How many US service people can go AWOL before the military can't keep up?

No one who works for the US government is getting paid right now. (Save those least deserving). A lot of military families are strapped for cash at the best of times. If a service person gets a job to support their families, they'll get arrested when they miss their next duty. But military police have the same problem.

How many people would have to go AWOL before there aren't enough well-off or fanatical service people (or federal law enforcement) to go after them?