and generally distributed, respectively. The reboot times are assumed to be exponentially distributed with parameter β. It is assumed that there is a significant probability q of a switching failure. Using the supplementary variable technique, we develop the explicit expressions for the steady-state availability, Av, for three configurations and perform comparative analysis for three various repair time distributions, such as exponential, gamma, and uniform. Under the cost/benefit criterion, comparisons are made based on assumed numerical values given to the distribution parameters, and to the cost of the primary and standby units.