An ROBDD Algorithm for the Reliability of Double-Threshold Systems

Alsayegh

2019

JAMCS

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.

Section: Discussionmentioning

confidence: 87%

Reliability Analysis of a Commodity-Supply Multi-State System Using the Map Method

Alsayegh

2019

JAMCS

“…The technique is illustrated by a probability map and an acyclic signal flow graph that resembles a reduced ordered binary decision diagram (ROBDD) [7,8,[24][25][26][27]. Similar ROBDD-like SFGs were earlier employed for handling related systems [28][29][30][31][32]. The recursive algorithm has two different implementations that stand for two distinct strategies.…”

Section: Discussionmentioning

confidence: 99%

“…We reiterate that threshold Boolean functions and coherent Boolean functions are two proper distinct subsets of unate Boolean functions [6]. Threshold Boolean functions might be coherent or non-coherent [6,30], and coherent Boolean functions might be threshold or nonthreshold [7,30]. The reader is referred to [30] for a taxonomy of CTS-related systems.…”

Section: Discussionmentioning

confidence: 99%

Reliability Evaluation of Rooftop Solar Photovoltaics Using Coherent Threshold Systems

Uswarman

2021

JERR

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.

“…The reduction rules [4] Therefore, the AR algorithm can be seen to be a subclass of ROBDD algorithms that is tailored specifically for the following equivalent purposes: Contrary to claims made in the literature (see, e.g., [20]), the AR algorithm is not only recursive but it has several iterative versions as well, that first appeared in [15] and reproduced in expository detail in [3,17]. There are ROBDD-like extensions of the AR algorithm, which handle (single or scalar) threshold systems [16,21], double-threshold systems [22], vector-threshold systems [23], and k-to-l-out-of-n systems [24].…”

Section: Comparison Of the Ar Algorithm With The Robdd Strategymentioning

confidence: 99%

Computation of k-out-of-n System Reliability via Reduced Ordered Binary Decision Diagrams

Alturki

2017

BJMCS

A prominent reliability model is that of the partially-redundant (k-out-of-n) system. We use algebraic as well as signal-flow-graph methods to explore and expose the AR algorithm for computing k-out-of-n reliability. We demonstrate that the AR algorithm is, in fact, both a recursive and an iterative implementation of the strategy of Reduced Ordered Binary Decision Diagrams (ROBDDs). The underlying ROBDD for the AR recursive algorithm is represented by a compact Signal Flow Graph (SFG) that is used to deduce AR iterative algorithms of quadratic temporal complexity and linear spatial complexity. Extensions of the AR algorithm for (single or scalar) threshold, double-threshold, vectorthreshold, and k-to-lout of -n systems have similar ROBDD interpretations.