Contents Telektronikk - Telenor

them gives similar formulas for delay
systems that was given for the pure loss
system in the previous section:
The congestion formula, E2,n , originally
given by Erlang in 1917, can be expressed:
E2,n =
(12)
From this formula Palm deduces the following
recursion relations:
E2,n =
E2,n−1 =
(13)
It is also possible to express E 2,n in terms
of E 1,n . and vice versa:
E2,n =
E1,n =
A n
(n−1)!
�n−1 Ai
i=0 (n − i) i!
A(n − 1 − A)E2,n−1
(n − 1)(n − A) − AE2,n
(n − 1)(n − A)E2,n
A(n − 1 − A + E2,n)
AE1,n−1
n − A + AE1,n−1
(n − A)E2,n
n − AE2,n
(14)
Palm next goes through the derivation of
the M/M/n and M/D/n FIFO queues.
Even though the derived formulas go
back to Erlang [7], Molina [8] and Crommelin
[9], Palm’s logical derivations and
interpreting comments are well worth
reading. He also compares the waiting
time distributions for constant versus
exponentially distributed holding times
for quite a large number of cases. He
concludes that “... it was found that the
differences in general are so small that
they must be considered as lacking significance
in the work of dimensioning.
The difference will be smaller the larger
the groups are. Only with very small
groups will the difference be of such a
magnitude that one possibly should take
them into account.”
It is also interesting to note that Palm in
the same paper even includes treatments
of queueing system with voluntary departure
of those waiting, i.e. queueing
systems where the subscribers get impatient
and replace the receiver. Whether
the deducted formulas originally go back
to Palm himself is a bit unclear, but the
only reference he gives is to a paper by
himself from 1937 [10]. It is, however,
rather typical for Palm to write long
papers where he includes several different
topics. The practice of today would
have been to chop them up into several
independent publications!
The last two papers in the series are:
“Waiting Times with random served
Queue” and “Duration of Congestion
States in Delay System”. They are both
long papers, perhaps a bit lengthy, with
detailed derivations and several numerical
calculations often presented graphically
in several curves. Personally,
I never used the results from the papers
so I renounce from making any further
comments.
5 Intensity variations in
telephone traffic
5.1 Palm’s dissertation and
Dr. Jacobæus
As mentioned in the introduction, Palm
wrote his famous doctor thesis with the
original German title: Intensitätsschwankungen
im Fernsprechverkehr in
1943. This fundamental work has since
been considered as the bible of teletraffic
theory. I will at this point refrain from
going through this work in detail and
rather limit myself to a few personal
comments connected to it. For an extensive
discussion I refer to the previously
mentioned work of Arne Jensen [2].
To me, the thesis of Palm will always be
closely associated with one of the great
pioneers of teletraffic theory, namely Dr.
Christian Jacobæus. I met Jacobæus for
the first time in 1976 at ITC 8 in Melbourne.
He was then still Technical
Director at L.M. Ericsson and was well
known to me through his famous works
within link systems that dated back to the
early fifties [11]. In fact, the so-called
‘chicken diagrams’, that is an instructive
way of depicting the hunting path
through a link system, go back to
Jacobæus and is still in use. Incidentally,
Palm himself acted as a tutor for
Jacobæus’ own thesis, just a year prior to
his (Palm’s) death.
Hence, it was with great respect I met
with the famous Jacobæus at that time,
myself being a newcomer in teletraffic
research. But Jacobæus was always interested
in new people and new ideas coming
into his field of interest, and did, in
fact, spend some time with us newcomers.
I still remember the nice evening
at the Sydney opera that culminated in a
dinner for the “Nordic group” hosted by
Jacobæus on behalf of L.M. Ericsson.
Since then, I happened to meet Jacobæus
several times over the years. He retired
from Ericsson just after the Melbourne
ITC but showed up regularly on the ITCs
and the Nordic Teletraffic Seminars
(NTS). He showed genuine interest in
any event that occurred within the teletraffic
field in Scandinavia. I remember
particularly in 1982, when Bjarne Helvik
at ELAB was about to defend his doctorate,
that Jacobæus showed up unexpectedly
the night before at the hotel in
Trondheim where Gunnar Lind and
myself were discussing the question to be
asked the next day. He firstly participated
actively in going through the dissertation
and secondly, acted as kind of a selfappointed
third opponent during the very
defense itself! I was told that something
similar happened the year before, in
1981, when Arne Myskja defended his
doctorate.
The reason for mentioning Jacobæus in
connection with the thesis of Palm is
because he was the one to translate it into
English. The translation was finished in
1987, the book was published in 1988
and thus made Palm’s work known to the
scientific community the world over. I
remember, though, that when I met
Jacobæus in Stockholm around 1981 he
told me already then that he was going to
translate Palm’s dissertation into English,
and whenever I met him in the following
years he talked about the progress with
his book. Hence, he must have spent
about 6 years on the translation, or at
least had it at the back of his mind during
these years. But the thesis [3], in its
translated version, contains more than
200 pages in double columns, and Palm’s
way of philosophizing definitely makes it
hard to translate. Besides, Jacobæus was
ill the last years before his death in 1988.
5.2 Intensity variations
As dwelt upon in the introduction,
Palm’s dissertation covers a broad range
of subjects, and may in itself be considered
as a textbook in teletraffic theory. His
style is not very modern, though, and he
likes to give lengthy arguments and
explanations before starting his mathematical
treatments. His terminology
might also differ somewhat from the one
of today. Hence, one has to read the text
carefully to fully grasp the underlying
idea. This certainly must have been a
challenge for the translator.
In telephony it is usually assumed that
the call arrivals are distributed randomly
over the time axis, and when one speaks
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