This lesson proves the general form of the formula used informally throughout the previous two lessons, and then proves one of queueing theory's most genuinely surprising results: the departure process of a stable M/M/s queue, in equilibrium, is itself a Poisson process — indistinguishable in distribution from the arrival process, despite individual departures depending intricately on how many servers are currently busy. The proof technique, time reversal, is a direct continuous-time descendant of the discrete-time reversibility theory built earlier in this unit.
Learning objectives. By the end of this note you should be able to:
- state Little's Law and explain its remarkable generality;
- state the time-reversal lemma for a stationary CTMC;
- state and explain Burke's theorem (departures from a stable M/M/s queue are Poisson);
- state the stronger independence result linking the departure process to the current queue state.
In this lesson
- Little's Law
- Time reversal and Burke's theorem
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