CIVL2700 · Transport Systems
Traffic-Stream Variables: Flow, Density, Speed
Week 7 of University of Sydney CIVL2700 Transport Systems introduces the three macroscopic descriptors of a traffic stream — flow q [veh/h], density k [veh/km] and speed v [km/h] — bound by the identity q = k·v. It links these aggregates to the microscopic gaps between vehicles (time headway and spacing) and to the two legitimate speed averages, and it isolates the examinable subtlety that q = k·v holds only for the space-mean speed. This foundation underpins the fundamental diagram, capacity and signal-timing chapters that follow.
What this chapter covers
- 01Flow q = n/T [veh/h], density k = n/L [veh/km], speed v [km/h] and the identity q = k·v (units: veh/km × km/h = veh/h)
- 02Time headway h and spacing s: q = 1/h̄ (mean headway), k = 1/s̄ (mean spacing), s = v·h
- 03Time-mean speed (arithmetic / flow-weighted mean of spot speeds at a fixed point)
- 04Space-mean speed (harmonic / density-weighted mean over a section) — the only speed that satisfies v_s = q/k
- 05The inequality v_t ≥ v_s and the relation v_t = v_s + σ²_s / v_s
- 06The inspection paradox: fast vehicles are over-represented at a point, biasing the time-mean speed upward
- 07Point (temporal) vs section (spatial) measurement, and why the two speeds differ
- 08Reading a time–space (trajectory) diagram: headway along the detector line, spacing along the snapshot line
Space-mean vs time-mean speed of a two-group stream
- +1Total density is the sum of the sub-stream densities: k = k₁ + k₂ = 40 + 20 = 60 veh/km.
- +1Flow is the sum of the sub-stream flows q_j = k_j·v_j: q = 40·20 + 20·80 = 800 + 1600 = 2400 veh/h.
- +1Space-mean speed is the density-weighted mean, equal to q/k: v_s = 2400 / 60 = 40 km/h. This is the speed that satisfies q = k·v.
- +1Time-mean speed is the flow-weighted mean: v_t = Σ(q_j v_j)/Σ q_j = (800·20 + 1600·80)/2400 = 144000/2400 = 60 km/h. So v_t (60) > v_s (40).
Key terms
- Flow (q)
- The number of vehicles passing a fixed point per unit time, q = n/T [veh/h]. Related to the mean time headway by q = 1/h̄.
- Density (k)
- The number of vehicles present on a unit length of road at an instant, k = n/L [veh/km]. Related to the mean spacing by k = 1/s̄.
- Space-mean speed (v_s)
- The harmonic (distance/travel-time, density-weighted) mean of vehicle speeds over a section. The only speed for which q = k·v is exact: v_s = q/k.
- Time-mean speed (v_t)
- The arithmetic (flow-weighted) mean of the spot speeds of vehicles passing a fixed point. Always at least as large as the space-mean speed: v_t ≥ v_s.
- Time headway (h)
- The time [s] between successive vehicles passing a point; the reciprocal of its mean is the flow, q = 1/h̄.
- Spacing (s)
- The distance between successive vehicles at an instant; the reciprocal of its mean is the density, k = 1/s̄. Linked to headway by s = v·h.
- Inspection paradox
- The upward bias in a point-based speed sample: a detector counts fast vehicles more often than slow ones, so the time-mean speed sits above the space-mean speed. The gap equals σ²_s / v_s.
Traffic-Stream Variables: Flow, Density, Speed FAQ
Why does q = k·v only work with the space-mean speed?
The identity is really flow = density × (distance covered per unit time by the vehicles on the road). That average distance-per-time is the space-mean (harmonic, section) speed. The time-mean speed averages spot speeds at a point, where fast vehicles are over-sampled, so it is biased upward and would over-estimate the flow. Substituting v_s = q/k is exact; substituting v_t is not.
What is the difference between time-mean and space-mean speed?
Time-mean speed is the arithmetic mean of the spot speeds recorded at a fixed point (a radar or loop detector). Space-mean speed is the harmonic mean over a length of road, equivalently distance divided by mean travel time, and it is density-weighted. They are equal only when every vehicle travels at the same speed; otherwise v_t = v_s + σ²_s/v_s ≥ v_s.
Can AI help me with traffic-stream variables in CIVL2700?
Yes, as a study aid. Sia can explain q = k·v step by step, show why the space-mean speed (not the time-mean speed) belongs in the identity, walk through headway/spacing conversions, and check your unit work on practice problems. Use it to understand the method and rehearse; it does not do your graded assignment or exam for you, and always confirm assessment details on Canvas.
Exam move
Anchor everything on q = k·v and drill the unit chain until it is automatic: q = 1/h̄ gives veh/s (multiply by 3600 for veh/h), k = 1/s̄ gives veh/km, and v = q/k returns the space-mean speed in km/h. Practise the time-mean vs space-mean distinction on sub-stream tables (list k_j, v_j and q_j = k_j v_j, then compute v_s = q/k and v_t = Σq_j v_j / Σq_j), and always state the ordering v_t ≥ v_s with the inspection-paradox reason. Week-7 material is assessed in In-class Test 2 (Weeks 5–8) and again in the comprehensive final exam (40%, 2.5 hours, a 40% hurdle), so make sure you can produce the sub-stream table, the correct speed in the identity, and the right units under time pressure. Confirm the exam date, room and open/closed-book status on Canvas.
Working through Traffic-Stream Variables: Flow, Density, Speed in CIVL2700? Sia is AskSia’s AI Engineering tutor — ask any CIVL2700 Traffic-Stream Variables: Flow, Density, Speed question and get a clear, step-by-step explanation grounded in how CIVL2700 is taught and assessed. Read this chapter free, then take your hardest questions to Sia.
