Models For Repairable Systems Research Articles

On the interval reliability of systems modelled by finite semi-Markov processes

In this paper, semi-Markov models of repairable systems are considered whose finite state space is partitioned into two sets, the set of ‘up’ states, U, and the set of ‘down’ states, D. The focus of attention is the system's interval reliability which is defined as the probability of the system being in U throughout a given time interval [t, t + x] where t, x ϵ [0, +∞) are fixed. Two results in the Laplace transform domain are obtained for the interval reliability. The first one is a closed form expression for the double Laplace transform with respect to both variables t and x. The second result is concerned with a closed form expression of the (single) Laplace transform of the interval reliability with respect to the variable t under the additional assumption that the modelling semi-Markov process is Markovian on U. As an example, the semi-Markov model of a two-unit repairable system is considered. The steady-state behaviour of the system's interval reliability is examined by a Tauberian theorem.

Modeling of repairable systems

The reliability of many stochastic repairable systems depends on several characteristics that are time dependent. In this paper, we develop general repair models for a repairable system by using auxiliary stochastic processes which describe physical characteristics of the system and derive various properties of resulting models. We also obtain an inference procedure to assess the number of failures and expected number of failures for our proposed models by observing auxiliary processes. Clearly, our inference procedure is different from the traditional approach, where successive times of occurrence of failures are observed.

Modeling of repairable systems

The reliability of many stochastic repairable systems depends on several characteristics that are time dependent. In this paper, we develop general repair models for a repairable system by using auxiliary stochastic processes which describe physical characteristics of the system and derive various properties of resulting models. We also obtain an inference procedure to assess the number of failures and expected number of failures for our proposed models by observing auxiliary processes.Clearly, our inference procedure is different from the traditional approach, where successive times of occurrence of failures are observed.

An analysis of a reliability model for repairable fault-tolerant systems

The ARIES reliability model, which models a class of repairable and nonrepairable fault-tolerant systems by a continuous-time Markov chain and uses the Lagrange-Sylvester interpolation formula to directly compute the exponential of the state transition rate matrix (STRM) that appears in the solution of the Markov chain, is discussed. The properties of the STRM for ARIES repairable systems are analyzed. Well-established results in matrix theory are used to find an efficient solution for reliability computation when the eigenvalues of the STRM are distinct. A class of systems that ARIES models for which the solution technique is inapplicable is identified. Several transformations which are known to be numerically stable are used in the solution method. The solution method also offers a facility for incrementally computing reliability when the number of spares in the fault-tolerant system is increased by one. >

An application of a bathtub failure model to imperfectly repaired systems data

AbstractIn this paper we use an exponential power distribution to develop a bathtub failure model for repairable systems. These results are compared with those using the more familiar Weibull distribution. The analysis is extended to three types of repair scenarios: good‐as‐new repairs (in which the repair following a failure completely refreshes the failure intensity), bad‐as‐old repairs (in which the repair has no effect on the failure intensity) and imperfect repairs (in which the failure intensity is only partially reset). We apply this analysis to hydro‐electric turbine data and find that an exponential power distribution produces a well‐defined bathtub failure model when used with an imperfect repair assumption. We furthermore find that this bathtub model produces an improved fit over the equivalent Weibull model.

Failure predictions in repairable multi-component systems

Multi-component repairable systems cannot be modeled by continuous distributions, such as the Weibull. Failures occurring in repairable systems are examples of a series of descrete events which occur randomly in a continuum. These situations which are stochastic point processes are analyzed using the statistics of event series. The Crow model [1], or power law nonhomogenous Poisson process, is recognized by the reliability community as being one of the best models for repairable systems. In this research, failure data from multiple versions of certain mechanical systems are modelled by Crow's nonhomogenous Poisson process, NHPP. The expected number of failures predicted for the respective system by the Crow model are compared to predictions using a Monte Carlo simulation that utilizes Weibull parameters of the major components of the system. The objective is to prove that a simulation based on Weibull parameters of the major component failure modes is able to duplicate the overall system prediction that the Crow NHPP model gives. The simulations, based on fitting component failure data to a continuous distribution such as the Weibull, are more valuable in that they give failure prediction results that can be traced to individual components. The advantage here is that only the major component failure modes will need to be included in the simulation and the NHPP model can be utilized as a gauge to determine when the simulation has the appropriate component failure modes included. The uniqueness in this research lies in the use of the simulation approach in conjunction with the Crow NHPP model so that the failure modes can be identified. The Crow model predicts when the overall system will be down and then the simulation predicts the number of failures from each of the included components. The simulation can identify a finite number of parts that contribute to the overall system downtime. This information can be used to design an optimum preventive maintenance program or guide research into more reliable components.

Modeling repairable systems with failure rates that depend on age and maintenance

A general repairable system model that includes age and preventive maintenance (PM) dependent failure rates is proposed. This model introduces the concept of system availability as a random variable, leading to availability measures for individual realizations, rather than for the average over many trials. Thus this model generalizes the classical definition, and serves as a performance index for system design. A design scheme suggested to maximize the probability of achieving a specified stochastic cycle availability with respect to the duration of the operating interval between PMs. Numerical computations of the results for Weibull-like distributions illustrate the design criteria. Asymptotic behavior of the performance with no PM has been studied. Extension to other distributions is straightforward. >

Optimal repair policies with general degree of repair in two maintenance models

Two maintenance models for repairable systems with exponential lifetimes are studied. A maintenance strategy in the proposed framework specifies the degree of repair to be taken as a function of the virtual age of the system upon failure. In Model I we restrict attention to maintenance strategies in which only minimal or perfect repair can be employed upon failure. In Model II the degree of repair may vary. For some age-dependent cost functions, optimal maintenance strategies (minimizing the expected discounted costs over an infinite horizon) are determined analytically.

A critical look at some point-process models for repairable systems

The use of a modeldriven approach to the analysis of repairable systems is considered and shown to be useful as a way of understanding the characteristics of such a system. However, considerable statistical problems arise from the use of a set of standard model-building elements. In particular, identification problems arise in many of the models. The argument is illustrated by examples from software reliability and mechanical reliability. The conclusion is that, in many cases, the exploratory data-analysis approach is as effective as the use of more sophisticated models.

On evaluating the cumulative performance distribution of fault-tolerant computer systems

Fault-tolerant computer systems may be evaluated by calculating their cumulative performance (e.g., number of processes jobs) over a finite mission time. A method for calculating the cumulative performance distribution assuming that the system fault-repair behavior can be modeled by a homogeneous Markov process is described. The method proposed for calculating the probability distribution of the accumulated reward over a finite mission is applicable to models of repairable and nonrepairable systems. The related solution algorithm shows a low polynomial computational complexity. The mathematical model is introduced, and the results already presented in the literature are surveyed. A comprehensive analysis of the time and space complexity of the proposed solution algorithm is presented. A numerical example is given. >

A maintenance model for repairable systems

Abstract A model using a sequence of distributions, with dependent parameters, for a repairable system is proposed. By using a set of failure data containing successive times to failures, the parameters of the model are estimated and functional forms relating the parameters to the state of the system are obtained. Simulation is then conducted to determine the optimal age maintenance scheme for such a system. Finally, a possible extension is suggested by introducing stabilizing rate and deterioration rate of a class of repairable systems so that a stochastic model for such repairable systems can be fully determined.

Some considerations on reliability theory and its applications

Abstract In this paper we discuss some main topics within reliability theory and its applications. These include for example, modelling of systems of dependent components, identification of critical components, modelling of repairable systems, and the use of multistate models. The starting point for the discussion is Bergman's review paper on reliability theory and its application.

Statistical Analysis of a Compound Power-Law Model for Repairable Systems

A compound (mixed) Poisson distribution is sometimes used as an alternative to the Poisson distribution for count data. Such a compound distribution, which has a negative binomial form, occurs when the population consists of Poisson distributed individuals, but with intensities which have a gamma distribution. A similar situation can occur with a repairable system when failure intensities of each system are different. A more general situation is considered where the system failures are distributed according to nonhomogeneous Poisson processes having Power Law intensity functions with gamma distributed intensity parameter. If the failures of each system in a population of repairable systems are distributed according to a Power Law process, but with different intensities, then a compound Power Law process provides a suitable model. A test, based on the ratio of the sample variance to the sample mean of count data from s-independent systems, provides a convenient way to determine if a compound model is appropriate. When a compound Power Law model is indicated, the maximum likelihood estimates of the shape parameters of the individual systems can be computed and homogeneity can be tested. If equality of the shape parameters is indicated, then it is possible to test whether the systems are homogeneous Poisson processes versus a nonhomogeneous alternative. If deterioration within systems is suspected, then the alternative in which the shape parameter exceeds unity would be appropriate, while if systems are undergoing reliability growth the alternative would be that the shape parameter is less than unity.

Two-Parameter Approximations For Multi-Echelon Repairable Inventory Models With Batch Ordering Policy

Abstract In this paper we present a model for multi-echelon repairable systems with batch ordering policy at the bases. Such an ordering policy is desirable when demand rates and/or the set up cost for ordering are relatively high. Operating characteristics of such a system are analyzed. A two-parameter approximation scheme for the distribution of the number of orders outstanding and consequently backorders is developed, and its performance evaluated. It is found that, under a variety of scenarios including both finite and infinite servers at the repair facility, the approximation scheme is very effective in providing a relatively simple means to determine the stocking levels at both the depot and the bases to minimize the cost of inventory holding and backorders. For the special case when the batch size is one, we confirm the effectiveness of the use of the negative binomial distribution as an approximation for the distribution of orders outstanding, as suggested by recent studies, under a much more general setting.

Unavailability evaluation in non-Markovian systems with spare components

Markovian models for repairable systems assume constant state transition rates. If the operating components are in their useful life phase then the assumption of a constant failure rate is valid. This assumption may not extend, however, to the repair and installation process. the inclusion of non-exponential residence times will make the process non-Markovian. This paper presents a range of studies for a repairable system containing an increasing number of spare components. The effect on system unavailability of various distributions is compared for a wide range of failure, repair and installation rates using Monte-Carlo simulation. The failure rate is assumed to be constant and the distributional variation is applied to the repair and installation processes. In order to facilitate comparison, the mean values associated with the repair and installation times are held constant. The change in unavailability is therefore due entirely to the change in the shape of the distribution.

Comments on: Reliability Discipline and Technology in Historical Perspective - 1930-1984

Too little attention is paid to accurate models for repairable systems.

Availability analysis of transit systems

This paper presents two mathematical models of repairable transit systems. State probability equations for both models are developed.

Repairable-system models : Balbir S. Dhillon. IEEE Trans. Reliab.R-30 (5), 492 (1981)

Repairable-system models : Balbir S. Dhillon. IEEE Trans. Reliab.R-30 (5), 492 (1981)

Transit vehicle reliability models

This paper presents two mathematical Markov models of repairable transit systems. Laplace transforms of the state probability equations are developed.

Repairable-System Models

This paper presents three repairable-system models. Laplace transforms of the state probability equations are developed. The paper describes the models. The derivations and results are in a separately available Supplement.