ASYMPTOTIC EVALUATION OF FLOW RELIABILITY FOR COHERENT REPAIRABLE NETWORKS UNDER PERIODIC DEMAND VARIATION
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概要
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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 φ(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 φ(X(t)) of the network is defined as a monotonic function of state-vector X(t) = (X_1(t), X_2(t), …, X_n(t)) with X_i(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 φ(X(s)) of the network is greater than or equal to demand φ(s) for all s ∈[0, t], i.e., R(t) = P{φ(X(s)) ≥φ(s) for all s ∈[0, t]}; and the demand function φ(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(-Λt) + Θ(t), where the parameter Λ is evaluated by expected llfe-times, repair-time distributions of the units, the structure / logic to define the flow of the network, and the demand function φ(i). It will also be proved that the error Θ(t) = R(t) - exp(-Λt) 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, R_0(t), is effective to characterize exp(-Λt) and Θ(t) in the present model.
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