This lesson specialises the birth-death stationary-distribution formula to the canonical queueing models — M/M/1 (Poisson arrivals, exponential service, one server) and M/M/s ( servers) — the "M" standing for "Markovian" (memoryless) on both the arrival and service side. These are the models applied probability is most famous for, and they give closed-form answers for queue length, waiting time, and (crucially) the exact stability condition separating a queue that reaches equilibrium from one that grows without bound forever.
Learning objectives. By the end of this note you should be able to:
- derive the M/M/1 stationary distribution and state Little's law application to it;
- derive the M/M/1-with-finite-waiting-room and balking variants;
- state the M/M/s stability trichotomy and compute its stationary distribution in concrete cases.
In this lesson
- The M/M/1 queue
- Variants: finite waiting room and balking
- The M/M/s queue
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