Ali M. Rushdi

Utilization of symmetric switching functions in the computation of k-out-of-n system reliability

Abstract Properties of symmetric switching functions are utilized in the development of an algorithm that computes the reliability or unreliability of a k-out-of-n system involving non-identical components. Round-off errors introduced in the computations are analyzed. The algorithm is compared to existing algorithms, and is shown to be of minimal computational complexity. As a bonus, all the intermediate results of the algorithm are meaningful numbers that can facilitate the economic assessment of redundancy.

Symbolic Reliability Analysis with the Aid of Variable-Entered Karnaugh Maps

A method for finding the symbolic reliability of a moderately complex system is presented. The method uses Bayesian decomposition to reduce the system into simpler series-parallel subsystems. The successes or failures of these subsystems are found by inspection and then recast into disjoint sum-of-product forms with the aid of the Karnaugh map. Subsequently, an almost minimal disjoint expression for the system success or failure is obtained with the aid of a variable-entered Karnaugh map (VEKM). This VEKM method is illustrated by applying it to some examples recently solved in the literature. The main advantage of the method is the pictorial insight it provides to the reliability analyst.

Indexes of a telecommunication network

Two recently proposed performance indexes for telecommunication networks are shown to be the s-t and overall versions of the same measure, namely, the mean normalized network capacity. The network capacity is a pseudo-switching-function of the branch successes, and hence it means value is readily obtainable from its sum-of-products expression. Three manual techniques of conventional reliability analysis are adapted for the computation of the proposed performance indexes: a map procedure, reduction rules, and a generalized cutset procedure. Four tutorial examples illustrate these techniques and demonstrate their computational advantages over the state-enumeration technique. >

How to Hand-Check a Symbolic Reliability Expression

Checking symbolic reliability expressions is very useful for detecting faults in hand derivations and for debugging computer programs. This checking can be achieved in a systematic way, though it may be a formidable task. Three exhaustive tests are given. These tests apply to unreliability and reliability expressions for noncoherent as well as coherent systems, and to cases when both nodes and branches are unreliable, or when the system has a flow constraint. Further properties of reliability expressions derived through various methods are discussed. All the tests and other pertinent results are proved and illustrated by examples.

Uncertainty Analysis of Fault-Tree Outputs

The multiaffine nature of the top-event probability as a function of component unavailability is recognized. This leads, under the gssumption of statistically independent failures, to the derivation of an exact formula relating the variance of the system unavailability to the variances of the component unavailability. Concise expressions for other central moments of the system unavailability are obtained. The variance formula partitions contributions due to the input variables and their interactions, and can be used to rank these variables by an importance that is related to well known measures of statistical importance. The variance formula is extended to handle linearly correlated input variables through the inclusion of certain joint central moment terms.

UNCERTAINTY PROPAGATION IN FAULT-TREE ANALYSES USING AN EXACT METHOD OF MOMENTS

Abstract An exposition is made of the exact method of moments which is based on the exact and finite Taylor expansion of the top-event probability in terms of the basic-event probabilities in a fault tree. This method allows calculation of the various moments with a readily quantifiable accuracy that can be arbitrarily improved. Typical approximations made in other versions of the method of moments are also discussed and their effects are empirically evaluated. The numerical results of the exact method of moments are in good agreement with those of the Monte Carlo method, and are superior to those of other existing methods.

On Reliability Evaluation by Network Decomposition

A simple algorithm for evaluating the symbolic terminal-pair reliability of a complex system is presented. The system graph is decomposed into two subgraphs through a minimal cut. The system success is expressed in terms of certain successes of these subgraphs, and then changed into an equivalent disjoint expression which is directly converted on a one-to-one basis into a reliability expression. It yields unusually simple reliability expressions. The algorithm can be computerized but has not been done. Three examples illustrate the algorithm and compare it with other algorithms.

Using variable-entered karnaugh maps to solve boolean equations

A new method for obtaining a compact subsumptive general solution of a system of Boolean equations is presented. The method relies on the use of the variable-entered Karnaugh map (VEKM) to achieve successive elimination through successive map folding. It is superior in efficiency and simplicity to methods employing Marquand diagrams or Conventional Karnaugh maps; it requires the construction of significantly smaller maps and produces such maps in a minimization-ready form. Moreover, the method is applicable to general Boolean equations and is not restricted to the two-valued case. Details of the method are carefully explained and then illustrated via a classical example.

Efficient computation of k-to-l-out-of-n system reliability

A space-economizer quadratic-time algorithm for computing the k-to-l-out-of-n: G system reliability is presented. The algorithm represents a dramatic reduction in computational complexity when compared to existing methods. The algorithm is suited for symbolic as well as numerical computations.

Improved variable-entered Karnaugh map procedures

An improved variable-entered Karnaugh map (VEKM) procedure for obtaining the minimal disjunctive form of a switching function and a dual procedure for obtaining its minimal conjunctive form are presented. These procedures apply to any general switching function of moderate complexity that can be incompletely specified with respect to all its variables. Existing VEKM procedures are compared to the new ones and are shown to be special cases of them. Various examples are given to illustrate the details of the new procedures and to demonstrate their efficiency and power.