Contents Telektronikk - Telenor

80
Similarly as for Poisson-traffic we note
that overflow traffic is defined for an
infinite group and may likewise be
named ‘offered traffic’. We can now calculate
the means M01 , M02 , ..., M0g and
variances V01 , V02 , ..., V0g overflowing
from direct links using Wilkinson’s formula.
Since the input streams A01 , A02 ,
..., A0g are independent Poisson traffic
processes we can conclude that the individual
overflow processes, too, are independent.
Thus the total traffic offered to
the upward transit link will have the
mean
M0 = M01 + M02 + ... + M0g + A0 and the variance
V0 = V01 + V02 + ... + V0g + A0 Our next step is to calculate the traffic
lost in link ko given that the offered traffic
has the mean M0 and the variance V0 .
This is done smartly be replacing the g
direct links to basic nodes by a single
imaginary link of n channels offered a
Poisson-traffic, A, as shown in Figure 4.
Thus we calculate A and n from the
Wilkinson formula given M = M0 and
V = V0 and we can use the same formulae
to estimate the moments M0 and V0 of
overflow traffic from the link k0 . Especially
we may estimate the average loss
in link k0 as
A n
B0 = M0/M0 = AEn+k0(A)
AEn(A)
Primary
group
Figure 3 ‘The Kosten system’
= En+k0(A)
En(A)
Owerflow
group
∞
A n M 0 , V 0 k 0 M 0 , V 0
Figure 4 The equivalent random principle
For the traffic A 0 offered directly to the
transit link a somewhat better loss estimate
can be obtained as the time congestion,
E 0 , encountered in that link [1]. The
results are approximate, of course, but are
generally considered rather acceptable for
dimensioning purposes. Though we have
omitted the network outside a single transit
centre and its basic nodes, this should
not hide any major difficulties as long as
we are contented with average losses of
combined traffic streams on upward links.
In the sequel, however, we will also
attempt to calculate individual losses
from one basic node to another.
3 Split traffics calculation
A natural approach to our problem of
individual loss calculations would be to
use the ERT concept shown in Figure 4,
where the imaginary poissonian traffic
intensity A is split as follows:
A = A0 + A1 + ... + Ag Ai = M0i /En (A), i = 1, 2, ..., g
Thus we have preserved the individual
overflow means from direct links. This
will hopefully lead to useful results also
when combining split overflow traffics
on downward transit links. The formula
to be used are obtained for the overflow
system of Figure 5 and may be found in
the paper Joint state distributions of partial
traffics [9]. They give estimates of
the means M 0i and variance V 0i of split
overflow traffics behind the (imaginary)
group n + k 0 offered g independent Poisson-processes
defined above. The results
are
M 0i = α i M 0
V0i
M0i
αi = Ai
A
i = 1, 2, ..., g
M0 =AEn + k0 (A)
V0 = M0
− 1=αi
� V0
M0
�
1 −M0 +
�
− 1
A
n + k0 − A + M0
�
It should be noted
that the overflow
processes behind a
transit link are
dependent. Thus
we did not obtain V0 as a sum of individual
variances. Instead we found ‘variance/mean
- 1’ to be an additive characteristic
of the partial overflow traffics.
The individual blocking probabilities in
link k0 are obtained as
B ′ 0i = M0i
=
M0i
Ai
· AEn+k0(A)/AiEn(A)
A
= En+k0(A)
En(A)
a result that is certainly approximate and
will be discussed below. Note that individual
variances V0i will not be used in
our simple routing structure with low
loss transit links.
Considering Figure 2 we could identify
g + 1 link patterns, each showing g links
to basic nodes, one upward transit link
and g downward transit links. Let us
focus on the downward link l1 , a group of
l1 channels. Among the overflow traffics
offered to k0 the only one that can reach
l1 is (M01 , V01 ). Similarly, it is seen that
among traffics offered to k2 the link l1 can only be reached by (M21 , V21 ), and
so on. The total traffic that can have
access to l1 may be written as the mean
M1 = M01 + M21 + M31 + ... + Mg1 + A1 and variance
V1 = V01 + V21 + V31 + ... + Vg1 + A1 Let us assume for simplicity that upward
links, k0 , k2 , ..., kg are all low loss links
that will allow almost all corresponding
overflow calls to reach link l1 . M1 and V1 define just a new combination of traffics
similar to the one offered to the upward
transit link k0 . Clearly the downward link
l1 can be treated in the same way by
ERT. Thus for the simple case of an isolated
network with one transit node and
with low loss transit links we may estimate
the loss from node 0 to node 1 as
B01 = En01 (A01 )(B’ 01 + B’’ 01 )
where B’ 01 and B’’ 01 are the individual
losses in the upward (k 0 ) and downward
(l 1 ) links, respectively.
4 Conclusion
The methods suggested above for estimating
point-to-point losses in hierarchical
alternative routing networks can be
judged only be extensive simulations. At
present, however, we may see some general
consequences of results obtained so
far: