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The exact analysis for one-linear server GI/G/1 can be set with use of the spectral decision. For the r calls arriving in system, we will designate a holding time - Xr, a waiting time - Wr, and a time interval between r and (r + 1) th calls - Yr+1 (fig. 13 see). From here we will define a service waiting time (r + 1) th call as formula (33)
Where formula which is equal z, if z ³ 0; or 0, if z <0.
It is known that the steady condition exists, if and only if server operating ratio r = lh <1 with rate of receipt l and an average holding time h. Having directed r ® ¥, we will receive Wr ® W and (Xr - Yr+1) ® U. Then from expression (33) we will have formula (34)
Having designated u (t) function of density from U, and W (t) - function of distribution of a waiting time, we will receive the equation of integral of Lindleja: formula (35)
Let's enter auxiliary function W _ (t), defined so that the equation (35) is fair for - ¥ <t <0, and we will receive expression: formula (36)
As U there is a difference between time облуживания X and a time interval between calls Y, LST from U is given by expression formula (37)
Where a * (θ) and b * (θ) is LSTs from X and Y, accordingly.
Having designated transformation of Laplasa (LT) from W (t) symbol W * (θ), and, having made LT transformations of expression (36), we will receive formula (38)
Let's assume that the equation (38) has spectral multipliers in a kind formula (39)
With which satisfies to following conditions:
(a) The spectral multiplier is analytical for Re (θ)> 0 c nonzero values and
(b) The spectral multiplier is analytical for Re (θ) <D c nonzero values and
Where D there is a positive real number.
Delivering spectral multipliers (39) in the equation (38), we will receive formula 40)
From conditions () and () both parties of the equation (40) are analytically defined in the field of 0 <Re (θ) <D, they can be equal to a constant, say, K.Tak kak LST from W (t) is defined as w * (θ) = θ W * (θ) the constant To is defined as formula (41)
Using the limiting relation w * (θ) ® 1 at θ ® 0, we will receive probability of that there is no expectation: formula (42)
And LST waiting time distributions it will be defined as formula (43)
Having made return transformation w * (θ), we can calculate waiting time distribution, and differentiation w * (θ) under entry conditions gives the waiting time moments. For example, we have an average waiting time
In one-linear model H2/G/1 with hyperexponential distribution of time intervals between arriving calls and with any holding time LST of service time (service) it is defined as formula (45)
Where l- rate of receipt;
Са2 - SCV time intervals between arriving calls;
Sk - asymmetry (the third central moment or dispersija3/2) time intervals between arriving calls.
In that specific case for symmetric conditions asymmetry will not be claimed also calculations become simpler:
At we will have: formula (46)
The spectral decision for model H2/G/1 yields following results. Believing that formula (47)
We can make sure that expressions (47) satisfy to conditions (39) with D = min (l1, l2, Re (θ0)). Here θ0 there is a root of the functional equation formula (48)
Where formula
The root θ0 can be calculated it is iterative with initial size θ0= l: formula (49)
Where formula
From expressions (42) - (44) it is received following results: formula (50)
Where WM - the average waiting time equivalent and for system M/G/1 (with the same rate of receipt of calls), also is defined from the formula of Poljacheka-Hinchina (21) with SCV distributions of service time.
Let's consider k-phase Erlang loading of port of the one-linear server with any holding time Ek/G/1. It is known that LST k-phase Erlang of distribution of time intervals between calls Ek it is defined as formula (51)
Applying the spectral decision to similarly earlier described way, we will receive following results:
formula (52)
Where θi, i = 1, 2, …, k - 1, are equation roots функционала with Re (θi)> 0: formula (53)
By putting in a functional (53) b * (θ) = g exp (jw) c j =, we will calculate these roots iteratively formula (54)
If k even number, we have one valid root and (k/2 - 1) interfaced pairs; if k odd number, we have (k - 1) interfaced pairs. From here ∏ θi and ∑1/θi become valid.
It is possible to notice that results (52) - (53) are fair only for whole k and asymmetry of time intervals between arriving calls is defined implicitly. In the expanded version for real numbers k the data of first three moments is valid.
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