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

Generalized semisynchronous stream of events which intensity is a piecewise constant stochastic process X(t) with two values X and X (X > 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 the process X(t) into the second is possible only at the moment of event occurrence, thus, the transition is carried out with probability p (0 < p < 1); with probability 1 - p process X(t) remains in the first condition. In this case the duration of process stay X(t) in the first state is a random variable with exponential distribution function F (t) = 1 - e . Transition from the second state of process into the first state can be carried out at any moment of time. Thus, duration of process stay X(t) in the second state is distributed according exponential law: (т) = 1 - e . By transition X(t) from the second state into the first one an additional event in the first state is initiated with probability 5 (0 < 5 < 1). The flow is considered in the condition of constant dead time. The dead time period of the fixed duration Т begins after every registered event at time t . During this period no other events are observed. When the dead time period is over, the first coming event causes the next Т -interval of dead time and so on (unprolonging dead time). 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: [0, 0 < x < T, T (X) = jy(T)V (X-T) + [1 - Y(T)] (а + X )e + 2 x-T), x > T, у (T) = 1 - я (T) , (T) = - [ - я (0 | T)] e- + X)T, X - X - a * 0. X1 - X 2 - a p (x , x ) = 0; 0 T, x > T, The conditions for the recurrence of generalized semisynchronous flow of events with unprolonging dead time are given.

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