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Internet Service Providers (ISPs) are expected to continuously provide services without interruptions. However, planning for this activity requires a number of tasks such as forecasting clients' demand that needs to be considered in fulfilling their objectives particularly in face of calamity and traffic load variations. We present a Software-Defined Networking (SDN) load distribution technique for a typical ISP. We have implemented an optimization method employing linear programming to achieve this goal. A multi-objective model has been developed for this purpose. The multiple objectives are maximization of link utilization, maximization of link availability and minimization of average round-trip time for the top visited websites by users. Furthermore, an automated load distribution system has been developed for carrying out these optimization related activities. In addition, an example is presented to illustrate traffic load distribution under different operational modes (normal, critical, and catastrophic). The findings of this study should be of great help to the service provider to better plan for load distribution between international links, and help in approving down times and maintenance windows requested by cable providers.

Natural and designed ecological corridors are key elements for the survival of a species, as they allow the species to avoid local extinction by migrating to more suitable habitat patches. This paper studies various reliability metrics for the process of migration in a metapopulation landscape network from a critical habitat patch to destination habitat patches via perfect stepping stones and imperfect (deletable) corridors. The work presented herein generalizes earlier work on the application of reliability theory in ecology by allowing corridors to be heterogeneous (of non-identical unreliabilities). The paper is a tutorial exposition of modern reliability techniques, which formulate a problem in the Boolean domain, manipulate formulas to achieve disjointness of logically added subexpressions and retain statistical independence of logically multiplied ones, and finally reach a probability-ready expression that is directly transformed back to the probability domain. Several metrics are covered including system unreliability, life expectancy (MTTF), and component importance measures. An interesting finding is that the life expectancy of a classical landscape network is more than double that of a single corridor. Extensions to quantification of uncertainty in the above metrics and to evaluation of more sophisticated metrics of landscape connectivity are also pointed out.

In a recent seminal paper, Rudeanu derived necessary and sufficient conditions for the most general form of the subsumptive general solution of a Boolean equation. The present paper investigates four prominent classes of the aforementioned most general form. These include the conventional (eliminants) class, the don't-care class, the Rudeanu class, and the complete-sum (CS) class. We show that each of these classes satisfies the Rudeanu conditions, and hence fits into the frame of the most general form of the subsumptive solution. The four classes are of variable but comparable computational complexities. The conventional (eliminants) class is the least likely to produce a compact solution, while the don't-care class is the most likely to produce such a solution. Three examples are presented to demonstrate solutions produced by the four classes and compare them form the point of view of compactness of solutions, which determines the ease of generating the tree of particular solutions. A remaining open question that has yet to be settled is whether these four classes are exhaustive, or in other words, whether there are general solution classes other than them. We demonstrate that the don't-care class is never inferior (and occasionally superior) to the other classes. In fact, this class seeks global minimality and has a better control on the pertinent variables and algebra generators. Both the CS class and the Rudeanu class have an advantage over the eliminants method. The CS class utilizes a canonical representation that explicitly shows all information in the simplest form. The Rudeanu class enhances the eliminants class by incorporating the consensus (Boolean resolution) concept in a way similar to that of the CS approach.

Weighted voting systems play a crucial role in the investigation and modeling of manyengineering structures and political and socio-economic phenomena. There is an urgentneed to describe these systems in a simplified powerful mathematical way that can begeneralized to systems of any size. An elegant description of voting systems is presentedin terms of threshold Boolean functions. This description benefits considerably fromthe wealth of information about these functions, and of the potpourri of algebraic andmap techniques for handling them. The paper demonstrates that the prime implicantsof the system threshold function are its Minimal Winning Coalitions (MWC). Thepaper discusses the Boolean derivative (Boolean difference) of the system thresholdfunction with respect to each of its member components. The prime implicants of thisBoolean difference can be used to deduce the winning coalitions (WC) in which thepertinent member cannot be dispensed with. Each of the minterms of this Booleandifference is a winning coalition in which this member plays a pivotal role. However,the coalition ceases to be winning if the member defects from it. Hence, the numberof these minterms is identified as the Banzhaf index of voting power. The conceptsintroduced are illustrated with detailed demonstrative examples that also exhibit someof the known paradoxes of voting- system theory. Finally, the paper stresses the utilityof threshold Boolean functions in the understanding, study, analysis, and design ofweighted voting systems irrespective of size.

The covering problem, a classical problem in switching theory, involves the selection of some prime implicants irredundantly to cover the asserted part of an incompletely specified switching function. This paper advocates the use of the variable-entered Karnaugh map (VEKM) to solve this problem. To set the stage for achieving this purpose, a novel tutorial exposition of the VEKM-related variable-entered cover matrix is presented. Subsequently, the paper offers two VEKM solutions of the covering problem. The first solution relies on the new concept of VEKM loops. By contrast to a loop on a conventional Karnaugh map which is characterized by a single product, a VEKM loop is identified by two products, namely loop coverage (which depends only on map variables) and loop depth (which depends only on entered variables). In the second solution, the VEKM is used directly as a cover map in such a way that it is not littered with unnecessary details or too many overlapping loops. All solution procedures are demonstrated via illustrative examples that help reveal their visual merits as well as their applicability to relatively large problems.

The probability mass function ( PMF ) and the cumulative distribution function ( CDF ) of the generalized binomial distribution ( E(k,n,p) and B(k,n,p) ) are shown to be governed by binary recursive relations similar to those of the binomial coefficient and the k-out-of-n system reliability/unreliability. It is possible to compute E(k,n,p) and B(k,n,p) via computer implementations of recursive functions that are directly based on the aforementioned recursive relations. However, such implementations are highly demanding in both time and space. Alternatively, the recursive relations of E(k,n,p) and B(k,n,p) are given nice interpretations in terms of very regular signal flow graphs, based on which efficient iterative algorithms for computing the set of values E(k,n,p), 0 ≤ k ≤ n, and B(k,n,p), 0 ≤ k ≤ (n−1), for any specific n ≥ 0, are developed. In the worst case, the temporal and spatial complexities of these algorithms are quadratic and linear, respectively, in the number of trials n pertaining to the underlying distribution.

The first-order sensitivity of system reliability is useful in evaluating several criticality measures, uncertainty measures, and the instantaneous failure rate of the system. Three new algorithms are described herein for the computation of the sensitivity of k-out-of-n system reliability. Generally, the numerical results of these algorithms check very well versus one another as well as versus those of known special cases. The computational complexities of these algorithms vary from almost double to slightly less than that of the best known algorithm for computing the k-out-of-n system reliability. Some observations are made on the important rankings of system components for different values of k, n and component reliabilities.

A switching expression is readily convertible to a reliability expression if (a) all ORed terms are disjoint, and (b) all ANDed sums are statistically independent. The usual approach of system reliability analysis makes a primary use of (a) and a secondary use of (b). An alternative approach reverses the roles of (a) and (b). Symbolic reliability expressions for the source-to-terminal reliability of a generalized Indra network (GIN) with nonidentical components are derived by the two approaches. For this particular case, the second approach leads to a shorter, more elegant derivation and simpler novel results. Typical plots of the GIN reliability functions are presented and their properties are discussed.

The region of useful redundancy for a k-out-of-n system of identical components is the interval in component reliability for which the reliability of the system is better than that of a single component. A table for the lower boundary of that region is presented for n ≤ 56. The table is compiled with a high precision and a minimal round-off error by using a highly efficient algorithm for computing the k-out-of-n system reliability. Certain symmetries and limiting values are noted, thereby allowing a cutting of the number of entries actually listed in the table by more than one half. If the component lifetime is assumed to be exponentially distributed, then the table entries can be used to compute the point that divides the time axis into intervals of a short mission and a long mission for k-out-of-n redundancy.

An exposition is made of the method of moments for the uncertainty analysis of fault trees with statistically correlated failure data. Exact formulas for the mean and variance of the top-event probability, as well as accurate formulas for its third and fourth central moments, are derived. An approximate analysis of partial correlation is also given. Two test problems are studied, the first of which demonstrates a good agreement between the present method and the Monte Carlo method for a case of total correlation, and also reveals that the effect of correlation can be quite significant. The second test problem studies the effect of partial correlation for different parameters of the basic events.

A threshold system is defined as a system whose success/failure is a threshold swithching function in the successes/failures of its components. Examples of such a system abound in applications involving decision mechanisms or involving supply-type components with fixed ratings for their capacity, flow, throughput, and the like. In general, a threshold system can be coherent or noncoherent. If a coherent threshold system is made to have identical component weights, then it reduces to the well known k-out-of-n or voting system. The paper lists some of the fundamental properties of a threshold system, and presents a recursive algorithm for computing the exact system reliability. Illustrative examples are given, and extension to the multi-threshold case is also discussed.

An explicit set of star-delta transformations is given for probabilistic flow networks whenever the initial star branches are bidirectional with the same branch flow capacity in both directions. These transformations, together with their (already known) dual transformations, can be used to simplify the given network before applying the various algorithms available for the solution of the network flow problem. The superiority of the present star-delta transformations to other existing transformations is pointed out, and their usefulness is illustrated via the example of a small power generation and transmission network.

An exposition of a fault-diagnosis procedure for general analog circuits is presented. This procedure can handle circuits of a wide variety of elements including the non-reciprocal, active, and two-port types. It can diagnose multiple soft faults as well as most catastrophic faults. Its characteristics approximate those of practical in situ testing, and it requires moderate linear computations that are mostly off-line.

An important problem in reliability analysis and in testing of digital circuits is the problem of deriving the partial derivative (Boolean difference) of a swithching (Boolean) function. This problem is tackled herein by a divide and conquer technique that relies on the use of a recently developed minimization procedure via the variable-entered Karnaugh map (VEKM). This technique is simple, systematic, and can handle switching functions of moderate complexity. Two detailed examples are given to illustrate the derivation of non conventional Boolean differences as well as single and multiple conventional ones.

Switching-algebraic techniques are utilized in a simple and instructive derivation of the linear-cost recursive relation of consecutive-k-out-of-n:F systems.

The concept of pseudo-switching (PS) functions has definite advantages in representing nonbinary discrete random functions, usually encountered in the study of flow networks. That concept is utilized in the development of star-delta and delta-star transformations that preserve the source-to-terminal (s−t) capacity function in a flow network. The usefulness of these transformations in reducing complex networks to equivalent series-parallel ones is illustrated by examples. The resulting series-parallel networks are easily solvable for the networks s−t capacity function, which is a compact expression of the probability mass function (p.m.f.) of the maximum s−t flow.

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.

In a strict consecutive-k-out-of-n:F system, strings of failed components of length less than k do not occur, and hence the system components are not totally statistically independent. A correct formulation for the reliability analysis of such a system is reported and then illustrated via several examples. The unreliability of a strict system is shown to be not necessarily smaller than or equal to that of an ordinary system. Finally, clear distinctions are drawn between the present model of a strict system, and a model of a noncoherent system that matches earlier mathematical treatments.

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.

An important problem in reliability is the problem of inverting the minimal sum representation of a switching (Boolean) function to obtain that of its complement. This problem is tackled by a divide-and-conquer inversion procedure that relies on the use of the variable-entered Karnaugh map (VEKM). An improved VEKM minimization procedure is presented, and then utilized as a part of the inversion procedure. Both procedures are illustrated by simple examples that cover the noncoherent as well as the coherent case.

This paper presents an entered-map version of the map factoring method for the logic design of NAND(NOR) circuits. The new version has a better variable-handling capability as well as pedagogical advantages. It is demonstrated by several examples, and is reported to have a favourable students' response.

The spectral coefficients of a switching function are computed in terms of the real transform of the function, or equivalently, in terms of a disjoint sum-of-products representation of the function. Such a representation can be cast into an almost minimal form through some existing algorithms, and its complexity can be significantly less than that of a minterm expansion, which it includes as a special case. The real transform is also utilized as a mechanism for the inverse transform from the spectral domain to the Boolean domain.

In a strict consecutive-k-out-of-n:F system, strings of failed components of length less than k do not occur. An earlier look [1] at such a system that viewed its components as statistically dependent has led to the conclusion that the unreliability of a strict system can be greater than that of an ordinary one. An alternative view adopted herein requires that the strict system unreliability be expressed as a conditional probability. This view confirms that the failure probability for a strict system is not necessarily smaller than that of an ordinary system.

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.

The real transform of an n-variable switching function is a multiaffine function of the same “truth table” as the switching function. Hence, it has a finite multivariable Taylor's expansion which is completely specified by 2 n coefficients. The value of this transform when all its arguments equal 0.5 is the syndrome of the switching function. Therefore, the problem of computing the syndrome is equivalent to that of computing the numerical reliability of a system whose success is the switching function under consideration. Several efficient methods of system reliability analysis are adapted to handle syndrome computation. Several examples are given to illustrate these methods and some remarks are made about the syndrome properties of certain special types of switching functions.

A simple procedure for evaluating the symbolic reliability of a complex system in presented. The switching (Boolean) functions of system success and system failure are expressed as the intersections of complements to the prime implicants of these functions. Shannon's expansion about suitable keystone variables is repeatedly applied to either or both functions until subexpressions with statistically independent terms are obtained. The procedure applies to a variety of reliability problems. In addition, it is relatively fast and yields simple symbolic expressions.

Switching-algebraic techniques are utilized in a simple and instructive derivation of the O(kn) recursive relation for a circular consecutive-k-out-of-n:F system with non-identical components. In addition, the boundary conditions associated with this recursive relation are developed. Consequently, an exact closed-form expression for system unreliability is obtained for the special case {n≤2k}. Finally, simple formulas are given for the Birnbaum importance of system components.

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.