Product Form
Network Empty Probability
The scoped tandem has product-form steady state. The chance that the whole network is empty is the product of the two node-empty probabilities. The important idea is that a theorem lets the stationary distribution factor even though jobs still flow through the stations in sequence.
Station A empty chance
The empty probability for station A is 2/3. Why: an M/M/one node is empty with one minus its utilization, so a lighter-loaded node is empty more often. This is a node-level probability before any claim is made about the joint network state.
Station B empty chance
The empty probability for station B is 1/2. Why: station B has its own utilization, and its empty probability follows from that local load. The tandem route couples the movement of jobs, but product form will let the steady-state distribution factor into node terms.
Both stations empty
The probability both stations are empty is 1/3. Why: product form multiplies the two node factors for this open Jackson tandem. The interpretation is subtle: the theorem gives a factorized stationary distribution as if node factors multiply, even though jobs physically move from one node to the next.
Diagram note
The network-empty value is recomputed as the product of the two exact node-empty probabilities. This product is a theorem-backed model result for this class, not a universal rule for every network of queues. 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.