Performance of queueing networks

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Complex manufacturing plants or telecommunication networks may be modelled as queueing networks.  Orders or packets progress from node to node and wait in queues to obtain service. The performance of the network is usually measured by the throughput or the rate at which orders or packets successfully travers the network. Other considerations include the rarity of network overloads and the mean time to transit the network. My first project involves a constructive large deviation theory for estimating the probability of rare events like network overload.  Rare events need a mathematical investigation since simulation can only generate such events with a lot of computational effort.  I am interested in the state of the network at the moment of overload since one can use this as the starting point for investigating a proposed overload recovery algorithm.  I am also interested in rare events in multitype networks and polling networks which defy current large deviations analysis. My second project involves a mean field analysis of the Internet protocols. We have described the evolution of the histogram of all the transmission window sizes of the sources transmitting through a bottleneck router.  The theoretical distribution has been confirmed by network measurement.  We want to use our tools to investigate some proposed control protocols.  The analysis also raises interesting questions about network turbulence.  There may simultaneous exist a fluid regime where HTTP sources transmit files without loss as well as a lossy regime with periodic burst of packet losses.  These regimes may be viewed as metastable states of a Markov chain.  Since the theory of large deviations for measure (or histogram) valued processes is incomplete we have lots of scope for new mathematical discoveries as well as interesting applications.