# Birth and death process with probability of staying

I have a birth and death process with states $$S=\{ 0,1 \}$$. If I'm in state $$0$$ the process change to state $$1$$ in a time with distribution $$\exp(\lambda)$$. If I'm in state $$1$$ I wait a exponential time $$\exp(\lambda)$$ then with probability $$1- \alpha$$ I change to $$0$$ and with probability $$\alpha$$ I stay in $$1$$. I need the rate of birth and death.

The rate of birth is $$\lambda$$ because I always change to $$1$$ when I wait a time $$\exp(\lambda)$$ but I don't know what to do with the death rate, sometimes I stay more time in the state $$1$$, this happens when after a time $$exp(\lambda)$$ I stay in $$1$$ with probability $$\alpha$$.

Your process is equivalent to one where state $$0$$ transitions to state $$1$$ with rate $$\lambda$$, and state $$1$$ transitions to state $$0$$ with rate $$(1-\alpha)\lambda$$.
The key insight here is that a process with a constant transition rate over time has an exponential waiting time until the next transition, and vice versa. Furthermore, randomly thinning the transitions from state $$1$$ to state $$0$$ by a constant factor of $$1-\alpha$$ still leaves the transition rate constant, and the waiting times thus exponential.
Ps. One useful corollary of this is that your process can be equivalently viewed as being derived from a simple Poisson process of "potential state transition events" with the constant rate $$\lambda$$, where each event in this Poisson process triggers a state transition from $$0$$ to $$1$$ with probability $$P_{0\to1} = 1$$, or from $$1$$ to $$0$$ with probability $$P_{1\to0} = 1-\alpha$$. In fact, any continuous-time Markov process with bounded transition rates can be reformulated in this way on top of an underlying state-independent Poisson process (with an event rate equal to the upper bound of the transition rates of the original process). This can be useful e.g. for simulating the trajectory of the process on a computer, as in the Gillespie algorithm. (In fact, some of my own PhD work is based on using similar techniques to simulate multiple coupled Markov processes with different transition rates at the same time.)