The joint probabilitydensity of duration of the intervals in a generalized semi-synchronous flow

Generalized semi-synchronous flow of events which intensity is piecewise constant stochasticprocess λ(t) with two values λ1 and λ2 (λ1 > λ2) is considered. During the time interval when λ(t) =λi , Poisson flow of events takes place with the intensity λi , i = 1,2. Transition from the first stateof the process λ(t) into the second is possible only at the moment of event occurrence, thus, thetransition is carried out with probability p (0 < p ≤ 1); with probability 1 - p process λ(t) remainsin the first state. In this case the duration of process stay λ(t) in the first state is a random variablewith exponential distribution function 11F( ) 1 ep− λ ττ = − . Transition from the second state of processinto the first state can be carried out at any moment of time. Thus, duration of process stay λ(t)in the second state is distributed according exponential law: 2F( ) 1 e−αττ = − . By transition λ(t)from the second state into the first one an additional event in the first state is initiated with probabilityδ (0 ≤ δ ≤ 1).We find the explicit form of the probability density p(τ) of duration of the interval betweentwo successive events in generalized semi-synchronous flow and the explicit form of p(τ1,τ2) -the joint probability density of the length of two adjacent intervals. The conditions forthe recurrence of generalized semi-synchronous flow of events are given.

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