Waiting Lines and Queuing Theory Models

Max Flow Tutorial

In this tutorial, we will cover the concept of maximal flow, or finding the maximum possible flow to get through a network. Note that this is not the shortest path through the network, but rather the route that allows, for example, the most vehicular traffic to pass through the network. It does not necessarily connect all nodes. In this example, we are looking at the pipes in a chemical processing plant and the numbers represent thousands of gallons per hour as the chemicals flow through the plant. Note that in this case, we need to be sure that we note the traffic going in both directions, as sometimes there are different flow rates when the direction is reversed.

Node 1 Node 2 Flow Reverse

Flow 1 2 4 0 1 3 1 1 1 4 3 3 2 5 1 0 3 6 3 3 3 7 2 1 4 8 1 1 5 9 4 4 6 10 3 1 6 13 6 0 6 7 2 1 7 11 2 2 8 11 4 0 9 12 3 1 9 10 1 0 10 13 5 2 11 13 5 0 12 14 5 1 13 14 6 0

Max Flow Tutorial

We have a total of 14 nodes, so open Excel QM, click on the Excel QM tab, alphabetical, network analysis, maximal flow. Here you must be very careful entering the flow and reverse flow. You will see 999999 entered from node 14 to node 1. That’s to tell Excel that there is no flow on that route and it’s automatically entered there. The solution is below:

Flow 1–2 1 1–3 1 1–4 1 2–5 1 3–6 1 4–8 1 5–9 1 6–13 1 8–11 1 9–12 1 11–13 1 12–14 1 13–14 2 Total maximum flow: 3.

In this case, our flow is in thousands, so our total flow along the route will be 3000 gallons per hour and will follow the route(s) noted above.