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As the Internet becomes increasingly widely used and as individuals and businesses come to depend on it ever more, disruptions of service are less and less tolerable. Disruptions occur either because of a natural disaster, e.g., a fire or equipment failure, or because of a deliberate attack. Protecting against natural disasters is generally easier than protecting against malicious attacks. Our focus here is on the latter. Malicious attacks fall broadly into two classes. One class is penetration, whereby an unauthorized user gains access to one or more machines. Another class is denial-of-service attacks. In this paper, we analyze the nature and impact of this latter class of attacks and propose a method to reduce their threat and impact.