The problem of how to control communication networks is both mathematically challenging and very important practically, with network operators, software vendors, network regulators and application developers actively seeking ways of delivering new services reliably and effectively. For example current controls in the Internet do not scale well, and are poorly suited to deal with new and emergent services and traffic patterns. The interplay between the user and the network has attracted the attention of economists as well as mathematicians and engineers.

Topics will be selected from amongst the following:

• Models of queueing networks and of loss networks (Markov process models of networks, equilibrium distributions, closed and open queueing networks, reversibility, loss networks).
• One-dimensional networks (Burke's theorem, Poisson flows, independence results for queueing networks, quasi-stationary distributions for loss networks).
• Approximations and asymptotics for loss networks (Erlang fixed point, uniqueness under fixed routing, possible non-uniqueness underdynamic routing, limit results for large capacities and for star networks).
• Capacity allocation and routing strategies in queueing and loss networks (Kleinrock's result, Gallager's algorithm, shadow prices, decentralization, convergence considerations, analogies with deterministic network flow). Dynamic routing in queueing and loss networks (sticky random routing in loss networks, Hajek's result for random graphs, state space collapse in queueing networks).
• Random access schemes (Aloha, Ethernet, critical rates for collision detect schemes, Aldous' result for binary exponential back-off).
• Models of broadband networks (statistical sharing, queueing and multiplexing models, effective bandwidths, connection acceptance, pricing).
• Stochastic network controls for loss and queueing networks. Loss networks: dynamic programming, reversible controls, trunk reservation. Queueing networks: control of a single queue; individual and social optimisation.
• Broadband controls: rate-based controls (leaky buckets) analysis and problems; Cruz's network calculus; heavy traffic analysis of queues in series. Multiple timescales, critical timescales and long-range dependence.
• `Internet Networks': control of elastic traffic: window-based flow control, modelling TCP (simple model and Ott's stochastic model of a single link); social, network and user optimisation; prices and tariffs as control structures; examples of control structures for the Internet.

References

1.

F.P. Kelly, S. Zachary and I.  Ziedins (eds), Stochastic Networks: Theory and Applications, RSS Lecture Note Series 4, Clarendon Press, Oxford (1996).

2.

K.W. Ross, Multiservice Loss Models for Broadband Telecommunication Networks, Springer-Verlag, London (1995).

3.

F.P. Kelly, Reversibility and Stochastic Networks, Wiley, Chichester (1979).

4.

P. Whittle, Optimization Over Time, Wiley, Chichester (1983).

5.

J.W. Roberts, U. Mocci and J. Virtamo (eds), Broadband Network Teletraffic, Springer, Berlin (1996).

6.

S.M. Ross, Introduction to Stochastic Dynamic Programming, Academic Press (1983).