hi i need some help to understand the following (from the general birth and death process).I'll give some context first , then i ask questions.
Consider general birth and death process with birth rates $\{\lambda_n \}$ and death rates,where $\mu_o = 0$, and let $T_i$ denote the time,starting from state $i$, it takes the process to enter $i+1$, $i\geq 0$. We will recursively compute $E[T_i]$,$i\geq 1$,by starting with $i = 0$. Since $T_0$ is exponential with rate $\lambda_0$ we have:
$E[T_0]= \frac{1}{\lambda_0}$
For $i> 0$, we condition wheater the first transition takes the process into state $i-1$ or $i+1$, that is let:
$I_i = \begin{cases} 1, &\mbox{if the first tranistion from $i$ is to $i+1$ } \\
0 & \mbox{if the first transition from $i$ is to $i-1$} \end{cases}$
and note that
$E[T_i\vert I_i = 1] = \frac{1}{\lambda_i +\mu_i}$
$E[T_i\vert I_i = 0] = \frac{1}{\lambda_i +\mu_i} + E[T_{i-1}]+ E[T_{i}] $
$\textbf{questions}$
$E[T_i\vert I_i = 1] = \frac{1}{\lambda_i +\mu_i}$
how do i interpret this? it is clear from the text above that $E[T_i\vert I_i = 1]$ is the expected time for the process to enter state $i+1$,given that the first transition is from $i $ to $i+1$.
but how is this equal to $\frac{1}{\lambda_i +\mu_i}$. Is this the mean for the minimum of two random variables with rates $\lambda_i$ and $\mu_i$?
Best Answer
Consider a general state $i>0$.
A transition from state $i$ to state $i+1$ means that the first birth (occurring after $i$) takes place before the first death (occurring after $i$).
Births occur with rate $\lambda_i$ and deaths with rate $\mu_i$, so if we consider two independent exponential random variables $B_i\sim\mathrm{Exp}(\lambda_i)$ and $D_i\sim\mathrm{Exp}(\mu_i)$, then
\begin{align*} \mathbb{E}\left[T_i\,|\,I_i=1\right]=\mathbb{E}\left[B_i\wedge D_i\,|\,B_i\le D_i\right]&=\mathbb{E}\left[B_i\,|\,B_i\le D_i\right]\\ &=\frac{1}{\mathbb{P}(B_i\le D_i)}\int_{u=0}^\infty\int_{v=u}^\infty u\lambda_i\mu_ie^{-(\lambda_i u+\mu_i v)}\,\mathrm{d}u\mathrm{d}v\\ &=\frac{1}{\mathbb{P}(B_i\le D_i)}\int_{u=0}^\infty u\lambda_ie^{-(\lambda_i+\mu_i)u}\,\mathrm{d}u\\ &=\frac{1}{\mathbb{P}(B_i\le D_i)}\frac{\lambda_i}{\lambda_i+\mu_i}\frac{1}{\lambda_i+\mu_i}\\ &=\frac{1}{\lambda_i+\mu_i}, \end{align*}
since you can easily check that $\mathbb{P}(B_i\le D_i)=\frac{\lambda_i}{\lambda_i+\mu_i}$.