The Mathematical Model and Numerical Solution of Instantaneous Availabilities for the Two-Unit Series Repairable System and Parallel Repairable System with Three States

Abstract

This paper is aimed to obtain the instantaneous availabilities (IAs) for the two-unit series system and parallel system with three states. By the compound S i m p s o n formula and the compound trapezoidal formula, we get the numerical solution of IA for the two-unit series system with three states based on the renewal process. With four-order R u n g e - K u t t a formula, the numerical solution of IA for the two-unit parallel system with three states is obtained based on the Markov process.

Fluctuation Analysis of Instantaneous Availability for the Parallel Repairable Systems

“…According to the Markov model in [19], the transfer rate matrix M is as (18), as shown at the top of this page. Let P i (t) be the rate that i-th unit are down at time t and P(t) = (P 0 (t), P 1 (t), • • • , P n (t)), so we can get ordinary differential equation…”

“…where P * (s) denotes the Laplace transformation of P(t) and M is the transform rate matrix in (18). So…”

“…For multiunits systems, the authors in [17] have concluded that there exists no fluctuation of IA for the two-unit series system with two states. The mathematical models for the two-unit series repairable system and parallel repairable system with three states have been analyzed in [18] and the numerical solutions of IAs have been obtained.…”