Warning: This comment is intuition (plus a few matlab simulations to verify).
For large p (the probability of 1 success), I'm not sure that this is an easy distribution to describe. At least, I don't see it.
As p get's smaller, I think that it may start to converge to a negative binomial. Suppose p is the probability of 1 success. For large N, the chance of a run of 2 is p times less likely than a run of 1. A run of 3 is p times less likely than a run of 2. Etc. BTW, a run of length 0 is a single failure.
p needs to be small so that you have lots of "trials" (chances for runs) compared to the length of the run lengths.
I should also point out that although I am waving my hands, I did run some simulations using matlab. For p = 0.5 and smaller, my experimental results matched the negative binomial distribution pretty well. I didn't actually try to run anything larger. So 'small' may not actually have to be too small.