Problem
Let us consider M/M/infinity queueing system. I would like to know the time distribution when $b$ customers received service. For this, I first showed that the distribution of the number of customers who received service until time $t$ ($M_t$) follows Poisson$(at+\exp(-at)-1)$. After that, I really have no idea how to calculate the pdf and the first-two moments of $T_b=\inf\{t: M_t\ge b\}$ from $M_t$.
How can I calculate the pdf of $T_b$ and its first-two moments (i.e. $E[T_b]$ and $E[T_b^2]$)?
My idea
Indeed, I have tried to use the optional stopping theorem. However, it seems to be difficult as I completely have no idea whether $M_t-E[M_t]$ is martingale or not. To my current feeling, $M_t-E[M_t]$ seems to be not a martingale. Then, how can we identify its pdf and moment...?