In the book adventures in stochastic process by Resnick define this :
${\tau}_{i}=inf\{n\ge 0:{X}_{n}=i\}$ and
${\tau}_{i}(n)=inf\{n>{\tau}_{i}(n-1):{X}_{n}=i\}$ assuming ${\tau}_{i}(n-1)<\infty$.
In a birth and death chain I would like compute :
${P}_{a}({\tau}_{a}(2)<{\tau}_{b})$ where $b$ is an absorbing state . I know how to compute ${P}_{c}({\tau}_{a}<{\tau}_{b})$ and I was thinking to proceed in the same way but I’m not sure.