Optimal estimate of the states of a generalized asynchronous event flow with an arbitrary number of states under conditions of unextendable dead time

In this paper, we consider a generalized asynchronous flow of events, the accompanying process (intensity) of which is a piecewise constant random process λ(t) with n states: λ(t) takes values from a discrete set of values {λ₁,...,λ_n} , λ₁>λ₂>...>λ_n ≥0. Let's say that that the i th state of the process λ(t) holds if λ(t) = λi, i = 1,n , n = 2,3,.... If the ith state takes place, then during the time interval when λ(t) = λ/ there is a Poisson flow of events with parameter (intensity) λ_i∙, i = 1, n . The duration of the process λ(t) (flow) in the i th state is distributed according to the exponential law with parameter α_i∙, i = 1, n . We consider the stationary mode of the flow functioning, therefore we can neglect transition processes on the observing semi-interval [t₀, t) , where t₀ - the beginning of observation, t - the ending of observation. In these premises, λ(t) is an accompanying stationary piecewise constant hidden transitive Markov process with an arbitrary number of states n ( n = 2,3,... ). At the moment of transition of the process λ(t) from the i th state to the j th, an additional event is triggered with probability p_ij , i, j = 1, n , j ≠ i; the transition occurs at an arbitrary time moment, not related to the moment of occurrence of the Poisson flow event with parameter λ_i, while initiating an additional event occurs in the j th state; transition and initiation of an additional event occur instantly. After each registered event at the time moment t_k ( both the event of the Poisson flow with parameter λ_i and the additional event), there is a period of fixed duration T (dead time), during which other events of the generalized asynchronous event flow are inaccessible to observation. An event that occurs during the dead time doesn't cause an extension of its period (unextendable dead time) At the end of the dead time period, the first event that occurred again creates a period of dead time of duration T and etc. It is required, on the basis of the moments of occurrence of events (from moment t₀ to moment t ) to estimate the state of the process λ(t) at the moment t. We denote the estimate of the state of the process λ(t) at the time moment t as λ(t). We found an explicit form for a posteriori probabilities w(λ_i ǀ t) = w(λ_i ǀ t ₁,..., t_m,t), i = 1,n , that at the time moment t the value of the process λ(t)= λ_i (m is the number of events that occurred at the time moments t₁,..., t_m at the observation interval (t₀ , t) ), here n ∑ w(λ_i ǀ t) = 1. The decision on the state of the process λ(t) is made according to criterion of a posteriori probability maximum. i=1. The algorithm for calculating a posteriori probabilities w(λ_j ǀ t)), j = 1, n , at any time moment t (t ≥ t0 = 0 ) was formulated. In parallel, as we calculate a posteriori probabilities w(λ_j ǀ t), j = 1,n , we can make a decision on the state of the process λ(t) at the current time moment t : λ(t) = λ_i , if w(λ_j ǀ t) = max w(λ_j ǀ t), j = 1,n .

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