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 of application of queueing theory. Doubly stochastic flows of events are adequate mathematical models of real information flows which functioning in ISDNs. In practice parameters of a flow of events are often unknown; only occurrence of events of the flow is observable. Therefore estimating of parameters by observing a sequence of events is an important problem. An asynchronous doubly stochastic flow of events with finite number of states is considered in the paper. The flow intensity is a piecewise stochastic process (<i>t</i>) with <i>n </i>states 1 > 2 > … > <i>n </i>> 0. Provided that at the instant <i>t </i>(<i>t</i>) = <i>i </i>(<i>i</i>= 1,<i>n</i>), the flow is said to be in the <i>i-</i>th state at this time <i>t</i>. While the flow is in the <i>i-</i>th state (<i>i</i>= 1,<i>n</i>), it is the Poisson flow with the intensity equals to <i>i</i>. The sojourn time in the <i>i-</i>th state is exponentially distributed: ( ) 1 <i>ii Fi </i> = −<i>e</i> , where 1, <i>n ii ij j</i>= <i>j</i><i>i </i> = −   (<i>i</i>= 1,<i>n</i>); <i>ij </i>> 0 ( <i>i</i>,<i>j </i>=1,<i>n </i>, <i>i </i> <i>j</i>) is the intensity of transition from the state <i>i </i>to the state <i>j</i>. Values of the parameters of the flow <i>i</i>, <i>ij </i>( <i>i</i>,<i>j </i>=1,<i>n </i>, <i>i </i> <i>j</i>) are unknown; the current state of the flow is unobservable. The number of the states <i>n </i>is known. The problem of optimal estimation of parameters of an asynchronous doubly stochastic flow of events is solved on the basis of sequence of the events observed. The obtained estimations are optimal in the sense of minimal mean square deviation from the true values of the parameters. The explicit form of the estimations 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 numerical experiments for estimation of parameters was carried out. The experiments confirm sufficient stability of the estimations obtained.