Engineering & Technology · Transport Infrastructure & Urban Systems
Traffic flow, capacity, demand and network models
A busy road can have many vehicles stored on it while relatively few pass a detector. Density, speed and flow describe different aspects of the same traffic stream. A network model needs those definitions and the locations where queues can accumulate.
Review forces, structural load paths, conservation balances, graph scales and rates. The worked examples use supplied values; they are not route-design or operating specifications.
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Use compatible flow definitions and a conservation model without confusing demand, observed flow and capacity.
Define compatible quantities
Flow q is vehicles passing a location per unit time; density k is vehicles occupying a length; speed describes their movement. For a compatible single-stream description, q=k u, where u is space-mean speed. Units such as vehicles/km multiplied by km/h give vehicles/h. An arbitrary average of spot speeds is not automatically the matching speed in this identity.
The identity does not say which combinations will occur. A speed-density or other behavioural relationship supplies additional information and needs evidence for its intended context. It is a mistake to treat q=k u alone as a law showing that higher density always increases throughput.
Separate arriving demand from service
Demand is the arrival requirement; observed flow is what passes the observation point; capacity is an estimate of the service attainable under stated conditions. A bottleneck can prevent arrivals from appearing as downstream flow, with the difference stored upstream. A low downstream count can therefore coexist with unmet demand.
For a point queue, inventory change equals arrivals minus departures while the queue remains nonnegative. This treats all waiting vehicles as a count at a point. It can answer a simple accumulation question but omits spatial storage, changing lane use and the interactions of real drivers.
Finite road space adds spillback. When a queue reaches an upstream junction, it may constrain another movement and change the original arrival and service assumptions. A network is therefore more than independent links with fixed travel times. Route choice, turning flows and junction interactions belong to its boundary.
Match evidence to the model purpose
State where and how counts and speeds were observed, the interval and the conditions. A short quiet interval is not evidence for a peak service limit. Weather, incidents and changing demand also complicate comparisons. Check a model against observations different from those used to adjust it.
An aggregate vehicle calculation does not determine people carried, accessibility or safety. Vehicle occupancy, walking conditions and complete journeys require separate evidence. Explain the question the model answers before extending it to a broader policy claim.
Check a uniform stream and a separate queue
For a constructed uniform single-lane stream, k=20 vehicles/km and compatible space-mean speed u=60 km/h give q=1200 vehicles/h. A 2 km segment contains 40 vehicles and takes 2 minutes to traverse at that speed. These values describe one supplied state, not its maximum possible flow.
For a separate bottleneck case, begin with an empty point queue. Supply constant arrivals 1400 vehicles/h and service 1200 vehicles/h whenever the queue is present. Inventory grows at 200 vehicles/h. After 15 minutes it is 50 vehicles; after one hour it is 200.
The two cases share a numerical flow but are distinct models. The queue calculation assumes unlimited storage and continuous service. It cannot turn 200 vehicles into a spatial length without vehicle spacing and geometry, and it cannot continue unchanged once real upstream storage constrains arrivals.
Try the reasoning
Does q=k u determine capacity?
Show the answer and reasoning
No. It relates compatible state quantities; a behavioural and service description is still needed.
How many vehicles accumulate in 30 minutes in the supplied point queue?
Show the answer and reasoning
100 vehicles: 200 vehicles/h multiplied by .5 h.
Why does spillback change a fixed-link model?
Show the answer and reasoning
A full upstream storage space can constrain other movements and alter arrivals or service.
What to carry forward
Follow conservation, definitions and storage before interpreting a traffic model.
References
Teaching resources and research supporting this lesson’s explanations and examples.