How do you actually prove that a M/M/1 queue is in fact
>a markov process
>a birth death process?
Almost every book and article I've read about queueing theory just assumes that M/M/1 queues can be modeled by a birth-death process, but I've never actually seen this justified. I just don't see how it follows if all we know is that arrival times and service times are exponentially distributed.
You should be able to prove that the time spent in a state is exponentially distributed (which is proven to be a necessary condition for a continuous-time Markov process), and that instantaneous transitions are only possible to neighboring states. But nobody does this, they just assume that a birth-death process is the right model and derive results from this assumption.
Did anyone actually prove it? Where? I refuse to believe it's just a big conjecture.