Asymptotic properties of first passage times for two tandem queues: Time to empty the system

We consider tow M/M/1 queues in series. Assuming that at time t = 0 there are n1 (n2) customers in the first (second) queue, we compute asymptotic expansions for the time to empty the system (residual busy peroid). We give explicitly the mean and variance for the emptying time, assuming that n1 and/or n2 are large. We derive recursion equations satisfied by the moments and analyze these equations using singular perturbation techniques, such as boundary layer theory and asymptotic matching.