Total time from Poisson process and exponential waiting time

Question

Suppose the number of active calls on a switch is given by a Poisson process $M(t)$ with rate $\lambda$ calls per minute. Each call is then active for an amount of time $T\sim Exp(\frac{1}{\mu})$. We are told in the question that the steady state number of active calls is $\frac{\lambda}{\mu}$ but we need to show why that is. I also need to find the distribution of the inter-departure times. But I'm confused about where to start. I have seen some answers for essentially the exact same question, but they use Markov chains and concepts I'm not familiar with. Is there a way to answer this without using Markov chains? I am coming from a signal processing background