For How Long Can a World with Nuclear Weapons Avoid Nuclear Annihilation?

We live in a world with nuclear weapons. We know there are enough nukes to blow up the world many times over. And we know that the more time passes the more opportunities there will have been for nuclear war to break out.

But how much time will have to pass for this possibility of nuclear war to become a near certainty?

Let's make four assumptions:

1. Only 10 governments have nuclear weapons at any given time.
2. A government exists for four years on average.
3. Only .1% (1 out of 1,000) of governments with nuclear weapons are stupid or suicidal/homicidal enough to accidentally or intentionally start a nuclear war.
4. Nuclear war will escalate into nuclear annihilation (either directly through the blasts or as an indirect effect in the form of fallout and nuclear winter).