Modeling a birth-death process that is not memoryless How does one approach the problem of modeling a "birth-death process" where the arrivals are dependent on the current state in the following way: if the population is above a certain point, the probability of an arrival decreases.
Basically, I'm interested in complicating (slightly) an existing model of "births" that just has Poisson distributed arrivals and thinking of adding in the idea that there's a "saturation point" above which arrivals are less likely (waiting until the population drops back below the point). 
Should I be reading about nth-order Markov processes? Or should I be looking at queueing theory?
 A: If this is for a particular application, I would first consider the question: "do you need to fit such a complex model?" which is code for, what other, more simple methods have you tried prior to this one?  Have you looked at a plot of the "births & deaths" data that you have?
I would advise that you look up some of the work on "state space" theory, and kalman filtering as a starter.  It sounds like you basically have an autoregressive process that isn't a simple random walk.  These can usually be dealt with via the Kalman filter (as long as you are willing to assume normality of the errors).
A simple way (I think) is to consider a "regression" of the births/deaths against the current state.  You could just use OLS as a simple start (to figure out what's going on), but this ignores that the errors from the regression are correlated (and not independent as in OLS).  This will have the effect of OLS standard errors being either too small or too big, depending on the direction of the correlation (the OLS estimates are still "unbiased")
A: Frankly, what you have is still a Markov Chain, as vqv has very rightly pointed out. (PS I tried to add this as a comment but where is the button??)
A: You might find Scott and Smyth '03 of interest: http://www.datalab.uci.edu/papers/ScottSmythV7.pdf.
They discuss markov modulated poisson processes, which varies the rate parameter based on which markov state it is in. You could have two states, one below and one above the given population of interest. 
