  This paper investigates a partially observable queueing system with $N$ nodes
in which each node has a dedicated arrival stream. There is an extra arrival
stream to balance the load of the system by routing its customers to the
shortest queue. In addition, a reward-cost structure is considered to analyze
customers' strategic behaviours. The equilibrium and socially optimal
strategies are derived for the partially observable mean field limit model.
Then, we show that the strategies obtained from the mean field model are good
approximations to the model with finite $N$ nodes. Finally, numerical
experiments are provided to compare the equilibrium and socially optimal
behaviours, including joining probabilities and social benefits for different
system parameters.
