  This paper investigates the limit behavior of Markov Decision Processes
(MDPs) made of independent particles evolving in a common environment, when the
number of particles goes to infinity. In the finite horizon case or with a
discounted cost and an infinite horizon, we show that when the number of
particles becomes large, the optimal cost of the system converges almost surely
to the optimal cost of a discrete deterministic system (the ``optimal mean
field''). Convergence also holds for optimal policies. We further provide
insights on the speed of convergence by proving several central limits theorems
for the cost and the state of the Markov decision process with explicit
formulas for the variance of the limit Gaussian laws. Then, our framework is
applied to a brokering problem in grid computing. The optimal policy for the
limit deterministic system is computed explicitly. Several simulations with
growing numbers of processors are reported. They compare the performance of the
optimal policy of the limit system used in the finite case with classical
policies (such as Join the Shortest Queue) by measuring its asymptotic gain as
well as the threshold above which it starts outperforming classical policies.
