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<div class="moz-cite-prefix">On 08/25/2014 10:09 AM, Detlef Bosau
wrote:<br>
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<blockquote cite="mid:53FB6DB6.2030805@web.de" type="cite">
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<div class="moz-cite-prefix">Am 23.08.2014 um 05:34 schrieb Jack
Haverty:<br>
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It's been a looonnggg time, but I still remember studying a
lot of mathematics about 50 years ago - queueing theory, graph
theory, etc. Used to be able to do it too.<br>
<br>
My recollection is that terms such as "flow control" and
"congestion control" were used in mathematics, well before
they were used in computer networks. <br>
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<br>
<br>
Hm. I read quite a lot of mathematical models used for computer
networks.<br>
<br>
However, I never happened to see, how flow control and congestion
control were modelled. The models were made that abstract, that
congestion control and flow control vanished. <br>
<br>
I really appreciate concrete pointers here. <br>
<br>
Detlef <br>
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<br>
Detlef,<br>
<br>
I suggest you research the literature of the "Operations Research"
branch of mathematics. That is where pure mathematics concepts of
queueing theory et al. were applied to real-world problems.<br>
<br>
I did a quick google search of ("queueing theory" "flow control")
and found this example of a mathematical paper discussing queues and
"flow control" in computer networks:<br>
<br>
<a class="moz-txt-link-freetext" href="http://www.jstor.org/discover/10.2307/2582975?uid=2&uid=4&sid=21104558206567">http://www.jstor.org/discover/10.2307/2582975?uid=2&uid=4&sid=21104558206567</a><br>
<br>
Here's another one, about flow control in "supply chain networks" in
the manufacturing environment:<br>
<br>
<a class="moz-txt-link-freetext" href="http://www.isr.umd.edu/~baras/publications/papers/2012/Ion_Asane_MTNS2012.html">http://www.isr.umd.edu/~baras/publications/papers/2012/Ion_Asane_MTNS2012.html</a><br>
<br>
What we might call "packet loss" they might call "supply chain
disruption".<br>
<br>
I suspect most of the material you may find online will be about the
use of mathematics in computer networks. Unfortunately, most
earlier work, before we had the Internet, is probably not available
online - it may only be in university libraries. So that's where
you may be able to find papers on flow control in pre-computer
environments like railroads.<br>
<br>
Also, the basic mathematical concept which we networking people call
"flow control" might have been described using different terminology
in, for example, the old railroad or other business examples.<br>
<br>
I suggest researching Bell Laboratories work from before 1970.
They did a lot of theoretical work modelling the telephone network,
and in particular the issues of managing many simultaneous voice
calls. The problem of designing the telephone network to minimize
the probability of busy signals is about the management of multiple
simultaneous flows and controlling congestion on circuits in the
interior of the network. But they may have used different
terminology.<br>
<br>
My recollection is also that, as you said, the mathematical models
were so abstract that they were not very useful in the real world of
computer networks. Mathematics could be used to model hypothetical
cases, and was useful to see how things might behave in theory.
The real-world was sufficiently chaotic and unpredictable that it
was difficult to model with sufficient accuracy. That's why the
Internet was built by a continuing series of experiments and
refinements.<br>
<br>
At one point, someone published a mathematical paper that proved
that the ARPANET would lock up and all traffic flows would stop.
This cause some great concerns among the users of the ARPANET who
were depending on it. Our analysts at BBN examined that paper in
great detail and concluded that it was mathematically correct -- but
one of the assumptions made in the model was that every packet
switch computer was started at the same moment in time, and all
those computers ran at exactly the same speed so instructions in all
machines were executed in total synchrony. We advised the users
that, if such a situation could be created, the ARPANET would crash,
but that the likelihood of that situation of perfect synchrony was
so tiny that there was no reason to worry. It was a mathematically
interesting theoretical problem, but not a real-world concern.<br>
<br>
I can't recall when I first encountered the terms "flow control" or
"congestion control" in computer networks, or seeing any formal
definitions of those terms. <br>
<br>
My personal view is that "flow control" refers to the management of
a single flow of information between two end-points. It could be a
TCP connection, or a telephone call, or a stream of railroad cars
between a factory and warehouse. There are mechanisms in the
endpoints, as well as in the interior, to manage that flow.<br>
<br>
Conversely, "congestion control" refers to the management of a set
of many flows as they compete for resources. If there are too many
flows going through a bottleneck, congestion happens and may result
in broken flows or "busy signals".<br>
<br>
These two phenomena interact in complex ways, so it was common in
the early Internet work to discuss them both when working on any
particular problem. For example, the TCP algorithms in host
computers would interact with the routing algorithms in switching
components, the error-control algorithms on individual circuits, the
load-levelling schemes of server farms, and almost anything else you
can imagine that's involved in regulating the flows through the
Internet.<br>
<br>
IMHO, the Internet is way more complex than we know how to model.
It's probably at the same level of complexity as other hard problems
- weather, astronomy, etc.<br>
<br>
We didn't model it. We just built it.<br>
<br>
Hope this helps,<br>
/Jack Haverty<br>
<br>
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