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We consider two parallel, infinite capacity, queues characterized by ( ) with denoting the unfinished work (buffer content) in queue . A new arrival is assigned to the queue with the smaller buffer content. We construct formal (as opposed to rigorous) asymptotic approximations to the Joint stationary distribution of the Markov process ( ), treating separately the asymptotic limits of heavy traffic, light traffic, and large buffer contents. In heavy traffic, the stochastic processes and become independent, with the distribution of identical to the heavy traffic waiting time distribution in the standard queue, and the distribution of closely related to the tail of the service time density. In light traffic, we obtain a formal expansion of the stationary distribution in powers of the arrival rate.