I had a bit of trouble deriving the following statement:
Setting: Consider an $M/M/\infty$ queue and a Poisson arrival process with rate $\lambda$ and exponential service time with rate $\mu$. Suppose that the arrival process starts in the distinct past and ends at time $t = 0$, i.e. the arrival rate is defined as $$ \begin{cases} \lambda(t) = \lambda & t \in (\infty, 0]\\ \lambda(t) = 0 & t \geq 0 \end{cases} $$ where $\lambda > 0$ is a constant.
Question: Let $m(\tau)$ be the mean number of customers in the system at time $\tau$. Verify that $m(0) = \frac{\lambda}{\mu}$ and $m(\tau) = \frac{\lambda}{\mu}e^{-\mu t}$.
My problem: In the solutions to this exercise they first state that $$ m(0) = \int_{-\infty}^{0} \lambda e^{-\mu(- t)} dt $$ I think this definitely makes sense, but I have trouble actually convincing myself that this is formally the case. I understand that expected number of customers that arrive between $t = -\infty$ and $t = \tau$ should be $\int_{-\infty}^{\tau} \lambda(t) dt$, since that is what it means to be rate.
From here I intuitively understand that we want to "discount" the amount of people in the system at every time by some factor so to get the actual number that are left in the system. It makes sense that this factor should be the exponential, but I do not exactly see how to formally arise to the exact expression: $m(\tau) = \int_{\infty}^{0} \lambda e^{-\mu(\tau - t)} dt$. I have tried multiple times to formally derive this myself, but unfortunately I couldn't figure it out. Could someone else help me? Thanks in advance! :)