Tied up detective (4)
I believe the answer is:
busy
I believe this clue is a double definition.
'tied up' is the first definition.
(similar in meaning)
'detective' is the second definition.
(I've seen this before)
(Other definitions for busy that I've seen before include "With much to do" , "Pot plant" , "Sedulous" , "Without leisure" , "detective" .)