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Model checking evaluation of airplane landing trajectories

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Abstract

Computing trajectories of a set of airplanes in their final descent is an important problem in air traffic control. It consists of deciding a trajectory, the runway, and the landing time for each airplane, such that several constraints are satisfied, while optimizing flying (fuel) costs, and minimizing waiting times. To solve this problem, we model it as a discrete game, the k-king puzzle, in which each airplane is represented (and it moves) as a king chess-piece on a chess-board. Although the model has already been introduced in the past, we propose several extensions, taking into account different aspects of the real problem, such as constrained airspaces, distinct airplane speeds, various separation distance among airplanes, and specific restrictions in the landing trajectories. Moreover, we model both static and dynamic cases for 2D and 3D airspaces. On these extensions, we describe an exact resolution method based on ideas and algorithms coming from the formal verification of correctness of hardware devices and software tools area. Furthermore, to improve the size and complexity of the models we are able to deal with, we propose a decomposition technique based on the divide-and-conquer paradigm. This solution, which we call Plane by Plane decomposition, trades-off between accuracy and efficiency, i.e., exact solutions degrade to non-optimal ones to maintain scalability. We finally propose an implementation of this algorithm based on (and taking advantages of) modern multi-core, multi-threaded, systems. We present a detailed description of the model and the algorithms, as well as our computational results for quite large static and dynamic 2D and 3D problems.

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Notes

  1. The term thermometric comes from old mercury thermometers, where one side is always filled with mercury and the other one is empty.

  2. NuSMV is a symbolic model checker originated from the re-engineering, reimplementation, and extension of CMU SMV, the original BDD-based model checker developed at CMU [19]. It is a well structured, open, flexible and documented platform for model checking, and is robust and close to industrial systems standards.

    Fig. 3
    figure 3

    Representing the running example of Fig. 2 as a game structure, using the NuSMV high-level description language

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  3. Notice that, to satisfy strong resource constraints, or solve larger problems, it is always possible to switch from exact formal verification strategies to heuristic methods. In any case, we do not explicitly adopt this technique in our experiments.

  4. For this reason, the wall clock time is also known as elapsed time. In this context, CPU time indicates the total time, i.e., the sum of all CPU times devoted to the task (i.e., to all threads) by each CPU running it.

  5. A cactus plot is a graph in which the \(x\) axis represents instances (whatever they are), and the data on the \(y\) axis are sorted in ascending order.

    Fig. 13
    figure 13

    CPU time results divided by chess-board size and ordered by increasing complexity

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    Fig. 14
    figure 14

    Memory results divided by chess-board size and ordered by increasing complexity

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  6. Beyond those board sizes, we report data for sake of completeness, even if our experiments would be physically unfeasible. In other words, beyond the inter-arrival time, we collect our data using a sort of “time machine”, i.e., whenever computation time is longer than inter-arrival time we stop the clock, compute the solution, and we move back to the past by displacing the airplanes following this solution and considering the new airplane on the board.

    Fig. 18
    figure 18

    Results for the dynamic case, obtained by freezing old solutions. Time results (left-hand side) and memory results (right-hand side) are reported for \(1\) and \(4\) runways as a function of the board size, varying from \(3\) to \(20\)

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Acknowledgments

We would like to thank our colleagues Gianpiero Cabodi and Sergio Nocco for some useful discussion and hints on the topic.

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  1. Politecnico di Torino, Dipartimento di Automatica e Informatica, Turin, Italy

    Stefano Quer

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Quer, S. Model checking evaluation of airplane landing trajectories. Int J Softw Tools Technol Transfer 16, 753–773 (2014). https://doi.org/10.1007/s10009-013-0273-2

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