Optimal state estimation of semi-synchronous events flow of the second order under conditions of unextendable dead time

We solve the optimal estimation problem for the states of a semi-synchronous events flow of the second order. We consider the stationary operation mode of the flow in conditions of an unextendable dead time, i.e. after each registered event at the moment t_k , there is a dead time period of fixed duration T, during which other events of the considered flow are inaccessible to observation. At the end of the dead time period, the first event that occurred again creates a period of dead time, etc. The accompanying random process of the flow is a piecewise constant process X(t) with two states: if X(t) = Xj , then there is the first process state, if X(t) = X₂ , then there is the second one. The process is unobservable in principle, and we can only observe moments t_t , t₂, ..., when events occur in the flow. We have to estimate the state of the process X(t) (flow) at moment t when the observations have stopped by observations t₁, t₂, ... of the events flow over the time interval (t₀,t), where 1₀ denotes the beginning of observations. The optimal estimation of states is performed using the maximum method of a posteriori probability. To make the decision regarding the state of the process X(t) at the moment t, we have to determine posterior probabilities w(X ₁11) = w(X, | t₁,...,t_m,t) = P(X(t) = Xj 11₁,...,t_m,t), i = 1, 2 , that at the moment t the value of the process X(t) = X_i (m is the number of events per time t), wherein w(Xj 11) + w(X₂11) = 1. The optimal estimation is as follows: if w(X 11) > w(X 11) , i, J = 1, 2 , i ф J , then the estimate of the process state is X(t) = X_;, otherwise X(t) = Xj , i, J = 1, 2 . In the paper we find an explicit form for posterior probabilities on the intervals of the flow observability and unobservability. Based on these formulas, the algorithms have been obtained for calculating the posterior probability w(X₁11) ( w(X₂11) = 1 - w(X₁11)) and for deciding on the state of the process X(t) at an arbitrary moment t. The algorithms were implemented by C# programming language in Visual Studio 2013. Statistical experiments were conducted on the imitational model to establish the frequency of making erroneous decisions about the state of the process X(t), the numerical results of which are given in the paper and illustrate an acceptable estimate of the total probability of making the erroneous decision.

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