A Call to Arms: Queue time fix.
A number of games already have a sort of "Call to Arms" for their queue types.
Would it not make sense for Riot to implement a system that detects the least desired role at a given time for Dynamic Queue and reward players who specifically choose that role as the main choice? It would get more people to queue for certain roles depending on the need. And the reward could be a simple IP boost. I'd queue for whatever was lowest in demand if it meant that I'd get more stuff to do.
Or, scratching that, give a call to arms that gives a bonus chest to open after playing X amount of games per week as a 'lower priority role' (EX. Support). It'd give people more incentive to queue for that role, subsequently reducing queue times.
World of Warcraft, (while not a Moba, this is a fine example), has a system in place that when it's short on healers or tanks in their queue system for content, it rewards people who queue up for those roles with bags that could contain a variety of rare items in the game, be they mounts, potions or pets.
Rift has the same system implemented. Again, not a Moba, but it feeds the same concept. It rewards players for attempting roles that might not be their best, but it speeds up the queues and allows them to learn a role they may not otherwise have chosen.