Reliability Evaluation of Multi-State Flow Networks Via Map Methods

Abstract

This paper examines two simple (albeit useful) methods used to evaluate the reliability of two-terminal multistate flow networks. These two methods involve two Karnaugh map versions, namely the Variable-Entered Karnaugh Map (VEKM) and the Multi-Valued Karnaugh Map (MVKM). These two versions are crucial in providing not only the visual insight necessary to write better future software but also adequate means of verifying such software. We assess these two versions of map methods versus the exhaustive search method, which guarantees conceptual clarity at the expense of lack of computational efficiency. Our target is the evaluation of the probability mass function (pmf) in a wide array of cases, in which we consider flow from a source node to a sink node in a capacitated network with a multistate capacity model for the links. Each network link has a varying capacity, which is assumed to exist in a mutually exclusive sense. The reliability of the system is wholly dependent on its ability to successfully transmit at least a certain required system flow from the source (transmitter) to the sink (receiver) station. The max-flow min-cut theorem is critical in obtaining all successful states. To demonstrate the proposed methods applicability, two demonstrative examples are given with ample details.

A Review of Flow-Capacitated Networks: Algorithms, Techniques and Applications

“…So, the providers need to constantly monitor the quality of their transmissions. Rushdi and Alsalami [36] examined two simples, albeit useful, methods used to evaluate the reliability of two-terminal multistate flow networks in communication systems. Their target is the evaluation of the probability mass function (pmf) in a wide array of cases, in which they consider flow in a capacitated network from a source node to a sink node with a multistate capacity model for the links.…”

“…Thulasiraman & Shen [35] Entails design of OFDMA based hybrid hierarchical wireless networks to solve interference aware resource allocation problems. Rushdi & Alsalami [36] Examines two simple (albeit useful) methods used to evaluate the reliability of two-terminal multistate flow networks in communication systems.…”

Checking Correctness of a Symbolic Reliability Expression for a Capacitated Network

“…A Boolean (Switching) function ( ): A mapping {0, 1} → {0, 1} , i.e., ( ) is any one particular assignment of the two functional values (0 or 1) for all possible 2 values of [18][19][20]. These values of are called the states of the system described by .…”

“…A mapping {0, 1} → where is the field of real numbers, i.e. ( ) is an assignment of a real number for each of the possible 2 values of [4,21,12,20,[22][23][24][25][26][27][28].…”

“…The minimal sum-of-products (s-o-p) expression for success (or failure) is unique since each of the products appearing in it represents an essential or core of the function [34,35]. For a noncoherent system, the minimal sum (minimal disjunctive (s-o-p) form) of the success or failure function is the union of certain of the function that may or may not be essential, so that the minimal sum is not usually unique [20,36].…”