Probabilities of rare events in stochastic processes are ipso facto hard to estimate by direct Monte Carlo simulations. In recent years there has been considerable interest in estimating such probabilities via the importance sampling or likelihood ratio method, which involves choosing to simulate some other process for which the desired event is not rare. In the context of queueing networks, the new process is usually chosen by modifying the arrival and service time parameters of the original process. Various heuristic rules for such modifications have been proposed, and asymptotic analyses of efficiency have been carried out in simple contexts. This paper gives a careful treatment of “perhaps the simplest case…in which the boundaries on the state space play a significant role.” The example concerns the chance, in a tandem network, of the network population reaching a large size before returning to 0.