Asymptotic Evaluation of Flow Reliability for Coherent Repairable Networks under Periodic Demand Variation

The purpose of the present paper is (i) to extend the stochastic models in the previous studies [1, 2 and 3] so as to analyze flow-reliability R(t) of an arbitrary coherent repairable network under periodically changing demand (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 (X(t)) of the network is defined as a monotonic function of state-vector X(t) = (X1(t), X 2(t), ... , Xn(t)) with Xi(t) = 1 in case of unit i being operative, and X;(t) = ° otherwise, at time t. Flow-reliability R(t) is introduced as the probability that flow (X(s)) of the network is greater than or equal to demand (s) for all s E [0, tJ, i.e., R(t) = P{(X(s)) ~ (s) for all s E [0, t]}; and the demand function (t) iH 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( -At) + 8(t), where the parameter A 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 (t). It will also be proved that the error 8(t) = R(t) exp( -At) of the approximation converges to ° 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, Ro(t), is effective to characterize exp( -At) and 8(t) in the present model.