Mixed-Fleet Runway Sequencing (3D)

What this is. A neutral scheduling model of one runway landing a batch of mixed aircraft. Each aircraft is Heavy, Medium or Light. A wake separation matrix says how many nautical miles must separate a follower from a leader of a given class; the page turns that into a time gap and then asks which landing order uses the runway best.

From distance to time

Leader i and follower j fly the common final path of length gamma at ground speeds v_i and v_j (approach speed minus headwind). The required distance s_ij must hold when the leader is at the entry gate and when it is at the threshold. Entry-gate condition: T >= s_ij / v_j. Threshold condition (binding when the follower is faster): T >= s_ij / v_j + gamma (1/v_i - 1/v_j). So

T_ij = max( ROT, s_ij / v_j + gamma max(0, 1/v_i - 1/v_j) )

with the runway occupancy time ROT as a floor. The closing-speed term is why a Light aircraft behind a Heavy one costs more than the matrix alone suggests when the follower is faster.

Capacity of a first-come-first-served stream

If the classes arrive at random with shares p_i, the mean gap is E[T] = sum_ij p_i p_j T_ij and the long-run rate is 3600 / E[T] per hour. Perfect grouping of equal classes would approach 3600 / sum_i p_i T_ii, the dashed bound in graph 3.

Sequencing a batch

For a batch of n aircraft the total time to land them all is the sum of the n-1 gaps. Minimising it over orders is a travelling-salesman-style problem, solved exactly here by dynamic programming over subsets (Held-Karp, 2^n x n states). Constrained position shifting allows each aircraft to move at most k places from its first-come-first-served position, a fairness limit used in practice; k = 0 is FCFS and k = n is the exact optimum. Grouped by class sorts into one block per class, taking the better of heavy-first or light-first.

Checks run in this page

• Monte Carlo over 200 000 random arrivals reproduces the formula for the FCFS rate.
• The dynamic programme equals a brute-force search over all 5040 orders of a 7-aircraft batch.
• Two aircraft flown in 0.05 s steps with the computed gap keep exactly the required distance at the binding point, for all nine class pairs.

What to try

• Heavy-rich hub: reordering saves about a quarter of the time.
• No wake rule: the gain shrinks, because only speed differences remain.
• Raise k from 0 to 12 and watch graph 4 flatten at the optimum.

Honest simplifications. The matrix values are illustrative round numbers in the style of a three-class rule, not an authority's current rules; real systems use finer classes, time-based rules and wind-dependent wake behaviour. All aircraft are assumed available at once (no earliest-time constraints), a single runway, constant speeds on final, no go-arounds. Changing a slider generates the same random batch again; the button draws a new one. The 3D view exaggerates aircraft size, runway width and distance scale.