Routing of Airplanes to Two Runways: Monotonicity of Optimal Controls

Abstract

:We consider the problem of routing incoming airplanes to two runways of an airport. Due to air turbulence, the necessary separation time between two successive landing operations depends on the types of the airplanes. When viewed as a queueing problem, this means that we have dependent service times. The aim is to minimise waiting times of aircrafts.We consider here a model where arrivals form a stochastic process and where the decision maker does not know anything about future arrivals. We formulate this as a problem of stochastic dynamic programming and investigate monotonicity of optimal routing strategies with respect e.g. to the workload of the runways.We show that an optimal strategy is monotone (i.e. of switching type) only in a restricted case where decisions depend on the state of the runways only and not on the type of the arriving aircraft. Surprisingly, in the more realistic case where this type is also known to the decision maker, monotonicity need not hold.

On the waiting time of arriving aircrafts and the capacity of airports with one or two runways

“…Our future work will concentrate on more complex routing decisions as it has been started in [1] and on different types of arrival patterns. Simulation should help to detect structures of optimal decisions which could be proven analytically.…”

“…In [1] the problem of minimizing the average waiting time of aircrafts being routed to two runways is formulated as a Markov decision program with discounted costs (it is mainly a technical question to extend these results to the average waiting times). Markovian routing policies δ are considered that assign aircrafts depending on the state of the runways and the type of the incoming aircraft.…”

“…In [1], the present authors formulated a complete dynamic programming model based on these states. It is shown there, that if the available information is restricted to the states of the two runways and nothing is known about the arriving type, then the optimal policy has a join-theleast-load (JLL) property: if the types at the end of the two queues are identical, then any optimal policy assigns the n−th aircraft to runway I if U I n < U II n and to runway II if U I n > U II n .…”

“…It is shown there, that if the available information is restricted to the states of the two runways and nothing is known about the arriving type, then the optimal policy has a join-theleast-load (JLL) property: if the types at the end of the two queues are identical, then any optimal policy assigns the n−th aircraft to runway I if U I n < U II n and to runway II if U I n > U II n . Surprisingly, this need no longer hold, if the assignment policy is allowed to take the type of the arriving aircraft into account as is shown in [1] by a counterexample for finite horizon.…”

“…Note that we do not address the problem of optimal assignment of aircrafts to runways here (see [1] for a qualitative result in that direction). Instead we only consider easy to implement heuristics like coin flipping, type splitting, Round Robin and variants of join-the-least-load.…”