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 replacem ent. A model assumes that the state of an industrial aggregate is desc ribed by a continuous positive variable X, which is a measure of its c omplexity. The failure probability exponentially decreases as the comp lexity 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 t he probability density of the lifetime of an aggregate has a long tail psi(t) similar to t(-1(1+H)) as t --> infinity where the fractal expo nent H is the ratio between the average complexity of an aggregate whi ch leaves the system and the average complexity of a new aggregate, Th e asymptotic behavior of all moments of the number N of replacement ev ents 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)] simil ar to t(H) and [Delta N-2(t)] similar to t(2H) as t --> as which outli nes the intermittent character of the fluctuations. A second model giv es a discrete description of industrial replacement. The aggregates ar e assumed to be made up of variable numbers of basic units. Each basic unit has a probability a to be associated in an aggregate and a proba bility beta of being in an active state. An aggregate can work if at l east a basic unit is in an active state. The mechanism of replacement is the same as in the first model, the number of basic units from an a ggregate playing the role of a complexity measure. The probability den sity of the lifetime has a long tail modulated by a periodic function in In t: psi(t) similar to t(-1(1+H))Xi(ln t), where H = ln alpha/ln(1 - beta) and Xi(ln t) is a periodic function of In t with a period - I n(1 - beta). A third model is related to the transmission errors in co mmunication networks, A network is made up of a large number of commun ication channels; each channel has a probability alpha to be open and a probability beta of transmitting a message. The number of open chann els is a random variable which is kept constant as far as the transmis sion is possible; if a failure occurs, then the number of open channel s is changed in a random way, We show that this model is approximately isomorphic with the second one. The probability density of the time b etween two successive errors has also an inverse power tail modulated by a periodic function in In t. The general implications of these mode ls for the physics of fractal time are analysed.