Cold standby redundancy optimization for non-repairable Series-parallel systems: Erlang time to failure distribution

In modeling a cold standby redundancy allocation problem (RAP) with imperfect switching mechanism, deriving a closed form version of a system reliability is too difficult. A convenient lower-bound on system reliability is proposed and this approximation is widely used as a part of objective function for a system reliability maximization problem in the literature. Considering this lower-bound does not necessarily lead to an optimal solution. In this study by assuming that working time of switching mechanism is exponentially distributed, exact value of system reliability is derived analytically through applying Markov process and solving a relevant set of differential-difference equations. The Runge-Kutta numerical scheme is also employed to verify the accuracy of the results. It is assumed that component time-to-failure follows an Erlang distribution which is appropriate for most engineering design problems by giving the possibility of modeling different increasing hazard functions. A new mathematical model is presented and its performance is evaluated through solving a well-known example in the literature. Results demonstrate that a higher level of system reliability is achievable through implementing the proposed model.