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Distribution of residual time. A population mean and SCV(coefficient of variation- оэффициент вариации случайной величины) residual time. The law of preservation of rate. Local balance of a network.
If time of interval Х between consecutive employment as authentic event, tell, the call has arrived, is independent and identically distributed (iid) - independently and identically distributed - with function of distribution F (x), process {X} is called as pulsing process. Puassonovsky process is a special case of pulsing process in which function of distribution F (x) экспоненциально is distributed.
For pulsing process (fig. 9 see) the interval of time X*ÎX from any way observable point xÎX* till the moment of following employment is called as residual (lives) time or the future time of repetition (forward). Thereupon the interval generated by employment X is called as life time.
Time from last employment to an observable point of an epoch is called as a century or last time of repetition (back). It is known that the century, as well as residual time X *, is distributed according to distribution function formula (13)
Where m = E {X} - there is a population mean of time of life X.
Basically believe that F (x) should be function of distribution of casual continuous variable Х ³ 0. Then define Laplace-Stieltjes transform (LST) from F (x) as formula (14)
Where formula - is function of density Х.
Transformations LST of function of distribution F (x) it is equivalent to transformations of Laplasa (LT) to density function f (x). Transformations LST of distribution of residual time R (t) are expressed as formula (15)
Where formula f * (θ) - is LST time of life of distribution F (x).
From here we receive average residual time formula (16)
Where formula E {X 2} - is the second moment Х,
σ2 - is dispersion Х,
Is square-law factor of a variation (SCV).
Let's consider system in which calls arrive in pulsing process with rate l, and we will demand экспоненциального a holding time with average size m-1. Believing that Pj the probability of that j calls exists in system in a steady condition, and Пj - probability just before call receipt, we have the law of preservation of rate: formula (17)
For systems with losses the law is interpreted as follows. We will designate a presence condition in system j calls symbol Sj. As Пj-1 there is a probability from condition Sj-1 just before call receipt, and l there is a rate of receipt of a call the left-hand side of the law (17) represents transition of rate from Sj-1 ® Sj. On the other hand, as jm there is a rate of clearing (an end rhythm) call, and Pj there is a probability of existence j calls the right party of the law (17) represents transition of rate from Sj ® Sj-1. Thus, both parties balance in a steady condition, providing local balance of a network: rate-up = rate-down.
The same interpretation is applied and to systems with delay unless rate of clearing (an end rhythm) call accepts value sm at j> s as only s served calls can come to the end.
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