The Network
The Tandem Network
A queueing network lets jobs move through more than one service station. This scoped book uses the exact two-node open tandem: jobs enter station A, route to station B, and then leave. The motivation is that a network is not just a larger single queue; routed departures from one node become arrivals to another.
Open tandem route
Jobs arrive from outside at rate 1, visit station A, then visit station B. Why: this is the smallest open network with routing, so it already has a feature a single M/M/one queue does not have. The output of the first station becomes the input to the next station, which means congestion must be tracked node by node rather than collapsed into one server.
Station A service
Station A has service rate 3. Why: its service capacity must exceed the arriving work for stability, but stability at this first station is only one part of the network story. A fast first station can still feed a slower downstream station, so the route matters as much as the local service rate.
Station B service
Station B has service rate 2. Why: every completed job from station A routes to station B in this tandem, so station B inherits the upstream flow instead of receiving an unrelated stream. The network question is whether both stations can absorb the same routed work in steady state.
Diagram note
The graph labels the arrival, route, and exit roles; the rates are recomputed from the pinned tandem parameters. It is a structural diagram, not a simulation trace: the arrows name how work moves, and the exact rates attached to those arrows are the quantities later used in product form. These are exact steady-state values for an idealized open Jackson tandem with memoryless service, infinite buffers, stable rates, and deterministic routing; real systems need measured arrivals, service times, and routing evidence. Pixel positions are rounded for layout; every number shown is exact.