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     

A condition based maintenance model for a two-unit series system

This paper discusses a condition based maintenance model with exponential failures and fixed inspection intervals for a two-unit system in series. The condition of each unit, such as vibration or heat, is monitored at equidistant time intervals. The condition indicator variables for each unit are used to decide whether to repair an individual unit or to overhaul the whole system. After a maintenance action is performed the monitored condition indicator variable takes on its initial value. Each unit can fail only once within an inspection interval and when one or both units fail the system fails. The probability of failure is exponential and the failure rate is dependent on the condition. The cost to be minimized is the long-run average cost of maintenance actions and failures. We study the optimal solution to this problem obtained via dynamic programming.     

General analysis of l-out-of-2: f system exposed to cumulative damage processes

This paper presents models in l-out-of-2:F system. In Model 1, one unit is exposed to cumulative damage process and the other unit lias a constant failure rate. In Model 2, the two units are exposed to cumulative damage processes. They have exponential thresholds and exponential inter-damage times. Introducing a repair facility which repairs ail the damages one by one after the system-failure, this paper treats the joint Laplace transforms of the up and the down times. Marginal down time distributions .are calculated when there exists a repair facility for every damage.     

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.     

Bilevel control of degraded machining system with warm standbys, setup and vacation

In this paper, we study (N, L) switch-over policy for machine repair model with warm standbys and two repairmen. The repairman (R₁) turns on for repair only when N-failed units are accumulated and starts repair after a set up time which is assumed to be exponentially distributed. As soon as the system becomes empty, the repairman (R₁) leaves for a vacation and returns back when he finds the number of failed units in the system greater than or equal to a threshold value N. Second repairman (R₂) turns on when there are L(>N) failed units in the system and goes for a vacation if there are less than L failed units. The life time and repair time of failed units are assumed to be exponentially distributed. The steady state queue size distribution is obtained by using recursive method. Expressions for the average number of failed units in the queue and the average waiting time are established.     

Renewal analysis of a replacement process

We consider a unit with a random lifetime which is replaced at renewal times by a new identical one regardless of whether it has failed before or not. For this random periodic replacement policy, we derive exact formulas for the cycle length, defined as the time between the replacements of two successive failed units, the stationary probability of the current unit to have failed, and the stationary and the transient distributions of the residual lifetime of the current unit.     

Maintenance optimization on parallel production units

The optimal preventive-maintenance schedule for a productionsystem consisting of N identical parallel production units isinvestigated. The lifetimes of the units are IFR-distributed,i.e. with an increasing failure rate, and are supposed to bestatistically independent. The relevant costs are due to productionlosses, which are increasing and convex in the number of unitsthat are out of operation simultaneously. Actual maintenancecosts (either preventive or corrective) are supposed to be negligibleas compared to the costs due to these production losses. First we consider the apparently trivial case of geometric (discrete-time)or exponential (continuous-time) lifetime distributions forthe units. In this situation, preventive maintenance cannotimprove the condition of a unit. Hence, apparently the onlyrelevant policy is to do corrective maintenance on failed units.However, the analysis reveals that this conclusion is not correct.It turns out that taking non-failed units out of operation deliberatelycan be better than restricting to corrective maintenance only. We first show that, in the case of geometrically distributedlifetimes and unit repair times, the optimal preventive–maintenancepolicy is characterized by a single control limit K. Wheneverthe number of working units is less than or equal to K, no unitsare taken out of operation, while i – K units are setapart whenever i ( > K) units are operational. Next we consider the case with exponentially distributed lifetimesand repair times. Moreover, we assume that the repair capacityis limited, in the sense that only s ( N) units can be underrepair simultaneously. We show that, also in this case, it canbe optimal to take a working unit out of operation until thenext decision epoch (which is either a failure epoch or a repaircompletion epoch). It is shown that the optimal policy has aweak monotonicity property: the number of units which remainin operation increases with the number of available units. However,it is not necessarily true that, under the optimal policy, thenumber of units in standby position increases with the numberof available units. Numerical examples are presented which illustrate that, fora wide range of parameter values, the easiest policy (only performcorrective maintenance on failed units) performs rather wellas compared to the overall optimal policy. Finally we consider the possible extension to the practicallymore interesting case of non-exponential lifetime distributions.In particular, we assume that the lifetimes are composed oftwo non-identical exponential phases. A unit in its first lifephaseis called ‘good’, while a unit in its second phaseis called "doubtful". In this situation, one has the optionto put a good or doubtful unit in standby position until thenext decision epoch or to perform preventive maintenance ona doubtful unit. The latter brings a unit back from the doubtfulinto the good state. An indication is given of the problemsthat arise in generalizing the results obtained for the exponentialcase.     

Optimal strategy policy in batch arrival queue with server breakdowns and multiple vacations

This paper considers a like-queue production system in which server vacations and breakdowns are possible. The decision-maker can turn a single server on at any arrival epoch or off at any service completion. We model the system by an M^[x]/M/1 queueing system with N policy. The server can be turned off and takes a vacation with exponential random length whenever the system is empty. If the number of units waiting in the system at any vacation completion is less than N, the server will take another vacation. If the server returns from a vacation and finds at least N units in the system, he immediately starts to serve the waiting units. It is assumed that the server breaks down according to a Poisson process and the repair time has an exponential distribution. We derive the distribution of the system size through the probability generating function. We further study the steady-state behavior of the system size distribution at random (stationary) point of time as well as the queue size distribution at departure point of time. Other system characteristics are obtained by means of the grand process and the renewal process. Finally, the expected cost per unit time is considered to determine the optimal operating policy at a minimum cost. The sensitivity analysis is also presented through numerical experiments.     

Exact and approximation methods for dependability assessment of tram systems with time window

The transportation system examined in this paper is the city tram one, where failed trams are replaced by reliable spare ones. If failed tram is repaired and delivered, then it comes back on work. There is the time window that failed tram has to be either replaced (exchanged) by spare or by repaired and delivered within. Time window is therefore paramount to user perception of transport system unreliability. Time between two subsequent failures, exchange time, and repair together with delivery time, respectively, are described by random variables A, E, and D. A/E/D is selected as the notation for these random variables. There is a finite number of spare trams. Delivery time does not depend on the number of repair facilities. Hence, repair and delivery process can be treated as one with infinite number of facilities. Undesirable event called hazard is the event: neither the replacement nor the delivery has been completed in the time window. The goal of the paper is to find the following relationships: hazard probability of the tram system and mean hazard time as functions of number of spare trams. For systems with exponential time between failures, Weibull exchange and exponential delivery (so M/W/M in the proposed notation) two accurate solutions have been found. For systems with Weibull time between failures with shape in the range from 0.9 to 1.1, Weibull exchange and exponential delivery (i.e. W/W/M) a method yielding small errors has been provided. For the most general and difficult case in which all the random variables conform to Weibull distribution (W/W/W) a method returning moderate errors has been given.     

A stochastic model for a general load‐sharing system under overload condition

A load‐sharing parallel system functions if at least one unit in the system is functioning and the surviving units share the load. In most of research on load‐sharing system, the performance of the system has been studied only for the case when the lifetimes of components in the system follow exponential distributions. In this paper a load‐sharing parallel system is considered when the lifetimes of the units in the system are any continuous random variables. The reliability function of the system is derived and the problem of load allocation is also considered. Copyright © 2009 John Wiley & Sons, Ltd.     

Performance modeling of state dependent system with mixed standbys and two modes of failure

The present investigation deals with a multicomponent repairable system with state dependent rates. For smooth functioning of the system, mixed standbys (warm and cold) are provided so that the failed units are immediately replaced by standbys if available. To prevent congestion in the system due to failure of units, permanent along with additional repairmen are provided to restore the failed units. It is assumed that the units may fail in two modes. The units have exponential life time and repair time distributions. The failed unit may balk in case of heavy load of failed units. The failed units may also wait in the queue and renege on finding the repairmen busy according to a pre-specified rule. The Chapman–Kolmogorov equations, governing the model in the form of matrix are constructed using transition flow rates of different states. The steady state solution of queue size distribution is derived using product formula. A cost function is suggested to determine the optimal number of warm and cold standbys units required for the desired level of quality of service. The numerical illustrations are carried out to explore the effect of different parameters on performance measures.     

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.     

Optimal replacement policies for two-unit machines with increasing running costs 1

A machine consists of two stochastically failing units. Failure of either of the units causes a failure of the machine and the failed unit has to be replaced immediately. Associated with the units are running costs which increase with the age of the unit because of increasing maintenance costs, decreasing output, etc.A preventive replacement policy is proposed under which, at failure points, we also replace the second unit if its age exceeds a predetermined control limit. It is proved that, for two identical units with exponential life-time distributions and linear running costs, this policy is optimal and the optimal control limit is calculated. In an additional model we take into consideration the length of time it takes to replace one unit or both units.The method of solution is a variation of dynamic semi-Markov programming. Analytical results are obtained and the influence of the various parameters on them is investigated. Finally, we study the saving due to our policy in comparison with a policy in which only failed units are replaced.     

两部件冷备系统的可靠性分析及其最优更换策略

本文研究了两个不同部件、一个修理工组成的冷贮备可修系统,假定它们的寿命分布和维修分布均匀为指数分布,但故障后均不能修复如新时,我们利用几何过程和补充变量法求得了一些可靠性指标,并以故障次数为策略,以长期运行单位时间内的期望效益为目标函数,确定了最优的故障次数,便得目标函数达到最大值,从而保证了系统的可用度。     

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.     

在常规故障和临界人为错误条件下具有易损坏储备部件复杂系统的可靠性分析

讨论了一个由两个部件和一个储备部件并且具有临界人为错误和常规故障的随机模型,研究了易损坏部件对系统的影响,故障系统的修复时间是任意分布的.运用泛函分析的方法,通过分析系统主算子的谱特征,给出了系统的可靠性分析的证明.     

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.     

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.     

Travel times in queueing networks and network sojourns

We first describe expected values of sojourn times for semi-stationary (or synchronous) networks. This includes sojourn times for units and sojourn times for the entire network. A typical sojourn time of a unit is the time it spends in a sector (set of nodes) while it travels through the network, and a typical network sojourn time is the busy period of a sector. Our results apply to a wide class of networks including Jackson networks with general service times, general Markov or regenerative networks, and networks with batch processing and concurrent movement of units. The results also shed more light on when Little's law for general systems, holds for expectations as well as for limiting averages. Next, we describe the expectation of a unit's travel time on a general route in a basic Markov network process (such as a Jackson process). Examples of travel times are the time it takes for a unit to travel from one sector to another, and the time between two successive entrances to a node by a unit. Finally, we characterize the distributions of the sojourn times at nodes on certain overtake-free routes and the travel times on such routes for Markov network processes.This research was supported in part by the Air Force Office of Scientific Research under contract 89-0407 and NSF grant DDM-9007532.     

Control policies for single-stage production systems with perishable inventory and customer impatience

We consider problems of inventory and admission control for make-to-stock production systems with perishable inventory and impatient customers. Customers may balk upon arrival (refuse to place orders) and renege while waiting (withdraw delayed orders) during stockouts. Item lifetimes and customer patience times are random variables with general distributions. Processing, setup, and customer inter-arrival times are however assumed to be exponential random variables. In particular, the paper studies two models. In the first model, the system suspends its production when its stock reaches a safety level and can resume later without incurring any setup delay or cost. In the second model, the system incurs setup delays and setup costs; during stockouts, all arriving customers are informed about anticipated delays and either balk or place their orders but cannot withdraw them later. Using results from the queueing literature, we derive expressions for the system steady-state probabilities and performance measures, such as profit from sales and costs of inventory, setups, and delays in filling customer orders. We use these expressions to find optimal inventory and admission policies, and investigate the impact of product lifetimes and customer patience times on system performance.