You should have an option in the settings that allows you to maintain X games running. If you complete a game and you have less than X then a new game is stated automatically. New players can only set X to something small perhaps 10 and you can use coffee to buy more. If you time out on a game then the system will not auto start new games until you start playing again.
Kind of related to this, i think there should be a next option on the end turn screen when there are other games waiting, rather than force you to go through the main screen each time.
Thanks for the game,
Martin