Search

This paper addresses two important useful extensions of binary reliability techniques to multi-state reliability techniques, namely: (a) the problem of complementation or inversion of the function of system success to that of system failure (or equivalently, of deriving the logical minimal cutsets in terms of the logical minimal paths), and (b) the associated problem of hand-checking of a symbolic reliability expression. The paper deals specifically with the reliability of a multi-state delivery network. It presents two complementation procedures, one via the application of multi-state De Morgan’s rules, and the other via the multi-state Boole-Shannon expansion. The paper also illustrates one case in which this complementation is needed, as it outlines a method for checking the reliability of the multi-state system in terms of its logical minimal paths and logical minimal cutsets.

This paper utilizes a modern regular and modular eight-variable Karnaugh map in a systematic investigation of cause-effect relationships modeled by partially-defined Boolean functions (PDBF) (known also as incompletely specified switching functions). First, we present a Karnaugh-map test that can decide whether a certain variable must be included in a set of supporting variables of the function, and, otherwise, might enforce the exclusion of this variable from such a set. This exclusion is attained via certain don’t-care assignments that ensure the equivalence of the Boolean quotient w.r.t. the variable, and that w.r.t. its complement, i.e., the exact matching of the half map representing the internal region of the variable, and the remaining half map representing the external region of the variable, in which case any of these two half maps replaces the original full map as a representation of the function. Such a variable exclusion might be continued w.r.t. other variables till a minimal set of supporting variables is reached. The paper addresses a dominantly-unspecified PDBF to obtain all its minimal sets of supporting variables without resort to integer programming techniques. For each of the minimal sets obtained, standard map methods for extracting prime implicants allow the construction of all irredundant disjunctive forms (IDFs). According to this scheme of first identifying a minimal set of supporting variables, we avoid the task of drawing prime-implicant loops on the initial eight-variable map, and postpone this task till the map is dramatically reduced in size. The procedure outlined herein has important ramifications for the newly-established discipline of Qualitative Comparative Analysis (QCA). These ramifications are not expected to be welcomed by the QCA community, since they clearly indicate that the too-often strong results claimed by QCA adherents need to be checked and scrutinized.

We use a regular and modular eight-variable Karnaugh map to reveal some technical details of Boolean minimization usually employed in solving problems of Qualitative Comparative Analysis (QCA). We utilize as a large running example a prominent eight-variable political-science problem of sparse diversity (involving a partially-defined Boolean function (PDBF), that is dominantly unspecified). We recover the published solution of this problem, showing that it is merely one candidate solution among a set of many equally-likely competitive solutions. We immediately obtain one of these rival solutions, that looks better than the published solution in two aspects, namely: (a) it is based on a smaller minimal set of supporting variables, and (b) it provides a more compact Boolean formula. However, we refrain from labelling our solution as a better one, but instead we stress that it is simply un-comparable with the published solution. The comparison between any two rival solutions should be context-specific and not tool-specific. In fact, the Boolean minimization technique, borrowed from the area of digital design, cannot be used as is in QCA context. A more suitable paradigm for QCA problems is to identify all minimal sets of supporting variables (possibly via integer programming), and then obtain all irredundant disjunctive forms (IDFs) for each of these sets. Such a paradigm stresses inherent ambiguity, and does not seem appealing as the QCA one, which usually provides a decisive answer (irrespective of whether it is justified or not).The problem studied herein is shown to have at least four distinct minimal sets of supporting variables with various cardinalities. Each of the corresponding functions does not have any non-essential prime implicants, and hence each enjoys the desirable feature of having a single IDF that is both a unique minimal sum and the complete sum. Moreover, each of them is unate as it is expressible in terms of un-complemented literals only. Political scientists are invited to investigate the meanings of the (so far) abstract formulas we obtained, and to devise some context-specific tool to assess and compare them.

Qualitative Comparative Analysis (QCA) is an emergent methodology of diverse applications in many disciplines. However, its premises and techniques are continuously subject to discussion, debate, and (even) dispute. We use a regular and modular Karnaugh map to explore a prominent recently-posed eight-variable QCA problem. This problem involves a partially-defined Boolean function (PDBF), that is dominantly unspecified. Without using the algorithmic integer-programming approach, we devise a simple heuristic map procedure to discover minimal sets of supporting variables. The eight-variable problem studied herein is shown to have at least two distinct such sets, with cardinalities of 4 and 3, respectively. For these two sets, the pertinent function is still a partially-defined Boolean function (PDBF), equivalent to 210 = 1024 completely-specified Boolean functions (CSBFs) in the first case, and to four CSBFs only in the second case. We obtained formulas for the four functions of the second case, and a formula for a sample fifth function in the first case. Although only this fifth function is unate, each of the five functions studied does not have any non-essential prime implicant, and hence each of them enjoys the desirable feature of having a single IDF that is both a unique minimal sum and the complete sum. According to our scheme of first identifying a minimal set of supporting variables, we avoided the task of drawing prime-implicant loops on the initial eight-variable map, and postponed this task till the map became dramatically reduced in size. Our map techniques and results are hopefully of significant utility in future QCA applications.

This paper deals with an emergent variant of the classical problem of computing the probability of the union of n events, or equivalently the expectation of the disjunction (ORing) of n indicator variables for these events, i.e., the probability of this disjunction being equal to one. The variant considered herein deals with multi-valued variables, in which the required probability stands for the reliability of a multi-state delivery network (MSDN), whose binary system success is a two-valued function expressed in terms of multi-valued component successes. The paper discusses a simple method for handling the afore-mentioned problem in terms of a standard example MSDN, whose success is known in minimal form as the disjunction of prime implicants or minimal paths of the pertinent network. This method utilizes the multi-state inclusion-exclusion (MS-IE) principle associated with a multi-state generalization of the idempotency property of the ANDing operation. The method discussed is illustrated with a detailed symbolic example of a real-case study, and it produces a more precise version of the same numerical value that was obtained earlier. The example demonstrates the notorious shortcomings and the extreme inefficiency that the MS-IE method suffers, but, on the positive side, it reveals the way to alternative methods, in which such a shortcoming is (partially) mitigated. A prominent and well known example of these methods is the construction of a multi-state probability-ready expression (MS-PRE). Another candidate method would be to apply the MS-IE principle to the union of fewer (factored or composite) paths that is converted (at minimal cost) to PRE form. A third candidate method, employed herein, is a novel method for combining the MS-PRE and MS-IE concepts together. It confines the use of MS-PRE to ‘shellable’ disjointing of ORed terms, and then applies MS-IE to the resulting partially orthogonalized disjunctive form. This new method makes the most of both MS-PRE and MS-IE, and bypasses the troubles caused by either of them. The method is illustrated successfully in terms of the same real-case problem used with the conventional MS-IE.

The reliability of a microgrid power system is an important aspect to analyze so as to ascertain that the system can provide electricity reliably over a specified period of time. This paper analyzes a home-scale model of a microgrid system by using the threshold system model (inadvertently labeled as the weighted k-out-of-n:G system model), which is a system whose success is treated as a threshold switching function. To analyze the reliability of the system, we first proved that its success is a coherent threshold function, and then identified possible (non-unique) values for its weights and threshold. Two methods are employed for this. The first method is called the unity-gap method and the second is called the fair-power method. In the unity-gap method, we utilize certain dominations and symmetries to reduce the number of pertinent inequalities (turned into equations) to be solved. In the fair-power method, the Banzhaf index is calculated to express the weight of each component as its relative power or importance. Finally, a recursive algorithm for computing system reliability is presented. The threshold success function is verified to be shellable, and the non-uniqueness of the set of weights and thresholds is demonstrated to be of no detrimental consequence, as different correct sets of weights and threshold produce equivalent expressions of system reliability.

The synchronous Boolean network (SBN) is a simple and powerful model for describing, analyzing, and simulating cellular biological networks. This paper seeks a complete understanding of the dynamics of such a model by employing a matrix method that relies on relating the network transition matrix to its function matrix via a self-inverse state matrix. A recursive ordering of the underlying basis vector leads to a simple recursive expression of this state matrix. Hence, the transition matrix is computed via multiplication of binary matrices over the simplest finite (Galois) field, namely the binary field GF(2), i.e., conventional matrix multiplication involving modulo-2 addition, or XOR addition. We demonstrate the conceptual simplicity and practical utility of our approach via an illustrative example, in which the transition matrix is readily obtained, and subsequently utilized (via its powers, characteristic equation, minimal equation, 1-eigenvectors, and 0-eigenvectors) to correctly predict both the transient behavior and the cyclic behavior of the network. Our matrix approach for computing the transition matrix is superior to the approach of scalar equations, which demands cumbersome manipulations and might fail to predict the exact network behavior. Our approach produces result that exactly replicate those obtained by methods employing the semi-tensor product (STP) of matrices, but achieves that without sophisticated ambiguity or unwarranted redundancy.

We utilize the electromagnetically-oriented LTI∅ dimensional basis in the matrix solution of dimensional-analysis (DA) problems involving mainly electromagnetic quantities, whether these quantities are lumped or distributed. Representations in the LTI∅ basis (compared with the standard MLTI basis) are more informative and much simpler. Moreover, matrix DA computations employing the LTI∅ basis are more efficient and much less error prone. Extensive discussions of two demonstrative examples expose technical details of a novel DA scheme, and clarify many important facets of modern dimensional analysis.

This paper deals with the reliability of a multi-state delivery network (MSDN) with multiple suppliers, transfer stations and markets (depicted as vertices), connected by branches of multi-state capacities, delivering a certain commodity or service between their end vertices. We utilize a symbolic logic expression of the network success to satisfy the market demand within budget and production capacity limitations even when subject to deterioration. This system success is a two-valued function expressed in terms of multi-valued component successes, and it has been obtained in the literature in minimal form as the disjunction of prime implicants or minimal paths of the pertinent network. The main contribution of this paper is to provide a systematic procedure for converting this minimal expression into a probability-ready expression (PRE). We successfully extrapolate the PRE concept from the two-valued logical domain to the multi-valued logical domain. This concept is of paramount importance since it allows a direct transformation of a random logical expression, on a one-to-one one, to its statistical expectation form, simply by replacing all logic variables by their statistical expectations, and also substituting arithmetic multiplication and addition for their logical counterparts (ANDing and ORing). The statistical expectation of the expression is its probability of being equal to 1, and is simply called network reliability. The proposed method is illustrated with a detailed symbolic example of a real-case study, and it produces a more precise version of the same numerical value that was obtained earlier by an alternative means. This paper is a part of an ongoing activity to develop pedagogical material for various candidate techniques for assessing multi-state reliability.

This paper presents a review of flow network concepts, including definition of some graph-theoretic basics and a discussion of network flow properties. It also provides an overview of some crucial algorithms used to solve the maximum-flow problem such as the Ford and Fulkerson algorithm (FFA), supplemented with alternative solutions, together with the essential terminology for this algorithm. Moreover, this paper explains the max-flow min-cut theorem in detail, analyzes the concepts behind it, and provides some examples and their solutions to demonstrate this theorem. As a bonus, it expounds the reduction and transformation techniques used in a capacitated network. In addition, this paper reviews one of the popular techniques for analyzing capacitated networks, which is the “decomposition technique”. This technique is centered on conditioning a complicated network on the possible states of a keystone element or on the possible combinations of states of many keystone elements. Some applications of capacitated network problems are addressed based on each type of problem being discussed.

This paper is a preliminary step towards the assessment of an alarming widespread belief that victims of the novel coronavirus SARS-CoV-2 include the quality and accuracy of scientific publications about it. Our initial results suggest that this belief cannot be readily ignored, denied, dismissed or refuted, since some genuine supporting evidence can be forwarded for it. This evidence includes an obvious increase in retractions of papers published about the COVID-19 pandemic plus an extra-ordinary phenomenon of inconsistency that we report herein. In fact, we provide a novel method for validating any purported set of the four most prominent indicators of diagnostic testing (Sensitivity, Specificity, Positive Predictive Value, and Negative Predictive Value), by observing that these indicators constitute three rather than four independent quantities. This observation has virtually been unheard of in the open medical literature, and hence researchers have not taken it into consideration. We define two functions, which serve as consistency criteria, since each of them checks consistency for any set of four numerical values (naturally belonging to the interval [0.0,1.0]) claimed to be the four basic diagnostic indicators. Most of the data we came across in various international journals met our criteria for consistency, but in a few cases, there were obvious unexplained blunders. We explored the same consistency problem for some diagnostic data published in 2020 concerning the ongoing COVID-19 pandemic and observed that the afore-mentioned unexplained blunders tended to be on the rise. A systematic extensive statistical assessment of this presumed tendency is warranted.

Aeronautical navigation systems installed at specific locations within Saudi Arabia report the actual observed data of the times to failure (TTFs) for many types of engineering elements made by some typical international highly-acclaimed manufacturers. There is a need to compare these observed data to the estimated data provided by the manufacturers themselves. The paper gathers a lot of data on the actual failure instances of many components produced by a variety of manufacturers and installed at different locations within the extensive area of Saudi Arabia. These data are used to calculate the mean times between failures for these components. The paper points to an inadvertent discrepancy between these data and the corresponding mean times between failures (MTBFs) suggested by some prominent manufacturers. Such suggested MTBFs are typically optimistic and unrealistically high irrespective of the elements, the location, and the manufactures. The work reported herein is a preliminary assessment of this phenomenon that might lead to its theoretical modeling and subsequent understanding.

The trend for use of rooftop solar photovoltaics (PV) is rising due to their promising economic potential as a source of clean renewable energy. In general, a source of renewable solar energy consists of solar PV, an automatic charge controller, a battery pack, and an inverter. The reliability of a rooftop solar PV system is evaluated herein as that of a coherent threshold system (CTS). First, we utilize the unit-gap method and the fair-power method to verify that a given Boolean function is a threshold one and to identify its threshold and component weights. Both methods utilize specific features of the Karnaugh map (K-map). The unit-gap method uses the map to list all necessary inequalities by inspection, and then reduce them significantly by omitting dominated ones. The fair-power method uses the Karnaugh map to compute Banzhaf indices by appropriate map folding followed by XORing of true cells and false cells. We evaluate the CTS reliability via a recursive algorithm based on the Boole-Shannon’s expansion in the switching domain, which is transformed via the real transform to the total probability law in the probability domain.

In the past few decades, the energy shortage and global warming problems became a serious concern for humanity. To solve these problems, many countries have evolved renewable energy sources (RESs) such as solar, wind, hydro, tidal, geothermal, and biomass energy sources. Solar energy is usually harvested via a solar panel that is connected to a boost converter to supply the loads. The converter has a key role in the system, since it controls the voltage at the DC bus. If any accidental fault occurs in the converter, the solar panel cannot supply electricity to the loads. Therefore, reliability evaluation of the converter is usually warranted. In this study, reliability evaluation of boost converters connected to a solar panel is carried out using the Markov technique. This technique is widely employed to evaluate the reliability and availability of a system with fixed failure and repair rates. Using the Markov method, we found that the reliability of the typical specific converter considered is 0.9986 for and that its life expectancy or Mean-Time-To-Failure (MTTF) is .

Prognostics is a term that engineering borrowed from medicine to refer to the discipline concerned with the Remaining Useful Life (RUL) of an engineering device. This paper surveys the RUL prediction techniques and classifies them into four categories of model-based techniques,knowledge-based techniques, experience-based techniques, and data-driven techniques. A comparative review is given for the main features, prominent advantages, potential shortcomings and main subcategories for each of these categories. The survey is supported by an extensive listfor up-to-date references.

Checking a symbolic reliability expression for a flow network is 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 when a reliability system or network has a flow constraint. These tests apply to unreliability and reliability expressions for non-coherent as well as coherent systems, and to cases when both nodes and branches are unreliable. Further properties of reliability expressions derived through various methods are discussed. All the tests and other pertinent results are proved and illustrated by examples.

A notable reliability model is the binary threshold system (also called the weighted-k-out-of-n system), which is a dichotomous system that is successful if and only if the weighted sum of its component successes exceeds or equals a particular threshold. The aim of this paper is to extend the utility of this model to the reliability analysis of a homogeneous binary-imaged multi-state coherent threshold system of (m+1) states, which is a non-repairable system with independent non-identical components. The paper characterizes such a system via switching-algebraic expressions of either system success or system failure at each non-zero level. These expressions are given either (a) as minimal sum-of-products formulas, or (b) as probability–ready expressions, which can be immediately converted, on a one-to-one basis, into probabilities or expected values. The various algebraic characterizations can be supplemented by a multitude of map representations, including a single multi-value Karnaugh map (MVKM) (giving a superfluous representation of the system structure function S), (m+1) maps of binary entries and multi-valued inputs representing the binary instances of S, or m maps, again of binary entries and multi-valued inputs, but now representing the success/failure at every non-zero level of the system. We demonstrate how to reduce these latter maps to conventional Karnaugh maps (CKMs) of much smaller sizes. Various characterizations are inter-related, and also related to pertinent concepts such as shellability of threshold systems, and also to characterizations via minimal upper vectors or via maximal lower vectors.

The synchronous Boolean network model is a simple and powerful tool in describing, analyzing and simulating cellular biological networks. This paper seeks a complete understanding of the dynamics of such a model by utilizing conventional matrix methods, rather than scalar methods, or matrix methods employing the non-conventional semi-tensor products (STP) of matrices. The paper starts by relating the network transition matrix to its function matrix via a self-inverse (involutary) state matrix, which has a simple recursive expression, provided a recursive ordering is employed for the underlying basis vector. Once the network transition matrix is obtained, it can be used to generate a wealth of information including its powers, characteristic equation, minimal equation, 1-eigenvectors, and 0-eigenvectors. These might be used to correctly predict both the transient behavior and (more importantly) the cyclic behavior of the network. In a short-cut partial variant of the proposed approach, the step of computing the transition matrix might be by-passed. The reason for this is that the transition matrix and the function matrix are similar matrices that share the same characteristic equation and hence the function matrix might suffice when only the partial information supplied by the characteristic equation is all that is needed. We demonstrate the conceptual simplicity and practical utility of our approach via two illustrative examples. The first example illustrates the computation of 1-eigenvectors (that can be used to identify loops or attractors), while the second example deals with the evaluation of 0-eigenvectors (that can be used to explore transient chains). Since attractors are the main concern in the underlying model, then analysis of the Boolean network might be confined to the determination of 1-eigenvectors only.

In order to assess the performance indexes of some practical systems having fixed channel capacities, such as telecommunication networks, power transmission systems or commodity pipeline systems, we propose various types of techniques for analyzing a capacitated network. These include Karnaugh maps, capacity-preserving network reduction rules associated with delta-star transformations, and a generalization of the max-flow min-cut theorem. All methods rely on recognizing the network capacity function as a random pseudo-Boolean function of link successes; a fact that allows the expected value of this function to be easily obtainable from its sum-of-products expression. This network capacity has certain advantages for representation of nonbinary discrete random functions, mostly employed in the analysis of flow networks. Five tutorial examples demonstrate the afore-mentioned methods and illustrate their computational advantages over the exhaustive state enumeration method.

Although corona discharge is notorious for its detrimental effects, it is also used in many beneficial practical applications. Despite the existence of a variety of sophisticated theoretical and experimental methods for investigating corona discharge, we explore yet a much simpler method that relies on the use of Dimensional Analysis (DA). The DA method does not demand profound knowledge of the underlying phenomenon or its governing equations, as it only needs the correct identification of the variables influencing the phenomenon, and the specification of their physical dimensions. The classical and well-known Gauss-Jordan elimination method is compared with other matrix-oriented computational approaches in analyzing the pertinent dimensional system. This method relies upon solution-preserving elementary row operations, i.e., operations that one can use on a matrix without spoiling the solution set for an associated matrix equation. A distinct advantage of this method is that it does not pre-suppose a particular value of the matrix rank but tackles the task of determining this rank while proceeding towards its ultimate result. Moreover, this method can also be used to find the inverse of a regular (invertible) matrix or to determine the solution (if any) of a system of linear equations. As a bonus, novel results of numerical investigations of bases, regimes, and dimensionless products are extensively presented in this paper. A remarkable observation made herein is that the set of variables pertinent for a particular phenomenon cannot be arbitrarily partitioned into basis (input) variables and regime (output or isolated) variables. The paper running example deals with expressing a specific variable, viz. ozone generation rate per unit length of wire (r0) through dimensionless products in terms of a set of determining or influencing variables.

We provide a novel method for validating any purported set of the four most prominent indicators of diagnostic testing (Sensitivity, Specificity, Positive Predictive Value, and Negative Predictive Value), by observing that these indicators constitute three rather than four independent quantities. This observation has virtually been unheard of in the open medical literature. We defined two functions, which serve as consistency criteria, since each of them checks consistency for any set of four numerical values claimed to be the four basic diagnostic indicators. Most of the data we came across in various Saudi medical journals met our criteria for consistency, but in a few cases, there were obvious unexplained blunders. We relate our present findings to the more general issue of detection and ramifications of flawed, fabricated or wrong data. We observe that the research field handling the detection of flawed data is still in its infancy, and hope that this field will reach maturity very soon.

This chapter deals with the reliability of a multi-state delivery network (MSDN) with multiple suppliers, transfer stations and markets (depicted as vertices), connected by branches of multi-state capacities, delivering a certain commodity or service between their end vertices. We utilize a symbolic logic expression of the network success to satisfy the market demand within budget and production capacity limitations even when subject to deterioration. This system success is a two-valued function expressed in terms of multi-valued component successes, and it has been obtained in the literature in minimal form as the disjunction of prime implicants or minimal paths of the pertinent network. The main contribution of this chapter is to provide a systematic procedure for converting this minimal expression into a probability-ready expression (PRE). We successfully extrapolate the PRE concept from the two-valued logical domain to the multi-valued logical domain. This concept is of paramount importance since it allows a direct transformation of a random logical expression, on a one-to-one one, to its statistical expectation form, simply by replacing all logic variables by their statistical expectations, and also substituting arithmetic multiplication and addition for their logical counterparts (ANDing and ORing). The running example used for demonstrating PRE derivation in this chapter is admittedly long and tedious. The statistical expectation of the expression is its probability of being equal to 1, and is simply called network reliability. The proposed method is illustrated with a detailed symbolic example of a real-case study, and it produces a more precise version of the same numerical value that was obtained earlier by an alternative means. This chapter is a part of an ongoing activity to develop pedagogical material for various candidate techniques for assessing multi-state reliability.

This paper considers the problem of solving a system of Boolean equations over a finite (atomic) Boolean algebra other than the two-valued one, referred to herein as a ‘big’ Boolean algebra...

The JK flip flop is a flexible type of bistable elements that has extensive uses in digital electronics and control circuits. It is usually described by its characteristic equation or next-state table (used for analysis purposes) and its excitation table (used for synthesis purposes). This paper explores a variety of novel characterizations of JK flip flops. First, equational and implicational descriptions are presented, and methods of logic deduction are utilized to produce complete statements of all propositions that are true for a general JK flip flop. Next, methods of Boolean-equation solving are employed to find all possible ways to express the excitations in terms of the present state and next state. The concept of Boolean quotient is used to impose certain requirements so as to find particularly useful expressions of the excitations. Relations of JK flip flops to other types of flip flops are also explored. This paper is expected to provide an immediate pedagogical benefit, and to help facilitate the analysis and synthesis of sequential digital circuits.

A multi-state k-out-of-n: G system is a multi-state system whose multi-valued success is greater than or equal to a certain value j (lying between 1 (the lowest non-zero output level) and M (the highest output level)) whenever at least km components are in state m or above for all m such that 1 ≤ m ≤ j. This paper is devoted to the analysis of a commodity-supply system that serves as a standard gold example of a non-repairable multi-state k-out-of-n: G system with independent non-identical components. We express each instance of the multi-state system output as an explicit function of the multi-valued inputs of the system. The ultimate outcome of our analysis is a Multi-Valued Karnaugh Map (MVKM), which serves as a natural, unique, and complete representation of the multi-state system. To construct this MVKM, we use “binary” entities to relate each of the instances of the output to the multi-valued inputs. These binary entities are represented via an eight-variable Conventional Karnaugh Map (CKM) that is adapted to a map representing four variables that are four-valued each. Despite the relatively large size of the maps used, they are still very convenient, thanks to their regular structure. No attempt was made to draw loops on the maps or to seek minimal formulas. The maps just served as handy tools for combinatorial representation and for collectively implementing the operations of ANDing, ORing, and complementation. The MVKM obtained serves as a means for symbolic analysis yielding results that agree numerically with those obtained earlier. The map is a useful tool for visualizing many system properties, and is a valuable resource for computing a plethora of Importance Measures for the components of the system.

Multi-State systems are systems whose outputs are multi-valued (due to multiple levels of capacity or performance) and (possibly) whose inputs are also multi-valued (due to multiple performance levels or multiple modes of failure). These systems are a generalization of binary or dichotomous systems that have binary or two-valued outputs and inputs. The multi-state reliability model generalizes and adapts many of the concepts and techniques of the binary reliability model, and naturally ends up with sophisticated concepts and techniques of its own. This paper explores the possibility of simply analyzing a multi-state system by reformulating or encoding its inputs in terms of binary inputs and evaluating each of its multiple output levels as an individual binary output of these alternative inputs. This means that we dispense with multiple-valued logic in the analysis of a multi-state system, since this system is now analyzed solely via switching algebra (two-valued Boolean algebra). The wealth of tools and techniques of switching algebra are now used (without any modification or adaptation) in the analysis of the multi-state system (at the cost of an expanded input domain). The paper makes its point though the analysis of a standard commodity-supply system, whose multi-valued inputs are expressed in terms of physically-meaningfully binary inputs. The analysis is made possible through the use of advanced techniques for deriving probability–ready expressions together with the employment of large-size Karnaugh maps and utilization of multiplication tables, symmetric switching functions, and Boolean quotients. Though the system studied involves twelve binary input variables, its manual analysis is completed successfully herein, yielding results that exactly agree with those obtained earlier via automated methods, and are possibly less prone to the notorious effects of round-off errors.

Given the contemporary importance of communications systems and infrastructures in general, networks need to be designed and operated in such a way that failures can be reduced, alleviated, or mitigated. Network components might for instance fail due to equipment failing, malicious attacks, planned or, unplanned activities, and so on. This paper presents a three-state system model (Good, Poor, and Fail). This model is demonstrated via two examples covering the proposed multistate model in detail. The first example studies a firewall system with two components in two operational modes, namely active-active and active-standby. The second example covers a simple private cloud setup with three layers, namely firewall, server, and service layer. The first example is solved by two methods (a) Tabular form of exhaustive enumeration; (b) Multi-state Decision Diagrams (MDD). The second Example is solved by (a) The exhaustive enumeration method, and (b) Dividing the system into 3 subsystems and integrating their results. The results of this study indicate that the firewall block modes of operation have a major impact on the studied systems. Firewalls in active-active mode significantly improve the probability of system’s Up state. The findings of this study should be of great help to designers to better design services to increase service availability.

This paper deals with the reliability of a multi-state delivery network with multiple suppliers, transfer stations and markets (depicted as vertices), connected by branches of multi-state capacities, delivering a certain commodity or service between their end vertices. We utilize a symbolic logic expression of the network success to satisfy the market demand within budget and production capacity limitations even when subject to deterioration. This system success is a two-valued function expressed in terms of multi-valued component successes, and it has been obtained in the literature in minimal form as the disjunction of particular prime implicants or minimal paths of the pertinent network. This paper provides a systematic procedure for converting this minimal expression into a probability-ready expression (PRE). We successfully extrapolate the PRE concept from the two-valued logical domain to the multi-valued logical domain. This concept is important since it allows a direct transformation of a random logical expression, on a one-to-one one, to its statistical expectation form, simply by replacing all logic variables by their statistical expectations, and also substituting arithmetic multiplication and addition for their logical counterparts (ANDing and ORing). The statistical expectation of the expression is its probability of being equal to 1, and is simply called reliability of the multi-state delivery network (MSDN).

An -bit comparator is a celebrated combinational circuit that compares two -bit inputs and and produces three orthonormal outputs: G (indicating that is strictly greater than ), E (indicating that and are equal or equivalent), and L (indicating that is strictly less than ). The symbols ‘G’, ‘E’, and ‘L’ are deliberately chosen to convey the notions of ‘Greater than,’ ‘Equal to,’ and ‘Less than,’ respectively. This paper analyzes an -bit comparator in the general case of arbitrary and visualizes the analysis for on a regular and modular version of the 8-variable Karnaugh-map. The cases 3, 2 and 1 appear as special cases on 6-variable, 4-variable, and 2-variable submaps of the original map. The analysis is a tutorial exposition of many important concepts in switching theory including those of implicants, prime implicants, essential prime implicants, irredundant disjunctive forms, minimal sums, the complete sum and disjoint sums of products (or probability-ready expressions).

A multi-state k-out-of-n: G system is a multi-state system whose multi-valued success is greater than or equal to a certain value j (lying between 1 (the lowest non-zero output level) and M (the highest output level)) whenever at least km components are in state m or above for all m such that 1 ? m ? j. This paper is devoted to the analysis of a commodity-supply system that serves as a standard gold example of a non-repairable multi-state k-out-of-n: G system with independent non-identical components. We express each instance of the multi-state system output as an explicit function of the multi-valued inputs of the system. The ultimate outcome of our analysis is a Multi-Valued Karnaugh Map (MVKM), which serves as a natural, unique, and complete representation of the multi-state system. To construct this MVKM, we use “binary” entities to relate each of the instances of the output to the multi-valued inputs. These binary entities are represented via an eight-variable Conventional Karnaugh Map (CKM) that is adapted to a map representing four variables that are four-valued each. Despite the relatively large size of the maps used, they are still very convenient, thanks to their regular structure. No attempt was made to draw loops on the maps or to seek minimal formulas. The maps just served as handy tools for combinatorial representation and for collectively implementing the operations of ANDing, ORing, and complementation. The MVKM obtained serves as a means for symbolic analysis yielding results that agree numerically with those obtained earlier. The map is a useful tool for visualizing many system properties, and is a valuable resource for computing a plethora of Importance Measures for the components of the system.

This chapter reviews various types of techniques utilized for analyzing capacitated networks, with an eye on assessing the performance indexes of practical systems having fixed channel capacities, such as telecommunication networks, power transmission systems or commodity pipeline systems. The techniques reviewed include Karnaugh maps, capacity-preserving network reduction rules associated with delta-star transformations, and a generalization of the max-flow min-cut theorem. All methods rely on recognizing the network capacity function as a random pseudo-Boolean function of link successes; a fact that allows the expected value of this function to be easily obtainable from its sum-of-products expression. This network capacity has certain advantages for representation of nonbinary discrete random functions, mostly employed in the analysis of flow networks. Five tutorial examples demonstrate the afore-mentioned methods and illustrate their computational advantages over the exhaustive state enumeration method. The chapter is supplemented with an appendix, which introduces the concept of a “probability-ready expression” for a Boolean-based coherent pseudo-Boolean function.

There is a definite need for representations of, and tools for, conditional probability that enhance understanding, simplify calculations, foster insight, and facilitate reasoning. Such representations and tools are desirable and useful in a wide variety of scientific disciplines, but their utility in medical contexts and applications are stressed herein, so as to address a clinical rather than a mathematical audience. We employ a plethora of time-tested pedagogical representations or tools of conditional probability including: (a) Visualization on Venn diagrams or Karnaugh maps, (b) Reformulation as natural frequencies, (c) Entity interrelations via Signal Flow Graphs, (d) Specification of certain problem formats such as the format of Trinomial Graphs, and (e) Use of a pair of complementary Karnaugh-map-like or eikosogram-like diagrams. Most of the new representations or tools have well known histories of pedagogical advantages, but are still to be tested in the specific realm of conditional probability. Further assessment of the novel representations or tools proposed herein is needed. Each of these is to be taught to a group of students, and a control group of students is to be instructed via the conventional representation. Detailed statistical analysis of the outcomes is warranted. Similar investigations were performed earlier, but they were on a limited scale. Therefore, the need arises for an explorative study that exhausts all proposed representations or tools.