M-M-infinity queue

An $M/M/\infty$ queue has Poisson arrivals of rate $\lambda$, independent exponential service times of rate $\mu$, and infinitely many servers. Its occupancy is a birth-death process with birth rate $\lambda$ and death rate $n\mu$ in state $n$.

For every $\lambda ,\mu >0$, an $M/M/\infty$ queue is positive recurrent and has the Poisson distribution with mean $\rho =\lambda /\mu$ as its unique stationary distribution:

In stationarity, the departure process of an $M/M/\infty$ queue is a Poisson process of rate $\lambda$. By time reversibility, the departures before any fixed time are independent of the occupancy at that time.

If $X_0=n$ and $p_t=e^{-\mu t}$, thenwith independent summands. The binomial term counts surviving initial customers; Poisson thinning gives the surviving later arrivals.

Writing $\rho =\lambda /\mu$, the stationary empty probability is $e^{-\rho }$. Alternating idle periods of mean $1/\lambda$ with busy periods and applying the renewal-reward theorem gives