The joint probability density of duration of the intervals in a generalized asynchronous flow of events with unprolonging dead time

Generalized asynchronous flow of events which intensity is a piecewise constant stochastic process X(t) with two states and (X > is considered. During the time interval when X(t) = X , Poisson flow of events takes place with the intensity X , i = 1,2. Transition from the first state of process X(t) into the second one (from the second state into the first one) is carried out at any moment of time. The sojourn time in the i-th state is exponentially distributed with parameter a,-, i = 1,2. The process of transition X(t) from the first state into the second one initiates with probability p (0< p <1) an extra event in the second state. Also the process of transition X(t) from the second state into the first one initiates with probability вероятностью q (0< q <1) extra event in the first state. We solve the problem of finding the explicit form of probability density p (т) of the interval between two events and the joint probability density p (т , т ) of the length of two adjacent intervals with unprolonging dead time.

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