Large deviations and rare events in the study of stochastic algorithms

New asymptotics formulas for the mean exit time from an almost stable domain of a discrete-time Markov process are obtained. An original fast simulation method is also proposed. The mathematical background involves the large deviation theorems and approximations by a diffusion process. We are chiefly concerned with the classical Robbins-Monroe algorithm. The validity of the results are tested on examples from the ALOHA system (a satellite type communication algorithm).