Stochastics and Statistics
Reliability function of a class of time-dependent systems with standby redundancy

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Abstract

By applying shortest path analysis in stochastic networks, we introduce a new approach to obtain the reliability function of time-dependent systems with standby redundancy. We assume that not all elements of the system are set to function from the beginning. Upon the failure of each element of the active path in the reliability graph, the system switches to the next path. Then, the corresponding elements are activated and consequently the connection between the input and the output is established. It is also assumed each element exhibits a constant hazard rate and its lifetime is a random variable with exponential distribution. To evaluate the system reliability, we construct a directed stochastic network called E-network, in which each path corresponds with a minimal cut of the reliability graph. We also prove that the system failure function is equal to the density function of the shortest path of E-network. The shortest path distribution of this new constructed network is determined analytically using continuous-time Markov processes.

Introduction

Many researchers in the second half of the last century have investigated system reliability evaluation and have developed a variety of methods in this regard (see Barlow and Proschan [2] or Grosh [9] for details). Yet, the existing analytical methods are usually constructed on the basis of some assumptions, which are quite restrictive and are not capable of analyzing all real systems. Thus, to calculate the systems reliability with special structures, it is still necessary to design new methods.

In this paper, a new approach is introduced to determine the reliability function of time-dependent systems with standby redundancy. The existing methods for time-dependent systems are developed with the assumption that all elements are set to function concurrently, from the beginning. However, this assumption is not true for many real cases. In fact, in practice not all elements are functioning at time zero, but whenever an element fails then some other one is activated. To illustrate this assumption is not true for all real cases, an example is presented in Section 4.

To calculate the reliability function of time-dependent systems when all elements are set to function at time zero, it is customary to apply the joint density or distribution functions techniques. The other approach is to apply state-transition models such as Markov chains.

The major obstacle in solving these models is the complexity, which arises from the large size of the first-order differential equations. For example, a system with n elements, modelled as a Markov process, may require a solution of as many as 2n first-order differential equations (see Shooman [20] for the details).

In this paper, we consider a time-dependent reliability system with standby redundancy. At the beginning, only the main elements work. No standby element is set to function unless one active element fails. In terms of graphs, at time zero only the elements of the first path are functioning. In other word, the reliability graph works because its input and output are connected through this path. As soon as one element of this path fails, the system is switched to the second path and consequently all elements of the second path are set to function. This process continues until all connections between the input and output are interrupted and as a result, the system fails.

We assume the lifetime of each element follows an exponential distribution function. Throughout its development, the theory of reliability has been based heavily on the exponential failure law, primarily because of its mathematical tractability. It is the appropriate model for used-good-as-new components, like fuses and many other electronic parts, because of the memoryless property of the exponential distribution (see Grosh [9] for more details). The objective is to determine the reliability function of this system.

The shortest path analysis of stochastic networks is applied in order to analyze this system. As a matter of fact, first we construct a directed network, from reliability graph of the system. This network is a stochastic one and we call it E-network (or equivalent network). In this network, each path corresponds with a minimal cut of the reliability graph. Then, we obtain the distribution function of the shortest path from the source to the sink node of this E-network using the method developed by Kulkarni [11]. The mean time to failure of the systems with the standby nature is always greater than that of the ordinary systems, in which all elements are set to function concurrently at time zero. Consequently, this system clearly works better, compared with the ordinary one. Therefore, what makes our research distinguished from the previous ones are the following:

  • a.

    We relax the assumption that all elements start working concurrently from the beginning.

  • b.

    The method is a new one, on the basis of the shortest path of stochastic E-networks.


For computing the probability that between two given nodes in the reliability graph of the system, there exists at least one operation path, Fishman [7] proposed a Monte Carlo sampling plan, which uses lower and upper bounds to increase its accuracy and efficiency. Manzi et al. [15] provided a detailed, clear exposition of the Fishman method, and its extension for computing the global network reliability (probability that the network is connected).

Exact evaluation of system reliability is extremely difficult and sometimes impossible. Once one obtains the expression for the structure function, the system reliability computations become straightforward. Attempts have been made to compute the exact system reliability of complex systems. For example, the algorithm in Aven [1] is based on minimal cut sets. Chaudhuri [4] overcame the problem of calculating system reliability in complex systems through a new representation of the structure function of a coherent system, and demonstrated that the well-known systems considered in the state-of-art follow this new representation. English et al. [6] presented a discretizing procedure for reliability prediction of complex systems. System reliability depends not only on the reliabilities of components in the system but also on their interactions or the dependencies among them. In recent years, the studies on the dependent failure theories have been widely developed. The main elements in research are the common-cause failures in redundant systems. Lin et al. [13] described the parallel redundant systems. Lesanovsky [12] proposed a multiple-state Markov model of the system with the dependent components, in which the system is a homogeneous continuous-time Markov process with discrete states. Humphreys and Jenkins [10] summarized developments of techniques for dealing with the dependent failures.

Fault-trees are also common modelling tools in system reliability analysis. Although fault-tree algorithms are useful to compute a good approximation of reliability for a very large system (see for example Rauzy [18] and Dutuit and Rauzy [5]) but the conventional fault-trees are not at all suited to modelling systems in which there are strong dependencies between components, like standby redundant systems. In order to be able to model component dependencies, one has to recur to dynamic models. The most popular are Markov processes, which we use in this paper. Stochastic Petri nets can also be used in complex systems dependability assessment (see Peterson [19]). The problem with this technique is that it is impossible to infer any interesting property of the Markov graph, that could be used to simplify its processing, from the model input by the user. Manian et al. [14] extended static fault-trees by adding some dynamic features. Bouissou and Bon [3] introduced a modelling formalism that enables the analyst to combine concepts inherited from fault-trees and Markov models in a new way. In this paper, instead of adding new kind of gates, they assign a new semantics to the traditional graphical representation of fault-trees, augmented only by a new kind of links.

In the area of determining the shortest path of stochastic networks, Martin [16] introduced a method to obtain its distribution function as well as its expected value. Frank [8] computed the probability that the duration of the shortest path in a stochastic network be smaller than a specific value, when the arc lengths are continuous random variables. Mirchandani [17] developed another method with the advantage that it is not necessary to solve multiple integrals. However, this method works only if the arc lengths are discrete random variables. Kulkarni [11] presented an algorithm for obtaining the distribution function of shortest path in directed stochastic networks, when the arc lengths are independent random variables with exponential distributions. This method is constructed in the framework of continuous-time Markov processes. Sigal et al. [21] used the uniformly directed cuts in their analysis of shortest paths.

The advantage of the proposed algorithm from the point of management implication roots in its assumptions. This new analytical approach is developed for obtaining the reliability function of time-dependent systems by considering the standby nature of the structure and it is not required that all elements start working concurrently at time zero. Since we relax this restrictive assumption, the proposed approach can be applied for many real world reliability systems, which cannot be solved by the existing methods.

The remainder of this paper is organized in the following way. In Section 2, we describe the reliability graphs and introduce E-networks, which is the basis of the proposed model. In Section 3, a method for obtaining the distribution function of shortest path in stochastic networks is presented. In Section 4, the method is illustrated through solving a numerical example, and finally we draw the conclusion of the paper in Section 5.

Section snippets

Reliability graphs

A very efficient method to compute the reliability of a system is to express it as a graph. Reliability graphs consist of a set of arcs. Each arc represents an element of the system, while the nodes of the graph tie the arcs together and form the structure. Corresponding with the ith arc of the reliability graph, i=1,2,…,n, there is an exponential random variable Ti, with parameter λi, which represents the lifetime of this element. These random variables are independent, due to the fact that

Distribution function of shortest path in stochastic networks

In this section, we present an analytical method for obtaining the distribution function of shortest path of E-network, or in fact the distribution function of the shortest path from the source to the sink node of a directed stochastic network, in which arc lengths are exponentially distributed. To do that, we need to apply a shortest path algorithm for stochastic networks.

Although there are many simple algorithms for solving the shortest path problem in deterministic networks, there are not so

Numerical example

To operate the accounting activities of a firm, either one computer or one calculator is needed. The calculator needs one battery to do the required operations. However, there are two batteries available in the system to function as standby. At the beginning, the system may start with the computer. If it fails, then, the calculator with one battery does the necessary operations. In that case, if the calculator fails so does the system. However, if the battery fails, the calculator works with

Conclusion

In this paper, we developed a new approach for obtaining the reliability function of time-dependent systems with standby redundancy. The elements lifetimes are assumed to be independent random variables and exponentially distributed.

In this type of system, all elements are not set to function concurrently, from the beginning. It is assumed that only the elements of the first path of the reliability graph of system work at time zero. Upon the failure of each element in one active path, the

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