Mathematical Theory Of Reliability Research Articles

Уравнения марковского процесса гибели в математической теории надежности

Уравнения марковского процесса гибели в математической теории надежности

A mathematical model of reliability of technical service subsystem of a grain processing plant

Relation between the theoretical research of mathematical reliability theory application, the increase of operating effectiveness and the period of reliable operation of grain processing plant equipment is given.

Generalized Chebyshev inequalities and their application in the mathematical theory of reliability

The paper considers a problem whose solution is generalized Chebyshev inequalities. Examples from the mathematical theory of reliability are given. General results and the results obtained by the author are briefly reviewed. A new problem for further research in this field is formulated.

Competing auctions with endogenous quantities

Two sellers decide on their discrete supply of a homogenous good. There is a finite number of buyers with unit demand and privately known valuations. In the first model, there is a centralized market place where a uniform auction takes place. In the second, there are two distinct auction sites, each with one seller, and buyers decide where to bid. Using the theory of potential games, we show that in the one-site auction model there is always an equilibrium in pure-strategies. In contrast, if the distribution of buyers values has an increasing failure rate, and if the marginal cost of production is relatively low, there is no pure-strategy equilibrium where both sellers make positive profits in the competing sites model. We also identify conditions under which an equilibrium with a unique active site exists. We deal with the finite and discrete models by using several results about order statistics developed by Richard Barlow and Frank Proschan [R. Barlow, F. Proschan, Mathematical Theory of Reliability, Wiley, New York, 1965; R. Barlow, F. Proschan, Inequalities for linear combinations of order statistics from restricted families, Ann. Math. Statist. 37 (1966) 1593–1601; R. Barlow, F. Proschan, Statistical Theory of Reliability and Life Testing, McArdle Press, Silver Spring, 1975].

Service Reliability Theory and Engineering, I: Foundations

The concepts and methods of reliability theory and engineering are applied to services. This has not yet been done in a systematic fashion because services are intangible and the primitives and concepts necessary to making the connection of intangibles and the mathematical theory of reliability have not yet been documented. A good example of the confusion this causes comes from the telecommunications services industry, where professionals often confuse “network reliability” with what is really the reliability of the services provided over the network: a phrase that could make sense when correctly applied to tangible engineering systems is applied (without proper preparation) to reliability of an intangible, the services. The purpose of this paper is to clarify the notions of service reliability theory and to describe service reliability engineering and the process of design for service reliability. The paper covers primitive terms in the theory, basic concepts, failure modes and failure mechanisms for services, and service reliability figures of merit and metrics. Many of the illustrative examples are drawn from the author’s experiences in the telecommunications industry. The paper offers a foundation for clear discussion and quantitative statements about the reliability of services, and brings clarity to service planning and development. The value for professionals in the quality sciences and quality engineering is in the systematic extension of reliability engineering concepts and techniques to intangibles, in the provision of a framework and language for quantitative study of service reliability, and in focusing quality attention on service industries, now the major part of the economies of many developed countries. A companion paper will describe mathematical models for service reliability and design for service reliability in the context of a comprehensive example drawn from the telecommunications service industry.

Some current problems in reliability theory

Reliability work occupies an increasingly important place in engineering practice. Although the details differ depending on whether mechanical, electrical, chemical, or other systems are under analysis, the reliability concepts and the mathematical foundations cut across the specific fields of application. Over the past 50 years thousands of papers and dozens of books on mathematical models of reliability have been published. A comprehensive survey alone on the current developments in the mathematical theory of reliability would fill a voluminous book. Based on importance both for theory and application and taking into account the interests of the author, current investigations in four important branches of reliability theory are considered: coherent systems, stochastic networks, software reliability, and maintenance theory.

Monotone Log-Odds Rate Distributions in Reliability Analysis

Monotone failure rate models [Barlow Richard, E., Marshall, A. W., Proschan, Frank. (1963). Properties of probability distributions with monotone failure rate. Annals of Mathematical Statistics 34:375–389, and Barlow Richard, E., Proschan, Frank. (1965). Mathematical Theory of Reliability. New York: John Wiley & Sons, Barlow Richard, E., Proschan, Frank. (1966a). Tolerance and confidence limits for classes of distributions based on failure rate. Annals of Mathematical Statistics 37(6):1593–1601, Barlow Richard, E., Proschan, Frank. (1966b). Inequalities for linear combinations of order statistics from restricted families. Annals of Mathematical Statistics 37(6):1574–1592, Barlow Richard, E., Proschan, Frank. (1975). Statistical Theory of Reliability and Life Testing. New York: Holt, Rinehart and Winston, Inc.] have become one of the most important models of failure time for reliability practitioners to consider and use. The above authors also developed models and bounds for monotone increasing failure rates (IFR) and for monotone decreasing failure rates (DFR). The IFR models and bounds appear to be especially useful for describing and bounding the hazard of aging. This article extends a new model for time to failure based onthe log odds rate [Zimmer William, J., Wang Yao, Pathak, P. K. (1998). Log-odds rate and monotone log-odds rate distributions. Journal of Quality Technology 30(4):376–385.] which is comparable to the model based on the failure rate. It is shown that in the case of increasing log odds rate (ILOR) in terms of log time (ln t), the model is less stringent than the IFR model for aging. The characterization of distributions of failure time by log odds rate is also derived. It is shown that the logistic distribution has the property of constant log odds rate over time and that the log logistic distribution has the property of constant log odds rate with respect to ln t. Some other properties of ILOR distributions are presented and bounds based on the relationship to the log logistic distribution are provided for distributions which are ILOR with respect to ln t. Motivational examples are provided. The ILOR bounds are compared to the more stringent bounds based on the IFR model. Bounds on system reliability are also provided for certain systems.

Mathematical Theory of Reliability of Time Dependent Systems with Practical Applications

Markov and Semi-Markov Models as a Basis for the Mathematical Analysis of System Reliability. Methods for Investigating Homogeneous and Non-Homogeneous Point Processes (Event Flows). Fault Trees--The Current State of Research. Theory of Redundant Systems. Monte Carlo Methods. Reliability Analysis using Perturbation Methods. Stiff Processes in Reliability Analysis. Variance Reduction Methods. Analytical-Statistical Methods for Rapid Simulation of Repairable Systems with Structure Redundancy. Measures of Reliability Importance of Components. Index.

Some reliability measures of a system of components sharing a common environment

A multivariate distribution for describing the lifespan of the components of a system which operates in an environment whose intensity is in constant overtime but different from that of the test bench (laboratory) environment. The properties of the reliability function of a system of components sharing a common environment are studied in this paper. Comparison would also give a sense of the robustness of independent assumptions in system reliability calculations. It is to be noted that the reliability function of parallel redundant systems whose components share a common unknown environment cannot be characterized by any of the well-known classes of distribution studied in the mathematical theory of reliability. The above conclusions suggest the need for defining and studying the properties of a new class of failure distribution in reliability theory. We have derived the mean time before failure, maintainability M( t), and mean time to failure of the system.

An approach to formalized description of the behavior dynamics of structurally and logically complex systems

A system of formal concepts is proposed, which is useful for describing the behavior dynamics of systems with multiple operating modes whose succession is determined by complex logical relations. The proposed apparatus is demonstrated by some applications to mathematical reliability theory.

On the reliability function of a system of components sharing a common environment

A multivariate distribution for describing the life-lengths of the components of a system which operates in an environment that is different from the test bench environment has been proposed by Lindley and Singpurwalla (1986). In this paper, the properties of the reliability function of such a system are studied and comparisons made with the reliability function obtained under the assumption of independence. It is interesting to note that the reliability function of parallel redundant systems whose components share a common unknown environment cannot be characterized by any of the well-known classes of distributions that have been proposed in the mathematical theory of reliability. This observation suggests the need for defining a new class of failure distributions. A formula for making Bayesian inferences for the reliability function is also given.

On the reliability function of a system of components sharing a common environment

A multivariate distribution for describing the life-lengths of the components of a system which operates in an environment that is different from the test bench environment has been proposed by Lindley and Singpurwalla (1986). In this paper, the properties of the reliability function of such a system are studied and comparisons made with the reliability function obtained under the assumption of independence. It is interesting to note that the reliability function of parallel redundant systems whose components share a common unknown environment cannot be characterized by any of the well-known classes of distributions that have been proposed in the mathematical theory of reliability. This observation suggests the need for defining a new class of failure distributions. A formula for making Bayesian inferences for the reliability function is also given.

Asymptotic analysis of a priority model with backup and repair

One of the problems in mathematical reliability theory is to estimate the reliability of systems with repair. In general, the problem can be stated in the following terms. Given is a system of n elements; one of the operational elements in the system should be busy at any given time. In case of failure, the failed element is routed to a repair unit, repaired, and returned to the system. The state of the system at any given moment is defined by a binary vector which shows the operational and the failed elements at that moment. The system state set is partitioned into two subsets -operational and failed states. Over time, the system repeatedly switches from operational to failed and back. A major problem in reliability theory is to compute or estimate the basic reliability characteristics, e.g., the time distributions of the operational and the failed states and their means. In mathematical terms, a model of this kind is a queueing model with customers represented by failed elements routed for repair and service represented by the repairs on these elements.

Mathematical theory of reliability: A historical perspective

The author reviews developments in mathematical reliability theory, especially those developments which have had or are likely to have impact on engineering reliability problems. The review to 1960 is based on a similar review which appeared in the 1965 book by R.E. Barlow and F. Proschan entitled Mathematical Theory of Reliability. It is pointed out that if the 1970s were the fault tree era, the 1980s may well turn out to be the era of network reliability, principally due to the importance of computer and computer network reliability.

Age Replacement Under Alternative Cost Criteria

Due to the complexity of optimum replacement problems over finite lime horizons various asymptotic criteria based upon fixed age replacement policies have been employed in the literature and in practice. In this paper the relationships between the optimum policies under three alternative cost criteria are considered. An ordering of the accounting costs under two of these is obtained, and for distributions with Increasing Hazard Rate an ordering of the optimum replacement time is derived. For a finite time horizon the policies are compared to the optimal sequential and fixed-age replacement polices through the example of a gamma distribution previously investigated by Barlow and Proschan (Barlow, R. E., F. Proschan. 1962. Planned replacement. K. J. Arrow, S. Karlin, W. L. Smith, eds. Studies in Applied Probability and Management Science. Stanford University Press, Stanford, Cal.; Barlow, R. E., F. Proschan. 1965. Mathematical Theory of Reliability. Wiley, New York.).

Optimal inspection when the system is repaired upon detection of failure

In Barlow and Proschan ( Mathematical Theory of Reliability, 1965, Section 3.2) a cost model is presented for a system subject to random failure and whose state is known only by inspection. Upon detection of failure repair (or replacement) is performed and the system is then as good as new. A method of determining the inspection schedule which minimizes the long run average (expected) cost per unit time is proposed. In this present paper we look closer into the problem of finding an optimal inspection schedule for this model. Some new results, which are useful in connection with the computation of the optimal inspection schedule, are given.

A cybernetical model of the internal cellular clock

A model of a cell-labeling mechanism for the developmental tree of a multicellular organism is suggested. The cell-labeling information is assumed to originate because the DNA strands are complementary, rather than identical, so a pair of replicated DNA can carry different information content to newly created pairs of chromosomes. This mechanism can serve as an internal cellular clock and as one of the factors involved in the control of cell differentiation. Some possible structures were analyzed by means of the mathematical theory of reliability and compared with the results of Hayflick's experiments on limited cells' lifetime. The external destruction of the clock mechanism can remove limitations on cell's reproductions and is considered as a cause of carcinogenesis; the model exhibits cumulative and threshold effects. The cell-labeling mechanism of the model is able to provide sometimes equivalent labeling for the initial zygote division as a necessary condition for the appearance of monozygote twins; it is most likely that this may occur only for pairs of monozygote twins with the probability equal to an integer power of 1 2 .

Theoretical studies of clonal selection: Minimal antibody repertoire size and reliability of self-non-self discrimination

Viewing the immune system as a molecular recognition device designed to identify “foreign shapes”, we estimate the probability that an immune system with N Ab monospecific antibodies in its repertoire can recognize a random foreign antigen. Furthermore, we estimate the improvement in recognition if antibodies are multispecific rather than monospecific. From our probabilistic model we conclude: (1) clonal selection is feasible, i.e. with a finite number of antibodies an animal can recognize an effectively infinite number of antigens; (2) there should not be great differences in the specificities of antibody molecules among different species; (3) the region of a foreign molecule recognized by an antibody must be severely limited in extent; (4) the probability of recognizing a foreign molecule, P, increases with the antibody repertoire size N Ab ; however, below a certain value of N Ab the immune system would be very ineffectual, while beyond some high value of N Ab further increases in N Ab yield diminishing small increases in P; (5) multispecificity is equivalent to a modest increase (probably less than 10) in the antibody repertoire size N Ab , but this increase can substantially improve the probability of an immune system recognizing a foreign molecule. Besides recognizing foreign molecules, the immune system must distinguish them from self molecules. Using the mathematical theory of reliability we argue that multisite recognition is a more reliable method of distinguishing between molecules than single site recognition. This may have been an important evolutionary consideration in the selection of weak non-covalent interactions as the basis of antigen-antibody bonds.

A method for estimating the reliability of burst-prevention measures

Rock bursts are random in character. To assess the efficiency of burst-prevention measures it is therefore desirable to use methods of the mathematical theory of reliability. Thus to calculate the reliability of procedures for preventing rock bursts we need data on the number of bursts which have occurred in the protected and unprotected zones, and the values of the face advance in these zones. Table 1 gives the results of estimates of the reliability of burst-prevention measures from the experimental data for a number of Donbass collieries. The reliability of a burst-prevention method like the drilling of advanced boreholes varies within the range 0.65 to 0.87 in different burst-prone seams. The variations of the reliability are due to the presence of geological faults, to deviations from the recommendations for the burst prevention procedure, and to violations of the coal extraction technology. The reliability of burst prevention by wetting of coal seams varies over a wide range from 0.2 to 0.9 indicating the pronounced influence of the geological and technical conditions on the efficiency of this method. Thus, the procedure developed for estimating the reliability of burst-prevention measures from the statistical character of bursts enables one to obtain accurate estimates ofmore » the efficiency of burst-prevention measures and to use them for future improvement of the methods.« less

A survey of maintenance models: The control and surveillance of deteriorating systems

AbstractThe literature on maintenance models is surveyed. The focus is on work appearing since the 1965 survey, “Maintenance Policies for Stochastically Failing Equipment: A Survey” by John McCall and the 1965 book, The Mathematical Theory of Reliability, by Richard Barlow and Frank Proschan. The survey includes models which involve an optimal decision to procure, inspect, and repair and/or replace a unit subject to deterioration in service.