Heuristic algorithm for estimating the number of states of asynchronous MC-flow events

The non-synchronous MC-flow of events, which is a mathematical model employed in Integrated Services Digital Networks (ISDNs), is considered. The flow intensity is a piecewise constant stationary random process A(t) with n states A ₁,A ₂,...,A _n (A ₁ > A ₂ > ... > A _n > 0 ). Process A (t) at a moment t is in the state i if A (t) = A _;- (i = 1, n ). During stationary state interval there is a Poisson flow of events with state rate A _i . Duration of staying in the i-th state is an exponentially distributed random value with the distribution function F _i (т) = 1 - e " , a _ii = - X a _ij (i = 1, n ), where a^ > 0 (i, j = 1, n , i * j ) is the transition rate from state i to state j. j=1, j *i A flow occurs under conditions of uncertainty when information about the parameters and the number of states is not available. The problem is to estimate the values of unknown parameters A _i, a^ (i, j = 1, n , i * j ), and n, using information from the moments t ₁,..., tk . According to the definition of a flow, trace includes time intervals with A (t) = A _;- (the stationary state intervals), where the flow behaves like a Poisson flow. We consider the histogram of all possible rate estimates of some stationary state interval, on which N +1 events of a Poisson flow with rate X were realized: t ₁,t ₂,...,t _N+ ₁ (A e A ₁,A ₂,...,A _n ). This histogram is an estimate of the mixture of distribution densities: N (N........- (-г f 1Г V¥ Pn ( )= N qfP ( ) Pk ( ) = A ^ _VA qkN )= ^^-/-И) , k = 2N ;Xq N )= 1. N (N - 1) k=2 It is proved that P _N (A) is an unimodal function, and arg max (P _N (A)) = A ₀ --_- > A . So, the histogram of all possible rate estimates of a Poisson process has the only maximum. The maximum value indicates the flow rate. Each stationary state interval in the trace of MC-flow is a Poisson flow segment of constant intensity. The histogram, based on this trace, has several maxima in the condition of large duration of stationary state intervals. The number of these maxima is the estimate of the number of states n. The estimates А _i, i = 1, n are defined as the maximum points of histogram envelope curve of all possible rate estimates. Another way to determine the estimates А _t, i = 1, n is to build n likelihood functions and to solve n optimization problems on maximization of the likelihood functions by the variable X.

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