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.

Reliability analysis of phased-mission systems (PMS) must consider the statistical dependences of element states across different phases as well as changes in system configuration, success criteria, and component behavior. This paper proposes a recursive method for the exact reliability evaluation of PMS consisting of nonidentical independent nonrepairable multistate elements. The method is based on conditional probabilities and the branch-and-bound principle. It is invariant to changes in system structure, demand, and the elements' state transition rates among phases. The main advantage of this method is that it does not require the composition of decision diagrams and can be fully automated. Both analytical and numerical examples are presented to illustrate the application and advantages of the proposed method. The computational performance of the proposed algorithm is illustrated through comprehensive experimentation on the CPU running time of the algorithm.

Phased-mission systems (PMSs) are systems supporting missions characterized by multiple, consecutive, and nonoverlapping phases of operation. Examples of PMSs abound in many practical applications such as aerospace, nuclear power, and airborne weapon systems. Reliability analysis of a PMS must consider statistical dependence of component states across different phases, as well as dynamics in system structure functions and component behavior. In this paper, we propose a recursive method for exact reliability evaluation of a binary-state or multistate PMS consisting of nonidentical, binary, and nonrepairable elements. The system elements can fail individually or due to common-cause failures (CCFs) caused by some external factors. The proposed method is based on the branch-and-bound principle, and can be fully automated. The method is applicable to PMSs with nonoverlapping or overlapping sets of elements that can fail as a result of CCFs. The method is illustrated using both analytical and numerical examples.

We study reliability characteristics of thek-out-of-nwarm 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 withinO(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 thek-out-of-nwarm 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 (Pólya) distribution, and it is log-concave inn. Similarly, we prove that the system reliability function is log-concave inn. Because thek-out-of-nsystem with active redundancy can be considered as a special case of thek-out-of-nwarm standby system, we indirectly provide some new results for the active redundancy case as well.

Many practical systems are phased-mission systems (PMS), where the mission consists of multiple, consecutive, and non-overlapping phases of operation. An accurate reliability analysis of a PMS must consider the statistical dependencies of component states across phases, as well as dynamics in system configurations, success criteria, and component behavior. In this paper, we propose a method for exact reliability evaluation of arbitrary binary or multi-state PMS consisting of non-identical binary non-repairable elements. The method is invariant to changes in system structure and demand among missions, and takes into account the time-varying and phase-dependent failure rates and associated cumulative damage effects. The proposed method is based on conditional probabilities, and an efficient recursive formula to compute these probabilities based on branch and bound. The main advantage of this method is that it does not require composition of decision diagrams, and can be fully automated. The method is illustrated using both an analytical example, and a numerical example.

A distinct characteristic of multistate systems (MSS) is that the systems and/or their components may exhibit multiple performance levels (or states) varying from perfect operation to complete failure. MSS can model behaviors such as shared loads, performance degradation, imperfect fault coverage, standby redundancy, limited repair resources, and limited link capacities. The nonbinary state property of MSS and their components as well as dependencies existing among different states of the same component make the analysis of MSS difficult. This paper proposes efficient algorithms for analyzing MSS using multivalued decision diagrams (MDD). Various reliability, availability, and performability measures based on state probabilities or failure frequencies are considered. The application and advantages of the proposed algorithms are demonstrated through two examples. Furthermore, experimental results on a set of benchmark examples are presented to illustrate the advantages of the proposed MDD-based method for the performability analysis of MSS, as compared to the existing methods.

The generalized multi-statek-out-of-nsystem model proposed recently provides more flexibility in describing practical systems. In this model, there arencomponents in the system where each component, as well as the system, can be in one ofM+1 possible states: 0, 1,hellip,m. The system is in statejor above if there exists an integer valuel(jlesllesM) such that at leastklcomponents are in statelor above. Although the model has several practical applications, existing methods for computing either the exact, or approximate reliability of these systems are computationally inefficient, and limited to very small systems. In this paper, we propose an efficient method, and a detailed algorithm to compute the exact reliability of multi-statek-out-of-nsystems. The method is based on conditional probabilities, and is applicable to all cases of multi-statek-out-of-nsystems: 1) constant, 2) decreasing, 3) increasing, and 4) non-monotonekvalues. The proposed algorithm is fast, and robust. The computational time complexity of the algorithm isO(knM), and the space complexity isO(n+M) wherek=mean{ki}. Using this algorithm, the reliability of very large multi-statek-out-of-nsystems can be computed in a short time. For example, the exact reliability of a system with 500 components with 200 possible states can be computed in less than one second. Several numerical examples, including the published examples, are considered to illustrate the effectiveness, and efficiency of the proposed method. In addition to the detailed algorithm, and theoretical background, we also provide the complete listing of the MATLAB code used in the calculations.

Almost every engineering and manufacturing system consists of several subsystems, which are in general nonidentical and are subjected to stochastic failures and repairs. The system success logic can be represented using a combinatorial reliability model in terms of the states of subsystems, where as the success logic of each subsystem can be represented using a k-out-of- n structure. The long run cost associated with the downtime can be lowered by adding additional spares in each subsystem, which in turn can increase the operational and maintenance costs. Thus, it is desirable to find the optimal number of components in each subsystem that minimizes the overall cost associated with the system. The main contributions of this paper are the following: 1) formulation of an average cost function of complex repairable systems and 2) development of a new method to obtain tighter bounds for the optimal number of spares for each subsystem. The tighter bounds are extremely useful to reduce the search space and hence improve the efficiency of the optimization algorithm. With the proposed bounds, for a series system consisting of m parallel subsystems, the computational complexity to find the near optimal solution, which is the optimal solution in most cases, is O(m)

Mean time between failures (MTBF) is a common reliability measure used to assess the failure behavior of repairable systems. In order to increase MTBF, in most systems, it is a common practice to perform preventive maintenance activities at periodic intervals. In this paper: We first discuss the validity of a commonly used equation for computing MTBF of systems subjected to periodic maintenance.) For complex systems where this equation is valid, we propose a simple and better approximation than the exponential approximation proposed in a recent paper. In addition, we prove that for systems with increasing failure rate on average (IFRA) distributions, the exponential approximation proposed in a recent paper always underestimates the MTBF; hence, it is a lower bound at best.) The proposed approximation and bounds are applicable for a wide range of systems because systems which contain components with exponential or any increasing failure rate (IFR) distribution (viz., Weibull with beta>1, gamma, Gumbel, s-normal, and uniform) follow an IFRA distribution. As a special case, the proposed bounds & approximations provide better results for systems that contain only exponential failure distributions

Algorithms for evaluating the reliability of a complex system such as a multistate fault-tolerant computer system have become more important. They are designed to obtain the complete results quickly and accurately even when there exist a number of dependencies such as shared loads (reconfiguration), degradation, and common-cause failures. This paper presents an efficient method based on ordered binary decision diagram (OBDD) for evaluating the multistate system reliability and the Griffith's importance measures which can be regarded as the importance of a system-component state of a multistate system subject to imperfect fault-coverage with various performance requirements. This method combined with the conditional probability methods can handle the dependencies among the combinatorial performance requirements of system modules and find solutions for multistate imperfect coverage model. The main advantage of the method is that its time complexity is equivalent to that of the methods for perfect coverage model and it is very helpful for the optimal design of a multistate fault-tolerant system.

In a recent paper, Keren & Mehrez have shown that in the case of a geometric probability mass function, the approximation to the cumulative hazard function converges to a finite value (1.606695), when the actual value of the cumulative hazard function should be diverging. In this commentary, we show that gives incorrect results, and that the corresponding results for the approximation are actually diverging.

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.

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.

This paper presents an efficient approach based on OBDD for the reliability analysis of a multi-state system subject to imperfect fault-coverage with combinatorial performance requirements. Since there exist dependencies between combinatorial performance requirements, we apply the multi-state dependency operation (MDO) of OBDD to deal with these dependencies in a multi-state system. In addition, this OBDD-based approach is combined with the conditional probability methods to find solutions for the multi-state imperfect coverage models. Using conditional probabilities, we can also apply this method for modular structures. The main advantage of this algorithm is that it will take computational time that is equivalent to the same problem without assuming imperfect coverage (i.e. with perfect coverage). This algorithm is very important for complex systems such as fault-tolerant computer systems, since it can obtain the complete results quickly and accurately even when there exist a number of dependencies such as shared loads (reconfiguration), degradation and common-cause failures.

In the paper on Generic rules to evaluate system-failure frequency, (see ibid., vol.49, p.85-7, 2000) the authora did not consider the case of shorter mission times while presenting the rules to evaluate system-failure frequency. Time-specific failure-frequency calculations are required for the systems with shorter mission times. One of the practical uses of time-specific failure-frequency is in finding reasonably accurate estimates of failure-rate and reliability of large systems consisting of repairable components, by using combinatorial methods (without using Markov models). This paper shows that, with some minor modifications, the rules in the original paper for evaluating steady-state failure-frequency can be used to find the time-specific failure-frequency.

Frequency of failure of a system with s-independent components can be obtained from the system availability (unavailability) expression and failure and repair rates of the components. Although, Grouped Variable Inversion is an efficient technique to find the system availability, there is no convenient method to convert the "availability expression obtained by this technique" into an "expression for system-failure frequency." This paper present generic rules to find system-failure frequency, particularly, when the availability or unavailability expression of a system is obtained using this technique. The rules are straightforward, and produce appreciably shorter expressions for system-failure frequency. Examples illustrate the simplicity and efficiency of the proposed rules.