The team announced the moves before Friday night's game against the Los Angeles Angels.
Miller's arrival gives Detroit an extra option in the bullpen on a night when left-hander Drew Smyly is making his first start of the season. Smyly is going to be part of the starting rotation, but he's pitched exclusively in relief so far this year because days off have allowed the Tigers to get by without needing all five starters.
The 26-year-old Miller has never pitched in the major leagues.
Collins went 2 for 14 in limited duty this season with the Tigers.
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