CIVL2700 · Transport Systems
Fundamental Diagram, MFD & Traffic Measurement
This University of Sydney CIVL2700 Transport Systems chapter is the Week-8 hinge of the traffic-engineering block: it ties a road's flow, density and speed together with the identity q = k·v and reads capacity and congestion off the fundamental diagram. You meet the course's default triangular FD (free-flow and congested branches, capacity qmax and the backward congestion-wave speed w), the Greenshields parabolic alternative, the Macroscopic Fundamental Diagram (MFD) that makes network production an inverted-U in accumulation n, and the loop-detector conversion of occupancy into density. It is examined in the second in-class test and again in the comprehensive final.
What this chapter covers
- 01The fundamental identity q = k·v and why only space-mean speed makes it exact
- 02The triangular fundamental diagram: free-flow branch q = v_f·k and congested branch q = w(k_j − k)
- 03Capacity q_max = v_f·k_c at the critical density k_c
- 04The backward congestion-wave speed w = q_max/(k_j − k_c) and what it physically means
- 05Greenshields parabolic model: q_max = v_f·k_j/4 at k_c = k_j/2 (use only when named)
- 06The Macroscopic Fundamental Diagram: production G(n) as an inverted-U in accumulation n
- 07The accumulation update and critical accumulation n* (congested when n > n*)
- 08Loop detectors: density from occupancy k = O/(L_v + L_d), then v = q/k
- 09Units discipline: veh/h, veh/km, km/h, veh/s, and the 1 ft = 0.3048 m conversion
Triangular fundamental diagram: capacity, congestion wave, and a congested state
- +2(a) Capacity. The peak of the triangle is at k = kc: qmax = vf·kc = 90 × 20 = 1800 veh/h.
- +2(b) Congestion-wave speed. It is the magnitude of the congested-branch slope: w = qmax/(kj − kc) = 1800/(140 − 20) = 1800/120 = 15 km/h — the queue tail travels upstream at 15 km/h.
- +2(c) State at k = 100. Since 100 > kc = 20, use the congested branch: q = w(kj − k) = 15 × (140 − 100) = 15 × 40 = 600 veh/h; then v = q/k = 600/100 = 6 km/h.
Key terms
- Fundamental identity (q = k·v)
- The exact relation between flow q [veh/h], density k [veh/km] and speed v [km/h] at a point on a road. It holds only when v is the space-mean (harmonic, section-based) speed; substituting the time-mean speed measured at a fixed detector gives the wrong flow because fast vehicles are over-sampled at a point.
- Fundamental diagram (FD)
- The flow–density curve q(k) for a road. CIVL2700's default is the triangular model: a straight free-flow branch q = vf·k rising to capacity, then a straight congested branch q = w(kj − k) falling to zero at jam density.
- Critical density (k_c) and capacity (q_max)
- kc is the density at which flow peaks; the peak value is the capacity qmax = vf·kc = w(kj − kc) [veh/h]. Below kc the road is free-flowing; above it, congested.
- Backward congestion-wave speed (w)
- The magnitude of the congested-branch slope, w = qmax/(kj − kc) [km/h]. It is the speed at which the boundary between free and jammed traffic travels upstream as vehicles join the back of a queue. The minus sign is already in q = w(kj − k), so w itself is a positive magnitude.
- Jam density (k_j)
- The density at which vehicles are bumper-to-bumper and flow is zero [veh/km]. Its reciprocal is the minimum spacing, smin = 1/kj.
- Greenshields model
- A parabolic FD from a linear speed–density law v = vf(1 − k/kj), giving q = vf·k(1 − k/kj) with capacity qmax = vf·kj/4 at kc = kj/2. Use it only when a question names Greenshields; otherwise use the triangular FD.
- Macroscopic Fundamental Diagram (MFD) and accumulation (n)
- A network-level relation between the accumulation n (vehicles currently inside the network) and the production G(n) (trips completed per second). G(n) is an inverted-U: production rises with n, peaks at the critical accumulation n*, then falls as gridlock spreads.
- Critical accumulation (n*)
- The accumulation that maximises production, found from dG/dn = 0 (for G(n) = a·n − b·n2, n* = a/(2b)). The network is uncongested for n < n* and congested for n > n*; perimeter (gating) control aims to hold n near n*.
- Loop-detector occupancy (O)
- The dimensionless fraction of an interval that the in-road loop is covered by a vehicle. Density follows from k = O/(Lv + Ld), where Lv is the mean vehicle length and Ld the detector length in the same units (1 ft = 0.3048 m). Speed is then v = q/k.
Fundamental Diagram, MFD & Traffic Measurement FAQ
Why must q = k·v use the space-mean speed and not the detector's spot speeds?
A loop or radar at a fixed point over-samples fast vehicles (they pass more often), so the arithmetic average of spot speeds — the time-mean speed — is biased high and is always ≥ the space-mean speed. Only the space-mean (harmonic / travel-time-based) speed makes q = k·v exact, which is why exam questions that hand you spot speeds expect a conversion before you multiply by density.
How do I know which branch of the triangular FD to use?
Compare the given density k with the critical density kc. If k ≤ kc the road is free-flowing, so q = vf·k; if k ≥ kc it is congested, so q = w(kj − k) with w = qmax/(kj − kc). A flow below capacity corresponds to two densities (one on each branch), so the wording — congested or not — tells you which one is meant.
Can AI help me with the fundamental diagram, MFD and traffic measurement in CIVL2700?
Yes — Sia can explain each idea step by step: it will walk you through picking the right FD branch, deriving capacity and the congestion-wave speed, stepping an MFD accumulation forward, or converting loop-detector occupancy into density, checking your units at each line. It is a study aid that builds your method and your own working step by step; it never hands over ready-made answers to submitted assessments, completes your work for you, or guarantees a grade; always confirm exam details on Canvas.
Exam move
Anchor everything to the two diagrams. For the triangular FD, sketch the q–k triangle first, mark kc, kj, qmax and the slopes +vf and −w, then let any question reduce to reading a point off it — and always test k against kc before choosing a formula. For the MFD, memorise the recipe: find n* from dG/dn = 0, compare n with n*, then step n forward with nnext = n + ΔT·[u·Iext + Iint − G(n)], watching seconds-versus-minutes on ΔT. Drill the loop-detector chain k = O/(Lv + Ld) then v = q/k with clean unit conversions (1 ft = 0.3048 m). Because most marks are procedural, write every substitution out so method marks survive an arithmetic slip. This is Weeks 5–8 content for the second in-class test and is also fair game in the final exam (40% of the unit, a 2.5-hour paper, with a 40% hurdle on the exam itself); confirm the exact date and whether it is open- or closed-book on Canvas.
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