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. A second model gives a discrete description of industrial replacement. The aggregates are assumed to be made up of variable numbers of basic units. Each basic unit has a probability α to be associated in an aggregate and a probability β of being in an active state. An aggregate can work if at least a basic unit is in active state. The mechanism of replacement is the same as in the first model, the number of basic units from an aggregate playing the role of a complexity measure. The probability density of the lifetime has a long tail modulated by a periodic function in ln t: ψ(t) t−(1+H)Ξ(ln t), where H = lnα/ln(1 − β) and Ξ(lnt) is a periodic function of ln t with a period − ln(1 − β). A third model is related to the transmission errors in communication networks. A network is made up of a large number of communication channels; each channel has a probability α to be open and a probability β of transmitting a message. The number of open channels is a random variable which is kept constant as far as the transmission is possible; if a failure occurs, then the number of open channels is changed in a random way. We show that this model is approximately isomorphic with the second one. The probability density of the time between two succesive errors has also an inverse power tail modulated by a periodic function in ln t. The general implications of these models for the physics of fractal time are analysed.