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Just a though for a game mode that was constant. If anyone has ever played the fps Battlefield (or other fps since that is what I am most familiar with) it would be like that. You swap out players who leave for new ones. This is just the basic Idea and I am sure many changes would need to occur to summoners rift or possibly even a new map for it to work.
This would be a regular game mode (not ranked) for players who may not have time to commit to a full game, or want the troubles that go along with
champ selection ,meta game, trolls, etc. You basically go in play with others to kill champs, complete objectives, learn your champ, or whatever (again just a jumping off point) but play and leave when you want.
Some people would say that custom games or bot games solves this. Well then I leave that up to debate.