I am struggling with the exact same problem.
Most of the time it works just fine, but some users report occasional 503 errors. I have also noticed this myself.. At one time "domain.com" loads just fine, then trying "www.domain.com" throws 503 error. Restarting LS solves the problem.
I have set CloudLinux and LS limits high so that is not the problem. Nothing like max entry processes or the obvious.
I have narrowed the issue down to be either APC, suEXEC daemon mode or a combination of the two.
It is extremely frustrating but very hard to reproduce, as it only happens occasionally.