Contents Telektronikk - Telenor

much less if a basic bandwidth unit can
be found.
This recursion formula has been extended
to Bernoulli and Pascal arrivals in [2],
but in these cases we only get approximate
values for the connection blocking
probabilities.
4.2.2 Convolution algorithm
This algorithm for calculating the global
state probabilities (1), is recursive in the
number of sources. The algorithm was
first published in [7]. In the paper it is
shown that it is allowed to truncate the
state space at K (K = the number of traffic
types) and renormalise the steadystate
probabilities. For Poisson traffic
this method is not so fast as the recursion
formula above. For Bernoulli and Pascal
traffic this method gives exact values, not
only for the time blocking probabilities,
but also for the connection blocking
probabilities.
The algorithm goes in three steps. In the
first step we calculate the state probabilities
pi (ni ) (ni = 0, ..., n max
i = D/di ) for
each traffic type i as if this traffic type
was alone in the system. For this we can
use the product form expression (1) with
K =1.
In the next step we calculate the global
state probabilities by successive convolutions:
Q1 (n1 ⋅ d1 ) = p1 (n1 ) for n1 = 0,...,n max
1
d/di �
Qi(d) = Qi−1(d − j · di) · pi(j)
j=0
for d = 0,...,D and i = 2,...,K
The global state probabilities are now
given by Q(d) = QK (d) for d = 0,...,D
after normalising.
The time blocking probability can then
be calculated as above, but for Bernoulli
and Pascal arrivals we need another step
to calculate the connection blocking
probabilities.
In this third step we deconvolute Q(d) by
finding Qi (d) such that
d/di �
Q(d) = Q i (d − j · di) · pi(j)
j=0
The connection blocking probability for
traffic type i is now given by:
Bi =
�D d=D−di+1
� D
d=0
�d/di j=0 λi(j) · Qi (d − j · di) · pi(j)
�d/di j=0 λi(j) · Qi (d − j · di) · pi(j)
Due to numerical problems it may be better
to calculate Q i by convolutions of
homogenous state probabilities p i .
4.2.3 Lindberger method
This is an approximate method.
Let dmax = max{d1 ,...,dK } and n = D -
dmax . We assume that dmax is much
smaller than n. The blocking states are
now approximated by [11]:
�
Q(n + k) ≈ Erl � n ′ , A
z
for k = 1,...,dmax , where
K�
A =
is the total bandwidth demand of the
offered traffic,
z =
is the peakedness factor (and a measure
of the mean number of slots for a connection
in the mixture (‘equivalent average
call’), see [3]),
n ′ =
i=1
Ai · di
� K
i=1 Ai · d 2 i
A
1 n + 2 1
−
z 2
z
and Erl(x,A’) is the Erlang formula (linear
interpolation between integer values
of x).
The validity of this method for ATM
should be tested and perhaps other values
of n’ should be chosen depending on the
traffic mix.
Now the idea is to divide the traffic types
into two classes and to dimension capacity
using the model above for each of
these classes separately. The first class is
for low capacity traffic, i.e. suggested for
connections with a highest peak rate /
effective bandwidth in the order of
0.5 Mb/s on a 155 Mb/s ATM link. For
this class the objective is to control the
average connection blocking probability
·
� � k
z A
D
E =
(only the low capacity traffic types are
considered now), and the highest individual
connection blocking probability in
the mixture Emax = max{Ei |i = 1,...,K}.
If dmin = min{d1 ,...,dK } approximate formulas
for these are
For the other class, i.e. suggested for
connections with peak rate/effective
bandwidth between 0.5 Mb/s and 5–10
Mb/s, the requirement for connection
blocking probability is not so strong. For
this class it is suggested to use the same
connection blocking requirement for all
traffic types in the class (trunk reservation,
see below). An approximation for
this blocking probability is given by
Traffic types with higher capacity
requirements will have to be treated
separately, perhaps on a prebooking
basis.
4.2.4 Labourdette & Hart method
This is another approximate method.
Let
the unique positive real root of the polynomial
equation
and
� K
i=1 Ai · di · Ei
A
�
D − z + dmin
E ≈ Erl
,
z
A
�
z
Emax ≈ dmax
z
·
� � (dmax−z)/2z
D
· E
A
�
D − dmax + dmin
E0 ≈ Erl
,
z
A
�
z
K�
K�
A = Ai · di, V =
i=1
K�
di · Ai · z di = D
i=1
ζ = log � D
A
log(α)
�
i=1
Ai · d 2 i , α,
for α ≠1. The blocking probability for
traffic type i is then by this approximation
method given by [10]:
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