### Optimization and Systems Theory Seminar

Friday, May 24, 2002, 11.00-12.00, Room 3721, Lindstedtsvägen 25

**
Luigi Iannelli **

Dipartimento di Informatica e Sistemistica

Universitā degli Studi di Napoli Federico II

Napoli, Italy

E-mail: luiannel@unina.it

####
Pricing techniques for network congestion control

Communications networks are systems becoming more and more complex to
analyse and to control. While in the past "good" design rules have
been used to avoid or at least to reduce problems due to the finite
bandwidth allocation to an increasing number of users, recently some
researchers from different fields have begun to investigate such
problems by using mathematical modeling and to solve them by some "ad
hoc" tools that could be easily implemented.
A promising technique to control the congestion in communications
networks is the pricing: the idea is that users pay a price in order
to obtain some bandwidth. The congestion can be controlled by
opportunely acting on the price and the allocated bandwidth.
Controlling the congestion means to maximize an overall utility
function. The decentralized constraint of the solution (networks are
intrinsically decentralized systems) makes the problem very
interesting since it involves different disciplines such as
optimization, algorithm theory and control theory. In fact the
optimization problem needs a decentralized solution and, at the same
time, the algorithm that implements the solution must converge (we
want a stable system).

In this talk a review of different approaches to the problem will be
given. Firstly we discuss the work by Kelly et al. [1] in which the
algorithm for solving the optimization problem by pricing is modeled
as a continuous-time dynamical system (a couple of nonlinear
differential equations) and its stability is proved by Lyapunov
techniques. Then we review the work by Low and Lapsley [2], where
they propose a decentralized algorithm modeled as a discrete-time
dynamical system. Finally our work in di Bernardo et al. [3] will be
presented as a particular case of resource allocation interpreted as a
competitive model. A comparison of the different approaches and
common points among the solutions will be highlighted.

[1] F.P. Kelly, A.K. Maulloo, D.K.H. Tan, *Rate control for
communication networks: Shadow prices, proportional fairness and
stability*, Journal of the Operational Research Society 49 (1998),
237-252.

[2] S.H. Low, D.E. Lapsley, *Optimization flow control - I: Basic
algorithm and convergence*, IEEE/ACM Transactions on Networking,
vol. 7, no. 6, Dec. 1999.

[3] M. di Bernardo, F. Garofalo, L. Iannelli, D. Manfredi, F. Vasca,
*A competitive model of user behaviour for resource allocation in
congested networks*, submitted to the CDC 2002.

Calendar of seminars

*Last update: May 13, 2002 by
Anders Forsgren,
anders.forsgren@math.kth.se.
*