Consider an $M_t/M/\infty$ queue where customers arrive as a NHPP with time-varying rate $\lambda(t)$ and the service time is exponentially distributed with parameter $\mu$ for each of the infinitely many providers.
Let $Z_t$ denote the number of customers waiting for service in the queue. I need to compute the probability distribution of the first passage time $T_m:=\inf\{t>0:Z_t\ge m\}$, that is the first time that at least $m$ customers are waiting in line. Any suggestions?