Switching-Algebraic Analysis of Multi-State System Reliability

Abstract

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.

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

“…The problem handled herein was solved via various techniques by Tian et al [34], Mo. et al [40], Rushdi [41], and Rushdi & Al-Amoudi [42,43]. In all cases, the results were tested by the following input matrix, in which the sum of entries in each row is 1, since such entries are the probabilities of mutually exclusive and exhaustive events.…”

“…A binary k-out-of-n: G system is uniquely defined as a dichotomous system that is successful if and only if at least k out of its n components are successful , By contrast, a multi-state k-out-of-n: G system does not possess a unique definition [24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43]. The definition adopted herein is that this system is a multi-state system (MSS) 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 k m components are in state m or above for all m such that 1 ≤ m ≤ j [34,[40][41][42][43].…”

“…In this paper, we a study a standard multi-state system, which was proposed and studied by Tian et al [34], and further studied by Fadhel et al [44], Mo et al [40], Rushdi [41], Rushdi & Al-Amoudi [42,43]. The system (shown in Fig.…”

“…We have recently reported several solutions of the aforementioned problem, and our present paper offers yet another solution of this problem. In our earlier solutions, we employed purely-algebraic methods of multivalued logic, in which we handled multi-valued variables either directly [41] or through some binary encoding [42,43], with various map versions used occasionally for verification. In this paper, however, we deliberately avoid the mathematically-demanding algebraic manipulations in [41][42][43] by employing the Karnaugh map [45][46][47][48][49][50] as the sole vehicle for our manipulations.…”

“…In our earlier solutions, we employed purely-algebraic methods of multivalued logic, in which we handled multi-valued variables either directly [41] or through some binary encoding [42,43], with various map versions used occasionally for verification. In this paper, however, we deliberately avoid the mathematically-demanding algebraic manipulations in [41][42][43] by employing the Karnaugh map [45][46][47][48][49][50] as the sole vehicle for our manipulations. There is a long history of utilization of the Karnaugh map as a probability map (or reliability map) in the binary case [51][52][53][54][55][56][57][58][59].…”