I am looking for a word which meaning is the opposite of "usage". The context is the use of devices of any sort. The best I've come up with so far is to look at the situation from the devices perspective and consider "usage periods" and "inactivity periods".
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Idle time or idle periods could be what you are looking for. The term is already in use in the computer and other industries to denote time when a machine could be working but isn't.
Neglect can fit a lot of cases, but has a negative connotation that may be best avoided.
Inactivity is indeed a good choice. The related "inactive periods" doesn't match noun-for-noun, but it does match noun-phrase for noun-phrase, and reads more naturally IMO.
Rest might also be considered, but it has a positive connotation that might be almost as well avoided as the negative connotations of neglect, especially if you want to reduce redundancy by reducing how often the devices are inactive.