M-M-1 queue with Bernoulli feedback

If a served customer returns with probability $p$, thinning service completions gives effective external departure rate $\mu (1-p)$ for the queue-length process.

In equilibrium, a Bernoulli-feedback $M/M/1$ queue has the same queue-length process as an ordinary queue with service rate $\mu (1-p)$, so its external departures form a Poisson process of rate $\lambda$.