find expected time and variance for state by using birth and death process

Question

Consider a birth and death process with birth rates $λ_n = (n + 1)λ$, $n \ge 0$, and death rates $μ_n = nμ$, $n \ge 0.$

(a) Determine the expected time to go from state 0 to state 3.

(b) Determine the variance in part (a).

Answer 1

Let $T_i$ denote the time, starting from state $i$, it takes for the process to enter state $i+1$.

$$ E[T_i] = \frac{1}{\lambda_i + \mu_i} + \frac{\mu_i}{\lambda_i + \mu_i}(E[T_{i-1}]+E[T_i]) $$

$$ Var[T_i] = \frac{1}{\lambda_i(\lambda_i + \mu_i)} + \frac{\mu_i}{\lambda_i}Var[T_{i-1}]+\frac{\mu_i}{\mu_i + \lambda_i}(E[T_{i-1}]+E[T_i])^2 $$

~ from Introduction to Probability Models by Sheldon Ross

Therefore, in your case

$E[T_0 + T_1 + T_2] = E[T_0] + E[T_1] + E[T_2] $

$E[T_0] = \frac{1}{\lambda}$

$E[T_1] = \frac{1}{2\lambda}(1 + \frac{\mu}{\lambda})$

$E[T_2] = \frac{1}{3\lambda}(1 + \frac{\mu}{\lambda} + (\frac{\mu}{\lambda})^2)$

Adding them up yields:

$E[T_0 + T_1 + T_2] = E[T_0] + E[T_1] + E[T_2] = \frac{1}{\lambda}(\frac{11}{6} + \frac{5}{6}\frac{\mu}{\lambda}+ \frac{1}{3}(\frac{\mu}{\lambda})^2)$

$Var[T_0 + T_1 + T_2] = Var[T_0] + Var[T_1] + Var[T_2] $ and you can plug in the values in the formula to get the answer too!