Maximum likelihood estimation of dead time value at a generalized semysynchronous flow of events

The generalized semisynchronous flow of events, which intensity is a piecewise constant stochastic process >(t) with two intensities > ₁ and > ₂ (> ₁ > >2), is considered. During the time interval when >(t) = > , the Poisson flow of events has the intensity > , i = 1,2. The transition from the first state of the process >(t) into the second state is possible only at the moment of occurring event, thus, the transition carries out with probability p (0 < p < 1); with probability 1 - p process >(t) remains in the first condition. In this case, the duration of process stay in the first state is a random variable with exponential distribution function F ₁ (т) = 1 - e pX1% . The transition from the second state of process into the first state can be carried out at any moment of time. Thus, the duration of process stay in the second state is distributed according exponential law: F ₂ (т) = 1 - e . By the transition 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 moment t _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). Process >(t) and the types of events (event from Poisson flows and additional events) are fundamentally unobservable and observable are only temporary moments of the events t ₁, t ₂,.... We consider the steady (stationary) mode of operation of the observed flow of events, so transients are neglected on the observation interval (t0, t), where t0 and t are the start and end of observations. It is necessary at the end of observations (at time t) to implement maximum likelihood estimate T of the dead time Т. Let т ₁ = t ₂ - t ₁, т ₂ = t ₃ - t ₂,... , т _/с = t _k+1 - t _k be the sequence of the measured (by observing the flow during the observation interval (0, t)) values of the length of intervals between adjacent flow events . We order quantities т ₁,... , тц ascending: т^ = т < т <...< т®. The objective is to estimate T the duration of the construction of the dead time (assuming that the other flow parameters > , p, а, 5 are known), by solving the optimization problem: L(T |x ,...,x k)) = Apt(x 1)) = П{[1 - f (T)]V X° -T) + 1=1 1=1 +f(T)(a + X2)e + -T)}^ max, 0< T m, Xm _m >0, T f ₍T) = P(a + X2)(X1 -X2 -a5)[X1 +(a + p^ -XQe (a+ K1' T ] (a + pX ₁ )(X ₁ -X ₂ - a)F(T) The value of T, at which L(T | X ,...,X )) reaches its global maximum, is the estimate T of the duration of the dead time. It is proved that the global maximum occurs at T = X _min .

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