6 Poisson distribution

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 t  0 as

1) Probability of that the call is generated in the range of time (t, t+Dt], aspires to t, irrespective of t where  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.

7 Distribution of a holding time

Determined time is set by sequence of sizes hk, characterising duration of service of k th call or k-й groups of calls. At hk = h the holding time is called to constants.

The casual holding time is set by the law or distribution density. Let's consider the elementary case. It assumes that the call comes to the end in a casual order. Taking the moment of generation of a call for the beginning, we will define probability of that the given call will come to the end in an interval (t, t+Dt], as m×Dt which does not depend on time t owing to an assumption about casual end. Additional function of distribution H (t) (probability of that the holding time is more than t) is equivalent to probability of that the call will not come to the end in the range of time (0, t]. We will divide an interval (0, t] on enough great number n pieces and we will put Dt = t / n. As last probability is equal (1 - m×Dt) n, that, directing n ® ¥, we will define required function of distribution as formula (9)

Thus, the holding time is distributed with a population mean m-1 where m is called as a rhythm of service or rate of clearings. It often carry to an indicative holding time in small and then transport loading (5) is expressed as a = l / m.

The assumption about an indicative holding time will well enough be co-ordinated with duration of telephone negotiations (fig. 7 see). Further for simplification of the theoretical analysis as we will see later, it will be widely used in the teletraffic theory.