Optimal estimate of the states of a generalized asynchronous event flow with an arbitrary number of states

In this paper, we consider a generalized asynchronous flow of events (hereinafter called a flow). The flow accompanying process (intensity process) is a piecewise constant random process x(t) with n states: x(t) takes values from a discrete set of values {X₁,...,X_n}, X₁ >X₂ > ... > X_n > 0 . Let's say that the i-th state of the process X(t) holds if X(t) = X_i, i = 1,n , n = 2,3,.... If the i-th state takes place, then during the time interval when X(t) = X_t there is a Poisson flow of events with parameter (intensity) X_i, i = 1, n . The duration of the process X(t) (flow) in the i-th state is distributed according to the exponential law with the parameter a_t: f (t) = 1 - e~^a'^T, t > 0 , i = 1, n . The process X(t) is a fundamentally unobservable process; we can only observe time moments of occurrence of flow events t₁, t₂,..., t_k,... We consider the stationary mode of the flow functioning. In these premises, X(t) is an accompanying stationary piecewise constant hidden transitive Markov process with an arbitrary number of states n (n = 2,3,...). A generalized asynchronous flow (a generalized MMPP-flow) is a generalization of the asynchronous flow. The generalization is as follows: at the moment of transition of the process X(t) from the i-th state to the j-th, an additional event is triggered with probability pj ; the transition occurs at an arbitrary time moment, not related to the moment of occurrence of the Poisson flow event with parameter X_t, while initiating an additional event occurs in the j-th state, i, j = 1, n , j ф i; transition and initiation of an additional event occur instantly. It is required, on the basis of the moments of occurrence of events observed from moment t₀ to moment t, to estimate the state of the process X(t) at the moment t . We denote the estimate of the state of the process X(t) at the time moment t as X(t). We found an explicit form for a posteriori probabilities w(X_t 11) = w(X_i 11₁,..., t_m,t) = P(X(t) = X_t | tj,..., t_m,t), i = 1,n , that at the time moment t the value of the process X(t) = X_t (m is the number of events that occurred at the time moments t₁,..., t_m at the observa- n tion interval (t₀, t)), here ^w(X_; 11) = 1. The decision on the state of the process X(t) obtained according to criterion of a posteriori i=1 probability maximum. The algorithm for calculating a posteriori probabilities w(Xj 11), j = 1, n , at any time moment t (t > t₀ = 0 ) was formulated according to analytical formulas. In parallel, as we calculate a posteriori probabilities w(Xj 11), j = 1, n , we can make a decision on the state of the process x(t) at the time moment t : XX(t) = X_;, if w(X,-11) = max w(X • 11) , j = 1, n .

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