Contents Telektronikk - Telenor

A(rl ,rs ) =
� �−L �
ζl KB(rl) PrlB
ζs
(3.29)
′ (rl)
KB(rs) − PrsB ′ �
(rs)
��K �
−1
k=1,k�=l ζs − ζ −1�
�
k
�K � �
−1
k=1,k�=l ζl − ζ−1
k
where ζi = ri /B(ri ) and L is the displacement
in slots between the backward busy
period and the busy period (for the periodic
source) (see Figure 3).
The generating function for the q M’s j in
this case may be expressed as:
K�
q M j x j−1
j=1
= P(1 – ρ)E 1 (I + W) -1 V(x) (3.30)
where E1 is the K dimensional row vector
E1 = (1,0,....,0), and W = (W(i,l)) is the
K × K matrix:
W (i, l) = �
� �M � �K−L−1 rl ζl
s=K+1
(3.31)
and V(x) is the K dimension column vector
V(x) = (V l (x)):
(3.32)
The overall cell loss probability for this
special case may be written:
(3.33)
(where E T
1 is the transposed of E1 ).
For large M we can expand (I + W) -1 in
(2.33), and by taking only the first term
in the expression for W we get the following
simple approximate formula for
the overall cell loss probability:
rs
� KB(rl) − PrlB ′ (rl)
KB(rs) − PrsB ′ (rs)
�K k=1,k�=l
ζs
(x − ζk)
Vl(x) = �K k=1,k�=l (ζl − ζk)
�
Vi(ζs)Vl(ζs)
(1 − ρ)
pL = −
ρ E1(I + W ) −1 WE T 1
pL ≈
(1 − ρ) 2
� �M � �K−L−1 1
1
rK+1 ζK+1
ρ � P
K (rK+1B ′ (rK+1) − B (rK+1)) �℘2
(3.34)
cell-loss
cell-loss
10 0
10 -5
10 -10
10-15 0
10 0
10 -5
10 -10
20
where ℘ is the product
40
60
buffer size
(3.35)
For most of the cases of practical interest
the formulas (3.34), (3.35) give satisfactory
accuracy.
80
rho=0.8
100
rho=0.9
5
10
15
20
speedup factor
Figure 4 Cell loss probability as a function of buffer size and speed up factor at load
0.9
10
0 10 20 30 40 50 50 60 70 80 100
-15
buffer size
Figure 5 Cell loss probability as a function of buffer size for different speed up factors
at load 0.8 and 0.9 (1 speed up = 1, 2 speed up = 5, 3 speed up = 10, 4 speed up = 15,
5 speed up = 20)
℘ =
� K
k=2 (ζK+1 − ζk)
� K
k=2 (1 − ζk) ζ = z
B(z)
5
1
25
As for the multiserver case the product
℘ may be given as an integral. We must,
however, substitute for
in the formulas (2.13) and (2.14) (and
also changing n to K and A(z) to B(z) P ):
(1 − ρB)
℘ = (3.36)
P (1 − ρ) (ζK+1 − 1) exp(I)
(ζK+1) K
1
5
215