On the shortest queue version of the Erlang loss model

We consider two parallel M/M/N/N queues. Thus there are N servers in each queue and no waiting line(s). The network is fed by a single Poisson arrival stream of rate lambda, and the 2N servers are identical exponential servers working at rate mu. A new arrival is routed to the queue with the smaller number of occupied servers. If both have the same occupancy then the arrival is routed randomly, with the probability of joining either queue being 1/2. This model may be viewed as the shortest queue version of the classic Erlang loss model. If all 2N servers are occupied further arrivals are turned away and lost. We let rho = lambda/mu and alpha = N/rho = N mu/lambda. We study this model both numerically and asymptotically. For the latter we consider heavily loaded systems (rho -> infinity) with a comparably large number of servers (N -> infinity with a = O(l)). We obtain asymptotic approximations to the joint steady state distribution of finding m servers occupied in the first queue and n in the second. We also consider the marginal distribution of the number of occupied servers in the second queue, as well as some conditional distributions. We show that aspects of the solution are much different according as a > 1/2, a approximate to 1/2, 1/4 < a < 1/2, a approximate to 1/4 or 0 < a < 1/4. The asymptotic approximations are shown to be quite accurate numerically.