Homeland security officials caution that universal exit monitoring is a daunting and costly goal, mainly because of the nation’s long and busy land borders, with more than one million crossings every day. The wrong exit plan, they said, could clog trade, disrupt border cities and overwhelm immigration agencies with information they could not effectively use.

Since 2004, homeland security officials have put systems in place to check all foreigners as they arrive, whether by air, sea or land. Customs officers now take fingerprints and digital photographs of visitors from most countries, instantly comparing them against law enforcement watch list databases. (Canadians and Mexicans with special border-crossing cards are exempt from those checks.)

But homeland security officials said that a series of pilot programs since 2004 had failed to yield an exit monitoring system that would work for the whole nation. They have not yet found technology to support speedy exit inspections at land borders. And airlines balked at an effort last year by the Bush administration to make them responsible for taking fingerprints and photographs of departing foreigners.

The current system relies on departing foreigners to turn in a paper stub when they leave.

Last year, official figures show, 39 million foreign travelers were admitted on temporary visas like Mr. Smadi’s. Based on the paper stubs, homeland security officials said, they confirmed the departure of 92.5 percent of them. Most of the remaining visitors did depart, officials said, but failed to check out because they did not know how to do so. But more than 200,000 of them are believed to have overstayed intentionally.

Immigration authorities have put in place a separate system for keeping track of foreigners who, unlike Mr. Smadi, come on student visas. That system has proved effective at confirming that the students have stayed in school and do not overstay their visas, officials said.

Image In Italy, Tex., Mr. Smadi was able to work at a restaurant. Credit... Courtney Perry/Dallas Morning News

Immigration analysts said that given the difficulties of enforcing the United States’ vast borders, it remains primarily up to law enforcement officials to thwart terrorism suspects who do not have records that would draw scrutiny before they enter the United States.