# Distribution to reflect situation where some waiting leads us to expect more waiting

In reading Blake Master’s notes on Peter Thiel’s lecture on start ups, I came across this metaphor of the technology frontier:

Picture the world as being covered by ponds, lakes, and oceans. You’re
in a boat, in a body of water. But it’s extremely foggy, so you don’t
know how far it is to the other side. You don’t know whether you’re in
a pond, a lake, or an ocean.

If you’re in a pond, you might expect the crossing to take about an
hour. So if you’ve been out a whole day, you’re either in a lake or an
ocean. If you’ve been out for a year, you’re crossing an ocean. The
longer [the] journey, the longer your expected remaining journey.
It’s true
that you’re getting closer to reaching the other side as time goes on.
But here, time passing is also indicative that you still have quite a
ways to go.

My question: is there a probability distribution or statistical framework that best models this situation, especially the bolded part?

Since I think the comparison with memorylessness is clearest, my first suggestion would be to look at other distributions for which the exponential distribution is a special case. That will allow you to control fairly intuitively the magnitude of this effect. The Weibull distribution with shape parameter $<1$ would be a good choice.