A multi-component standby system subject to inspection and truncated normal failure time distribution

This paper investigates a stochastic model of a single server, two identical unit cold standby system. Each unit consists of n separately maintained independent components arranged in series. After each failure, an inspection is required to detect which component of the unit has failed. The failure time distribution of a unit is truncated normal while all the other time distributions are negative exponential. Using the regenerative point technique, we obtained various measures of system effectiveness to carry out the profit function analysis.     

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     

Reliability models of a bridge system structure under incomplete information

Most methods of system reliability analysis assume that the precise probability distributions of the component lifetime to failure are available, and the system components are independent. However this assumption may be unreasonable in a wide scope of cases. Therefore, imprecise reliability modes of a bridge system structure are put forward in this paper, for cases when the above assumptions are violated. The analysis is based on only some partial information about the component lifetime distributions including points of unknown distributions, and probabilities on nested intervals. The effect of the component independence condition on the reliability of systems is studied. The formulas of the unreliability of the bridge system structure are obtained in the explicit form under different conditions.     

Estimators for Reliability Measures in Geometric Distribution Model Using Dependent Masked System Life Test Data

Masked system life test data arises when the exact component which causes the system failure is unknown. Instead, it is assumed that there are two observable quantities for each system on the life test. These quantities are the system life time, and the set of components that contains the component leading to the system failure. The component leading to the system failure may be either completely unknown (general masking), isolated to a subset of system components (partial masking), or exactly known (no masking). In the dependent masked system life test data, it is assumed that the probability of masking may depend on the true cause of system failure. Masking is usually due to limited resources for diagnosing the cause of system failures, as well as the modular nature of the system. In this paper, we present point, and interval maximum likelihood, and Bayes estimators for the reliability measures of the individual components in a multi-component system in the presence of dependent masked system life test data. The life time distributions of the system components are assumed to be geometric with different parameters. Simulation study will be given in order to 1) compare the two procedures used to derive the estimators for the reliability measures of system components, 2) study the influence of the masking level on the accuracy of the estimators obtained, and 3) study the influence of the masking probability ratio on the accuracy of the estimators obtained     

Probability of Component or Subsystem Failure Before System Failure

A method of determining the probability of having a given set of components failed and another set of components working at the time of system failure is based on the notion of boundary probability. The method is simple and easily applied to any s-coherent system for which the reliability structure is known. The application of this method is limited to the case of continuous probability distributions of time to failure because no simple method of computing boundary probabilities in the discrete case could be found. This is not, however, a major limitation of the method since in the majority of practical applications continuous random variables representing times to failure are used. The method can be extended to the case of non-s-independent random variables. Several examples illustrate the procedure.     

Availability and MTTF analysis of a three-state parallel redundant multi-component system under critical human failures

This paper deals with the availability and mean time to failure of a single server complex system made up of two classes A and B under critical human errors. Sub-system A has two identical components arranged in parallel whereas B has N non-identical components arranged in series. The complex redundant system has three states, viz. good, degraded, failed and suffers two types of failures, viz. unit failure and failure due to critical human errors. The failure and repair times for the system follow exponential and general distributions respectively. Laplace transforms of the probabilities of the complex system being in various states have been obtained along with steady state behaviour of the equipment. A numerical example has also been appended in the end to highlight the important results. There is only one repair facility, which is availed only when the system is in failed state due to failure of sub-system B.     

Optimal allocation of s-identical, multi-functional spares in aseries system

This paper considers the problem of allocating statistically-identical, multi-functional spares to subsystems of a series system. The objective is to maximize the system reliability for mission time T which can be deterministic or stochastic. Several problems which are conceptually similar to this one have been discussed in the literature in different contexts. An algorithm is provided for obtaining standby redundancy allocation, and sufficient conditions are derived for optimality of the resulting allocation for general T. The algorithm is equivalent to a simple allocation rule under the sufficient conditions. The allocation rule gives an optimal allocation for the special cases: the PDF's of component lifetimes are log concave (which implies increasing failure rate), and T is deterministic; the components have exponential failure times, and T follows a gamma distribution; and component lifetime distributions are general, and T follows an exponential or a mixture of exponential distributions. No simpler method is available for latter two cases     

A Monte Carlo simulation algorithm for finding MTBF

Prediction of mean time between failures (MTBF) is an important aspect of the initial stage of system development. It is often difficult to predict system MTBF during a given time since the component failure processes are extremely complex. The authors present a Monte Carlo simulation algorithm to calculate the MTBF during a given time of a binary coherent system. The algorithm requires the lifetime distributions of the components and the minimal path sets of the system. The MTBF for a specific time interval, e.g. a month or a year, can be estimated. If the component lifetime distributions are unknown, then a lower bound of system MTBF can be estimated by using known constant failure rates for each component     

Maximum likelihood analysis of component reliability using maskedsystem life-test data

Life data from multicomponent systems are often analyzed to estimate the reliability of each system component. Due to the cost and diagnostic constraints, however, the exact cause of system failure might be unknown. Referring to such situations as being masked, the authors use a likelihood approach to exploit all the available information. They focus on a series system of three components, each with a constant failure rate, and propose a single numerical procedure for obtaining maximum-likelihood estimations (MLEs) in the general case. It is shown that, under certain assumptions, closed-form solutions for the MLEs can be obtained. The authors consider that the cause of system failure can be isolated to some subset of components, which allows them to consider the full range of possible information on the cause of system failure. The likelihood, while presented for complete data, can be extended to censoring. The general likelihood expressions can be used with various component life distributions, e.g., Weibull, lognormal. However, closed-form MLEs would most certainly be intractable and numerical methods would be required     

Failure Probability of Strict Consecutive-k-out-of-n:F Systems

A system with n components in sequence is a strict consecutive-k-out-of-n:F system if and only if it fails when at least k consecutive components are failed, but isolated strings of component failures of length less than k do not occur. This paper gives the failure probability function of a strict linear consecutive-k-out-of-n:F system in a closed form. The calculation of the failure probability of a strict circular consecutive-k-out-of-n:F system is reduced to the linear case.     

Comparison of Monte Carlo Techniques for Obtaining System-Reliability Confidence Limits

Digital computer techniques are developed using a) asymptotic distributions of maximum likelihood estimators, and b) a Monte Carlo technique, to obtain approximate system reliability s-confidence limits from component test data. 2-Parameter Weibull, gamma, and logistic distributions are used to model the component failures. The components can be arranged in any system configuration: series, parallel, bridge, etc., as long as one can write the equation for system reliability in terms of component reliability. Hypothetical networks of 3, 5, and 25 components are analyzed as examples. Univariate and bivariate asymptotic techniques are compared with a double Monte Carlo method. The bivariate asymptotic technique is shown to be fast and accurate. It can guide decisions during the research and development cycle prior to complete system testing and can be used to supplement system failure data.     

Fault-Tree Analysis by Fuzzy Probability

In conventional fault-tree analysis, the failure probabilities of components of a system are treated as exact values in estimating the failure probability of the top event. For many systems, it is often difficult to evaluate the failure probabilities of components from past occurrences because the environments of the systems change. Furthermore, it might be necessary to consider possible failure of components even if they have never failed before. We, therefore, propose to employ the possibility of failure, viz. a fuzzy set defined in probability space. The notion of the possibility of failure is more predictive than that of the probability of failure; the latter is a limiting case of the former. In the present approach based on a fuzzy fault-tree model, the maximum possibility of system failure is determined from the possibility of failure of each component within the system according to the extension principle. In calculating the possibility of system failure, some approximation is made for simplicity.     

Build better diagnostic decision trees

Enhancing the ability to perform diagnostics on a system that has failed can significantly impact maintenance and repair costs. A good diagnostic tool enables a user to analyze a failed system and identify the failed components. While the field of diagnostics is not a modern one, the way in which system diagnostics are performed is continuously changing. The automatic diagnosis based on reliability analysis (ADORA) methodology utilizes reliability information developed during the design phase to build a diagnostic map. Previous work on ADORA demonstrated how a diagnostics procedure can be performed on a system that has been analyzed using a static reliability model, particularly a fault tree (Assaf and Dugan, 2003). In this article, we extend the ADORA methodology to utilize reliability analysis of dynamic fault trees (DFTs), which are reliability models that capture sequences and combinations of component failures that induce system failure. DFTs are particularly well suited for analyzing computer-based systems.As an example the common rail fuel injection system is discussed.     

MTBF of a complex binary coherent system

It is noted that there has yet been no detailed study of the relationships between the MTBF (mean time between failures) of a system and the sequences of component failures, except for the case of a series system where every component failure causes a system failure. The author defines MTBF anew and derives relationships between the properties of the MTBF of a binary coherent system and the properties of the sequences of component failures, assuming that the lifetime distributions of the components are either new-better-than-used (NBU) exponential or increasing failure rate (IFR). Lower bounds of MTBF that can be used to predict the MTBF and to decide whether the system would satisfy the MTBF requirement are derived     

A multicomponent two-unit cold standby system with three modes

This paper investigates a mathematical model of a two-unit cold standby redundant system with three possible states of each unit—normal, partially failed and failed. Each unit has n components, each having a constant failure rate and a repair rate, an arbitrary function of the time spent. These vary from component to component. Steady-state probabilities, steady-state pointwise availability, mean time to system failure and Laplace transforms of various transient probabilities have been obtained. Several earlier results are verified as special cases.     

Distribution of Failures in an Array of Cells

Consider a system which is an array of cells in which the system function is performed by the rows. The level of performance of the system depends on the failure rate of the cells and on the distribution of the failed cells in the array. For example, a system may operate at only 50 percent efficiency if 3 or more cells in a row are failed. This paper presents a procedure for computing the probabilities of various failure distributions in the array rows for a given number of failed cells. Combined with a failure model for the individual cells, the results can be used to compute probabilities about performance of the array. Cell failures are assumed to be statistically independent and identically distributed.     

A two unit series system with correlated failures and repairs

This paper investigates a stochastic model of a system having two identical units, each with two components, arranged in series configuration. Upon failure of a component of the unit, the standby unit replaces the failed unit instantaneously. It is assumed that the joint distribution of the repair and failure times of a component are bivariate exponential. Several reliability characteristics of interest to system designers as well as operations managers have been evaluated and relevant results obtained earlier are verified as a particular case. The MTSF and availability have also been studied through graphs.     

Studies in Cascade Reliability?I

An n-Cascade system is defined as a special type of standby system with n components. A component fails if the stress on it is not less than its strength. When a component in cascade fails, the next in standby is activated and will take on the stress. However, the stress on this component will be a multiple k times the stress that acted on its predecessor. The system fails if due to an initial stress, each of the components in succession fails. The stress is random and the component strengths are independent and identically distributed variates, with specified probability functions; k is constant. Expressions for system reliability are obtained when the stress and strength distributions are exponential. Reliability values for a 2-cascade system with Gamma and Normal stress and strength distributions are computed, some of which are presented graphically.     

Availability for repairable components and series systems

The failure pattern of repairable components is often modelled by an alternating renewal process which implies that a failed component is perfectly repaired. In practice, repair is often imperfect. This paper proposes a generalized availability model for repairable components and series systems. The lifetime of a repaired component has a general distribution which can be different from that of a new component. Availability and some asymptotic quantities in these models are derived. An example illustrates the application of these models     

Assessing throughput and reliability in communication and computernetworks

System performance and reliability are jointly assessed for highly reliable communication/computer networks. The model assumes that at most a small number of components can be down at a time and that the average repair/replacement time of a failed component is small when compared with the average failure times of network components. The system performance is measured in terms of network throughput at steady-state operation