CATESBY: . . . His horse is slain, and all on foot he fights,
Seeking for Richmond in the throat of death.
Rescue, fair lord, or else the day is lost!
KING: A horse, a horse! My kingdom for a horse!
That famous Shakespearean plea is universal in its expression of angst over an out-of-reach goal. It is a feeling familiar to anyone who is about to begin another long haul, international trip and starting what will surely be a long, arduous day wishing — indeed, almost praying — to have slept a few hours more. It's a feeling of despair every bit equal to King Richard III's.
The science of sleep has long been far ahead of the federal aviation regulations regarding its application to crew rest. In recent years, after several tragic accidents in which pilot fatigue was a factor, the FAA set about codifying that body of science as it applies to scheduling aircraft and aircrew movement. Most of the statistics I quote here are from that work.
Accident rates for pilots who have been on duty for 10-12 hr. are 1.7 times greater than that of the general control group. Meanwhile, the accident rates for pilots who have been on duty for 13 or more hours are 5.5 times higher than the general group.