PASTA
Let Pj - probability of that j calls exist during any moment of time in a steady condition, and Пj - corresponding probability just before approach of an epoch of a call (fig. 2 see). And generally speaking, these two probabilities are not equal each other. However for systems with экспоненциальным distribution of time intervals between calls (the Poisson stream of calls) the above-stated probabilities are identical, i.e. Pj = Пj. (2)
The relation (2) is called PASTA (Poisson arrivals see time average,т.е это Пуассоновское поступление вызовов, наблюдаемое за среднее время) - average time of supervision of the Poisson stream. The relation (2) follows from exponential distributions on property марковости. Term PASTA was generated from this the fact that probability Pj is equal to an average (expected) window of time in which it will be observed j calls if supervision was carried out throughout enough long period of time.
Value of the given remark consists that in specified average time of supervision other process, except the Poisson stream can be observed. In this case average time of supervision of process is called ASTA (arrivals see time average,т.е это поступление вызовов, наблюдаемое за среднее время). Such processes can create problems in networks with package switching.
If we unite n independent Poisson streams with rates lj, j=1,2, …, n (fig. 3а see) результирующий the stream becomes again Poisson with rate l=l1+l2+ … +ln. Such occurs because convolution of Poisson distributions gives again Poisson distribution.
If the Poisson stream with rate l goes on a route j with probability pj (fig. 3б see) the stream in a direction j becomes again Poisson. These properties are used at the analysis of systems with Poisson loading.
Formula Littla
Let the system is in a steady condition (fig. 4 see). We will enter following designations:
l- rate of receipt of calls;
- An average waiting time;
- Average of expecting calls.
Then we will receive the basic parity which is called as the formula of Littla: formula (3)
The equation (3) is interpreted as follows: as there is average time of stay of a call in turn this size can be taken into consideration as average time of deduction of expecting call in turn. At the same time the right party of this equation represents transport loading of expecting call which is equal to average of expecting calls according to property of 4 loadings on a theme 2.1 Modules 2. Can be noticed that there is an average size which will be noticed by extreme observers, and time interval W is defined experimentally on one skilled проходке turns.
System time is defined as the general time of stay (a waiting time + a holding time), spent by a call. Having designated average system time a symbol, and average of the calls existing (expecting + served) in system a symbol, we will receive a variant of the formula of Littla: formula (4)
The formula of Littla is applicable to any systems G/G/s, despite entrance process, the service mechanism and discipline of service of turn.
Дата добавления: 2015-02-16; просмотров: 84 | Поможем написать вашу работу | Нарушение авторских прав |