# birth and death processes

Suppose we have a system of N balls, each of which can be in one of two boxes. A ball in box I stays there for a random amount of time with exponential(lambda) distribution and then moves instantaneously to box II. A ball in box II stays there for a random amount of time with exponential(lambda) distribution and then moves instantaneously to box I. All balls act independently of each other. Let Xt be the number of balls in box I at time t.

a) I'm trying to show that X is a birth and death process and specify the birth and death rates. b) How can we find the stationary distribution of the process.

For part a, if we can show it satisfies a Yule process, this is essentially what we're trying to do. And for part b, I also want some clarification on the detailed balanced equations for this problem.

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 this is the ehrenfest urn model in continuous time. At the event times of a poisson $\lambda$ process $X_t$ goes up or down by 1. – mike Apr 13 '12 at 11:57 Like @mike said, except the intensity of the Poisson process is $N\lambda$. (Unrelated: Yule processes are pure birth hence they cannot model bounded populations like here.) – Did Apr 14 '12 at 9:20