Optimal control of an M/E<Subscript><Emphasis Type="Italic">k</Emphasis></Subscript>/1 queueing system with removable service station subject to breakdowns

In this paper we deal with a single removable service station queueing system with Poisson arrivals and Erlang distribution service times. The service station can be turned on at arrival epochs or off at departure epochs. While the service station is working, it is subject to breakdowns according to a Poisson process. When the station breaks down, it requires repair at a repair facility, where the repair times follow the negative exponential distribution. Conditions for a stable queueing system, that is steady-state, are provided. The steady-state results are derived and it is shown that the probability that the service station is busy is equal to the traffic intensity. Following the construction of the total expected cost function per unit time, we determine the optimal operating policy at minimum cost.     

修理设备可更换且修理延迟的两相同部件冷储备可修系统(Ⅱ)

在文[1]的基础上,本文研究了修理有延迟和修理设备可更换的两单元冷储备可修系统.在假定单元的寿命服从指数分布、修理时间和延迟时间服从一般分布、修理设备的寿命和故障后的更换时间服从指数分布下,通过定义修理设备的"广义忙期",使用更新过程理论和全概率分解技术,提出一种新的分析技巧,讨论了修理设备的一些可靠性指标,获得了如修理设备的可用度和故障次数等可靠性结果.     

修理设备可更换和修理延迟的两类故障状态并联可修系统

考虑两同型部件组成的并联可修系统,每个部件有两类故障状态,部件故障后修理有延迟,且修理设备在修理故障部件的过程中也可能发生故障.假定部件的寿命和修理设备的寿命服从指数分布,部件发生故障后的修理延迟时间、修理时间和修理设备故障后的更换时间均服从一般分布,利用马尔可夫更新过程理论和拉普拉斯变换工具,求得了系统有关的可靠性指标.     

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.     

修理设备可更换且有修理延迟的N部件串联系统分析

假定部件的寿命服从指数分布,修理延迟时间和修理时间均服从任意分布,并且修理设备的寿命服从指数分布,其更换时间服从任意分布的情况下,利用马尔可夫更新过程理论和拉普拉斯变换工具,研究了修理有延迟且修理设备可更换的n部件串联可修系统,求得了系统的可用度和(0,t]时间内的平均故障次数.进一步,在定义修理设备“广义忙期”下,利用全概率分解,提出了一种新的分析技术,讨论了修理设备的可靠性指标,得到修理设备的一些重要可靠性结果.     

Stochastic analysis of a n-unit repairable system with slow switch

This paper deals with the costn–benefit analysis of a cold standby system composed of n identical repairable units, subject to slow switch. Two models of system functioning are studied in this paper. In model 1, the repair time of a unit is assumed to follow exponential distribution and the other time distributions as arbitrary, while in model 2, the repair time of a unit is assumed to be arbitrarily distributed and the other time distributions follow exponential law. For both the models, the system characteristics, namely

(i) the expected upn–time of the system during the period (O,t]

(ii) the expected busyn–period of the repair facility during the period (0,t] and

(iii) the expected time the units spend in the switchover/installation state during the period (O,t]

are studied by identifying the system a t suitable regeneration epochs. The cost-benefit analysis is carried out using these characteristics     

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 control of a removable and non-reliable server in an M/M/1 queueing system with exponential startup time

We study a single removable and non-reliable server in the N policy M/M/1 queueing system. The server begins service only when the number of customers in the system reaches N (N1). After each idle period, the startup times of the server follow the negative exponential distribution. While the server is working, it is subject to breakdowns according to a Poisson process. When the server breaks down, it requires repair at a repair facility, where the repair times follow the negative exponential distribution. The steady-state results are derived and it is shown that the probability that the server is busy is equal to the traffic intensity. Cost model is developed to determine the optimal operating N policy at minimum cost.     

Cost-benefit analysis of one-server n-unit adaptive system with slow switch

This paper considers a shared parallel system consisting of n-units supported by single service facility to carry out both installation and repair of a unit. Initially, all the n units share the total load equally and when one or more units fail, they go for repair while the other surviving units share the entire load equally till the failed units are ready for operation after installation. The installation time (switchover time) of a repaired unit is assumed to be non-negligible and random. The system will be down when all the units are non-operative , Assuming that the failure rates are different when the units function under varying loads, the system characteristics, namely, (1) the expected up-time of the system during (0, t], (2) the expected repair time of the units which failed due to varying failure rates during (0, t] and (3) the expected time spent by the units in the installation state during the period (0, t], are obtained by identifying the system at suitable regeneration epochs. The repair time and the switchover time of the units are arbitrarily distributed. The failure rate of unit is assumed to be constant. It depends on the number of surviving units at any instant. The cost-benefit analysis is also carried out using these system characteristics     

The Optimum Repair Limit Replacement Policies

This paper considers a single unit system which is first repaired if it fails. If the repair is not completed up to the fixed repair limit time then the unit under repair is replaced by a new one. The cost functions are introduced for the repair and the replacement of the failed unit. The optimum repair limit replacement time minimizing the expected cost per unit of time for an infinite time span is obtained analytically under suitable conditions. Two special cases where the repair cost functions are proportional to time and are exponential are discussed in detail with numerical examples.     

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.     

Optimal design of unreliable production–inventory systems with variable production rate

In this article, we study an economic manufacturing quantity (EMQ) problem for an unreliable production facility where the production rate is treated as a decision variable. As the stress condition of the machine changes with the production rate, the failure rate of the machine is assumed to be dependent on the production rate. The unit production cost is also taken as a function of the production rate, as the machine can be operated at different production rates resulting in different unit production costs. The basic EMQ model is formulated under general failure and general repair time distributions and the optimal production policy is derived for specific failure and repair time distributions viz., exponential failure and exponential repair time distributions. Considering randomness of the time to machine failure and corrective repair time, the model is extended to the case where certain safety stocks in inventory may be useful to improve service level to customers. Optimal production policies of the proposed models are derived numerically and the sensitivity of the optimal results with respect to those parameters which directly influence the machine failure and repair rates is also examined.     

Reliability analysis of 1-for-2 shared protection systems with general repair-time distributions

In this paper, we investigate the reliability of a type of 1-for-2 shared protection systems. The 1-for-2 shared protection system is the most basic fault-tolerant configuration with shared backup units. We assume that there are two working units each serving a single user and one shared protection (spare) unit in the system. We also assume that the times to failure and to repair are subject to exponential and general distributions respectively. Under these assumptions, we derive the Laplace transform of the survival function (the cdf that the system will survive beyond a given time) for each user as well as the user-perceived Mean Time to First Failure (MTTFF) by combining the state transition analysis and the supplementary variable method. We also show the effect of the repair-time distribution, the failure rates and the repair rates of the units through the case study of small-sized two enterprises that share one spare device for backup purpose. The analysis reveals what is important and what should be done in order to improve the user-perceived reliability of shared protection systems.     

Reliability analysis of a warm standby repairable system with priority in use

In this paper, a warm standby repairable system consisting of two dissimilar units and one repairman is studied. In this system, it is assumed that the working time distributions and the repair time distributions of the two units are both exponential, and unit 1 is given priority in use. After repair, both unit 1 and unit 2 are “as good as new”. Moreover, the transfer switch in the system is unreliable, and the function of the switch is: “as long as the switch fails, the whole system fails immediately”. Under these assumptions, using Markov process theory and the Laplace transform, some important reliability indexes and some steady state system indexes are derived. Finally, a numerical example is given to illustrate the theoretical results of the model.     

Transient analysis of a queue with system disasters and customer impatience

A single server queue with Poisson arrivals and exponential service times is studied. The system suffers disastrous breakdowns at an exponential rate, resulting in the loss of all running and waiting customers. When the system is down, it undergoes a repair mechanism where the repair time follows an exponential distribution. During the repair time any new arrival is allowed to join the system, but the customers become impatient when the server is not available for a long time. In essence, each customer, upon arrival, activates an individual timer, which again follows an exponential distribution with parameter ξ. If the system is not repaired before the customer’s timer expires, the customer abandons the queue and never returns. The time-dependent system size probabilities are presented using generating functions and continued fractions.     

Availability and failure frequency of a Gnedenko system

This paper considers anN-unit series system supported by a warm standby unit and a single repair facility. Suppose that the operating units and the standby unit have constant failure ratesa anda ₁, respectively. When the system is down, all the operable units have constant failure ratea ₂. The repair time of a failed unit has an arbitrary distribution. Using Takács' method and a Markov renewal process, we discuss the stochastic behavior of this system and obtain the explicit formulae of the system availability and failure frequency.Project supported by the National Natural Science Foundation of China.     

Probabilistic analysis of a system with dependent units having a single repair facility subject to preventive maintenance

The paper deals with the availability and the reliability analysis of a system with dependent units having a single repair facility subject to preventive maintenance. The system initially consists of n-identical units (connected in parallel) each with failure rate λ_n. The failure rate of a unit at any given instant of time depends upon the number of units operating at that instant. The time to repair of a failed unit and the time for maintenance of the repair- facility are arbitrarily distributed whereas the time to failure of a unit is exponentially distributed. The results obtained have been compared with those obtained when the repair facility is not subject to preventive maintenance.     

On modeling failure and repair times in stochastic models of manufacturing systems using generalized exponential distributions

Failures of machines have a significant effect on the behavior of manufacturing systems. As a result it is important to model this phenomenon. Many queueing models of manufacturing systems do incorporate the unreliability of the machines. Most models assume that the times to failure and the times to repair of each machine are exponentially distributed (or geometrically distributed in the case of discrete-time models). However, exponential distributions do not always accurately represent actual distributions encountered in real manufacturing systems. In this paper, we propose to model failure and repair time distributions bygeneralized exponential (GE) distributions (orgeneralized geometric distributions in the case of a discretetime model). The GE distribution can be used to approximate distributions with any coefficient of variation greater than one. The main contribution of the paper is to show that queueing models in which failure and repair times are represented by GE distributions can be analyzed with the same complexity as if these distributions were exponential. Indeed, we show that failures and repair times represented by GE distributions can (under certain assumptions) be equivalently represented by exponential distributions.This work was performed while the author was visiting the Laboratory for Manufacturing and Productivity, Massachusetts Institute of Technology, Cambridge, MA 02139, USA.     

带有N策略,启动时间和服务台故障的M/M/1排队的均衡策略

本文考虑带有N策略,启动时间和服务台故障的M/M/1排队的顾客的策略行为.当系统为空时服务台关闭,并且只有当系统中的顾客数达到一个给定的阈值时才会被激活,启动时间服从指数分布.服务台在工作时可能会故障,一旦发生故障,它立即被维修,维修的时间服从指数分布.我们得到了不同状态的均衡到达率并且给出了均衡社会收益函数.最后对均衡到达率和均衡社会收益进行了数值研究.     

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.