Bullet Chess is a Chess game that is played very rapidly. At the beginning of the game each player gets a timer set to a specific number of minutes that runs down towards zero while it is his move. For the sake of this discussion, let's limit Bullet Chess to two types of time control: with and without increment. In a game without increment, each player has a specific number of minutes (typically 2 or less) to make all of his moves. The time control for this sort of game is usually denoted by minutes/0, so a game where each player has one minute on his clock at the beginning of the game would be denoted 1/0. In a game with increment, in addition to his starting time each player gets a specific number of seconds added to his clock every time he makes a move. Such time controls are denoted by two numbers x/y where x is the number of minutes each player has at the start of the game, and y is the number of seconds each player gets per move. Typically, the longest games considered bullet are 2/1 or 1/2. Games that are 3/0, 2/2, or 1/3 are usually too long to be considered bullet games.
Now that I've defined what a bullet game is, my question is what sort of distribution would best map to the total time taken for a bullet game. The distributions for games without increment would be sharply truncated -- for example, the longest time a 1/0 game could go would be 2 minutes. On the other hand, a game with increment could theoretically go on more or less endlessly. So perhaps there are two different sorts of distributions that describe these two different situations?
Empirically, the total time distribution is high near the upper end of the distribution for games with no increment, as might be expected. As an example, for 1/0 games I found a mean total time of 92.3 seconds, with a median of 104 seconds, a standard deviation of 28.3 seconds, and a mode of 117 seconds. The longest game was 119.9 seconds. To the untrained eye, for games with no increment the total time distribution looks almost like a truncated normal distribution, but I have no idea if this is correct or not.
For 1/1 games I found a mean of 148.2 seconds, a median of 155.5 seconds, a standard deviation of 52.7 seconds, and a mode of 153 seconds. The longest 1/1 game on the other hand was 497 seconds. I'm not entirely sure what the distribution for games with increment looks like, but if I had to guess it looks normalish (though obviously it's truncated on one side -- a game can't be shorter than 0 seconds!).
Anyway, my question is, are there any distributions that would theoretically fit this sort of data, and if so, which ones should I look at?
Edit
At the suggestion of @Schach21 I have added histograms for 1/0, 1/1 and 2/1 games below
