Chaoticity on path space for a queueing network with selection of the shortest queue among several

We consider a network with N infinite-buffer queues with service rates lambda, and global task arrival rate N nu. Each task is allocated L queues among N with uniform probability and joins the Least loaded one, ties being resolved uniformly. We prove Q-chaoticity on path space for chaotic initial conditions and in equilibrium: any fixed finite subnetwork behaves in the limit N goes to infinity as an i.i.d. system of queues of law Q. The law Q is characterized as the unique solution for a non-linear martingale problem; if the initial conditions are q-chaotic, then Q(0) = q, and in equilibrium Q(0) = q(rho) is the globally attractive stable point of the Kolmogorov equation corresponding to the martingale problem. This result is equivalent to a law of Large numbers on path space with limit Q. and implies a functional law of large numbers with limit (Qt)t greater than or equal to 0. The significant improvement in buffer utilization, due to the resource pooling coming from the choices, is precisely quantified at the limit.