Maximum likelihood estimation of unextendable dead time period duration in the modulated semi-synchronous generalized flow of events

In this paper we consider the modulated semi-synchronous generalized flow of events, which is one of the mathematical models for incoming streams of events in computer communication networks and which is related to the class of doubly stochastic Poisson processes (DSPPs). The flow intensity process is a piecewise constant stationary random process Я(/) with two states 1, 2 (first, second correspondingly). In the first state Я(/) = Я₁ and in the second state Я(/) = Я₂ (Я₁ >Я₂ > 0 ). During the time interval of a random duration when the process Я(/) is in state Я_г- (Я(/) = Я_г- ), a Poisson flow of events with intensity Я_г-, i = 1,2, arrives. The transition of the process ) from the first state to the second state is possible at any moment of a Poisson event occurrence in state 1 of the process h(t) , herewith the process %(t) can change its state to the second one with probability p (0 < p < 1) or continue to stay in state 1 with complementary probability 1 - p . The transition of the process Я(^) from state 1 to state 2 is also possible at any moment that does not coincide with the moment of a Poisson event occurrence, herewith the duration of the process Я(^) staying in the first state is distributed according to the exponential law with parameter P : F(x) = 1 - e, x > 0 . Then the duration of the process Я(^) staying in the first state is distributed according to the exponential law with distribution function F₁(t) = 1 - e' +, т> 0 . The transition of the process X(t) from the second state to the first state at the moment of a Poisson event occurrence in state 2 is impossible and can be done only at a random time moment. In this case the duration of the process X(t) staying in state 2 is distributed according to the exponential law with parameter а : F₂ (т) = 1 - е, т> 0 . At the moment when the state changes from the second to the first one, an additional event is assumed to be initiated with probability 5 (0 < 5 < 1) . The registration of the flow events is considered in conditions of a constant (unextendable) dead time. The dead time period of a constant duration T begins after every registered at the moment t_k , k > 1, event. During this period no other events are observed. When the dead time period is over, the first coming event causes the next interval of a dead time of duration T and so on. This paper contains analytical results that are devoted to finding the maximum likelihood estimate T of the dead time period duration on monitoring the time moments of the events occurrence. We assume that the flow parameters Xj >X₂ > 0, 0 < p < 1, Р> 0, а> 0, 0 < 5 < 1 are known and the duration of the dead time period T is not known. According to the maximum-likelihood technique the likelihood function L(T | т,...,x) is maximized and the following task of optimization is solved: L(T | т,..., x) = J\p_T (j ^ max, 0 < T <т_шт, т_шт > 0, j=1 where p_T (тj-) is the one-dimensional probability density function of the interval length between two consecutive flow events. Finally, we obtain that the likelihood function L (T |т,..., тk > ) reaches its global maximum at the point T = т _min, where x_min = min x_k (k = 1 , n ); %k = tk+1 tk , Tk > 0 , k - 1, n - the sequence of the values of the intervals lengths between consecutive flow events measured during the interval of observation (0, t ], i.e. the solution of optimization problem is the estimate of the dead time period dura T = шШ .

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