10 Markovsky models with expectation

Condition of that the steady condition in system with delay does not exist, is P₀ = 0, i.e. the system is overloaded and a ³ s. It means that turn in due course only will be infinite to grow.

Use factor is defined from a condition  = a / s and defines switched loading at the rate on one line (the switching device). It is known that the steady condition exists, if and only if r <1.

The probability of expectation is defined as probability of that the message will wait, is designated as M (0), estimates probability of that the waiting time exceeds 0, and pays off under the formula of Erlanga C.

The order of service, such as FIFO, RSO, LIFO etc. in which expecting a call are served, irrespective of their service time, is classified as discipline of service without последействия.

The order of service, such as SSTF (shortest service time first- Наименьшее время обслуживания), depends on service time, is classified as discipline of service with последействием for which the formula of Littla is not valid.

11 Expanded Markovsky models

Stochastic process. Birth- destruction process. Structural parametres and indicators of quality of model M (n)/M/s (m, x).

As stochastic process is called integer function of casual variables {N (t); t ³ 0} with parametre t, tell number of the calls which are in system during time t. The parametre t is often considered as time and if N (t) = j it is said that process has to is in a condition j during time t. If stochastic process possesses property марковости such process is called as Markovsky process.

As birth- destruction process is called Markovsky process in which the transition condition occupies only one step. At Dt®0 birth- destruction process is described by a parity formula (11)

Where  - is called as rate of a birth;

 - is called as a  destruction rhythm, m0 = 0.

Expanded model M (n)/M/s (m, x) produces all basic Markovsky models in special cases. The parametre x = g / m should be considered as that additional loading which forms turn. Hence, the parametre g should be considered as rate of clearing of turn (a rhythm of deportation of a call from turn for service on СМО(Система массового обслуживания –Queuing System)).

Indicators of quality of functioning of the expanded Markovsky model are waiting time W, probability of blocking B, probability of expectation M (0) and function of additional distribution of a waiting time (probability of that the waiting time will exceed t) M (t).

12 System overflow

In model of transport overflow the Poisson stream arrives with rate l and if finds all servers s primary group occupied the blocked traffic flows in secondary group with infinite number of servers. It is necessary to familiarise with model, in detail to study its elements, to build the diagramme of conditions of transition for model of overflow and on a condition of global balance of a network to receive the equation of a steady condition.

The secondary group is infinite also the process occurring in it, is characterised by sequence of calls of overflow. As the characteristic of this sequence n th factorial of moment Mn which is calculated under the formula of Heffiza serves. Time intervals between переполнениями pulse, LST (Laplasa-Stajltesa of transformation) which are set under the formula of Desklouksa.

Population mean of loading from calls of overflow and its dispersion are set under the formula of Vilkinsona. At building СМО(Система массового обслуживания –Queuing System) the decision of a return problem is required: it is necessary to define loading and неполнодоступный a bunch of lines on the set population mean and a dispersion of the overflowed calls. To the specified purpose apply the formula of Reppa.