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Statistical
Distributions in
Telecommunications
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The use of statistical models is essential to
describe:
non-guided propagation in random
environments;
users mobility; phone calls and data connections;
users influence in network performance.
Basic NotionsBaNo (1/4)
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Given the probability density, p(x), and cumulative
distribution functions, P(x), i.e., PDF and CDF,
one has
or
where
P(x) = Prob(X x)
One has
Prob(|X| x) = P(x) - P(-x)
Basic NotionsBaNo (2/4)
dx
dPxp =)(
dttpxPx
=
)()(
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The main parameters are:
average,
mean square,
median, xm
Basic NotionsBaNo (3/4)
dxxpxx =
)(
dxxpxx =
)(22
x
2x
2/1)()( ==
dxxpdxxpm
m
x
x
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mode, mx
p(mx) = max[p(x)]
moments, n
variance, x2, and standard deviation, x,
x2 =2
Chebyshevs inequality allows to quantify thedispersion of the random variable:
Basic NotionsBaNo (4/4)
( ) dxxpxx nn =
)(
( )2
1Prob
kkxx x
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CDF and PDF:
Uniform DistributionUnDi (1/2)
=
o.c.,0
,
1
)( bxaabxp
++
vvv
ev
v
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Occurrence intervals
Normal (Gauss) DistributionNoDi (3/5)
u 1 - P(u) u 1 - P(u)
0 5.00010-1 1.282 10-1
1 1.58710-1 2.326 10-2
2 2.27510-2 3.090 10-3
3 1.35010-3 3.719 10-4
4 3.16710-5 4.265 10-5
5 2.86710-7 4.753 10-6
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Diagrams in Gauss Scale
Normal (Gauss) DistributionNoDi (4/5)
[Source: Boithias, 1983]
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It is used to describe fluctuations around a mean
value.
It cannot be used to describe entities that cannot be
negative.
Normal (Gauss) DistributionNoDi (5/5)
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PDF e CDF:
Log-Normal DistributionLNDi (1/3)
0,121)( 2
)ln(2
>=
xex
xp
xx
l
l
l
2
2)ln(erf1
)(
+
= ll
xx
xP
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Parameters:
It is used to describe entities signal power or
amplitude, namely slow fading.
Log-Normal DistributionLNDi (2/3)
2/2ll
+
=
x
ex 222 ll +=x
ex
lx
m ex =2ll
=
x
x em
2/22
1lll
+
=
x
x ee
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PDF and CDF:
Log-Normal DistributionLNDi (3/3)
[Source: ITU-R, Vol. V, Rep. 1007]
log-normal
normal
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PDF and CDF:
Rayleigh DistributionRaDi (2/3)
[Source: ITU-R, Vol. V, Rep. 1007]
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It is associated to the magnitude of the sum of
vectors having amplitudes with a normal
distribution and phases with a uniform one.
It is used to describe intense fast fading.
Rayleigh DistributionRaDi (3/3)
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PDF and CDF:
It describes joint slow and fast fading.
Suzuki DistributionSuDi (1/2)
22/
2
2/),( vx
Re
v
xvxp =
2
2
)ln(
1
2
1)(
= ll
l
LN ep
)0(2
11)(
2/222
==+
l
l
vdtexP
text
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CDF:
Suzuki DistributionSuDi (2/2)
[Source: ITU-R, Vol. V, Rep. 1007]
l
2/ xx
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PDF:
describing the sum of a fixed vector (amplitude x0)
with a Rayleigh distributed vector (power ). It is used to describe non-intense fast fading.
Usually, the Rice parameter is used
Rice DistributionRiDi (1/3)
( )
=
+
2/I
2
)( 20
0
2
2
20
2
R
xxx
R x
xx
ex
x
xpR
220]dB[ log10 RxxK =
2
Rx
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PDF:
Rice DistributionRiDi (2/3)
[Source: Parsons, 1992]
22Rx 0x
K0
K1
K>>1
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PDF and CDF:
Parameters:
Exponential DistributionExDi (1/2)
0,e1
)( /- >= xx
xp xx
xx
exP/-
1)( =
22 2 xx =
0=xm
)2ln(xxm =
22 xx =
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It is used to describe the duration of various
phenomena, namely associated to signal fading and
phone calls.
Exponential DistributionExDi (2/2)
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Mass probability function
where q is the probability of occurring 1.
Parameters are
It is used to describe the occupation of a
telecommunications channel.
Bernouli DistributionBeDi (1/1)
1,0,)1()( == sqqsp ss
qs =)1(2 qqs =
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Mass probability function:
where q is the occurrence probability for each of
the n times. Parameters
Binomial DistributionBiDi (1/2)
nkqqk
nkp knk ,,0,)1()( K=
=
qk =
qnmk )]1int[( +=
)1(2 qqnk =
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It is used to describe phone calls, where q = t/n, t
being the sampling time interval and the average
phone calls generation.
Binomial DistributionBiDi (2/2)
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Mass probability function:
Parameters:
It is used to describe the generation of phone calls
(= t).
Poisson DistributionPoDi (1/1)
k
k
kp
= e!)(
k =k
2 =
>