8 Classification of transport models

The system connecting the inputs and exits, is called as switching system. If any input of system can be connected to any free exit of the same system such system is called полнодоступной as system, otherwise - неполнодоступной system.

As overload is called the condition when connection cannot be executed because exits or intermediate lines of switching system are occupied. At overloads if any call is blocked such system is called as system with losses or system without delay. If the call can expect a connection establishment such system is called as system with expectation or system with delay. Graphic representation of the specified systems is resulted on fig. 8.

Full accessible systems are described as follows:

1) The entrance stream describes an order of generation or receipt of calls. Though the Poisson stream has been considered earlier, there are various other streams which will be discussed later;

2) The service mechanism describes quantity of exits (lines, servers etc.), distribution of time of deduction (service) etc. As a rule, экспоненциальное distribution is applied to the telephone traffic. However other distributions (for example, fixed or determined) can be demanded for transfer of packages of the data;

3) The discipline of service specifies principles of processing of calls during an overload (losses or delay). In systems with delay the order of service of expecting calls should be specified, such as First In, First Out (FIFO- первым пришёл - первым ушёл); last In, first Out (LIFO- последним пришёл — первым ушёл), a random order of service (RSO- случайный порядок обслуживания) etc.

For classification полнодоступных systems designations on Kendallu are used: formula

Where A - represents distribution of an entrance stream;

B - distribution of service time (service);

s - quantity of servers.

Instead of positions A and B the following symbols specified in tab. 1 are used.

Table 1

Designation A distribution Kind

M Exponential (Markovsky)

Ek K-phase Erlangovyj (curtailed from k exponential with identical population means)

Hn GiperExponential of an order n (alternative n-Exponential)

D Determined (fixed)

G General (any)

GI Any independent (pulsing)

MMPP Markovsky modulation of Poisson process (not pulsing -> raging)

For example, the system with Poisson receipt, экспоненциальным service time and with s servers is expressed as M/M/s. At final number of inputs (sources) we will receive M (n)/M/s, and at the expecting complete set from m positions model СМО(Система массового обслуживания –Queuing System) will register as M/M/s (m) or M/M/s + m. Therefore the system with losses is expressed as M/M/s (0) or M/M/s/s. Without such additional symbols designate system with the delay, serving infinite turn in the order of FIFO.

9 Markovsky models with losses

Distribution of Erlanga sometimes name the truncated distribution of Poisson. It is necessary to deduce this distribution to an example of consideration of model M/M/s (0).

As statistical balance is called the probability of steady position (balance). It is known that the steady condition exists, if the unique final probability of distribution {Pr} such that takes place at t ® ¥ Pr (t) ® Pr irrespective of entry conditions.

It is necessary to learn to build the diagramme of conditions of transitions in model M/M/s (0) and on it to receive the equations of local and global balance of a network.

It is necessary to consider model M (n)/M/s (0), to construct the diagramme of conditions of transitions and to receive distribution of Engseta and binomial distribution; to compare among themselves various distributions: Пуассона®Эрланга®Энгсета® Binomial.

With switching of channels it is necessary to carry probability of blocking to indicators of quality of work of a network. Within the limits of degree of service GOS(Grade of Service- класс обслуживания) distinguish probability of an overload on calls, probability of an overload on time and к.п.д. Lines or employment. It is necessary to deduce the formula of losses of Engseta and the formula of losses of Erlanga (the formula of Erlanga B).

The general switched loading does not depend on a way of linear search. Distinguish LTC (capacity of last line) and ATC (capacity of an additional line). Parametres LTC and ATC are used at building and optimization of systems of alternative routeing.