Contents Telektronikk - Telenor

The corresponding distribution function is
F(t|i ≥ n) = 1 – e –µ(n – A) ⋅ t (83)
The unconditional distribution function, taken over all states
0 ≤ i ≤∞, is
F(t) = 1 – E2 ⋅ e –µ(n–A) ⋅ t (84)
It should be noted that F(0) = 1 – E2 . That indicates the probability
of zero waiting time, being equal to p(i < n). Thus there
is an infinite probability density in t = 0. It may look like a
lucky – and remarkable – coincidence, that the waiting time
distribution, as given by an infinite sum of geometrically
weighted Erlang-distributions turn out to give an exponential
distribution.
The mean waiting time in queue for those waiting can be
taken directly from the distribution function
The next attempt is not random, but conditioned
on the previous one encountering
a failure state. Given a calling rate of
λ for repetitions it can be shown by convolution
that the following failure rates
will be given by
fi>1 = (74)
This is the interdependence effect, causing
a jump from first to second attempt.
The selection and interdependence
effects are superimposed in the failure
rate, whereas only selection effect
applies to the persistence. The indexing
of δ and γ implies the selection effect,
while the indexing of λ is just in case
there is a change of repetition interval
with rank number. A calculated example
is given in Figure 34.
Observations show that when the causes
“subscriber busy” and “no answer” are
compared, “busy” has greater persistence
and shorter repetition intervals. Both
have a distinct jump in failure rate, but
not in persistence, from first to second
call. The selection effect of failure probability
seems to be stronger for “no
answer” than for “busy”.
λi + δi
λi + δi + γi
15 Waiting systems and
queues
Up till now we have made only sporadic
mention of waiting times and queues. In
real life those are too well known concepts.
Obviously, there are two parts in a
service situation, the part offering service
and the part seeking service. Both parts
want to optimise their situation, that is to
maximise benefit and minimise cost.
Particularly on the user (service seeking)
side convenience and inconvenience
come into the calculation beside the
purely economic considerations. A simple
question is: What is most inconvenient,
to be waiting or to be thrown out
and forced to make a new attempt? The
answer depends on the waiting time and
the probability of being thrown out.
There is also the issue of predictability
and fairness. In an open situation like
that of a check-in counter most people
prefer to wait in line instead of being
pushed aside at random, only with the
hope of being lucky after few attempts at
the cost of somebody else. A similar case
in telecommunication is that of calling a
taxi station. To stay in a queue and get
current information of one’s position is
deemed much better than to be blocked
in a long series of attempts.
In telephone practice blocking is used on
conversation time related parts like lines,
trunks, junctors and switches, whereas
waiting is used on common control parts
like signal senders and receivers, registers,
translators, state testing and connection
control equipment, etc. Blocking is
preferred because of the cost of tying up
expensive equipment and keeping the
user waiting. (Waiting time is proportional
to holding time.) This requires a
low blocking probability, less than a few
percent. The high probability of noncompletion
caused by “busy subscriber”
or “no answer” (10 – 70 %!) is a significant
problem. It is alleviated by transfer
and recall services, voice mail, etc.
w = 1/µ(n – A) = s/(n – A) (85)
Similarly, averaged on all customers, waiting time will be
W = E2 ⋅ w = E2 ⋅ s/(n – A) (86)
Alternatively, waiting times can be found by using Little’s
formula. (Note that the expression of L q2 , not L q1 , as defined
above, must be used):
w = Lq2
λ =
A s
=
(n − A) · λ n − A
W = L
λ = E2 · A
(n − A) · λ = E2
s
·
n − A
Another aspect of queuing is that of service
integrity. For real time communication
the admissible delay and delay variation
is very limited, lest the signal integrity
suffers, which severely affects the
use of queuing. On a different time scale
interactive communication also permits
only limited delay. However, it is not a
matter of signal deterioration, but rather
the introduction of undue waiting for the
user.
15.1 Queuing analysis
modelling
The loss systems treated up till now are
sensitive to the arrival distribution, and
we have mostly assumed Poisson arrivals.
Traffic shaping, in particular overflow,
changes the arrival distribution,
with significant consequences for the loss
behaviour. With Poisson arrivals a full
availability loss system is insensitive to
the holding time (service) distribution.
Queuing systems, on the other hand, are
in general sensitive to arrival distribution
and holding time distribution as well. It
should be distinguished, therefore, between
various cases for both elementary
processes.
Kendall introduced a notation of the
basic form A/B/C, where A indicates the
arrival distribution, B the service distribution
and C the number of servers. If
nothing else is said, an unlimited queue
with no dropouts is assumed. The simplest
model for analysis is the M/M/1-system,
with Markov input and service and a
single server. The notation G/G/n indicates
that any distributions are applicable.
Since dependence may occur, it
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