I am looking for the PDF of a Poisson variable.
The setup is that pit crews are working on racecars on the track after the drivers reports trouble. The pit crew's expected work time is exponentially distributed with an expected value of $\frac 1{\lambda}$. The drivers arrive as a Poisson process with rate $\mu$
So far I have tried using a convolution integral to get the PDF of the wait. I set up the PDF of the arrival as $\mu e^{-\mu t}$ and the PDF of the expected work time as $\lambda e^{-\lambda t}$. Then I took the convolution as follows.
$P_W(t) = \int^{\infty}_{-\infty} \mu e^{-\mu\tau}\lambda e^{-\lambda (t-\tau)}d \tau$
I got $-\frac{\mu\lambda }{\mu-\lambda }e^{(\lambda-\mu)\tau- \lambda t)}|^{\tau \to\infty}_{\tau \to-\infty}$ which seems to diverge.
Am I completely on the wrong track or am I missing something?
You might want to read about an M/M/1 queue, though I think Wikipedia has your $\lambda$ and $\mu$ reversed.
The probability that any one time there are $N$ cars being serviced or waiting is $\left(\frac{\mu}{\lambda}\right)^N\left(1-\frac{\mu}{\lambda}\right)$ so when a new car arrives, the expected number of cars in front is $\frac{\mu}{\lambda-\mu}$ and the expected waiting time until its service starts is $\frac{\mu}{\lambda(\lambda-\mu)}$ while the expected waiting time until its service finishes is $\frac{1}{\lambda-\mu}$.
The queue is unstable, with infinite expected waiting times, if $\mu \ge \lambda$.