Do ants or other insects sleep, and if so why?
A quick search on Web of Science yields "Polyphasic Wake/Sleep Episodes in the Fire Ant, Solenopsis Invicta" (Cassill et al., 2009, @Mike Taylor found an accessable copy here) as one of the first hits. 

The main points from the abstract:


Yes, ants sleep.
indicators of deep sleep: 

ants are non-responsive to contact by other ants and antennae are folded
rapid antennal movement (RAM sleep)

Queens have about 92 sleep episodes per day, each 6 minutes long.
Queens synchronize their wake/sleep cycles.
Workers have about 253 sleep episodes per day, each 1.1 minutes long.
"Activity episodes were unaffected by light/dark periods."


If you study the paper you might find more information in its introduction or in the references regarding why ants sleep, although there doesn't seem to be scientific consens. The abstract only says that the shorter total sleeping time of the workers is likely related to them being disposable.