Contents Telektronikk - Telenor

26
n *
ooo---o
A * M, V
stream is not identical to a single overflow
stream with exactly the same mean
and variance. However, for practical
application the overflow given by the
pair (M,V), as calculated above, is assumed
to be equivalent to a single overflow
stream with the same (M,V) - pair. The
principle is shown in Figure 28, where M
and V are calculated sum values and the
pair (A*,n*) represents the equivalent
primary group to be determined. Riordan’s
variance formula gives an explicit
expression A* = f(n*) or n* = g(A*). In
principle A* and n* are found by insertion
in Erlang’s formula. This formula,
however, is only implicit, and the solution
can only be found by an approximation
formula, by tables or by numerical
calculation.
When A* and n* are determined, the
overall lost traffic is given by m:
m = A* ⋅ E (n* + k, A*)
and the loss probability
g�
B = m/A = m/
Note that the loss B is the ratio between
the calculated lost traffic and the total
offered real traffic, m/A, and not related
to the fictitious traffic A* by m/A*.
The method is called the ERT (Equivalent
Random Traffic) method and often
referred to as Wilkinson’s equivalence
method [10]. It can be applied in a stepwise
manner for progressive gradings.
Another, and at present more interesting
area, is that of networks with alternative
routing, where primary circuit groups
k
ooo---o m
Figure 28 A fictitious equivalent group for generation of a given overflow M,V
A 1primary
A 2primary
n 1
ooo---o
⇓
A 1overflow
Reserve part of n 2 forA 2primary
Figure 29 Trunk reservation for primary traffic in order to balance loss
i=1
Ai
n 2
oooooo---
may have overflow to a common secondary
group.
14.5.2 The IPP (Interrupted
Poisson Process) method
Another method for overflow analysis
was introduced by Kuczura [11], and is
based on the distinction between the two
states S1 = [Primary group is busy] and
S2 = [Primary group not busy]. With the
primary input being a Poisson process,
the secondary group will see a switching
between Poisson input and no input. The
method is thus named IPP = Interrupted
Poisson Process. The approximation is
the assumption that the dwelling times in
the two states are exponential, which we
know is not true. The process has essentially
three parameters, the Poisson
parameter λ, and the S1 and S2 exponential
parameters α and β. A three-moment
match is based on the mentioned three
parameters. An alternative method by
Wallstrøm/Reneby is based on five
moments in the sense that it uses the first
moment and the ratios of 2nd/3rd and
7th/8th binomial moments. [12].
The IPP method is widely used and has
been generalised to the MMPP (Markov
Modulated Poisson Process) method,
where the switching occurs between two
non-zero arrival rates.
The methods presented do not assume that
all primary groups are equal or near equal.
The independence assumption assures the
correctness of direct summation of means
and variances. Thus, calculation of the
overall loss is well founded. In symmetrical
cases the overall loss B as calculated
above will apply to each of the single pri-
⇓
mary groups. However, with substantial
non-symmetry the single groups may
experience quite different loss ratios. This
has been studied by several authors, but
will not be treated in this paper. Reference
is made to [13], this issue.
A particular dimensioning problem of
overflow arises in network cases when
overflow from a direct route to a final
route mixes with a primary traffic on the
final, Figure 29. The A2 traffic may experience
a much higher blocking than A1 .
The problem can be solved by splitting
n2 and reserving part of it to A2 only, to
obtain a more symmetric grading.
14.6 Access limitations by link
systems
Link systems were developed on the
basis of crossbar switches and carried on
for crosspoint matrices in early electronic
switching systems. Like many grading
methods, link systems had their motivation
in the limitations of practical constructions,
modified by the need for
improved utilisation of quite expensive
switching units.
This can be explained as follows: A traditional
selector has one single inlet and,
say, 100 to 500 outlets. When used as a
group selector, 10 to 25 directions with
10 to 20 circuits per direction would be
feasible. All this capacity would still only
permit one connection at a time, and a
large number of selectors with common
access to the outlets (in what is termed a
multiple) would be necessary to carry the
traffic. A crossbar switch could be organised
in a similar way with one inlet and,
say, 100 to 200 outlets, and with the
same limited utilisation. However, the
construction of the crossbar switch (and
likewise a crosspoint matrix), permits a
utilisation whereby there are, say, 10
inlets and 10 – 20 outlets, with up to 10
simultaneous connections through the
switch. This tremendous advantage carries
with it the drawback of too few outlets
for an efficient grouping. The solution
to this is a cascading of switches in 2
– 3 stages. That, of course, brings down
the high efficiency in two ways: an increase
in the number of switches, that
partly “eats” up some of the saving, and
introduction of internal blocking, with a
corresponding efficiency reduction. Still,
there is a considerable net gain.
The selection method through two or more
stages is called conditional selection, as
the single link seizure is conditioned on
the whole chain of links being available,
and so are seized simultaneously.