Classes Of Limit Reliability Functions Research Articles

ON COMPONENT FAILURES' DEPENDENCY INFLUENCE ON SYSTEM'S LIFETIME

In the paper the results of the reliability investigation of multi-state homogeneous parallel-series systems with independent and dependent components are presented. The multi-state reliability functions of such systems and other reliability characteristics in both cases are determined under the assumption that their components have exponential reliability function. Moreover the asymptotic approach to the reliability evaluation of these systems is also presented and the classes of limit reliability functions for the considered systems in both cases are fixed. Finally, the presented theoretical results are applied to the reliability evaluation of the shipyard rope transportation system. The comparison of the multi-state exact and limit reliability functions of the considered transportation system under the assumption that its components are independent and under the assumption that its components have failure dependency is performed and illustrated graphically.

Asymptotic Approach to Reliability Analysis of Large Systems with Degrading Components

Basic notions of the multistate systems reliability analysis are discussed. The multistate homogeneous and non-homogeneous series, parallel, m out of n, series-parallel and parallel-series systems with degrading components are defined and their multistate reliability functions, risk functions and other characteristics are determined. The asymptotic approach to reliability analysis of large systems is defined and the notion of the multistate limit reliability function of the system is introduced. Auxiliary theorems on limit reliability functions of large multistate systems which are necessary to their approximate reliability evaluation are presented. The classes of limit reliability functions for the considered systems are fixed. Applications of the asymptotic approach to reliability evaluations of two large technical systems are presented and its accuracy is illustrated+.

Asymptotic approach to reliability evaluation of large multi-state systems with application to piping transportation

The essentials of multi-state system reliability analysis and an asymptotic approach to system reliability evaluation are introduced. The classes of limit reliability functions for multi-state series, parallel, series–parallel and parallel–series systems composed of different components are fixed. Moreover, auxiliary theorems useful in reliability evaluation of real technical large systems are presented. These are applied to reliability evaluation of an oil transportation system. Approximate mean values and standard deviations of this system lifetimes in the state subsets and in particular states are determined. This system risk function and the moment when the risk exceeds an admissible level are evaluated as well. The accuracy of the approximation is illustrated numerically and graphically.

On the class of limit reliability functions of homogeneous series-“ k-out-of- n” systems

The problem of finding possible limit reliability functions of basic systems is well known. For instance Gnedenko [Ann. Math. 44 (1943) 432–453] has found a class of possible limit reliability functions for series and parallel systems. Smirnov [Trudy Matem. Inst. im. V.A. Stelkova, 1949] has solved the problem for “ k-out-of- n” systems. In this paper these two areas are brought together and a class of limit reliability functions of homogeneous series-“ k-out-of- n” systems is presented. Moreover some applications are given as well.

Domains of attraction of asymptotic reliability functions

Theorems about domains of attraction of limit distribution of extreme statistics of independent identically distributed random variables are well known and included in [R.E. Barlow, F. Proshan, Statistical Theory of Reliability of Life Testing, Probability Models, Holt. Reinehart and Winston, New York, 1975; H. Chernoff, H. Teicher, Trans. Amer. Math. Soc. 116 (1965) 474–491; B.W. Gnedenko, Ann. Math., 44 (1943) 432–453; B. Kopocinski, Zarys Teorii Odnowy i Niezawodnosci, PWN, Warszawa, 1973]. In [K. Kolowrocki, Appl. Math., 36 (1993) 56–69] a new four-element class of limit reliability functions for a regular, homogeneous series–parallel system in which the number of series components of the system has the order of logarithm of the number of parallel components is fixed. This paper presents the solution of the problem of domains of attraction for these asymptotic reliability functions. Moreover, some examples are given.

A remark on of limit reliability function of large series–parallel systems with assisting components

In the reliability investigation of large scale systems the problem of the complexity of their reliability functions arises. This problem may be approximately solved by assuming that the number of the system components tends to infinity and finding the limit reliability function of the system. The results for series, parallel, series–parallel and parallel–series systems are given in R.E. Barlow, F. Proschan, Statistical Theory of Reliability and Life Testing, Probability Models, Holt Rinehart and Winston, New York, 1975; K. Kołowrocki, On a Class of Limit Reliability Functions for Series–Parallel and Parallel–Series Systems, Monograph, Gdynia Maritime Academy Press, 1993; K. Kołowrocki, Reliability Engineering and System Safety 39 (1993) 11–23. In the paper as a partial extension of these results, theorems which determinate necessary and sufficient conditions for a reliability function to be an asymptotic reliability function of the series–parallel system with assisting components are given. The problem is solved for the system which the number of its series components is of order of the number of its parallel components. Moreover, some examples are presented.

Limit reliability functions of some homogeneous regular series–parallel and parallel–series systems of higher order

In the reliability investigation of large-scale systems, the problem of the complexity of their reliability functions arises. This problem may be approximately solved by the assuming that the number of system components tends to infinity and finding the limit reliability function. It is well known that there exist only three possible nondegenerate limit reliability functions for regular, homogeneous, series and parallel systems [1]. The same three-element classes of limit reliability functions for series–parallel and parallel–series systems of order 1 has been fixed by Chernoff and Teicher [2]. These classes have been extended to ten-element classes of limit reliability functions by Kołowrocki [3–6]. This paper presents the generalization of the results given in [6].

On applications of asymptotic reliability functions to the reliability and risk evaluation of pipelines

In the reliability investigation of large systems the problem of the complexity of their reliability functions appears. This problem may be approximately solved by assuming that the numer of the system components tends to infinity and finding the limit reliability function of the system. The solution for two-state series and parallel systems is well known and given for instance in `Statistical theory of reliability and life testing' by Barlow and Prochan. Limit reliability functions of more complex two-state series-parallel and parallel-series systems with identical components are discussed by Kołowrocki (Reliability Engineering and System Safety 1993;41:251–7). The results on limit reliability functions of two-state series-parallel and parallel-series systems with different components are given by Kołowrocki (Reliability Engineering and System Safety 1994;46:179–88). Systems with multistate components are more general and play an important role in the reliability practice, specially in system safety analysis. Therefore in the paper, besides of the classes of limit reliability functions for non-homogeneous two-state systems, the classes of such functions for any large homogeneous multistate series and series-parallel systems are given. Moreover, some applications of the results to the reliability and risk evaluation of pipelines are presented.

Asymptotic reliability functions of some nonhomogeneous series-parallel and parallel-series systems

In the reliability investigation of large-scale systems the problem of the complexity of their reliability functions appears. This problem may be approximately solved by assuming that the number of the system components tends to infinity and finding the limit reliability function. In this paper three-element classes of limit reliability functions for series-parallel and parallel-series systems with different components are presented. The series-parallel system is such that the number of its series components is of greater order than the logarithm of the number of its parallel components. The parallel-series system is such that the number of its parallel components is of greater order than the logarithm of the number of its series components. The fixed classes are different from the known three-element classes of asymptotic distributions of maximin and minimax statistics of independent random variables with a common distribution function. All results are obtained under the assumption that lifetimes of the particular components are independent random variables. Moreover, an example of the considered systems and its limit reliability function is given. The results may be useful in the reliability evaluation of large systems with unalike components.

On a class of limit reliability functions for series-parallel and parallel-series systems

In the reliability investigation of large-scale systems, the problem of the complexity of their reliability functions arises. This problem may be approximately solved by the assuming that the number of system components tends to infinity and finding the limit reliability function. This paper presents the solution of this problem and some applications. The classes of asymptotic reliability functions for series-parallel and parallel-series systems with identical and unlike components are presented. They are ten-element and more extensive than the known three-element classes of asymptotic distributions of maximum and minimum statistics of independent random variables with a common distribution function. The results may be useful in the reliability evaluation of large scale systems.

Limit reliability functions of some series-parallel and parallel-series systems

In the reliability investigation of large-scale systems, the problem of the complexity of their reliability functions appears. This problem may be approximately solved by the assumption that the number of the system components tends to infinity and by finding the limit reliability function of this system. In this paper, some three-element closed classes of limit reliability functions for series-parallel and parallel-series systems with identical components are fixed. The series-parallel system is such that the number of its series components is of greater order than the logarithm of the number of its parallel components. The parallel-series system is such that the number of its parallel components is of greater order than the logarithm of the number of its series components. Moreover, an example of the considered systems and its limit reliability function is given. The results may be useful in the reliability evaluation of large systems with identical components and may be the origin in the reliability investigation of large systems with different components.

Limit reliability functions of some non-homogeneous series-parallel and parallel-series systems

Abstract In this paper, some three-element classes of limit reliability functions for series-parallel and parallel-series systems with non-identical components are considered. The series-parallel system is such that the number of its series components is of less order than the logarithm of the number of its parallel components. The parallel-series system is such that the number of its parallel components is of less order than the logarithm of the number of its series components. Moreover, an example of the considered systems and its limit reliability function are given. The results may be useful in the reliability evaluation of large systems with non-identical components. In particular they can be useful to reliability practitioners dealing with design problems.

The classes of asymptotic reliability functions for series-parallel and parallel-series systems

Abstract In this paper ten-element classes of limit reliability functions for series-parallel and parallel-series systems with identical components are presented. Next, the results are transferred to the systems with non-identical components. These systems are such that at least the number of their series or the number of their parallel components tends to infinity. The classes are more extensive than the known classes of limit distributions of minimax and maximin statistics of independent random variables with the same distributions. The results can be useful in the reliability evaluation of large systems, especially in design problems.

On a class of limit reliability functions of some regular homogeneous series-parallel systems

In the investigation of large-scale systems the problem of the complexity of their reliability functions arises. This problem may be solved approximately by assuming that the number of system components tends to infinity and finding the limit reliability function. In this paper a four-element class of limit reliability functions for a regular homogeneous series-parallel system is presented. The number of series components of the system has the order of the logarithm of the number of its parallel components. The result is obtained under the assumption that the lifetimes of the particular components are independent and identically distributed random variables. The class presented is different from the known class of limit distributions of minimax statistics of independent random variables with the same distribution. Moreover, some examples of the systems considered and their limit reliability functions are given. The results can be useful in the investigation of the reliability of large systems.

On a class of limit reliability functions of some regular homogeneous series-parallel systems

Abstract In the reliability investigation of large-scale systems, the problem of the complexity of their reliability functions arises. This problem may be approximately solved by the assumption that the number of system components tends to infinity and finding the limit reliability function of this system. In this paper, a three-element closed class of limit reliability functions for a regular homogeneous series-parallel system is fixed. The system is such that the number of its series components is of less order than the logarithm of the number of its parallel components. Moreover, some examples of the considered systems and their limit reliability functions are given. The results may be useful in the reliability evaluation of large homogeneous systems and may be the origin in the reliability investigation of large nonhomogeneous systems.

On asymptotic reliability functions of series-parallel and parallel-series systems with identical components

Abstract In the investigations of large scale systems the problem of the complexity of their reliability functions appears. This problem may be approximately solved by assuming that the number of system components tend to infinity and finding the limit reliability function of this system. In this paper 10-element closed classes of limit reliability functions for regular homogeneous series-parallel and parallel-series systems are fixed. These systems are such that at least the number of their series components or the number of their parallel components tends to infinity. The result is obtained under the assumption that lifetimes of the particular components are independent, identically distributed random variables. The fixed classes are more extensive than classes hitherto known of limit distributions of minimax and maximin statistics of independent random variables with the same distributions. The results can be useful in the reliability evaluation of large regular homogeneous systems and may originate reliability investigations of large nonregular nonhomogeneous systems.