A $G/G/1$ queue initially has $M$ customers in the line, the probability distribution function (pdf) of inter-arrivals are $f_{a}(x)$ and the pdf of processing times is $f_p(x)$. It starts to process the packets at time $t=0$. What is the probability that the queue length becomes zero, for the first time, when the queue processes the $\ell^{\mathrm{th}}$ customer, with $\ell, M \to\infty$?
I solved it for a $M/M/1$ queue, i.e., $f_a(x)=f_p(x)=\lambda e^{-\lambda x}$, where the memorylessness of exponential random variables makes this problem equivalent to a symmetric random walk. Then, we only have to find the probability distribution function of the first hitting time. Any idea how to solve it generally? How about when $f_a(x)=f_p(x)$?