Contents Telektronikk - Telenor

Frame 2
Useful time distributions
Negative exponential distribution (ned)
The most interesting time distribution is the exponential distribution
F(t) = 1 – e –λt ; G(t) = e –λt ; f(t) = λe –λt (16)
with the Laplace transform of f(t)
L{s} = f*(s) = λ/(λ + s) (17)
and the nth derivative
(−1) n L (n) {s} =
(18)
A particular property of the exponential distribution is that the
coefficient of variation c = σ/µ = 1 and the form factor
ε = v/µ 2 + 1 = 2. This is used to distinguish between steep
(c < 1) and flat (c > 1) distributions. It has a positive skewness
s = 2.
Hyperexponential distribution
The hyperexponential distribution is obtained by drawing at
random, with probabilities p 1 , p 2 , p 3 , ... , p k , from different
exponential distribution with parameters λ 1 , λ 2 , λ 3 , ... , λ k
For a true distribution we must have
k
∑ pi = 1
i=1
The n th moment similarly becomes (as the Laplace transform
of a sum is the sum of the Laplace transforms)
Hyperexponential distributions are always flat distributions
(c > 1).
Erlang-k distribution
(19)
(20)
(21)
If k exponentially distributed phases follow in sequence, the
sum distribution is found by convolution of all phases. If all
phases have the same parameter (equal means) we obtain the
special Erlang-k distribution (which is the discrete case of the
Γ-distribution),
f (t) =
λ (λt)k −1
(k −1)! e−λt
n!⋅λ
(λ + s) n+1 ;⇒ M n!
n =
λ n
k
F(t) = 1− ∑ pi ⋅e
i=1
−λ i t ; f (t) = piλ ie −λ
k
∑ i
i=1
t
k
k
n
Mn = ∑ pi ⋅ Mni = n!⋅ ∑ pi / λi i=1
i=1
(22)
Since the Laplace transform of a convolution is the product of
the Laplace transforms of the phases, we have for the special
Erlang-k distribution
k
⎛ λ ⎞
L{s} = ⎜ ⎟
⎝ λ + s ⎠
(23)
and the nth moment is found by
(-1) n ⋅ L (n) {s} = k ⋅ (k + 1) ... (k + n - 1) ⋅ λk /(λ + s) k+n
⇒ Mn = k ⋅ (k + 1) ... (k + n - 1)/λn (24)
The mean value is µ = k/λ. A normalised mean of µ = 1/λ is
obtained by the replacement λ ⇒ kλ or t ⇒ kt in the distribution
and the ensuing expressions. If k → ∞ for the normalised
distribution, all central moments approach zero, and the
Erlang-k distribution approaches the deterministic distribution
with µ = 1/λ.
For an Erlang-k distribution it applies in general that
k
µ = M1 = ∑µ
i
i=1
k
mn = ∑ mni for n = 2or3
i=1
k
mn ≠ ∑ mni for n = 4,5,...
i=1
For the general Erlang-k distribution the phases may have different
parameters (different means). The expressions are not
quite so simple, whereas the character of the distribution is
similar. The particular usefulness of the Erlangian distributions
in a traffic scenario is due to the fact that many traffic
processes contain a sequence of independent phases.
β-distribution
The β-distribution is a two-parameter distribution with a variable
range between 0 and 1, 0 ≤ x ≤ 1. Thus, it is useful for
description of a population with some criterion within this
range, for example average load per destination, urgency of a
call demand on a 0 – 1 scale, etc. The distribution density
function is
f (x) =
Γ(α + β )
Γ(α )⋅Γ(β ) xα −1 ⋅ 1− x
( ) β −1 ;α,β > 0
The n th moment is given by
n−1 (α + i)
Mn = ∏
i=0 (α + β + i)
Typical of the above distributions is
Special Erlang-k: c = 1/√k < 1 (steep, serial)
Exponential (ned): c = 1
Hyperexponential: c > 1 (flat, parallel)
β-distribution: c > 1 for α < 1 and β > α(α + 1)/(1 – α)
c < 1 otherwise
(25)
(26)
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