What does limiting fraction of time mean in Poisson processes and renewal processes?

Question

I'm reading on Poisson and Renewal processes and I encountered the term limiting fraction of time. Although it was somewhat defined I couldn't really grasp the meaning of it.

Edit

Here is an example of what I mean: The weather in a certain locale consists of alternating wet and dry spells. Suppose that the number of days in each rainy spell is a Poisson distribution with mean 2, and that a dry spell follows a geometric distribution with mean 7. Assume that the successive durations of rainy and dry spells are independent. What is the long-run fraction of time that it rains?

Answer 1

This refers to the limit of the fraction of time when the process is in such or such state (and is probably an incorrect formulation, as far as English language is concerned).