Departure Queue and Pushback Metering (2D map)

What this is. A neutral queuing model of one departure runway. Flights push back at random (Poisson) times, taxi for a fixed unimpeded time to the hold line, queue first-in-first-out and use the runway one at a time. The runway is the single server; its service time is Gamma-distributed with mean Ts and coefficient of variation CV.

Waiting time without control

With arrival rate lambda, service rate mu = 1/Ts and utilisation rho = lambda Ts, the Pollaczek-Khinchine formula for an M/G/1 queue gives the mean wait

Wq = lambda E[S^2] / (2 (1 - rho)) = rho (1 + CV^2) Ts / (2 (1 - rho))

so the taxi-out time is Tu + Wq. It is flat at low demand and explodes as rho approaches 1 (the hockey stick in graph 1); above rho = 1 the queue grows without bound. Variable service (larger CV) raises the wait in proportion to 1 + CV^2.

Little's law

For any stable queue, the time-average number in the system equals the throughput times the mean time in it: L = lambda W. It holds for the runway queue and for the whole taxi-out zone.

Pushback metering (N-control)

The controller releases flight k from the gate only when fewer than N* flights are taxiing or queued: release_k = max(demand_k, release_(k-1), start_(k-N*)). Waiting then happens at the gate with engines off instead of on the taxiway with engines running. If the runway never starves, throughput is unchanged and the taxi-out time of a saturated runway becomes N* Ts (Little's law again). The runway starves if N* is below the bandwidth-delay product mu Tu: the aircraft in transit cannot reach the hold line in time, and throughput falls to about N*/Tu. That gives the knee in graph 2 and the shallow minimum of fuel in graph 3.

Fuel is taxi time times the fuel flow you set (default 12 kg/min for all engines, an illustrative figure); CO2 uses 3.16 kg per kg of fuel.

Checks run in this page

• A 200 000-departure simulation (utilisation capped at 75 %) reproduces the Pollaczek-Khinchine wait to within 5 %.
• Little's law: a sweep over arrival and departure events integrates the queue length and equals lambda times the mean wait.
• Overload with deterministic service: taxi-out time equals N* Ts and throughput equals 3600/Ts.

What to try

• Busy bank, no metering then N = 14: same departures per hour, far less taxi time and fuel.
• Too tight, N = 3: the runway starves and departures per hour collapse.
• Raise the service variability and watch the queue lengthen at the same average demand.

Honest simplifications. One runway used only for departures, FIFO order, one gate-to-runway route, identical taxi speed, no arrivals crossing the runway, no sequencing by weight class or departure fixes, Poisson demand (real demand comes in banks), fixed unimpeded taxi time, fixed-seed random numbers so runs repeat. Fuel and CO2 figures are illustrative. Changing a slider restarts the run. The 3D view exaggerates aircraft size and runway width.