Suprasad V. Amari

Computing system failure frequencies and reliability importance measures using OBDD

The recent literature showed that, in many cases, ordered binary decision diagram (OBDD)-based algorithms are more efficient in reliability evaluation compared to other methods such as the inclusion-exclusion (I-E) method and the sum of disjoint products (SDP) method. We present algorithms based on OBDD to compute system failure frequencies and reliability importance measures. Methods are presented to calculate both steady-state and time-specific frequencies of system-failure as well as system-success. The reliability importance measures include the Birnbaum importance, the criticality importance, and other indices for the risk evaluation of a system. In addition, we propose an efficient approach based on OBDD to evaluate the reliability of a nonrepairable system and the availability of a repairable system with imperfect fault-coverage mechanisms. The powerful capability of OBDD for reliability evaluation is fully exploited. Further, we extend all of the proposed algorithms to analyze systems with imperfect fault-coverage.

Reliability Characteristics of

We study reliability characteristics of the k-out-of- n warm standby system with identical components subject to exponential lifetime distributions. We derive state probabilities of the warm standby system in a form that is similar to the state probabilities of the active redundancy system. Subsequently, the system reliability is expressed in several forms that can provide new insights into the system reliability characteristics. We also show that all properties and computational procedures that are applicable for active redundancy are also applicable for the warm standby redundancy. As a result, it is shown that the system reliability can be evaluated using robust algorithms within O(n-k+1) computational time. In addition, we provide closed-form expressions for the hazard rate, probability density function, and mean residual life function. We show that the time-to-failure distribution of the k-out-of-n warm standby system is equal to the beta exponential distribution. Subsequently, we derive closed-form expressions for the higher order moments of the system failure time. Further, we show that the reliability of the warm standby system can be calculated using well-established numerical procedures that are available for the beta distribution. We prove that the improvement in system reliability with an additional redundant component follows a negative binomial (Polya) distribution, and it is log-concave in n. Similarly, we prove that the system reliability function is log-concave in n. Because the k -out-of-n system with active redundancy can be considered as a special case of the k-out-of-n warm standby system, we indirectly provide some new results for the active redundancy case as well.

k

Systems subjected to imperfect fault-coverage may fail even prior to the exhaustion of spares due to uncovered component failures. This paper presents optimal cost-effective design policies for k-out-of-n:G subsystems subjected to imperfect fault-coverage. It is assumed that there exists a k-out-of-n:G subsystem in a nonseries-parallel system and, except for this subsystem, the redundancy configurations of all other subsystems are fixed. This paper also presents optimal design polices which maximize overall system reliability. As a special case, results are presented for k-out-of-n:G systems subjected to imperfect fault-coverage. Examples then demonstrate how to apply the main results of this paper to find the optimal configurations of all subsystems simultaneously. In this paper, we show that the optimal n which maximizes system reliability is always less than or equal to the n which maximizes the reliability of the subsystem itself. Similarly, if the failure cost is the same, then the optimal n which minimizes the average system cost is always less than or equal to the n which minimizes the average cost of the subsystem. It is also shown that if the subsystem being analyzed is in series with the rest of the system, then the optimal n which maximizes subsystem reliability can also maximize the system reliability. The computational procedure of the proposed algorithms is illustrated through the examples.

-out-of-

In this paper, an efficient method is proposed for the exact reliability evaluation of k-out-of-n systems with identical components subject to phased-mission requirements and imperfect fault coverage. The system involves multiple, consecutive, and non-overlapping phases of operation, where the k values and failure time distributions of system components can change from phase to phase. The proposed method considers statistical dependencies of component states across phases as well as dynamics in system configuration and success criteria. It also considers the time-varying and phase-dependent failure distributions and associated cumulative damage effects for the system components. The proposed method is based on the total probability law, conditional probabilities and an efficient recursive formula to compute the overall mission reliability with the consideration of imperfect fault coverage. The main advantages of this method are that both its computational time and memory requirements are linear in terms of the system size, and it has no limitation on the type of time-to-failure distributions for the system components. Three examples are presented to illustrate the application and advantages of the proposed method.

n

Many practical systems are phased-mission systems (PMSs), where the mission consists of multiple, consecutive, and non-overlapping phases of operation. An accurate reliability analysis of a PMS must consider statistical dependence of component states across phases, as well as dynamics in system configurations, success criteria, and component behavior. This paper proposes a new method based on multiple-valued decision diagrams (MDDs) for the reliability analysis of a non-repairable binary-state PMS. Due to its multi-valued logic nature, the MDD model has recently been applied to the reliability analysis of multistate systems. In this work, we present a novel way to adapt MDDs for the reliability analysis of systems with multiple phases. Examples show how the MDD models are generated and evaluated to obtain the mission reliability measures. Performance of the MDD-based method is compared with an existing binary decision diagram (BDD)-based method for PMS analysis. Empirical results show that the MDD-based method can offer lower computational complexity as well as a simpler model construction and improved evaluation algorithms over those used in the BDD-based method.

Warm Standby Systems

Investigations conducted in several industries indicate that there is no direct relationship between equipment failure and equipment age in the majority of cases. Most failures are caused by events or conditions that occur during component operation and manufacturing processes. Therefore, optimal maintenance decisions should be based on the actual deterioration conditions of the components. Condition-Based Maintenance (CBM) is a methodology that strives to identify incipient faults before they become critical to enable more accurate planning of preventive actions. For the ultimate success of CBM methodology, we must have sound methods for modeling deterioration (the propagation of faulty conditions), the conditions and their effects, and the optimal selection and scheduling of inspections and preventive maintenance actions (the right action at the right time). In this paper, we present a generalized CBM model that can be applied to a wide range of applications. The CBM model includes a stochastic deterioration process, a set of maintenance actions and their effects, and a scheduled inspection policy that identifies the condition of deterioration. Using Markov Decision Processes (MDP), we provide an optimal cost-effective maintenance decision based on the condition revealed at the time of inspection. In addition, we present a procedure for finding optimal inspection schedules

Optimal design of k-out-of-n:G subsystems subjected to imperfect fault-coverage

Load-sharing systems have several practical applications. In load-sharing systems, the failure of a component will result in a higher load on each of the surviving components, thereby inducing a higher failure rate for them. This introduces failure dependency among the load-sharing components, which in turn increases the complexity in analyzing these systems. Therefore, in spite of a wide range of applications for load-sharing systems, the methods for computing the reliability of load-sharing systems are limited. In this paper, we first discuss the modeling concepts of load-sharing systems and explain the role of accelerated life testing models in analyzing these systems. We also describe existing analysis methods and their limitations in analyzing load-sharing systems. In modeling load-sharing systems with general failure distributions, it is important to consider an appropriate model to incorporate the effects of loading history. In this paper, we explore using the cumulative exposure model to account for the effects of loading history. We present an efficient method to compute the reliability and mean life of k-out-of-n load-sharing systems with identical or non-identical components following general failure distributions. The method can solve large k-out-of-n systems in a short time. Further, we show how to use the existing computational procedures for solving stochastic reward models for solving load-sharing models. In addition to the exact solutions, we also propose efficient approximations and bounds that can be computed easily. The computational procedure and the bounds proposed in this paper help reliability engineers to accurately model the load-sharing systems that arise in many practical situations.

Reliability of k-out-of-n systems with phased-mission requirements and imperfect fault coverage

This paper describes and demonstrates a solution methodology that determines optimal design configurations that maximize the reliability of a wide range of non-repairable systems. The problem formulation considers the generic case of warm-standby redundancy and extends state-of-the-art reliability optimization techniques in several dimensions: (1) non-constant component hazard functions, (2) warm standby components including cold and hot standby situations, (3) imperfect switches, (4) k-out-of-n redundancy structures, (5) multiple component choices, and (6) redundancy strategy choices. The problem involves selection of components, redundancy strategies, and redundancy levels to maximize system reliability subject to constraints. Optimal solutions are determined based on an equivalent binary integer programming formulation. Compared to other available methods, the proposed methodology more accurately models many engineering design problems with both active and standby redundancies. Previously, it has been difficult to determine optimal solutions for this class of problems or to calculate system reliability efficiently. The methodology is successfully demonstrated on a large problem with 14 subsystems with arbitrary failure distributions.

A Multiple-Valued Decision Diagram Based Method for Efficient Reliability Analysis of Non-Repairable Phased-Mission Systems

Abstract The paper introduces a new model of fault level coverage for multi-state systems in which the effectiveness of recovery mechanisms depends on the coexistence of multiple faults in related elements. Examples of this effect can be found in computing systems, electrical power distribution networks, pipelines carrying dangerous materials, etc. For evaluating reliability and performance indices of multi-state systems with imperfect multi-fault coverage, a modification of the generalized reliability block diagram (RBD) method is suggested. This method, based on a universal generating function technique, allows performance distribution of complex multi-state series–parallel system with multi-fault coverage to be obtained using a straightforward recursive procedure. Illustrative examples are presented.

Cost-effective condition-based maintenance using markov decision processes

In this chapter, a state-of-the-art review of various analytical modeling techniques for reliability analysis of phased-mission systems (PMS) is presented. The analysis approaches can be broadly classified into three categories: combinatorial, state-space oriented, and modular. The combinatorial approaches are computationally efficient for analyzing static PMS. A combinatorial binary decision diagram based method is discussed in detail. Methods to consider imperfect fault coverage and common-cause failures in the reliability analysis of PMS will also be presented.