What is the expected duration of a game of War?
War is actually not a game but an Automata, as players don't have any options.

Wimpy Programmer already made this simulation, he found that when shuffling the winning cards, the mean number of turns is 262, the mode is 84, and the max (on a sample of 100,000 trials) is 2,702 turns.  He also found that without shuffles of the winning cards, the game might be endless.

He used N=2 while you play with N=3, higher N shortens the game, however I don't think that it will make a big difference.