# Problem about non-homogeneous poisson process

$$\textbf{Problem}$$ Suppose customers arrive at a system according to the poisson process with rate $$\lambda$$. Every customer stays in the system for an exp($$\mu$$) amount of time and then leaves. Customers behave independently of each other. Show that the expected number of customers in the system at time $$t$$ is $$\frac{\lambda}{\mu} (1-e^{-\mu t})$$

Hint: Calculate the probability that a customer arrive at time $$s( is still in the system at time $$t$$ and apply the result of non-homogeneous Poisson process.

I don't have any clue about the problem.... Any help is appreciated ...

Thank you!

• Could you clarify -- does every customer stay for a constant period of time (exp($\mu)$) or that period is distributed according to the exponential distribution with parameter $\mu$? – dnqxt Apr 6 '19 at 23:09
• A period is distributed according to the exponential distribution with parameter $\mu$! – user453447 Apr 7 '19 at 1:59

The problem describes an $$M/M/\infty$$ queue.