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

Abstract

We consider a network with N infinite-buffer queues with service rates λ, and global task arrival rate Nν. 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 ρ 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 (Q t ) t≥0. The significant improvement in buffer utilization, due to the resource pooling coming from the choices, is precisely quantified at the limit.