  We consider a system of N queues with decentralized load balancing such as
power-of-D strategies(where D may depend on N) and generic scheduling
disciplines. To measure the dependence of the queues, we use the clan of
ancestors, a technique coming from interacting particle systems. Relying in
that analysis we prove quantitative estimates on the queues correlations
implying propagation of chaos for systems with Markovian arrivals and general
service time distribution. This solves the conjecture posed by Bramsom et. al.
in [*] concerning the asymptotic independence of the servers in the case of
processor sharing policy. We then proceed to prove asymptotic insensitivity in
the stationary regime for a wide class of scheduling disciplines and obtain
speed of convergence estimates for light tailed service distribution.
  [*] M. BRAMSON, Y. LU AND B. PRABHAKAR, Asymptotic independence of queues
under randomized load balancing, Queueing Syst., 71:247-292, 2012.
