Product Form
Product-Form Grid
A product-form distribution turns each joint state into a product of node factors. The lesson displays a small exact grid, not a simulated estimate. Each cell should be read as a stationary probability for a network state inside the displayed window.
Both empty cell
The base grid cell is 1/3. Why: it is the probability both stations are empty and the anchor for nearby joint states. Once this base is known, product form grows the grid by multiplying by the relevant node utilization for each additional job at a station.
One job at station B
The cell with station B one step above empty is 1/6. Why: multiply the base cell by station B utilization because the state changes only at station B. The calculation is local in the formula even though the queueing story is a routed network.
One job at station A
The cell with station A one step above empty is 1/9. Why: multiply the base cell by station A utilization because this cell adds work at station A instead. Comparing the two one-job cells shows how different node utilizations shape the joint distribution.
One job at each station
The cell with one job at each station is 1/18. Why: product form multiplies both utilization factors, one for each occupied station. The grid is therefore an exact stationary table for the displayed state window, not a Monte Carlo estimate. Diagram note: this is a finite window into the product-form distribution, and the factorization is valid for the idealized open Jackson tandem with stable nodes, not for arbitrary queueing networks. The exact display stops at that model boundary. Pixel positions are rounded for layout; every number shown is exact.