Overloading parallel servers when arrivals join the shortest queue

We consider two parallel servers with an arrival stream of customers who join the shorter of the two servers' queues. We use the theory of Markov additive processes to prove exact asymptotic results on the mean time for the queue at the first server to attain a high level l. We also give the limiting behaviour of the second queue at this hitting time as well as the steady state of the second queue when the first contains l customers.