Industrial replacement, communication networks and fractal time statistics

Three models for the fractal time statistics of renewal processes are suggested. The first two models are related to the industrial replacement. A model assumes that the state of an industrial aggregate is described by a continuous positive variable X, which is a measure of its complexity. The failure probability exponentially decreases as the complexity of the aggregate increases. A renewal process is constructed by assuming that after the occurrence of a breakdown event the defective aggregate is replaced by a new aggregate whose complexity is a random variable selected from an exponential probability law. We show that the probability density of the lifetime of an aggregate has a long tail ψ(t) ∼ t−(1+H) as t → ∞ where the fractal exponent H is the ratio between the average complexity of an aggregate which leaves the system and the average complexity of a new aggregate. The asymptotic behavior of all moments of the number N of replacement events occurring in a large time interval may be evaluated analytically. For 1 > H > 0 the mean and the dispersion of N behave as 〈N(t)〉 ∼ tH and 〈ΔN2(t)〉 ∼ t2H as t → ∞ which outlines the intermittent character of the fluctuations.