Busy period requires routine with no time off, hospitals admitted (4,4)
I believe the answer is:
rush hour
'busy period' is the definition.
(I've seen this before)
I don't understand the rest of the clue.
(Other definitions for rush hour that I've seen before include "Traffic-snarl period" , "Time of maximum traffic" , "Time when traffic is at its peak" , "having battled through this?" , "Time of day when traffic is busiest" .)