Optimal states estimation of the modulated semisyncronous integrated flow of events in the condition of constant dead time

In this paper, the estimation problem of states of the modulated semisyncronous integrated flow of events is solved. Such flows of events are the mathematical models that may be used in Integrated Services Digital Networks (ISDNs). The flow intensity is a piecewise constant stationary random process X(t) with two states X , X (Xj > X ) . In the time interval by X(t) = X , (during time interval) there is a Poisson flow of events with intensity X , i = 1,2 . Transition of the process X(t) from the state to another state can be realized at any time. The staying time of process X(t) in the first state is distributed by the exponential law with a parameter p. At the same time, after the event occurrence, the process X(t) can change the state to X with probability р or stay in the first state X with probability 1 -р. While the process X(t) is in the second state, the staying time is distributed by the exponential law with a parameter a. At that, in the moment of changing state from the second state to the first one, an additional event is initiated in the state X with probability 5. The flow is considered in the condition of constant dead time. The dead time period of the fixed duration T begins after every registered event at time tk . During this period no other events are observed. When the dead time period is over, the first coming event causes the next T -interval of dead time and so on (constant dead time). The process X(t) and possible events (events of the Poisson flows with intensities X , i = 1,2 and additional events) are unobservable in principle. We observe only the moments t , 1 ,... of occurring events. So, using this information, we estimate the state of the process X(t) at the stop time of observations. Note that the process X(t) is only considered in stationary conditions. To make the decision on the state of process X(t) at moment t, we found the explicit formula for a posteriory probability w(X 11) = w(X 11 ,...,t ,t), i = 1,2 , where X(t) = X (m is the number of observable events in the time period t), and w(X 11)+ w(X 11)= 1. The obtained formulas w(X 11) provide with the calculation algorithm for a posteriory probability w( 11), and the decision algorithm on a state of process X(t) at a moment t (the algorithm of the optimal estimator for flow states). By the calculation of the probability w( 11) at a moment t, the decision on a state of process X(t) is given according to the criterion of a posteriory probability maximum: if w( 11) > w(X 11) , then X(t) = , else X(t) = X . The algorithm of the optimal evaluation for flow states is a basis for simulations. The first stage assumes the flow simulation in condition of constant dead time and obtaining the real path of process X(t) and the moments t , 1 ,... for observable events. The second stage is the computation of probabilities w( 11) and estimation X(t) . For the same values of parameters X , X , p, p, a, 5, the probability estimation of the erroneous decision P0 increases when the dead time period T increases as well. This results are expected, since the increase of the dead time period always generates the loss of the useful information on flow events.

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