Mathematics of Bandwidth Sharing and the Future Internet

F. Baccelli.

* Nobody has ever produced a comprehensive quantitative
  model for the dynamics of the competition between flows
  that takes into account 1) the congestion control algorithms
  used or to be used in the Internet; 2) the statistical
  structure of typical Internet applications; 3) the basic
  queueing/scheduling mechanisms used in access/core
  networks. The design of new distributed congestion control
  algorithms with predictable statistical behavior will
  require some mathematical breakthrough on the modeling
  of flow competition within this context.

* The dynamics of large overlay networks can be seen as yet
  another type of interaction between a large collection of
  point to point flows, which involves competition, but also
  collaboration (e.g. client-server, split etc.), with as a
  corollary new phenomena such as starvation, overflow at
  overlay nodes etc. The understanding of the interplay
  between these local point to point mechanisms, the overlay
  control and classical applications is a major challenge.

* From a given geographic point, the Internet will more and
  more be accessed through a variety of coexisting wireless
  technologies: WIFI meshes, WIMAX, 3G/LTE, cellular etc.
  Current routing and congestion control mechanisms succeed
  very well in sharing dynamically the wired resources of the
  Internet in a fully distributed way. There is no analogue of
  this for efficient spectrum sharing in heterogeneous wireless
  access networks yet. There is a need for scalable user
  association/split algorithms, power control algorithms, access
  scheduling algorithms and congestion control mechanisms within
  this context that take into account the specificities of
  wireless channels.

For addressing these questions, new mathematical tools will
be needed from such disciplines as probability theory,
statistics, control, discrete event dynamical systems,
statistical physics. Scalable simulation methodologies
will also be essential within this framework.