Optimal estimation of parameters of an asynchronous doubly stochastic flow of events with arbitrary number of the states

Designing and constructing of integrated services digital networks is an important sphere ofapplication of queueing theory. Doubly stochastic flows of events are adequate mathematicalmodels of real information flows which functioning in ISDNs. In practice parameters of a flow ofevents are often unknown; only occurrence of events of the flow is observable. Therefore estimatingof parameters by observing a sequence of events is an important problem.An asynchronous doubly stochastic flow of events with finite number of states is consideredin the paper. The flow intensity is a piecewise stochastic process ƒ(t) with n states ƒ1 > ƒ2 > … >ƒn > 0. Provided that at the instant t ƒ(t) = ƒi (i= 1,n), the flow is said to be in the i-th state at thistime t. While the flow is in the i-th state (i= 1,n), it is the Poisson flow with the intensity equalsto ƒi. The sojourn time in the i-th state is exponentially distributed: ( ) 1 iiFi ƒ = −eƒ ƒ , where1,nii ijj= jiƒ = − ƒ ƒ (i= 1,n); ƒij > 0 ( i,j =1,n , i  j) is the intensity of transition from the state i tothe state j. Values of the parameters of the flow ƒi, ƒij ( i,j =1,n , i  j) are unknown; the currentstate of the flow is unobservable. The number of the states n is known. The problem of optimalestimation of parameters of an asynchronous doubly stochastic flow of events is solved on the basisof sequence of the events observed. The obtained estimations are optimal in the sense ofminimal mean square deviation from the true values of the parameters. The explicit form of theestimations is obtained; it allows to find the estimations sufficiently fast without numerical methods.The approximate numerical algorithm for real-time estimation is developed.Using the simulation model of an asynchronous doubly stochastic flow a number of numericalexperiments for estimation of parameters was carried out. The experiments confirm sufficient stabilityof the estimations obtained.

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