Suppose we have an M/G/1/c loss system, with equilibrium distribution $\;\pi,\;$ service times $\;S_i\;$ and arrival rate $\;\lambda.\;$ I'm trying to show that $\;(1-\pi_0)=(1-\pi_c)\lambda \mathbb{E}S_1.\;$ The question suggests use of Little's formula.
Now, I can see that $\;(1-\pi_c)\lambda\;\;$ is the effective arrival rate, and that $\displaystyle 1-\pi_0=\lim_{t\to \infty}\frac{\int_0^t Q_t dt}{t}$, where $\;Q_t\;$ is the number of items in the queue at time $\;t\;$, but I'm not sure where to go from here. It seems that we would need to know the expected sojourn time, or the expected wait time to use Little's formula?
Thank you.