Approximation Formulas for Reliability with Repair

A system consists of n-identical parallel subsystems, each having an exponential distribution of times to failure and an exponential distribution of times to repair. The system reliability with repair is the probability of no more than q out of n subsystems being simultaneously in a failed state during time t. Under conditions frequently met in practice, system reliability with repair R(t) can be approximated by: R(t) ? exp [?t/Tm] where Tm is the mean time for the system to pass for the first time from zero to (q + 1) simultaneous subsystem failures. Exact and approximate methods of calculating Tm are developed. A detailed error analysis is presented showing the limitations of using Tm to calculate system reliability with repair.     

Assignment of Priority in Improving System Reliability

In many maintenance situations for certain weapon systems, such as anti-aircraft systems, the problems confronted are: 1) which priority for repair is to be assigned, and 2) which type of component should be assigned priority for repair. This can be done on the basis of mean time to system failure. This paper discusses the reliability characteristics of a system of two paralleled radars working in conjunction with two paralleled computers. The system is in up stage even if one computer and/or one radar fails. The system failure takes place only when both the computers or both the radars are in failed condition. The distribution of time to system failure and its expectation have been derived assuming that the failures occur following Poisson distribution and the repair times follow the negative exponential distribution for these two types of subsystems imposing head-of-the-line priority and preemptive resume priority for the repair process. The results are discussed with reference to numerical examples. It has been observed that the mean time to system failure is higher when the head-of-the-line priority discipline is adopted for repair of components, especially when the repair times are shorter.     

A Two-Unit Cold Standby Repairable System with One Replaceable Repair Facility and Delay Repair: Some Reliability Problems

1 Model and Assumption In reliability analysis of repairable systems, it is usually assumed that the repair facility neither fails nor deteriorates as well as the repairman is instantaneously available. So that the repair is started immediately upon the failure of a unit provided that he is not busily repairing another unit. However, in actual practice, the repair facility in a repairable system is subject to failure and can be replaced (or can be repaired) after it fails, and certain delay ac…     

Reliability measures for two-unit systems with a dependent structure for failure and repair times

This paper deals with reliability measures for two-unit systems with a repair facility assuming that the failure times and the repair time follow a trivariate exponential distribution of Marshall and Olkin, J. Amer. Statist. Assoc., 1967, 62, 30–44. The case where the system down-time is observed, is also discussed. The system reliability and system mean-time before failure are evaluated for standby and parallel systems. When the down-time is observed the system availability, steady-state availability and the system mean down-time are evaluated for standby, parallel and series systems.     

Initial Provisioning of a Standby System with Deteriorating and Repairable Spares

A technique is developed for finding the time dependent operating probabilities used by reliability systems designers for provisioning a system with N + k identical units, k of which are called spares and N called operating units, and s repair facilities. System failure occurs when less than N units are operational. Units fail with exponential interfailure times and are repaired with exponential service time. Idle spares fail due to deterioration at a rate possibly different from that of the operating units. Graphs are presented which show the minimum numbers of spares needed to achieve system reliabilities of 0.90 and 0.99 as a function of time. The technique is applicable for finding, numerically, the first passage time distribution for any system modeled by birth and death processes.     

Reliability analysis of two-unit warm standby system with partial failure mode and inspection time

A two-unit warm standby system is discussed, in which units are identical. Partial failure and complete failure of a unit can be detected by inspection from time to time. The inspection time follows an exponential distribution, whereas the repair and failure time follows an exponential and arbitrary distribution. Several reliability characteristics of interest to system designers and operation managers have been evaluated.     

Reliability analysis of a two-unit cold standby redundant system with two operating modes

This paper considers a two unit cold standby redundant system subject to a single repair facility with exponential failure and general repair time distribution. Each unit can work in three different modes — normal, partial failure and total failure. There is a perfect switch to operate the standby unit on total failure of the operative unit. The system has been analysed to determine the reliability parameters e.g. mean time to system failure (MTSF), steady state availability, mean recurrence to a state and expected number of visits to a state, first two moments of time in transient state, by using the theory of Semi-Markov Process. Howard's reward structure has been super-imposed on the Semi-Markov Process to obtained expected profit of the system. A number of results obtained earlier are derived as particular cases.     

Reliability and MTTF evaluation of a repairable complex system under waiting

This paper deals with the reliability and mean time to failure (MTTF) evaluation of a complex system under waiting incorporating the concept of hardware failure and human error. Failure rates of the complex system follow exponential time distributions, whereas repair follows a general repair time distribution. Laplace transforms of various state probabilities have been evaluated and then reliability is obtained by the inversion process. A formula for variance of time to failure has also been developed. A particular case is also given to highlight some important results. Moreover, various plots have been sketched at the end.     

Profit evaluation of a two unit cold standby redundant system with two operating modes

This paper considers a two unit cold standby system subject to a single repair facility with exponential failure time and arbitrary repair time distribution. Each unit has three modes—normal (N), partial (P) and total failure (F). By using the regenerative point technique the system has been analysed to determine mean time to system failure and profit earned by the system. A numerical example is used to highlight the important results.     

Phased-mission system reliability under Markov environment

The authors show how to determine the reliability of a multi-phase mission system whose configuration changes during consecutive time periods, assuming failure and repair times of components are exponentially distributed and redundant components are repairable as long as the system is operational. The mission reliability is obtained for 3 cases, based on a Markov model. (1) Phase durations are deterministic; the computational compact set model is formulated and a programmable solution is developed using eigenvalues of reduced transition-rate matrices. (2) Phase durations are random variables of exponential distributions and the mission is required to be completed within a time limit; the solution is derived as a recursive formula, using the result of case 1 and mathematical treatment-a closed-form solution would be prohibitively complex and laborious to program. (3) Phase durations are random variables and there is no completion time requirement; the solution is derived similarly to case 1 using moment generating functions of phase durations. Generally, reliability problems of phased-mission systems are complex. The authors' method provides exact solutions which can be easily implemented on a computer     

关于可修复系统的MTBF和MTTR

在可修复系统中，可用性作为一种可靠性测度，其指标可用度是基本的；然而从实际应用角度来说平均无故障工作时间和平均修复时间有时显得更为重要，却又往往难以得知，本文首次提供了计算一般的ＭＴＢＦ和ＭＴＴＲ的有效公式，此系统具有负指数分布失效和修复时间部件。     

On Failure-Time Distributions for Systems of Dissimilar Units

Certain reliability problems of systems of dissimilar units with repair are described. The mean time to system failure is based on the relation of mean first passage times between states of the system. The failure-time distribution is obtained from an integral equation of the renewal type. The two approaches can be also applied to a system of dissimilar units under an overload. Finally, it is shown that these results include many earlier results as special cases.     

A note on the statistical characteristics of a two-unit system

Expressions for the Laplace transforms of reliability and availability functions are obtained for a two-unit system, with different repair times for the units which have failed from online and standby states, and a dead time value for the repair facility by the use of regeneration point technique. The system consists of two-units with one repair facility. The repair facility is not available for a random time immediately after each repair completion. From the Laplace transforms of reliability and availability functions the steady state availability, reliability and mean time to system failure can be obtained.     

Modeling and performance analysis for wireless mobile networks: a new analytical approach

In wireless mobile networks, quantities such as call blocking probability, call dropping probability, handoff probability, handoff rate, and the actual call holding times for both complete and incomplete calls are very important performance parameters in the network performance evaluation and design. In the past, their analytical computations are given only when the classical exponential assumptions for all involved time variables are imposed. In this paper, we relax the exponential assumptions for the involved time variables and, under independence assumption on the cell residence times, derive analytical formulae for these parameters using a novel unifying analytical approach. It turns out that the computation of many performance parameters is boiled down to computing a certain type of probability, and the obtained analytical results can be easily applied when the Laplace transform of probability density function of call holding time is a rational function. Thus, easily computable results can be obtained when the call holding time is distributed with the mixed-Erlang distribution, a distribution model having universal approximation capability. More importantly, this paper develops a new analytical approach to performance evaluation for wireless networks and mobile computing systems.     

Failure Time for a Redundant Repairable System of Two Dissimilar Elements

The distribution of time to failure for a system consisting of two dissimilar elements or subsystems operating redundantly and susceptible to repair is discussed. It is assumed that the times to failure for the two system elements are independent random variables from possibly different exponential distributions, and that the repair times peculiar to each element are independently distributed in an arbitrary fashion. For this basic model a derivation is given of the Laplace-Stieltjes transform of the distribution function of time to system failure, i.e, the time until both elements are simultaneously down for repair, measured from an instant at which both are operating. An explicit formula is given for the mean or expected time to system failure, a natural approximation to the latter is exhibited, and numerical comparisons indicate the quality of this approximation for various repair time distributions. In a second model the possibility of system failures due to overloading the remaining element after a single element failure is explicitly recognized. The assumptions made for the basic model are augmented by a stochastic process describing the random occurrence of overloads. Numerical examples are given. Finally, it is shown how the above models may be easily modified to account for delays in initiating repairs resulting from only occasional system surveillance, and to account for random catastrophic failures.     

Human error analysis of a standby redundant system with arbitrarily distributed repair times

This paper presents human error analysis of a (two units working and one on standby) system with arbitrarily distributed repair times. The supplementary vairables method is used to develop the system availability expressions. A general formula for the system steady-state availability is developed when the failed system repair times are gamma distributed. Time-dependent availability, system reliability with repair, mean time to failure and variance of time to failure formulae are developed for some particular cases. Selective plots are shown to demonstrate the impact of critical human error on system availability and reliability.     

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.     

A non-homogeneous Markov model for phased-mission reliabilityanalysis

Three assumptions of Markov modeling for reliability of phased-mission systems that limit flexibility of representation are identified. The proposed generalization has the ability to represent state-dependent behavior, handle phases of random duration using globally time-dependent distributions of phase change time, and model globally time-dependent failure and repair rates. The approach is based on a single nonhomogeneous Markov model in which the concept of state transition is extended to include globally time-dependent phase changes. Phase change times are specified using nonoverlapping distributions with probability distribution functions that are zero outside assigned time intervals; the time intervals are ordered according to the phases. A comparison between a numerical solution of the model and simulation demonstrates that the numerical solution can be several times faster than simulation     

Reliability prediction for repairable redundant systems

It is assumed that the mean time to failure and mean repair time are known for each of the subsystems of a system. The subsystems conform to the usual exponential failure (and repair) laws and their behaviors are mutually independent. The system includes redundant subsystems in active standby status. Whenever, after a system failure, repair of a failed subsystem re-establishes an adequate configuration, the system as a whole is returned to active status while repair of other failed subsystems (if any) continues. Under this set of assumptions, equations are developed which permit prediction of mean time to failure and mean down time for the system. The development differs somewhat from the use of birth-and-death equations which has been customary for similar problems in the past.     

Evaluation of Repairable System Reliability Using the ``Bad-As-Old' Concept

It is usually assumed that the underlying distribution of times to failure of systems is the exponential distribution. This is justified on the basis of the bathtub curve or Drenick's theorem, but the bathtub curve is merely a statement of plausibility and conflicts with Drenick's theorem. Even if exponentiality is not assumed, it is usually assumed that a system under study is as-good-as-new after repair. This is not a plausible assumption to make for a complex system. If failure data are available they should be tested for trend among successive failure times. If a trend exists, a time dependent (nonhomogeneous) Poisson process (called bad-as-old model in this paper) should be fitted and tested for adequacy. This paper is not intended to provide a rigorous, definitive treatment of bad-as-old models. Rather, it has three main purposes: 1) to point out the glaring, but somehow usually overlooked, inconsistency between the commonly accepted concept of wearout of repairable systems and the a priori use of renewal processes for modeling these systems; 2) to outline basic procedures for evaluating data from repairable systems and for formulating bad-as-old probabilistic models; and 3) to present the results of Monte Carlo simulations, which illustrate the grossly misleading results which can occur if independence of successive failure times is invalidly assumed.