Contents Telektronikk - Telenor

24
200
184
150
100
50
0
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70
60
50
40
30
20
10
0
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2
1,5
1
0,5
0,5% loss line
Server group size
0
25 26 27 28 29 30
Figure 26 Observed undisturbed (Poisson) traffic, smoothed traffic and overflow
The traffic offered to those j servers is
thus the overflow from the initial i
servers:
A ⋅ E (i,A) = A'
If this traffic were fresh Poisson traffic,
the carried traffic would be
Aj' = A' ⋅ [1 – E(j,A')]
Example:
A = 5 Erlang, i = 5, j = 1
Aj = 5 ⋅ [E(5, 5) – E (5 + 1, 5)]
= 5 ⋅ (0.284868 – 0.191847)
= 0.465 Erlang
A' = 5 ⋅ E (5, 5) = 5 ⋅ 0.284868
= 1.42434
Aj' = 1.42434 ⋅ [1 – E (1, 1.42434)]
= 1.42434 ⋅ [1 – 0.587]
= 0.588 Erlang
The example illustrates that the traffic
carried by a single server is smaller when
the offered traffic is an overflow than
when it is fresh traffic (Aj /Aj' = 0.465/
0.588 = 0.79). The offered traffic mean is
the same in both cases. As this applies to
any server in the sequence, it will also
apply to any server group. In other
words, overflow traffic suffers a greater
loss than fresh traffic does.
This property of overflow traffic can be
explained by the greater peakedness
(variance-to-mean ratio) of the overflow.
This is the opposite property of that of
the carried traffic in the primary group,
as shown in equation (61). Calculation of
the variance is not trivial. The formula is
named Riordan’s formula and is remarkably
simple in its form:
1− M + A
V = M ⋅
n +1− A + M
(62)
where
A = traffic offered to the primary
group (Poisson type)
n = the number of servers in the
primary group
M = A ⋅ E (n,A) = mean value of the
overflow.
We note that n and A are the only free
variables, and M must be determined
from Erlang’s loss formula, so there is no
explicit solution where V is expressed by
n and A only.
The variance-to-mean ratio = peakedness
= y = V/M = 1 – M + A/(n + 1 – A + M)
of an overflow stream is an important