Well.....Realistically,how do you know you've JUST fallen asleep, Alastar?
Read my hub, "Dreams Don't lie."....that was a plug. Oops!
Seriously, Once you're sleeping, which means your "conscious mind" is at rest....shut down....not functioning as when awake....we can't possibly know we've just fallen asleep.
It may seem to you as though you dreamt All night long, but the truth is, our dreams last only minutes. Yes, we can have more than one dream, but they usually run non-stop. This is why often we can't make heads or tails of our dreams and they seem chaotic and disorganized.
In order to dream, we must be in R.EM sleep. The amount of time it takes to reach REM varies for numerous reasons. It may occur quickly for you.
Basically, we can dream while we're wide awake....ever heard of day dreaming? I do it constantly............Peace, Paula