The purpose of the present paper is (i) to extend the stochastic models in the previous studies so as to analyze flow-reliability R(t) of an arbitrary coherent repairable network under periodically changing demand ø(t), and (ii) to prove that the flow-reliability can also be evaluated asymptotically as an exponential function under mild assumptions. In the model, flow ℝlsquo;(X(t)) of the network is defined as a monotonic function of state-vector X(t)=(X1(t),X2(t),...,Xn(t)) with Xi(t)=1 in case of unit i being operative, and Xi(t)=0 otherwise, at time t. Flow-reliability R(t) is introduced as the probability that flow ℝlsquo;(X(s)) of the network is greater than or equal to demand ø(s) for all s∈[0,t], i.e., R(t)=P∈ℝlsquo;(X(s))∈ø(s) for all x∈[0,t]∈; and the demand function ø(t) is given arbitrarily as a nonnegative periodic function with a certain period T∈0. It will finally be proved that the flow-reliability R(t) of the network is asymptotically exponential, i.e., R(t)=exp(¸-ℝt)+¦](t), where the parameter ℝ is evaluated by expected life-times, repair-time distributions of the units, the structure/logic to define the flow of the network, and the demand function ø(t). It will also be proved that the error ¦](t)=R(t)-exp(¸-ℝt) of the approximation converges to 0 as the expected lifetimes of the units increase indefinitely under certain assumptions. In the course of the proof an extended renewal equation is first derived, from which a variational equation is yielded, and its unique solution, R0(t), is effective to characterize exp(¸-ℝt) and ¦](t) in the present model.
