Contents Telektronikk - Telenor

gestion, as determined from the distributions
by p(i = n). Call congestion is
given by the ratio of all calls that arrive
in state i = n. The main cases when call
congestion is different from time congestion
are
- arrivals are Markovian, but state
dependent (binomial, Engset)
- arrivals are non-Markovian.
An illustrating intuitive example of the
latter case is when calls come in bursts
with long breaks in between. During a
burst congestion builds up and many
calls will be lost. The congested state,
however, will only last a short while, and
there will be no congestion until the next
burst, thus keeping time congestion
lower than call congestion. The opposite
is the case if calls come more evenly distributed
than exponential, the extreme
case being the deterministic distribution,
i.e. constant distance between calls.
These cases are of course outside our
present assumptions.
The four cases in Frame 5 are summarised
in Table 3. Some comments:
The Poisson case:
1 With the assumptions of an unlimited
number of servers and a limited overall
arrival rate, there will never be any
calls lost.
2 The memoryless property of the Poisson
case for arrivals does not apply to
the Poisson case for number of customers
in the system. Thus, if we look
at two short adjacent time intervals, the
number of arrivals in the two intervals
bear no correlation whatsoever, whereas
the number in system of the two
intervals are strongly correlated. Still
both cases obey the Poisson distribution.
3 The departure process from the unlimited
server system with Poisson
input is a Poisson process, irrespective
of the holding time distribution. (The
M/G/∞ system.)
4 The Poisson distribution has in some
models been used to estimate loss in a
limited server case (n). Molina introduced
the concept “lost calls held”
(instead of “lost calls cleared”), assuming
that a call arriving in a busy state
stays an ordinary holding time and
possibly in a fictitious manner moves
to occupy a released server. The call is
still considered lost! The model
implies an unlimited Markov chain, so
that the loss probability will be
∞
− A
P{loss} = ∑ p(i) = e A
i=n
i ∞
∑ / i!
i=n
(60)
The Molina model will give a higher loss
probability than Erlang.
The Erlang case:
1 E(n,A) = B(n,A) is the famous Erlang
loss formula. In the deduction of the
formula it is assumed that holding
times are exponential. This is not a
necessary assumption, as the formula
applies to any holding time distribution.
2 If the arrival process is Poisson and the
holding times exponential, then the
departure process from the Erlang system
is also Poisson. This also applies
to the corresponding system with a
queue. (The M/M/n system, Burke’s
theorem.)
The Bernoulli case:
The limited number of sources, being
less than or equal to the number of
servers, guarantees that there will never
be any lost calls, even though all servers
may be busy. The model applies to cases
with few high usage sources with a need
for immediate service.
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The Engset case:
This case may be considered as an intermediate
case between Erlang and
Bernoulli. There will be lost calls, with a
loss probability less than the time congestion,
since B(N,n,b) = E(N–1,n,b)
< E(N,n,b). A curious feature is the analytic
result that offered traffic is dependent
on call congestion. The explanation is
the assumption that the call rate is fixed
for free sources, and with an increasing
loss more calls go back to free state
immediately after a call attempt.
Tabulations and diagrams have been
worked out for Engset, similar to those of
Erlang, however, they are bound to be
much more voluminous because of one
extra parameter.
In principle the Erlang case is a theoretical
limit case for Engset, never to be
reached in practice, so that correctly
Engset should always be used. For practical
reasons that is not feasible. Erlang
will always give the higher loss, thus
being on the conservative side.
A comparison of the four cases presented
in Frame 5 is done in Figure 23, assuming
equal offered traffic and number of
servers (in the server limited cases).
Bernoulli
Engset
Erlang
Poisson
0
0 2 4 6 8 10 12
Figure 23 Comparison of different traffic distributions
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