WASHINGTON >> Publicly fuming yesterday, the FAA chief collected the resignation of the head of the U.S. air traffic system, doubled controller staffing at more than two dozen airports and ordered a sweeping review of the entire system that ensures planes fly safely, as the government sought to reassure the public that air travel is safe despite at least four instances of controllers sleeping on the job.

But present and former controllers said that grueling work schedules and the design of the job itself — sitting in a dark room at night waiting for pilots to call — have made taking naps on the job necessary, even if unauthorized by the FAA. One whistle-blower complained to the Transportation Department that cots can be found in one radar center, most often with controllers asleep in them.

The National Transportation Safety Board warned FAA after a deadly 2006 air crash that controllers’ schedules were creating unsafe situations in which they were going into work after only a few hours of sleep. But little had changed until this week when Federal Aviation Administrator Randy Babbitt said he was immediately adding a second controller on overnight shifts at 26 airports and a radar facility that had been staffed with a lone controller. Presumably the second controller provides a margin of safety if the first falls asleep.

Babbitt’s order came hours after the pilot of a plane transporting a critically ill passenger was unable to raise the single controller working at 2 a.m. Wednesday in the tower of the Reno-Tahoe International Airport in Nevada. The FAA said the controller, who was out of communication for 16 minutes, was sleeping.