Contents Telektronikk - Telenor

t1 t2 t3 t4 t5 t6
T1 T2 T3 T4 T5 T6 T7
equal size as the one around 10. One
observation is that the residential subscribers
cause a better utilisation of the
system than do the business subscribers,
due to the greater traffic concentration of
the latter.
An example of data traffic observations
[3] is shown in Figure 10, a), b), and c).
The observations are given in the form of
bits per second from a workstation over a
three hour period. Three degrees of resolution
are given, 1 second, 10 seconds
and 1 minute, clearly indicating the loss
of detail by integration over longer time.
Indicative is the reduction of peak values
from 750 via 350 to 160 kb/s for the
same set of data. (Note different ordinate
scales.)
8 Traffic modelling
τ1 τ2 τ3 τ4 τ6 τ5
s1 s2 s3 s5
Figure 11 The double process of arrivals and departures
Up till now we have focused on the traffic
load on a set of servers and on different
types of variation. Apart from an
indication that the traffic is generated by
calls from traffic sources we have not
studied how traffic is created.
In Figure 3 the call arrivals create all the
+1 steps. In the long run there are equally
many -1 steps, and these all stem from
departures from the system. Earlier or
later any arrival must result in a departure.
This double process may be depicted
as in Figure 11. If we start by observing
only arrivals, we have the points T1 to T7 with the intervals t1 to t6 .
Arrivals and departures are always connected
in pairs. This implies that the longterm
averages of arrival rate λ and departure
rate γ must be equal. That again
secures a limitation of the traffic load as
long as λ and the holding times (service
times si ) are limited. It is seen from the
figure that the traffic measured during the
period T = T7 – T1 is a little less than 1
Erlang. This is seen by adding the holding
times and dividing the sum by T: A = (s1
s4 s6
+ s2 + s3 + s4 + s5 + s6)/T. By reducing
the intervals T i – T i–1 , thereby increasing
λ, the traffic increases. The same effect is
obtained by increasing s i .
9 Little’s formula, the
“Ohm’s law” of teletraffic
It is well known from electrophysics that
the three quantities voltage v, current i
and resistance r are connected by the formula
v = i ⋅ r, and that in an electric network
this formula may be applied to the
whole network or any part of it. This is
Ohm’s law. In a similar way the three
quantities traffic A, arrival rate λ and
service (holding) time s are connected by
the formula
A = λ⋅s (3)
This is Little’s formula, and like Ohm’s
law in an electric network, Little’s formula
applies to the whole or any part of a
traffic network. A difference is that for
Ohm’s law constant values are assumed,
whereas Little’s formula applies to mean
values, including the constant case (given
co-ordination of arrivals and departures,
which is a bit artificial). The only condition
is a stationary process, which will
be discussed later. In simple terms stationarity
means that the statistical properties
of the involved processes remain
unchanged over time. There are no other
conditions laid on the statistical distributions
of the variables.
With reference to the traffic models of
Figures 1 and 2 an alternative model is
shown in Figure 12.
The following indexes can be used:
o ⇒ offered, relating to sources
l ⇒ lost
q ⇒ queue
c ⇒ carried, relating to servers.
One could also use corresponding indexing
on holding times, however we use,
according to common practice, the following:
s ⇒ service time
w ⇒ waiting time (in queue).
Very often h is used for holding time. It
can be indexed to show which system
part it is related to. For instance hl may
be used to indicate that even a lost call
occupies a source for a non-zero time.
According to Little’s formula we obtain:
Traffic load on queue:
Aq = λq ⋅ w = λc ⋅ w
Traffic load on server group:
Ac = λc ⋅ s
Traffic load on source group (encompassing
the whole system):
Ao = (λo – λl ) ⋅ (w + s) + λl ⋅ hl = λc ⋅ (w + s) + λl ⋅ hl We see that, if the lost traffic holding
time hl is zero, the traffic load on the
sources, Ao , is actually equal to the sum
of the queue traffic and the server traffic.
In this case – but not in general – the
“non-empty” call rate is identical for all
parts of the system, whereas the holding
times are different.
This would be different if the calls waiting
in queue had a limited patience, so
that some of them would leave the queue
without being served. That would reduce
the mean holding time on the source
group and the queue, and the arrival rate
on the server group. Thus the load would
be reduced on all parts of the system.
The linear property of Little’s formula
can be expressed by
A = A1 + A2 = λ1 ⋅ s1 + λ2 ⋅ s2 , (4)
N sources
X X X X
Traffic requests
X X X X X
X X X X
Overflow
Carried calls
n servers
O O
O O O
O O
q waiting positions
Figure 12 Traffic model with overflow from queue
9