I have done a good bit of research on the subject, and cannot seem to find many materials. I was just wondering if you all knew of a good resource regarding chains which are Markov excepting the fact that time spent in the state matters. For instance, an example of what I am talking about is the following:
Suppose there is a birdfeeder $F$ and you want to know at any given time $t$ how many birds are at $F$. Now clearly, in this process, things like the number of birds at the feeder would influence the chain. But, the length of time which birds have spent at the feeder does also (i.e. a bird which has been there 45 minutes is much more likely to leave than one that has been there 1 minute). What is the mathematical way to model such behavior?
These are called semi-Markov processes. The specific example in your post might even require the use of a hidden semi-Markov model. You might want to check the introduction of this book on the subject.