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Large Deviations at Equilibrium For a Large Star- Shaped Loss Network

Graham, Carl; O'Connell, Neil


Keyword(s): large networks; stationary distribution; partition function; large deviations

Abstract: Please Note. This abstract contains mathematical formulae which cannot be represented here. We consider a symmetric network composed of N links, each with capacity C. Calls arrive according to a Poisson process and each call requires L distinct links (chosen at random). If each of these links has free capacity, the call is held for an exponential time; otherwise it is lost. The semi-explicit stationary distribution for this process is similar to a Gibbs measure: it involves a normalizing factor, the partition function, which is very difficult to evaluate. We consider the limit N * * with the offered arrival rate to a link fixed. We use asymptotic combinatorics and recent techniques involving the law of large numbers to obtain the logarithmic equivalent for the partition function, and deduce the large deviation principle for the empirical measure of the occupancies of the links. We give an explicit formula for the rate function and examine its properties. Notes: Carl Graham, CMAP, Ecole Polytechnique, 91128 Palaiseau, France

21 Pages

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