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Tandem Jackson networks and more sophisticated variants have found widespread application in various domains. One such variant is the tandem queue with server slow-down, in which the server of the upstream queue reduces its service speed as soon as the downstream queue eXceeds some prespecified threshold, to provide the downstream queue some sort of `protection'. This paper focuses on the overflow probabilities in the downstream queue. Owing to the Markov structure these can be solved numerically, but the resulting system of linear equations is usually large. An attractive alternative could be to resort to simulation, but this approach is cumbersome due to the rarity of the event under consideration. A powerful remedy is to use importance sampling, i.e. simulation under an alternative measure, where unbiasedness of the estimator is retrieved by weighing the observations by appropriate likelihood ratios. To find a good alternative measure, we first identify the most likely path to overflow. For the standard tandem queue (i.e. no slow-down) this path was known, but we develop an appealing novel heuristic which can also be applied to the model with slow-down. The knowledge of the most likely path is then used to devise importance sampling algorithms, both for the standard tandem system and for the system with slow-down. Our eXperiments indicate that the corresponding new measure is sometimes asymptotically optimal, and sometimes not. We systematically analyze the cases that may occur. [ABSTRACT FROM AUTHOR]