Queueing Systems Large Deviations Analysis of the Generalized Processor Sharing Policy

In this paper we consider a stochastic server modeling a multiclass communi cation switch fed by a set of parallel bu ers The dynamics of the system evolve in discrete time and the generalized processor sharing GPS scheduling policy of is implemented The arrival process in each bu er is an arbitrary and possibly autocorrelated stochastic process We obtain a large deviations asymptotic for the bu er over ow probability at each bu er In the standard large deviations method ology we provide a lower and a matching up to rst degree in the exponent upper bound on the bu er over ow probabilities We view the problem of nding a most likely sample path that leads to an over ow as an optimal control problem Using ideas from convex optimization we analytically solve the control problem to obtain both the asymptotic exponent of the over ow probability and a characterization of most likely modes of over ow These results have important implications for traf