Real service systems are rarely a single queue — a customer at the DMV passes through vision testing, then payment, then a photo station; an order might route through several processing stages before settlement. This lesson uses Burke's theorem and departure-process independence to show that chaining M/M/1-type queues together produces a remarkably simple joint stationary distribution — the queue lengths are independent, each with its own M/M/1-shaped marginal — a result that extends, with real but manageable extra bookkeeping, to networks of arbitrary size and routing structure (Jackson networks).
Learning objectives. By the end of this note you should be able to:
- state and use the tandem-queue independence result;
- set up and solve the traffic-rate equations for a network with feedback;
- state the Jackson network product-form theorem and apply it to a concrete multi-station example.
In this lesson
- Tandem queues
- A two-station network with feedback
- Jackson networks: the general case
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