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Probability II / Continuous-Time Markov Chains and Queueing Theory

Derivation — Stationary Distributions of M/M/1 and M/M/s via Detailed Balance

This note carries out, in full algebraic detail, the detailed-balance derivations behind the previous lesson's M/M/1 and M/M/s stationary distributions — both direct instances of the birth-death formula, but worth working through explicitly because the M/M/s case requires splitting the birth-death recursion into two regimes (k≤sk\le s and k>sk>s), a technical step easy to get wrong.
Learning objectives. By the end of this note you should be able to:
  • derive the M/M/1 geometric stationary distribution term by term from detailed balance;
  • derive the M/M/s two-regime stationary distribution and its normalising constant;
  • compute the M/M/s blocking-probability (Erlang-C-type) formula in a concrete case.

In this lesson

  1. M/M/1: the geometric derivation
  2. M/M/s: the two-regime derivation

The full lesson, with worked examples and exercise solutions, is for members.

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