  A multiclass queue with many servers is considered, where customers make a
join-or-leave decision upon arrival based on queue length information, without
knowing the scheduling policy or the state of other queues. A game theoretic
formulation is proposed and analyzed, that takes advantage of a phenomenon
unique to heavy traffic regimes, namely Reiman's snaphshot principle, by which
waiting times are predicted with high precision by the information available
upon arrival. The payoff considered is given as a random variable, which
depends on the customer's decision, accounting for waiting time in the queue
and penalty for leaving. The notion of an equilibrium is only meaningful in an
asymptotic framework, which is taken here to be the Halfin-Whitt heavy traffic
regime. The main result is the identification of an $\epsilon$-Nash equilibrium
with probability approaching 1. On way to proving this result, new diffusion
limit results for systems with finite buffers are obtained.
