Industrial replacement, communication networks and fractal time statistics

Marcel Ovidiu Vlad and Michael C. Mackey

Physica A: Statistical Mechanics and its Applications, 1996, vol. 229, issue 3, 295-311

Abstract: 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.

Date: 1996
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Persistent link: https://EconPapers.repec.org/RePEc:eee:phsmap:v:229:y:1996:i:3:p:295-311

DOI: 10.1016/0378-4371(95)00402-5

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