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Lecture Series |
In this series of lectures, we will review
recent results on the modeling of flow control in stochastic networks, based on
algebraic representations of the pathwise dynamics of these networks.
The simplest networks to be considered are
those of the so-called max-plus linear
class, with dynamics admitting a representation in terms of products of
random matrices in this semi-ring. We
will first show that the regulation of such a network by an adaptive flow
control preserves max-plus linearity.
This reduction to iterates of random
non-expansive maps will then be used to:
q assess the stability of such networks via the
notion of a max-plus Lyapunov exponent;
q derive representation and computational
formulas for the steady state characteristics (e.g. throughput, stationary
waiting times or queue sizes).
We will also give a few examples of flow
control on more general classes of networks (e.g. multiclass networks) which are not amenable to such a linear
representation, and we will show that counter-intuitive phenomena may then
occur.
DETAILED PROGRAM OF THE LECTURES AND RELATED REFERENCES
1. The max-plus semi ring, spectral theory of matrices, and window
flow control.
F. Bacelli, G. Cohen, G.J. Olsder, J.P.
Quadrat, Synchronization and Linearity, Wiley 1992.
2. Products of random matrices, Lyapunov exponents, stochastic
eigenvalue problems, stochastic affine equations.
F. Bacelli & J. Mairesse, Ergodic
Theorems for Stochastic Operators and Discrete Event Systems, in Idempotency,
J. Gunawardena Ed. Cambridge University Press, 1998.
and the references therein.
3. Mathematical modeling of TCP.
F. Bacelli & D. Hong, TCP is max-plus
linear, Sigcomm 2000.
4. Analyticity and expansions for max-plus Lyapunov exponents.
F. Bacelli, S. Gaubert \& D. Hong,
Representation and Expansions of max-plus Lyapunov Exponents, Proceedings of
the Allerton Conference (1999)
and the references therein.
5. Window flow control and
multiclass networks.
F. Bacelli & T. Bonald, Window Flow
Control in FIFO Networks with Cross Traffic,
Questa, 32 (1999) 195-231.
TRANSPARENCIES FROM THE LECTURES