Important Definitions

A number of definitions are introduced here and will be used in the remainder of this chapter.

6.3.1 Traversal Paths

The concept of a traversal path and the importance of the path allocation were introduced in section 2.8. When the holding area is defined as a directed graph, such as in figures 6.2, 6.4 and 6.6, the traversal path followed by an aircraft can be uniquely defined in terms of the ordered sequence of nodes passed.

Figure 6.8 shows a graph for a simplified holding area. Assuming the entrance nodes are A and B, there are four possible paths through this graph: ACER, ADER, BCER and BDER.

Figure 6.8: Example simplified holding area graph

6.3.2 Path Suffix

The concept of a path suffix is important for the algorithm to perform the feasibility test. A path suffix is an ordered set of nodes beyond the current node which one or more paths traverse to reach the runway. A path suffix defines a section of a path from a given node to the runway. Any nodes on the path before the given node are ignored for the purposes of defining a path suffix. Multiple paths often, therefore, share the same path suffix beyond a given node.

For example, in figure 6.8 there are two suffixes for A, which are CER and DER. The relationships between paths, suffixes and nodes are illustrated in figure 6.8. Suffixes effectively relate nodes and paths together. For example, there are two suffixes at nodes A and B and only one at nodes C, D and E. There is a one-to-many relationship between nodes and suffixes. There is a many-to-many relationship between paths and suffixes.

6.3.3 Aircraft Priority

The concept of a priority for an aircraft is introduced here, to simplify explanations later. The priority of an aircraft for the feasibility check is defined by the take-off order. If aircraft A must leave before aircraft B then its priority is said to be greater than that of aircraft B. By assigning a numerical priority to all aircraft (from the position in the take-off sequence), the number of higher priority aircraft on any suffix can be quickly counted.

Figure 6.9: The relationship between paths, nodes and suffixes

6.3.4 Nodes of convergence

A ‘node of convergence’ is a node which can be entered from more than one other node, so it is a node in the holding area graph where multiple paths converge. An example node of convergence is node E in figure 6.10. Here the paths ABEF G and CDEF G converge at node E. Nodes of convergence can be quickly identified as they are the only nodes with multiple incoming arcs. The possible nodes of convergence in figure 6.2 are K, N , R, V and Y .

Nodes of convergence are vitally important for the feasibility check as it is at these points that a decision must be made by a runway controller about which aircraft should pass first. The sequence in which aircraft pass a node of convergence will restrict the possible take-off sequences. The sequence in which aircraft pass any other node is uniquely determined by the sequencing at the nodes of convergence.

Figure 6.10: Path convergence Figure 6.11: Path divergence

6.3.5 Nodes of divergence

‘Nodes of divergence’ are nodes at which two paths diverge. The diverging paths both use this node but the next node after the node of divergence differs between the paths. In figure 6.11, node B is a node of divergence.

Two paths will have different path suffixes at any node, N , if they diverge at node N itself or at any node between N and the runway. Given a holding area graph that has arcs only for paths that will be used, the nodes of divergence are the nodes with multiple outgoing arcs.

The possible nodes of divergence in figure 6.2 are D, G, N , R and V . The nodes that are actually nodes of convergence or divergence in practice will depend upon which paths are actually used at the time.

6.4

The General Path Allocation Algorithm

As was explained in section 2.8, the allocation to aircraft of paths through the holding area is a very important aspect of the problem considered in this thesis. The method presented here for allocating these paths is derived from the observations about sensible path allocations that were made in section 2.8.

The designed solution method requires that a path allocation heuristic is produced for each holding area. The production of the path allocation heuristics is discussed in this section. The detailed path allocation heuristics, which were used for the Heathrow holding areas considered in this research are detailed in section 6.5 and were produced by applying the principles which are explained in this section.

6.4.1 Identification of the types of paths

Before the path allocation process can take place, it is important to evaluate the different paths that are available for each holding area entrance and the circumstances under which they are used. A number of path categories have become apparent through discussing this problem with controllers and examining the ways in which holding areas are used. The ways in which these paths are used (for example when a slow path is used instead of a fast path) can be observed in the general path allocation algorithm, given by algorithm 2 and the specific algorithms for each holding area entrance, given by algorithms 3 to 10.

The fast paths

These are the most direct paths through the holding area, and are often preferred for that reason. The fastest paths in the 27R holding area graph, given in figure 6.2, are the paths DU V Y Z, KLM N Y Z and GOP QRZ for the holding area entrances D, K and G respectively.

The slower but good paths

These paths are slower but are fast enough and easy enough for an aircraft to traverse that they are worth allocating to aircraft. These paths are commonly used, but usually only when there is a reason why the fast paths cannot be used. The slower paths in the 27R holding area are DU V XY Z, KLM N Y Z and GOP QRST Z.

The default path

It is important to identify the default path for each holding area entrance. This is the path that controllers will use if there is no restriction upon the re-sequencing of the aircraft. This will often be the fast path, but may be a slower path in cases where the controllers view this to be almost as fast and easy to navigate as the fast path, but more useful for maintaining flexibility for later re-sequencing. The default paths in the 27R holding area are the DU V XY Z, KLM N Y Z and GOP QRST Z paths. The DU V XY Z path is a slower path but is almost as fast to use as the DU V Y Z path. The GOP QRST Z path is longer than the GOP QRZ path but is useful for allowing later aircraft to overtake those at S or T and is very easy for a pilot to use.

Overtaking and short-cut paths

There may be one or more paths which are useful for overtaking aircraft already within the holding area, but which would not otherwise be used. For example, in the 27R holding area, it is useful to use the DU V Z path to overtake an aircraft that has already been assigned to the DU V Y Z path, or to allow more than one aircraft from entrance D to be overtaken. The overtaking paths in the 27R holding area are the paths DU V Z and KLM N V Z, for entrances D and K respectively.

Sometimes the overtaking path may enter the runway away from the end. Any path which enters the runway from node V in the 27R holding area in figure 6.2 is an example of such a path. These can be considered to take a short-cut to the runway, allowing additional overtaking, however the fact that the full runway cannot be used for the take-off means that they are not ideal. Since larger aircraft may need the full runway, the path allocation algorithms used for this research prevented the decision support system from assigning a heavy weight class aircraft to a path which would not allow it to use the full runway.

6.4.2 Overview of the general path allocation heuristic

Prior to applying the path allocation heuristic, the aircraft must first be divided according to the holding area entrance at which they arrive. It is then necessary to determine the overtaking that is required between the aircraft which arrive at the same holding area entrance. Relative overtaking between aircraft which arrive at different entrances is ignored. The paths that are allocated to some aircraft may have been fixed, as discussed in section 2.12.2. For these aircraft, the fixed path is retained and is allocated prior to using the path allocation heuristic. Finally, variants of algorithm 2 are applied to allocate the paths to the aircraft which arrive at each of the holding area entrances, in turn.

Algorithm 2 uses the concepts of a fast, slow, default and shortcut path which were introduced above. Steps 1 to 5 allocate the default path to any aircraft for which it is appropriate. Steps 6 to 10 allocate the slow path to aircraft which need to be overtaken, and do not need

Algorithm 2 General path allocation heuristic algorithm

1: for each aircraft a arriving at the entrance, which has not had a path allocated do

2: if a is not overtaken and does not overtake then

3: assign the default path to aircraft a.

4: end if

5: end for each

6: for each aircraft a arriving at the entrance, which has not had a path allocated do

7: if a is overtaken but does not overtake then

8: assign the slow path to aircraft a.

9: end if

10: end for each

11: for each aircraft a arriving at the entrance, which has not had a path allocated do

12: if a only overtakes aircraft which have been allocated to the slow path, and these are few enough that it can overtake using the fast path then

13: assign the fast path to aircraft a.

14: end if

15: end for each

16: for each aircraft a arriving at the entrance, which has not had a path allocated do

17: if paths have been allocated to all of the aircraft that a overtakes and it is valid for a to use the overtaking or shortcut path then

18: assign the shortcut path to aircraft a.

19: end if

20: end for each

21: if each there exists an aircraft a which has not had a path allocated then

22: declare the re-sequencing to be infeasible

23: end if

to overtake, in order to move them out of the way of the overtaking aircraft. Steps 11 to 15 then allocate the fast path to aircraft which only need to overtake those aircraft which have been allocated to the slow path. Note that, in order to overtake using the fast path it is important to know the number of aircraft overtaken and the number of positions at which aircraft can be parked on the slow path to be overtaken by aircraft which are on the fast path. It may be necessary to use the shortcut path to overtake if there is insufficient space to park aircraft in order to overtake using the fast path. Finally, steps 16 to 20 allocate the overtaking or shortcut path to the remaining aircraft where possible, as long as an overtaking path exists for that holding area entrance and the aircraft is permitted to use it. For example, heavy aircraft are constrained by the implemented algorithms to only be allocated paths which will allow them to use the full runway for take-off.

6.4.3 Reasons for a per-entrance allocation of paths

As discussed in section 1.3, although the intention is for the decision support system to present a controller with a suggested take-off sequence, it is important that the method to achieve the desired re-sequencing is obvious to controllers. This means that a controller must be able to easily identify the paths to allocate to aircraft and the sequence in which aircraft should pass different points within the holding area. As the size of the control problem increases, this identification becomes more difficult and it becomes necessary to decompose the problem into smaller sub- problems. The obvious and intuitive way to do this is to consider the paths allocated to aircraft

on a per-entrance basis. The designed decision support system uses this approach.

The intuitiveness of the per-entrance decomposition of the problem is a major advantage of the approach, enabling the controllers to consider a much smaller set of aircraft and paths in isolation. The controllers can then use a simple method to determine the paths to allocate to aircraft. Moreover, the controllers can be sure that such a simplistic consideration of the paths to allocate to aircraft will be sufficient to achieve any suggested take-off sequence as the designed decision support system provides a guarantee that the suggested sequences are achievable using a per-entrance path allocation method.

A per-entrance path allocation has been adopted, despite the fact that it is possible to construct scenarios where a per-entrance path allocation will not allow the aircraft to be suc- cessfully re-sequenced but where there is an alternative path allocation that could be assigned to aircraft which would allow the re-sequencing to be achieved, since it works well for the Heathrow holding areas and has a number of important advantages. The advantages can be summarised as follows:

1. The per-entrance path allocation ensures that the default path is allocated whenever pos- sible. Furthermore it ensures that paths will be allocated in a manner which will seem sensible to controllers and will be acceptable.

2. If the per-entrance path allocation fails but a different path allocation would work then, when considering the aircraft arriving at each entrance in isolation, the path allocation will not be as sensible as a per-entrance path allocation would be for at least one of these entrances (otherwise it would be the per-entrance path allocation). It would be necessary in that case to verify that the path allocation would still be acceptable to controllers. 3. If the per-entrance path allocation is not used then there is no guarantee that the controller

will be able to identify how the re-sequencing is to be achieved. As explained above, the decomposition of the problem by holding area entrance makes it easier for controllers to determine how a take-off sequence can be attained. Without this, the controller is less likely to see the sequence as sensible, so less likely to accept the sequencing advice. This could undermine long term controller confidence in the system as well as meaning that the controller ignores the immediate advice.

4. If the system ever suggests any take-off sequences which are not achievable using a per- entrance allocation method then the controller can no longer use the fact that the system only uses such per-entrance allocated paths in order to simplify the determination of the paths necessary to achieve any suggested sequence. Consistently using such a system allows the controller to rely upon this fact in determining how the sequence will be attained and ensures that the method will be obvious to the controller.

Heathrow holding areas, since the number of points at which aircraft can conflict with each other is usually quite small and, in most practical cases, where a per-entrance allocation fails to achieve the re-sequencing, either the re-sequencing would not have been possible with any other path allocation, or the path allocation that would have been necessary would not have been acceptable. The observations below give some insight into the reasons for this.

Reasons why the per-entrance path allocation is effective

Firstly, in some cases, such as the 09R holding area, the independence of path allocation between aircraft arriving at different entrances is assured as the holding area structures beyond the en- trances are disjoint. In this case, consideration on a per-entrance rather than global basis can make no difference.

Secondly, problems can only occur when aircraft are moved from the queue from one entrance into the queue for another entrance, potentially interfering with the re-sequencing that is occurring with the aircraft from that second entrance. The only time an aircraft cannot be allocated to the preferred path (and thus may need to be moved into the queue from another entrance) is when it overtakes or is overtaken by another aircraft which arrives at the same entrance. When this is not the case, re-sequencing merely requires the interleaving of the queues from the different entrances at the node at which the queues converge. There is an objective to control the inequity in the take-off sequence and this helps to control the total amount of re- sequencing that is required. Since so much re-sequencing can be performed by the re-sequencing of aircraft which arrive at different entrances relative to each other, the amount of re-sequencing that is usually required between aircraft in the same queue is even lower.

The effect of moving an aircraft between queues often depends upon how long the moved aircraft will be present for, since it will block a path from the other entrance for the duration for which it is present. The bias towards more equitable sequencing helps to ensure that an aircraft is not usually delayed for long in the holding area, so reduces the length of time for which any path will be blocked. As will be seen in chapter 8, aircraft only have long holds when they are waiting for the start of a CTOT take-off slot. In practice, the controllers would know about any excessive required delay for a late CTOT in a live situation and (if an earlier CTOT could not be negotiated) would release the aircraft from the stand later (when possible) or park the aircraft at a remote holding location, in order to reduce the duration of the delay in the runway holding area. (As will be seen in chapter 8, where long delays are observed with the test data, the controllers had actually re-negotiated the CTOTs to allow aircraft to take off earlier.) This is important since it helps to limit the duration for which parked aircraft block paths through the holding area.

Note that it is possible to estimate the take-off time of an aircraft prior to performing the feasibility check, as discussed in section 6.9.3. If large delays are expected, then the length of time for which the path would be blocked could be predicted and used in the path allocation

system for the affected entrances to avoid allocating to the blocked paths aircraft which need to take off before the blocking aircraft does.

6.4.4 Controllers are not obliged to use the predicted paths

It should be noted that controllers are not obliged to achieve the take-off sequence in the same way as the system predicts they will, although the focus upon using the preferred paths should mean that they usually will. It is useful to be aware of this, since it allows the path allocation system