Reliability analysis of a repairable parallel system with standby involving human error and common-cause failures

This paper presents two analytical models of special multiple-state devices with repair. The failure rates are constant and the repair rates between failure states are constant, while the repair rate times between failure state and good state are arbitrarily distributed. Laplace transform of the state probabilities and steady-state availability are derived.     

Reliability analysis of multiple-state devices

This paper presents two analytical models of special multiple-state devices with repair. The failure rates are constant and the repair rates between failure states are constant, while the repair rate times between failure state and good state are arbitrarily distributed. Laplace transform of the state probabilities and steady-state availability are derived.     

A k-out-of-N:G redundant system with dependent failure rates and common-cause failures

This paper presents a k-out-of-N: G redundant system with dependent failure rates, common-cause failures and r repair facilities. The failure rates of the components increase as the number of components failed increases, while the repair rates are constant. Common-cause failure is not considered in Model I. In Model II the common-cause failures are involved. Steady-state probabilities and steady-state availability 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.     

Approximate steady-state solutions for Markov systems based upon the transition-rate matrix deviation concept

This paper derives an approximate method for solving the steady-state Markov equations. The approximate solutions are determined using the known or easily obtainable solutions of a Markov system whose transition-rate matrix slightly differs from that of the original system. The method suggested makes it possible to derive approximate analytic expressions for the steady-state probabilities of system states and associated reliability indices and/or to calculate their numerical values. The expressions for the error bounds for the method are provided. The method is applied to derive expressions for the state probabilities of two transmission lines sharing the same right of way and exposed to common cause failures, as a demonstration.     

Cost analysis of a 3-state 2-unit reparable system

This paper presents the cost analysis of a 2-unit system with 3 states: good, degraded and failed. The units suffer from two types of failure: partial and catastrophic. The partial failure brings a unit to degraded state, whereas the catastrophic failure breaks down a unit completely. There is one repair facility, which is availed only when the system is either degraded or failed. The failure and repair times for the system follow exponential and general distributions respectively. Laplace transforms of various probability states have been obtained along with steady-state behaviour of the system. Inversions have also been computed so as to obtain time dependent probabilities, which determine expected profit as well as availability of the system at any time.     

Stochastic behaviour of a two-unit cold standby redundant electronic equipment under human failure

This paper presents a mathematical model to evaluate availability and M.T.T.F. of a two-unit cold standby system with three possible states of each unit; viz. good, partially failed and failed, incorporating the concept of human failure. The model has been developed for exponential failures and general repairs. Single service facility is available in each state during the operational stage of the electronic equipment. Laplace transforms of the various state probabilities have been obtained. Steady-state probabilities, steady-state availability and mean time to failure have been derived.     

An inversion algorithm to compute blocking probabilities in lossnetworks with state-dependent rates

The algorithm developed in Choudhury et al. (1994) for computing (exact) steady-state blocking probabilities for each class in product-form loss networks is extended to cover general state-dependent arrival and service rates. This generalization allows to consider, for the first time, a wide variety of buffered and unbuffered resource-sharing models with non-Poisson traffic, as may arise with overflows in the context of alternative routing. As before, the authors consider noncomplete-sharing policies involving upper-limit and guaranteed-minimum bounds for the different classes, but in the present paper both bounds are discussed simultaneously. These bounds are important for providing different grades of service with protection against overloads by other classes. The algorithm is based on numerically inverting the generating function of the normalization constant, which is derived in the present paper. Major features of the algorithm are: dimension reduction by elimination of nonbinding resources and by conditional decomposition based on special structure, an effective scaling algorithm to control errors in the inversion, efficient treatment of multiple classes with identical parameters and truncation of large sums. The authors show that the computational complexity of the inversion approach is usually significantly lower than the alternative recursive approach     

Solving dependability/performability irreducible Markov models using regenerative randomization

Markov models are commonly used to asses the dependability/performability of fault-tolerant systems. Computation of many dependability/performability measures for repairable fault-tolerant systems requires the transient analysis of irreducible Markov models. Examples of such measures are the unavailability at time t and the s-expected interval unavailability at time t. Randomization (also called uniformization) is a well-known Markov transient analysis method and has good properties: numerical stability, well-controlled computation error, and ability to specify the computation error in advance. However, the randomization method is computationally expensive when the model is stiff, as is the case for Markov models of repairable fault-tolerant systems when the mission time of interest is large. Steady-state detection is a technique proposed to speedup randomization when the model is irreducible. This paper points out that another method, regenerative randomization, which has the same good properties as randomization, also covers irreducible models, and compares, for the important class of irreducible failure/repair models with exponential failure and repair time distributions and repair in every state with failed components, the efficiency of the regenerative randomization method with that of randomization with steady-state detection. In the frequent case in which the initial state is the state without failed components the regenerative randomization method can be faster than randomization with steady-state detection, especially when the model is large and the failure rates are much smaller than the repair rates. For other initial probability distributions, the regenerative randomization method seems to perform worse than randomization with steady-state detection.     

Graph Theory Concepts in Frequency and Availability Analysis

Steady-state availability and failure frequency are two key indices in a repairable system. They are generally evaluated from Markov models with constant transition rates. Numerical solutions can be found for relatively large systems using computer programs. It is more difficult to obtain general equations for a specific system using transition rate symbols. The determination of these equations usually involves linear simultaneous equations, either directly or using Cramer's Rule. This paper describes a general purpose graphical approach for obtaining steady-state availability and frequency expressions from a flow graph based on the Markov model. A set of generalised formulae is developed and applied to several configurations. This technique avoids the need to use matrices for developing general purpose equations. Its direct approach is useful to practising engineers and students of reliability concepts.     

Cost analysis of a system with common cause failure and two types of repair facilities

A mathematical model to predict the cost involved to run an n-component single unit system which can fail in n-mutually exclusive ways of total failure or due to common cause, has been developed. Each component has two modes (normal and failure) with two types of repair facilities. Repair rates are arbitrary functions of the time spent. All other transition rates are constant. Laplace transform of the state probabilities are developed along with steady-state behaviour of the system. Inversions are computed to determine the expected profit and availability of the system at any time.     

A k-out-of-N:G redundant system with cold standby units and common-cause failures

This paper presents a k-out-of N:G redundant system with M cold standby units, r repair facilities and common-cause failures. The constant failure rates of the operating and cold standby units are different. Failed system repair times are arbitrarily distributed. The system is in a failed state when (N+M?k+1) units failed or a common-cause occurred. Laplace transforms of the state probabilities, the availability of the system and the system steady-state availability are derived.     

Computable Bounds for Conditional Steady-State Probabilities in Large Markov Chains and Queueing Models

A method is presented to compute, for the steady-state conditional probabilities of a given Markov subchain, the best lower and upper bounds derivable from the submatrix of transition probabilities between the states of that subchain only. The bounds can be improved when additional information, even fragmentary, on the entire chain is available. When the submatrix has a special structure, analytical expressions of the bounds can be obtained. The method is shown to be useful and economical to bound performance measures in large nonproduct-form queueing network models of computer communication systems.     

Operational behaviour of a three state standby redundant electronic equipment under critical human errors

This paper deals with the operational behaviour of a cold-standby redundant system incorporating the concept of three states, with four types of failures, namely major unit failure, minor unit failure, partial failure due to critical human errors and complete failure due to critical human errors, under only one repair facility. Failure and repair times for the complex system follow exponential and general distributions, respectively. Repair is undertaken only when the system is either in degraded state or in failed state. Laplace transforms of the probabilities of being in various states as well as in up and down states are computed, along with the steady state behaviour of the system. A particular case of such a system has also been appended to highlight the important results.     

Dependability Under Priority Repair Disciplines

We have studied the operational behaviour of a complex system consisting of two classes of units (with standby redundancy in one class) under two repair disciplines, viz., preemptive-resume and preemptive-repeat. Units have constant failure rate, while repair follows general probability distributions. Supplementary variable and Laplace transform techniques have been used to obtain the transient state probabilities for such a system. From these, steady state behaviour of the complex system has been examined.     

Splitting-based importance-sampling algorithm for fast simulationof Markov reliability models with general repair-policies

Markov chains with small transition probabilities occur while modeling the reliability of systems where the individual components are highly reliable and quickly repairable. Complex inter-component dependencies can exist and the state space involved can be huge, making these models analytically and numerically intractable. Naive simulation is also difficult because the event of interest (system failure) is rare, so that a prohibitively large amount of computation is needed to obtain samples of these events. An earlier paper (Juneja et al., 2001) proposed an importance sampling scheme that provides large efficiency increases over naive simulation for a very general class of models including reliability models with general repair policies such as deferred and group repairs. However, there is a statistical penalty associated with this scheme when the corresponding Markov chain has high probability cycles as may be the case with reliability models with general repair policies. This paper develops a splitting-based importance-sampling technique that avoids this statistical penalty by splitting paths at high probability cycles and thus achieves bounded relative-error in a stronger sense than in previous attempts     

Approximate Markov Modeling of High-Reliability Telecommunications Systems

The reliability model of a system with redundancy but only a single repairman is Markov only if the component failure rates and the repair rate are constants. This paper introduces a method with which a reliability analyst can formulate an approximate (time-homogeneous) Markov model for a system with 1-out-of-2 redundancy when repair is nonexponential (repair rate is time-dependent). This approximate model yields accurate steady-state predictions of system reliability when time-to-repair is orders of magnitude smaller than timeto-component-failure, as is typical in high-reliability telecommunications systems. Transition rates and error bounds for the approximate model are given based on the first three moments of the repair time distribution. An application of the method is shown for the system in which repair time is composed of a "next day" parts delivery phase followed by an on-site repair phase.     

Cost analysis of a three-state repairable redundant complex system under various modes of failures

In this paper investigations have been carried out for the availability and mean time to failure analysis of a three unit repairable electronic equipment having three states; viz; good, degraded and failed under critical human errors. The three states three units repairable electronic equipment suffers two types of failures; viz; unit failure and failure due to critical human errors. Entire system can fail 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 are obtained along with steady state behaviour of the equipment. A numerical example has also been appended to highlight the important results. Three graphs have also been given in the end. There is only one repair facility, which is availed only when the system is in either degraded or failed state due to unit failure.     

False alarm and correct detection probabilities over a time interval for restricted classes of failure detection algorithms

The statistical analysis of failure detection decisions in terms of the instantaneous probabilities of false alarm and correct detection for a specified failure magnitude at each check-time have previously been performed for several different failure detection techniques that utilize a Kalman filter. By performing a discrete-time specialization of a result of Gallager and Helstrom on a tightened upper bound for continuous-time level-crossing probabilities, upper bounds on the probabilities of false alarm and correct detection over a time interval have been obtained for the specific technique of CR2 tailnre detection (to allow an accounting for the effect of time correlations of the filter estimates). When these upper bounds are optimized to be as tight as possible to the desired probabilities, the resulting optimization problem for discrete-time is a collection of quadratic programming (QP) problems, which may easily be solved exactly without recourse to approximate solutions as were resorted to in the continuous-time formulation. This technique for evaluating tightened upper bounds on the false alarm and correct detection probabilities may be of general interest, since it can be applied to any failure detection technique or signal detection technique that can relate an exceeding of the deterministic decision threshold by the test statistic directly to a deterministic level being exceeded by a scalar Gaussian random process.     

Cost analysis of a multi-component screening system in the paper industry

The paper discusses the screening system in the Paper Industry with three states—good, reduced and failed. The failure rates are constant while the repair rates are general. Formulation of the problem is carried out using simple probability consideration. Various probabilities are obtained along with steady state probabilities of the system. Availability and MTTF tables and graphs for various parameters are given which are useful to the designer and the management for improvement in design.