= Solution
An <M M 1 queue> has <Poisson process> arrivals of rate $\lambda$, independent <exponential distribution>[exponential] service times of rate $\mu$, and one server. The two letters $M$ indicate the <memoryless property> of the interarrival and service-time distributions. Its queue length is a <birth-death process> with birth rate $\lambda$ and death rate $\mu$ away from zero. When $\rho=\lambda/\mu<1$, its normalized <stationary distribution> is
$$
\pi_n=(1-\rho)\rho^n;
$$
the powers $\rho^n$ quoted in the question form the corresponding unnormalized invariant measure.
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