This capstone closes both the topic and the unit by widening the lens: the birth-death machinery built for queues applies, essentially unmodified, to population models with a genuinely different flavour — a population replenished from outside (immigration) and a population where the "birth rate" is driven by interaction between two sub-populations (a nonlinear epidemic). Neither has a closed-form stationary distribution as clean as M/M/1's geometric, but both are worked far enough to show exactly where the birth-death toolkit's assumptions (in particular, linear-in-population rates) start to break down — and why that breakdown is itself informative.
Learning objectives. By the end of this note you should be able to:
- set up a birth-death-with-immigration chain and describe its stabilising mechanism;
- set up a simple M/M/1-style queue example as a birth-death chain (a direct review application);
- set up a nonlinear (density-dependent) epidemic model as a CTMC and identify why it resists closed-form stationary analysis.
In this lesson
- A simple queue, reviewed
- Birth-death with immigration
- A nonlinear epidemic
The full lesson, with worked examples and exercise solutions, is for members.
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