I have the following scenario, where I am trying to set up a birth and death process.
There are $10$ bulbs, and the bulbs have independent Exponential $(\lambda)$ lifetimes. If a bulb stops working, the amount of time required to replace it is distributed as independent Exponential $(\mu)$. The bulbs are replaced one by one, in the order in which they failed.
Here, I am trying to formulate the evolution of the number of working bulbs as a birth and death process, and trying to find the rates.
My attempt is as follows:
In order to formulate the evolution of the number of working bulbs as a birth and death process, I did the following.
Let $X(t)$ be the number of working bulbs at time $t$.
The state space is $S$ $=$ $(0, 1, 2, ...., 10)$, where any $i$ in the state space refers to the number of working bulbs.
$Death$ in this context refers to a bulb breaking down.
$Birth$ in this context refers to the replacement of a bulb that broke down.
Here, the lifetime of each bulb is distributed as Exponential $(\lambda)$.
Therefore, the Death Rate is $\lambda_i$ $=$ $i\lambda,$ for all $i$ $=$ $0, 1, 2, ...., 10$
And the time required to replace one bulb is distributed as Exponential $(\mu)$.
Therefore, the Birth Rate is $\mu_i$ $= 0$ when $i$ $= 10,$ and $\mu_i$ $= \mu$ when $i$ $= 0, 1, 2, ...., 9$.
Am I doing this right? Is there anything else I should calculate, in order to set up the evolution of the number of working bulbs as a birth and death process and to find the rates? Any insights will be very helpful. Thank you very much!