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Due to lack of detention capacity (the U.S. government measures capacity by the number of detention beds), tens of thousands of apprehended illegal aliens are released into the U.S. interior each year, instead of being removed from the country. This vulnerability can be exploited by terrorist groups wanting to enter the United States. We construct a queueing model of the U.S. detention and removal operations, and derive approximate analytical expressions for key performance measures, including a simple normal approximation for the required number of beds. Due to shortcomings in the U.S. government's data collection procedures, we cannot directly estimate all of the model's parameter values. Consequently, we use the approximate analytical expressions and the 2003 U.S. government data quantifying these key performance measures to estimate several unknown parameter values. Although current funding is for approximately 21,000 detention beds, we estimate that approximately 34,500 beds are needed to remove all potential detainees (this does not include nonviolent, noncriminal Mexicans, who are returned to Mexico within several hours) based on 2003 data. The dramatic increase in the arrivals of potential detainees since 2003 suggests that approximately 50,000 beds are currently required, although the estimation of future arrival rates is very difficult due to uncertainties about the future direction of U.S. immigration policy. Our estimated bed requirements are approximately 25% higher than naive estimates that fail to account for right censoring of residence times due to some detainees being released from detention before removal to make way for higher-priority detainees.