Suppose we have a queue with parameters M/G/1 where M denotes a Markovian input describing the arrival times of customers into a system. Since M is Markovian, this implies that the inter-arrival times of customers into a system has a poisson process with rate $\lambda$. Now, the queue follows a general input describing the service times for any ith customers. Denote the service time for any ith customer to be $s_{i}$. The service time is a poisson process with rate $\mu$.
Now, for the case where $\frac{\lambda}{\mu}$:
For any unit time, the server clears the queue at a rate faster than the arrival of customers into the system. From this, it is then reasonable to argue that there exists a duration of time where the servers are idle. This suggests that the Markovian process alternate between an idle and busy state.
From here, I am told that the idle duration follows a poisson process with rate $\lambda$.
I am unable to understand why the occurence of the idle duration $I_{n}$ has a rate $\lambda$.
Any help is appreciated.