We study how rare events happen in the standard two-node tandem Jackson queue and in a generalization, the socalled slow-down network, see . In the latter model the service rate of the first server depends on the number of jobs in the second queue: the first server slows down if the amount of jobs in the second queue is above some threshold and returns to its normal speed when the number of jobs in the second queue is below the threshold. This property protects the second queue, which has a finite capacity B, from overflow. In fact this type of overflow is precisely the rare event we are interested in. More precisely, consider the probability of overflow in the second queue before the entire system becomes empty. The starting position of the two queues may be any state in which at least one job is present.