13 Methods of approximation in systems with overflow

The method of a casual equivalent - equivalent random theory (ERT) is widely applied at designing of alternative systems of routeing. Here the equivalent of casual transport loading a* and number of fictitious lines s* in a primary bunch is defined uniquely. From here also the required average probability of blocking of the overflowed direction is calculated. However the average size does not provide individual probability of blocking for transport loadings an and a1, accordingly, on highly-used and alternative route of the designed scheme of linear reservation.

For estimations of individual possibilities of the scheme of linear reservation in directions the overflowed process is often approximated by Interrupted Poisson process (IPP- Прерывистый Пуассоновский процесс). Here the Poisson input with rate l interrupts the casual switchboard with экспоненциальными on and of intervals with sizes g-1 and w-1, accordingly.

Process of superposition of the overflowed traffic and the casual background traffic in a roundabout route does not pulse. However at GI-approximation superposition process is considered as pulsing, and the alternative route is modelled as GI/M/s1 (0). At last, using the law of preservation of loading and PASTA(Poisson arrivals see time average,т.е это Пуассоновское поступление вызовов, наблюдаемое за среднее время), we define individual required probabilities of blocking.

14 Optimum designing of alternative routeing

For an estimation of system of alternative routeing in telephone systems the so-called conditional method is used. Thus the cost proportion of an alternative route to highly-used k = ATC/LTC minimises system of estimations.

Optimum designing for minimisation of cost of system on the set probability of blocking B in a roundabout way (a bunch of lines s1) is reduced to a problem of nonlinear programming for criterion function minimisation f = s+ks1.

The scheme of linear reservation is used for protection of degree of service GOS(Grade of Service- класс обслуживания) at disbalance of individual probabilities of blocking. At studying of the block diagramme of linear reservation it is necessary to acquire appointment of making elements, to pay attention to the distinctions connected with a kind of priorities of arriving streams of ordinary and not ordinary calls.

15 Numerical analysis of the equations of conditions

The equation of a condition of an alternative route. Iteration of Gaussa-Shidela. The decision of the equations of a steady condition in the scheme of linear reservation.

Having made system of the equations of a steady condition for the scheme of linear reservation, we meet difficulties with the analytical decision owing to a considerable quantity of variables. In such cases apply the numerical analysis of the equations of conditions.

In practice the recurrent algorithm with use of initial weight of probability is used formula (12)

Where C - a normalising constant.

Consecutive calculations proceed until the error will not decrease to the predetermined error. Depending on value of the entered weight factor w the algorithm carries the name iteration of Gaussa-Shidela (w = 1) or a super-relaxation method (w> 1).