I appreciate all the replies and feedback to this post.
My logic is based on the average play time of, in this case, EotS. They are usually around 20 minutes long. I'm not saying there aren't 5 caps that last 5-10 minutes or deadlocks that last 30.
I'll try to further explain what I'm thinking here:
1. Open BG window, Take note of which BG's are there. Notice that EotS #1 is not there, but EotS #2 is.
2. Join a bg, for this example EotS #2
3. Hopefully EotS #2 will be midway or close to being over.
4. In theory the que time you have to wait will be enough time for EotS #1 to have begun while you are in que or right as your que pops.
5. You play out EotS #2 win or lose.
6. Exit and que up for EotS #1, which should have started roughly 10-15 minutes before you join.
7. Join EotS #1. Once again, in theory now EotS #2 should just be starting by the time #1 pops.
8. Play out EotS #1, Rinse and repeat.