Contents Telektronikk - Telenor

22
14 Disturbed and shaped
traffic
Up till now we have looked at traffic
generation as the occurrence of random
events directly from a group of independent
free sources, each event leading to
an occupation of some server for an independent
time interval. Any call finding
no free server is lost, and the source
returns immediately to the free state. The
traffic generated directly by the set of
sources is also called fresh traffic.
All sources in the group are assumed to
have equal direct access to all servers in
the server group. This is a full availability
or full accessibility group. (The term
availability is used in a reliability context
as a measure of up-time ratio. In a traffic
context availability is usually synonymous
with accessibility.) The four models
discussed previously are the most feasible
ones, but not the only possible.
In a linear Markov chain of n + 1 states
(a group of n servers) all states have two
neighbours, except states 0 and n that
have only one. The dwelling time in any
state i is exponential with mean 1/(λi +
µ i ), where µ 0 = λn = 0. (Arrivals in state
n get lost and do not influence the system,
and there can be no departures in
state 0.) When the system changes to
state i from i – 1 or i + 1, it may go direct
to i + 1 after an exponential interval, or it
may first go to i – 1 and possibly jump
forth and back in states i, i – 1, i – 2, ...
until it eventually returns to state i + 1. It
is thus obvious that the interval elapsed
between a transition to i (i > 0) and a
transition to i + 1 is not exponential. It is,
however, renewal, since it is impossible
to distinguish between different instances
of state i. There is no memory of the
chain of events that occurred before an
arrival in state i.
Assume a group of n servers split in a
primary group P of p servers and a secondary
group S of s = n – p servers (Figure
24). In a sequential search for a free
server an s-server will only be seized if
A
(P)
ooo------o
P
(S)
ooo------o
s=n-p
Figure 24 A server group split in a primary
group (P) and a secondary group (S)
the whole p-group is occupied. There
will be two distinct situations in the
moment of a new arrival:
1 arrivals and departures in P keep the
number of busy servers in the primary
group < p (there may be ≥ 0 busy
servers in S)
2 all servers in P are busy.
During situation 1) all calls go to P,
while in situation 2) all calls go to S. The
primary group P will see a Poisson
arrival process and will thus be an Erlang
system. By the reasoning above, however,
the transitions from situation 1) to
situation 2), and hence the arrivals to
group S, will be a non-Poisson renewal
process. In fact, in an Erlang system with
sequential search, any subgroup of
servers after the first server will see a renewal,
non-Poisson, arrival process.
The secondary group S is often termed an
overflow group. The traffic characteristic
of the primary group is that of a truncated
Poisson distribution. The peakedness
can be shown to be
yp = 1 – A [E(p – 1,A) – E(p,A)], (61)
where yp < 1, since E(p – 1,A) > E(p,A)
The overflow traffic is the traffic lost in
P and offered to S:
As = A · E(p,A)
and the primary traffic is
Ap = A – As = A[1 – E(p,A)]
For As the peakedness is ys > 1. (To be
discussed later.) A consequence of this is
a greater traffic loss in a limited secondary
group than what would be the case
with Poisson input. The primary group
traffic has a smooth characteristic as
opposed to the overflow that is peaked. It
is obvious that it is an advantage to handle
a smooth traffic, since by correct
dimensioning one can obtain a very high
utilisation. The more peaked the traffic,
the less utilisation.
The main point to be noted is that the
original Poisson input is disturbed or
shaped by the system. If the calls carried
in the P-group afterwards go on to
another group, the traffic will have a
smooth characteristic, and the loss experienced
will be less than that of an original
traffic offer of the same size.
There are many causes for deviation
from the Poisson character, of which
some are already mentioned:
- The source group is limited
- Arrivals are correlated or in batches
(not independent)
- Arrival rate varies with time, non-stationarity
- Access limitation by grading
- Overflow
- Access limitation by link systems
- Repeated calls caused by feedback
- Intentional shaping, variance reduction.
We shall have a closer look at those conditions,
though with quite different emphasis.
14.1 Limited source group
The case has been discussed as Bernoulli
and Engset cases. The character is basically
Poisson, but the rate changes stepwise
at each event.
14.2 Correlated arrivals
There may exist dependencies between
traffic sources or between calls from the
same source. Examples are e.g.:
- several persons within a group turn
passive when having a meeting or
other common activity
- in-group communication is substantial
(each such call makes two sources
busy)
- video scanning creates periodic (correlated)
bursts of data.
The last example typically leads to a
recurring cell pattern in a broadband
transmission system.
14.3 Non-stationary traffic generation
Non-stationarity are essentially of two
kinds, with fast variations and slow variations.
There is a gradual transition between
the two. Characteristic of the fast
variation is an instantaneous parameter
change, whether arrival rate or holding
time. A transition phase with essentially
exponential character will result.
An example of instantaneous parameter
change is when a stable traffic stream is
switched to an empty group. Palm has
defined an equilibrium traffic that
changes exponentially with the holding
time as time constant. If thus the offered
traffic changes abruptly from zero to
λ⋅s, then the equilibrium traffic follows