Reliability of various 2-out-of-4:G redundant systems with minimal repair

In safety critical applications, it is becoming quite common to improve reliability through the use of quadruplexed, redundant subsystems organized with 2-out-of-4:G (2oo4:G) logic. As the subsystems involved will generally be complex, comprising many individual components, they are likely to fail at a time-dependent rate λ(t). The maintenance schedule here is assumed to require failed subsystems to be repaired and workable within a given time τ of failure; and repairs are assumed to be minimal so that all functioning subsystems possess the same rate λ(t). Within this context, we consider the dependence of the system reliability measures on the allowed repair time τ by providing solutions to two integro-differential-delay equations (IDDEs) which bound the exact solution above and below; these bounds may be tightened by iterating the IDDEs to higher order. Results for the stationary system are used to investigate the order required to provide sufficiently tight bounds for the general case. In addition, we consider examples in support of the conjecture of Solov'yev and Zaytsev (Engineering Cybernetics, 1975) that if λ(t)τ is small, then the (asymptotic) instantaneous hazard function of the system hs(t), with a time-varying λ(t), will approach (λ(t)τ → 0) the limit hso(λ(t),τ), where hso(λ, τ) is the asymptotical hazard rate for the same system with constant failure rates. This method then allows for a simple analysis of the case of arbitrary time-varying λ(t) in terms of the much simpler stationary case.