Traffic Flow

Traffic flow theory describes vehicle movement on a roadway using three microscopic-to-macroscopic variables: flow rate q (veh/h), density k (veh/km), and space-mean speed v (km/h). Fundamental relation: q = k·v. Definitions: Flow q = n/Δt vehicles counted past point per time. Density k = n/L vehicles per unit length (or occupancy). Space-mean speed vs = L / (1/n Σ ti) = harmonic mean, used in q=k·vs; time-mean speed vt = arithmetic mean of spot speeds, vt ≥ vs. Headway and spacing: Average time headway h = 1/q (h = 3600/q in seconds if q in veh/h). Average spacing s = 1/k. Relationship vs = s/h. Fundamental diagram: Empirical q-k curve. Greenshields linear model assumes v = vf·(1 - k/kj) where vf = free-flow speed, kj = jam density. Then q = k·vf·(1 - k/kj) = vf·k - vf·k²/kj, parabola max at k* = kj/2, v* = vf/2, capacity qm = vf·kj/4. Capacity and LOS: Capacity is max sustainable q under ideal conditions (base ~1900-2400 pc/h/ln). Level of Service A-F based on density (HCM). Breakdown occurs when demand > capacity, queue forms with shockwaves. Assumptions: homogeneous traffic, steady state for relation, lane discipline, no intersection effects. Limitations: Greenshields is simplified; real diagrams have triangular or multi-regime shapes, hysteresis, and stochastic breakdown. Engineers use q=k·v for estimating capacity, signal timing, shockwave speed ω = (q2-q1)/(k2-k1).