25-08-2017, 09:32 PM
Where Mathematics Meets the Internet
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Introduction
The Internet has experienced a fascinating evolution
in the recent past, especially since the early
days of the Web, a fact well documented not only
in the trade journals but also in the popular press.
Unprecedented in its growth, unparalleled in its heterogeneity,
and unpredictable or even chaotic in
the behavior of its traffic, “the Internet is its own
revolution,” as Anthony-Michael Rutkowski, former
executive director of the Internet Society, likes to
put it. At the same time, folklore has it that mathematics
lies at the heart of Internet operation.
After all, the argument goes, mathematics is the
language of computers, and the Internet is currently
connecting tens of millions of them and
still doubling every year [Lo98]. Yet the Internet is
a new world, one where engineering reality wins
over tradition-conscious mathematics and requires
“paradigm shifts” that favor a combination of
mathematical “beauty” and high potential for contributing
to pragmatic Internet engineering.
Teletraffic Theory and Internet
Engineering
The term “teletraffic theory” originally encompassed
all mathematics applicable to the design,
control, and management of the public switched
telephone networks (PSTN): statistical inference,
mathematical modeling, optimization, queueing
and performance analysis. Later its practitioners
would extend this to include data networks such
as the Internet too. Internet engineering, an activity
that includes the design, management, control,
and operations of the global Internet, would thus
become part of teletraffic theory, relying on the
mathematical sciences for new insights into and
a basic understanding of modern data communications.
However, from its early days the Internet
emphasized engineering and experimentation and
was less concerned with mathematics and theory.
In fact, some in the Internet community are quick
to point out that today’s Internet “works” because
it ignored mathematics—in particular, teletraffic
theory—and herein lies an interesting tale.
Mathematics and POTS
For someone living in an industrialized country,
what is the likelihood of not getting a dial tone
when trying to make a phone call?2 Now, what
about not being able to connect to a popular Web
server over the Internet?
The Changing Internet
The first, basic element of change concerning
the Internet is that of growth. Simply put, the
network grows exponentially, has done so for
well over a decade, and shows no signs of
slowing down. Figure 4 illustrates one growth
statistic: the volume of traffic in bytes per day
flowing through the USENET bulletin board
system. The data start in 1984 and continue
to 1994. The measurements fit beautifully a
straight line, reflecting sustained exponential
growth of about 80% per year for over a decade
(note log-linear scale). Clearly, Internet growth
is nothing new—it in no way began with the
Web—and current statistics are consistent with
the growth continuing completely unabated.
Should Mathematicians Care?
The original finding of fractal scaling phenomena
in Internet traffic was greeted with skepticism by
many mathematicians. They considered it as yet
another example of a “fad” that comes and goes,
with ultimately nothing to show for it, similar to
what had happened in other areas in the natural
or social sciences such as hydrology, economics,
or biophysics, where the fractal “craze” proved to
be short lived and had absolutely no impact beyond
some philosophical discussions about the general
purpose of modeling.