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The basic terms of the theory of distribution of the information are such initial concepts, as the message, a call, employment, clearing and a stream of homogeneous events.
Call - the requirement of a source of the establishment of the connection which have arrived in a communication network, of switching system, of an input of a step of search, in the actuation device for the purpose of message transfer.
The set of the consecutive moments of receipt of calls forms a stream of calls. The stream of calls is called as determined if the sequence of the moments of receipt of calls is in advance defined, known (for example, the program of telecasts), and casual if the given sequence is casual.
As it will be shown later, there are various models of generation of a call. We will consider casual generation which is modelled at Dt ® 0 as
1) Probability of that the call is generated in the range of time (t, t+Dt], aspires to l×Dt, irrespective of t where l is a constant (stationarity property);
2) The probability of that two or more calls is generated in the range of time (t, t+Dt], aspires to zero (property of ordinariness);
3) Calls are generated independently one from another (no aftereffect).
In the given model let's calculate probability pk (t) that k calls are generated in the range of time (t, t+Dt]. As is shown in fig. 6, we will divide time interval (0, t] into enough great number of pieces in the size Dt = t / n.
As probability of that is exact k calls are generated in k pieces, it is set by expression (l×Dt) k × (1-l×Dt) n - k at Dt®0 appears such exclusive events. From here probability formula
The probability (6) is called as distribution of Poisson with a population mean equal l×t where l is called as rate of receipt or rate of generation of a call. That fact that l is a constant and does not depend on time, is one of features of casual generation. The given model concerns Poisson receipt (- calls, - loadings, - generations, - processes etc.).
As average of the calls generated in the range of time (0, t], is l×t l it is interpreted as average of receipts of calls in unit of time which actually is size with from expression (5) on the first property of loading. Rate of receipt of calls depends on unit of used time, and, if hour is used, l is measured in BHCA (busy hour call attempts) – т.е число попыток установления соединений в час наибольшей (телефонной) нагрузки.
From distribution of Poisson (6) probability of that time interval (0, t] it is empty (are not present the generated calls), is set as
p0 (t) = e-l× t. (7)
From here function of distribution of time intervals between calls (probability of that time intervals between calls no more t) is set as
A (t) = 1 - e-l× t. (8)
Function of distribution A (t) exponentially is distributed with a population mean l-1. Thus, exponentially distribution of time intervals between calls is other feature of casual generation.
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