Contents Telektronikk - Telenor

Figure 3.3.b shows a model outline for a
slow receiver. For a slow sender the
transfer will never have to stop and wait,
and the transfer may be modelled by the
first two system states.
Figure 3.3.c shows how the system states
are detailed to fit the item size distribution
by a phase type distribution. Empirical
data from [28] shows that FTP-item
distribution does not follow a negative
exponential distribution. It is therefore
necessary to refine the Send … system
states to fit the empirical data. A closer
examination of the distribution revealed
that approximating it with a five branch
hyperexponential distribution gave an
acceptable fit, however, with deviations
from the empirical data for the protocol
specific “outliers” like items 512 bytes
long.
The transfer of items “belonging” to the
two branches of the hyperexponential
distribution representing the largest items
are chopped into smaller segments by the
Window full state. For the simplicity of
the example it is assumed that an
acknowledgement is received after a negative
exponentially distributed time with
mean λ-1 and that the transfer time of a
segment is negative exponentially distributed
with mean µ -1 . These distributions
may easily be replaced by more
realistic Erlang distributions. Note also
that the “chopping” due to waits for
acknowledgement, the two branches
must be kept separate.
3.1.2 Continuously varying activity
level
Some source types do not have distinct
system states. An example of such a
source type is variable bitrate coded
video. The procedure below indicates
how such a source may be approximated
by a state model by making the process
discrete. The procedure processes an
available sample output from the source.
A video source model with good performance,
defined according to a procedure
similar to the one below, is reported
in [29].
Smoothing
First, regard the information/bit-stream
from the source as a continuous process.
If it is not already smooth, the process
should be smoothed, for instance over a
moving window of a duration similar to
the typical time constant reflecting the
burst level, e.g. 20 – 200 ms. Such a
smooth stream is depicted in
Figure 3.4.a.
D
C
B
A
λ(t)
T
B
a) The information rate from the source with identification of information rate levels
A B C D
≈P{f(λ)∈C}
f(λ)
b) The probability density (pdf) f (λ) of the
source having bitrate λ at a random
instant and the approximation level
probabilities.
Level identification
The next step is to identify any levels in
the information rate from the source.
Such levels are recognised in Figure
3.4.a. An aid in identifying these levels
is to plot the frequency of the various
information rates. This is illustrated in
Figure 3.4.b, where recognised levels are
seen as peaks.
In cases where the source does not exhibit
levels and corresponding frequency
peaks, the range of the rate should be
split into appropriate levels. The higher
the number of levels, the higher the
potential modelling accuracy.
Each information generation level corresponds
to one state in the source type definition.
This is illustrated in Figure 3.4.c.
The detailed behaviour of the source
when it is in a certain state (within a
level) is encompassed in the pattern of
this state as discussed in the next section.
λ
#(B→A)
#(B→A,C,D)
A B
C D
Sojourn time
E[T B ]
c) The state diagram derived from source
behaviour. State corresponds to rate
level
Figure 3.4 Definition of a state diagram from source types without distinct states
Intralevel transition statistics
The relative numbers of jumps between
the various levels correspond to the state
transition probabilities. When a level is
only passed, i.e. the duration of the level
is significantly shorter than the average,
the level should be omitted. For instance
in Figure 3.4.a we take the following
jumps/transitions into account: B → C →
D → C → A → B → C, and the estimate
of the transition probability from B to A
based on this small sample becomes
pBA =
#(B → A)
#(B → A, C, D) =0
Correspondingly, the transition probability
from B to C is one. A real estimation
requires a far larger number of transitions.
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