Pick a point in time. The probability that any one door is open at that point in time is q. Think of a door's being open as a "success"; then the number of doors open is the total number of successes. Since the successes are independent, the total number of successes (i.e., the total number of doors open) has a Binomial distribution with n=N and probability of success q (which is a little confusing because we usually write q as the probability of failure.)
I'm assuming you are familiar with the Binomial distribution, if not let us know.