Arrivals are a Poisson process at rate λ, each job needs an exponential service (rate μ) from one server, and queuing theory calls the result M/M/1. It’s also exactly a birth–death CTMC — the same species as the immigration–death chain — with a stationary distribution you can solve on a napkin: P(N = n) = (1 − ρ)·ρⁿ, where ρ = λ/μ is how busy the server is.

Keep λ below μ (the page clamps it for you): push ρ toward 1 and the average line grows without bound — the one-point-nothing cliff every café and call center eventually discovers. The form below shows the exact steady-state distribution and the classical performance numbers.