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The telegraph process is a two state stochastic process defined by the master equation

$$ \dot{\pi}_0(t) = \tau^{-1} \pi_1(t) - \sigma^{-1} \pi_0(t) $$ $$ \dot{\pi}_1(t) = \sigma^{-1} \pi_0(t) - \tau^{-1} \pi_1(t) .$$

In steady state, the solution is $$ \pi_0 = \frac{\sigma}{\sigma + \tau}$$ $$ \pi_1 = \frac{\tau}{\sigma+\tau}.$$

I would like to determine the distribution of times spent in state $0$ or state $1$ -- the residence times. Can anyone offer guidance? I have read this is an exponential distribution but the things I've read kind-of brush over this point. How can I derive it for myself? (See Gillespie 1992 or Gardiner 1983)

Attempt:

If the process is in state $1$ at $t_0$, the probability of remaining in state $1$ with no transitions at time $dt+t_0$ comes from

$$ \text{prob}(1 \text{ at } dt + t_0) = \text{prob}(1\text{ at }t_0)\text{prob}(\text{no transitions in } dt) $$ which means $$\pi_1(dt+t_0) = \pi_1(t_0)(1- dt \sigma^{-1}) $$ which means $$ \dot{\pi_1}(t_0) = -\sigma^{-1}\pi_1(t_0),$$ which is the correct differential equation for an exponential distribution, but I'm sure my notation is all messed up. How should I write this in terms of a lifetime distribution?

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