1. INTRODUCTION
World air traffic is forecast to grow considerably in the coming decades, but the price of jet kerosene will vary in the same way as that of crude oil. This has led to concerns about the ability of the Air Traffic Management (ATM) system to provide sufficient capacity to handle increased traffic volumes both safely and economically – e.g. to deliver (climate change benefiting) fuel efficiency improvements. Past improvements have mainly been achieved through new operational procedures, such as airspace design, enhanced coordination between airline operators and ground-based air traffic control (ATC), and better planning and management of traffic flows. But the fundamental ATC concepts have not changed markedly. The core ATC technologies have not developed as much as have aircraft navigational systems, and there is a belief that ATM has not undergone the information revolution seen in other technological industries.
Thus, it is now widely recognised that a paradigm shift in ATC concepts is needed. This has to be supported by state-of-the-art innovative technologies, making much better use of the information in the ATM system. These paradigm shifts go under the names of NextGen in the USA and SESAR in Europe. Arbuckle et al (Reference Arbuckle, Rhodes, Andrews, Roberts, Hallowell, Baker, Burleson, Howell and Anderegg2006) and Ky & Maillier (Reference Ky and Miaillier2006) provide comparable background on the two concepts: the NextGen and SESAR websites have much more detailed up-to-date descriptions.
A vital part of moving from an existing system to a new paradigm is what happens during the transition process. This note is about one important aspect of the transition: delays to aircraft arriving at airports. The problem is that the transition process will tend to introduce different classes of user, which will each get different kinds of priority service from the ATM system. Different kinds of service then imply different levels of delay. The aim here is to provide some rough estimates of the magnitude of these delay effects, and the method used is analytical queuing theory. This kind of model has been used to identify and derive rough quantitative estimates of airport congestion effects for at least 60 years (Bell, Reference Bell1949).
2. SKETCH OF CURRENT ARRIVAL PROCESS
An airport's capacity is not simply the rate at which its runways operate. There have to be sufficient efficient runway exits, taxiways and apron space to move aircraft to and from terminals, and sufficient capacity in terminal buildings to process passengers. But runway capacity is a necessary ingredient, and usually the leading throughput constraint. The key element in determining this capacity is the regularly achievable safe separation between aircraft. The main example used here is mixed mode, which means mixing departures (D) and arrivals (A) on the same runway, typically a sequence A-D-A-D-A-D-… Mixed mode delivers high capacity because it allows a departure to line up for takeoff whilst an arrival is using the runway (Table 1).
Table 1. Simplified A-D-A sequence model.
A major airport using mixed mode, such as Gatwick, can achieve an A-D-A cycle of better than 180 seconds, the Service Time, which corresponds to an hourly movement rate of 40 – the maximum sustainable throughput (MST). So, the problem is solved: schedule arrivals at 180-second intervals and deliver 40 movements an hour regular as clockwork? Alas, airline schedules are not precise. Even European short haul flights of 60 minutes are considered punctual when they are within 15 minutes of schedule. It is quite usual to have up to 10 flights scheduled at an airport to depart or arrive at the same time. Airline operations, even on good days, can be subject to considerable disruption. The disruptions arise for mundane reasons: missing check-in passengers, late connecting passengers and baggage, knock-on effects of previously delayed flights by the aircraft, etc. Flights only achieve the kinds of punctuality observed in practice by padding out schedules, i.e. by embedding buffer times to schedules. Some examples: Wu (Reference Wu2005) discusses the reliability of airport schedules; Eurocontrol PRC (2008) presents a picture of European punctuality; Young (Reference Young2008) models USA data – where bad weather is a major additional factor (his Figures 4.1 and 4.1 show trends over a ten-year period).
The usual consequence of a high demand for a runway and large variations in arrival times is a queue of delayed aircraft. These delays are an intrinsic consequence of the way that the ATM system is designed to feed aircraft into and out of airports, not the product of inefficient ATC. Traditionally, most airborne delays are taken in holding stacks. Figure 1 is a sketch of the position of a holding stack. A typical holding pattern is a straight leg of 1-minute duration, a 180-degree turn of 1-minute duration, another straight leg and another 180-degree turn. This racetrack circuit usually takes 4 minutes to complete. Several aircraft can hold by each flying one above another – separated vertically by 1000ft – until directed to make their final approach to the runway. As shown, some delays are taken up by path stretching, with the controller vectoring the aircraft's flight-path. The vertical profile of the arrival is a sequence of level flight and descent segments, typically with the final ten miles using the Instrument Landing System (ILS) guidance to fly the aircraft to the runway on an ILS 3° glide-slope.
Figure 1. Plan-view of traditional arrival process for a busy runway.
Computer simulations of a runway's operation, over a day or a busy period, produce something like Figure 2. These are average figures, so on good days the delays are less, but on bad days they can be much larger. Average delays are small when traffic is much less than the MST, but there is a sharp increase in the average delay for demands near to it. Thus, there is a trade-off between delay and movement rate. Runway capacity for busy airport schedules requires agreement on an acceptable average delay – shown here at 10 minutes.
Figure 2. Average aircraft delay versus traffic demand.
3. TRAJECTORY-BASED AIR TRAFFIC MANAGEMENT
What is wrong with traditional arrival processes? What would a better system look like? Aircraft use up fuel when stacking and flying non-direct flight-paths. Passengers sometimes suffer large delays. Runway capacity could be higher – allowances that need to be made for the variations in arrival times and the large safety buffer time mean that valuable seconds are wasted.
SESAR and NextGen offer solutions to these kinds of problems as part of a new paradigm ATM system. A nice explanation of the thinking behind this solution is given in Wichman et al (Reference Wichman, Lindberg, Kilchert and Bleeker2003). This identifies the two critical ingredients as:
“System Wide Information Management: All actors supply and receive the latest pertinent information from all other actors; air and ground.
4D Trajectory Management: The aircraft's 4D trajectory is shared among all actors and managed for flow and conflict resolution purposes only. This includes establishing an arrival time ‘contract’ with Air Traffic Service Providers (ATSPs)…”
An actor here is an entity contributing to the system operation – airlines, ATSPs, etc. 4D means the three dimensions of space plus one of time. This trajectory is generated very accurately by the aircraft Flight Management System (FMS) from the aircraft's current state, throughout the remainder of the flight. Trajectories are shared and updated from the source(s) best suited to the prevailing operational circumstances. Precision positioning, intent and performance combine to deliver time-based operations – a contract. A contract is an agreement that both parties are supposed to meet. In contrast, current non-4D-equipped aircraft receive traditional speed, altitude and heading clearances from ATC.
How a 4D trajectory ATM system works in practice for both airports and airspace is outside the present scope. The key point is that ATM moves from what is essentially a space-based system to a time-based one. Figure 3 illustrates the kinds of timing that would be achieved – there are several different versions of these kinds of timing metrics in the literature, but the overriding feature is that aircraft cross runway thresholds at very precise planned times. This means that both the standard deviation and the safety buffer time periods shown in Table 1 can be markedly reduced.
Figure 3. Elevation-view of SESAR/NextGen approach – with illustrative time metrics for approach (±1 standard deviation). Plan view would be a smooth turn from a direct routeing onto final approach. Note: SESAR/NextGen might offer better en route performance than shown here – and the current ‘Top of Descent’ might be difficult to interpret for a fuel-efficient profile.
It is unlikely that the system transition to a 100% 4D-equipped aircraft population would be a ‘big bang’, i.e. with the new paradigm not being brought into use until all aircraft were equipped. First, there would need to be business incentives for aircraft fitment and the new ground systems: early adopters would very much want to get operational benefits from the changes. Second, experience shows that new technologies and operational concepts that can be introduced gradually probably have a better chance of risk-minimising success than a switch-on to a wholly new kind of operation. Gradual changes can be accompanied by extra layers of safety protection until the new system has proved that it does indeed deliver a much safer system. Third, researchers have already proposed potential transition steps to the new system, making use of largely extant technology.
With transitional steps, it would not be necessary for the entire ATM infrastructure to be in place before changing to a time-based arrival system. An example of a transitional – 3D+trajectory – system for suitably equipped aircraft is shown in Figure 4 (Haraldsdottir et al, Reference Haraldsdottir, Scharl, Berge, Schoemig and Coats2007). The main ideas are that:
• ATC has new automation support tools to aid descent process.
• Ground computer assistance computes schedule approach fixes, based on required runway threshold time.
• Selection of cruise/descent speeds and lateral path stretch to meet schedule.
• Voice based clearances by ATC.
• Path stretching required when speed control is insufficient to achieve spacing – thus a 3-minute delay would usually require a path stretch.
• Aircraft FMS flies optimal vertical profile when executing its flight path.
The Figure shows Triangular Delay Paths via waypoints. The FMS executes the clearance taking advantage of P-RNAV (precision area navigation) capabilities.
Figure 4. 3D Paths in arrival management (after Haraldsdottir et al, Reference Haraldsdottir, Scharl, Berge, Schoemig and Coats2007).
Figure 4 is obviously not the only possibility. Figure 5 is adapted from Morris (Reference Morris2004). Here, there is a policy to spread noise – not current UK government noise policy, which in general is to concentrate traffic on specific routes. There is a potential to develop multiple P-RNAV Initial Approach Procedures, with different times from the fix to the runway, so spreading out the noise. But note that P-RNAV is at present focused on lateral deviations from track rather that longitudinal time keeping on approach – which is a much more difficult system to develop – e.g. see Ballin et al (Reference Ballin, Williams, Allen and Palmer2008).
Figure 5. Plan-view of P-RNAV dispersed arrival process (sketch adapted from Morris, Reference Morris2004).
Whatever the form of the transition to a 4D-trajectory system, there is one unifying feature: priority for the time-based aircraft.
“an arrival time contract with Air Traffic Service Providers … which, if met, ensures some level of priority servicing” (Wichman et al, Reference Wichman, Lindberg, Kilchert and Bleeker2003)
“Today's system is based on binary access (where users meet all of the requirements for access or are denied admission), one level of service (first come, first served), and a regulatory structure largely built around specific equipment types … Clearly defined service tiers will allow the service provider to create service guarantees for given performance levels so that users can determine appropriate investments to meet their needs.”
(Arbuckle et al, Reference Arbuckle, Rhodes, Andrews, Roberts, Hallowell, Baker, Burleson, Howell and Anderegg2006)Thus, there has to be an intention that 4D-trajectory aircraft, or transitional – 3D+ trajectory – aircraft get priority. This priority ATM system delivers system efficiency benefits, but has consequences for non-equipped aircraft.
4. QUEUING THEORY
To analyse some of the consequences of a priority ATM system, an analytical Queuing Theory model is used. This model is explained in standard textbooks and lecture notes (e.g. Adan and Resing, Reference Adan and Resing2002). The gains from using this kind of model rather than a computer simulation are that it is much easier to make approximate quantitative estimates and the key variables can be explicitly identified. It is a simple what if model.
Queuing theory models describe systems in which there is some kind of service subject to comparatively high utilisation, to the extent that simultaneous demands cannot always be met. Figure 6 shows the basic queuing structure and language. Although these models are mainly used to solve telecommunications and computing problems, the terminology usually follows that of supermarkets – shown at the top of the Figure. The equivalent ATM system elements are at the bottom of the Figure.
Figure 6. Queuing structure terminology.
Queuing theorists have had to invent a great deal of jargon to describe the problems they analyse, mainly to ensure that there is complete clarity about what is being modelled. The model here is:
M/G/1 Priority queue with two classes, non-preemption and FCFS.
What that means is easy to explain using the supermarket analogy. M means that customers arrive randomly rather than at regular intervals: technically these are Poisson arrivals. G means that the time to serve a customer varies according to a general statistical distribution, i.e. it is not specified (e.g. as constant service time). The 1 says that there is a single server, rather than a set of tills. Two classes of priority says that there are actually two queues rather than a single one, one having Priority 1 customers and the other having Priority 2 customers. Priority 1 customers are served in preference to Priority 2 customers. Thus, the customer at the head of the Priority 2 queue will be pushed aside if a Priority 1 customer arrives. Non-pre-emptive means that service is not interrupted: a customer's service is always completed before the next customer can be served. FCFS is first come first served, i.e. the earliest arrivals within a queue are served first (which is largely the current ATM discipline when dealing with arriving aircraft (although see Brentnall and Cheng (Reference Brentnall and Cheng2009) on the possible benefits of different sequencing arrangements).
Table 2 sets out a mathematical description of the model. Table 3 then lists the main outputs from it, in terms of the average time spent queuing by the two classes and on average. Note here that these estimates are for a steady state queue. This is the long run or average state of the system. In reality, the average delay (=waiting time) for an M/G/1 queue examined over short periods will vary widely: often there will be no queue, and other times the queue will be two or three times the very long-run average.
Table 2. M/G/1 system with customers with two priority classes and non-pre-emptive.
Table 3. Steady state queuing expressions derived from Table 2.
5. PRIORITISATION, TRANSITION AND DELAYS
What does the queuing model indicate about aircraft delays? First, how significant are the general effects of this kind of prioritisation? Figure 7 shows how average delay increases for a 50%:50% mixture of the two priority classes, with an equal constant service time of 180 seconds. For low utilisation, the average delays are very similar. As the intensity of traffic demand grows there are more occasions when a Priority 2 arrival finds Priority 1 arrivals already queuing, and while those are being served, other Priority 1 arrivals may well arrive – and hence get served – before the head of the Priority 2 queue. At high intensity demand, the situation deteriorates badly for Priority 2 arrivals. Thus, at a total utilisation of 0·9, the relative average delay between the two priority classes is a factor of 10.
Figure 7. Two-class, equal proportion, non-pre-emptive M/G/1 queue delays as total utilisation increases.
What do these differences imply for transitions? Figure 8 is the first simple version of the transition process. In this scenario, the service time for Priority 2 aircraft is still 180 seconds, but that of Priority 1 arrivals is reduced to 85% of that, i.e. 153 seconds. This is simply an arbitrary reduction to account for the reductions in buffer time and standard deviations of arrival time for 4D-equipped aircraft. Some pairings of aircraft will both be 4D-equipped, but others will have the 4D aircraft as the follower or leader in a pair. What air traffic controllers would in practice do to adjust pair separation times is speculative at present. The simplest mental model is to focus on 4D-equipped aircraft as pair leaders, with the follower then being at a reduced separation (=service) time. This assumes that the variability in timings attaches solely to the follower rather than by combining the leader and follower variabilities (summing two variances in statistical language). A cautious assumption, one which assumes no information is available about the arrival sequence of aircraft, is that the follower is a Priority 2 aircraft – it is obviously possible to construct other models. The utilisation is set at 87%, which is the figure that produces in this model an average delay of 10 minutes for 100% Priority 2 aircraft.
Figure 8. Two-class, non-pre-emptive M/G/1 queue delays as priority 1 replaces priority 2.
The delay picture in Figure 8 assumes a one-for-one shift between the two classes of aircraft during the technology transition period: each non-4D aircraft is replaced by a 4D-equipped aircraft. The average delay for the Priority 1 flights is always small, increasing from about 1 minute to about 4 minutes over the range 0% to 100% 4D-equipped. The average delay for all flights goes down from 10 minutes to ~4 minutes. The Priority 2 delay reduces from 10 minutes to about 9 minutes, but then up to ~14 minutes when there are few of these aircraft arriving. The main reason for these patterns is that the intensity of the traffic demand reduces, from 87% to 74%, matching the reduction in the average service time, which largely compensates for the priority queuing effects caused by simultaneous demands.
A large drop in airport arrival utilisation would be unlikely in practice. Airport slots are very valuable, so operators would desire to use what they would see as new slots. Thus, a second scenario might be as shown in Figure 9. This grows the intensity of demand by Priority 1 aircraft: each Priority 2 arrival is replaced by (180/153) Priority 1 arrivals, so the intensity of demand remains constant at 87%. Now, there is little reduction in average delay for all flights – one or two minutes over the whole range. Priority 1 aircraft delays increase up to about 2 minutes at the 50% point, but then increase markedly to about 9 minutes at the 100% Priority 1 level. The most noticeable feature is the rapid increase in the Priority 2 delays: rising from 10 minutes to 30 minutes at the 80% Priority 1 level, and then at an even faster rate to the order of an hour for just less than 100% 4D aircraft.
Figure 9. Two-class, non-pre-emptive M/G/1 queue delays in priority 1 traffic growth scenario.
The Priority 2 delay picture of Figure 9 would be very unlikely to happen in reality. These levels of delay would wreck the airline schedules involved. The problem is not so much the average delay but rather the frequent delays of two or three times that level. The airlines would be less and less able to deal with the delays by scheduling buffers. Delays and costs to airlines would escalate for the aircraft involved: fuel, crew scheduling, passenger compensation, etc, plus reactionary delays from previously delayed flights. The airlines would have to decide to cancel some flights in order to preserve the majority of their schedule. Total economic costs over typical peak months would be substantial: Eurocontrol (2007) presents some estimates of the various delay and cancellation costs.
The example discussed above is a mixed-mode runway, chosen because it is common configuration and because the transition must work for all kinds of runway modes. Thus, it is a necessary case which the transition must pass successfully. A similar model works for arrival-only runways. Arrivals generally require varying service times. This is because the time-separation between successive arrivals depends on the relative aircraft weights of the leader and follower aircraft, in order to reduce the risk from strong wake vortices from the leader seriously affecting the aerodynamic stability of the follower. There is also an absolute minimum separation required between any successive pairing of aircraft, which is mainly a function of ATC radar performance. Airport runways that do not have the most effective rapid-exit taxiway systems can add a further constraint on the separation required, i.e. when the leader aircraft is very large and hence takes a comparatively long time to clear the runway. It is not safe to have two active aircraft on the runway simultaneously.
To model an arrival runway using queuing theory methods, it is therefore necessary to use a service-time distribution that matches actual variations resulting from the variety of aircraft leader/follower combinations. This will depend on the nature of the airline operations, ie depend on the proportion of long-haul and very long-haul flights. The approximation of a deterministic service used above would not be appropriate.
6. DISCUSSION
The analysis above is model-dependent. How good is the model? There are more focused questions. If the Priority 1 aircraft have a contract time, how can they be delayed for several minutes? Could some kind of different queuing discipline reduce the delays for Priority 2 aircraft? Is there a better analytical queuing model? These are addressed in turn.
6.1. If the Priority 1 aircraft have a contract arrival time, how can they be delayed?
The delays to Priority 1 aircraft are apparent from Figures 8 and 9: for the highest Priority 1 traffic intensities, the average delays are about 4 and 9 minutes respectively. The queuing answer is that these large delays are simply the product of the competition for service by Priority 1 aircraft alone. These delays are real – and would be real in a new paradigm system. In practice, they would be realised by speed control and minor route extensions – as in Figures 4 and 5. They would also be realised by ground delays at the originating airports. This is why the ATM system for delay analysis actually needs to cover the gate-to-gate progress of each flight. It is far cheaper operationally to delay aircraft on the ground than to have them use up surplus time in the air by holding or controller vectoring. Ground Delay Programmes (GDP) are already an integral part of current system planning, especially if there are atypical disruptions – e.g. see Ball et al (Reference Ball, Barnhart, Nemhauser, Odoni, Barnhart and Laporte2006). More sophisticated ways of optimising operational performance – whilst retaining the basic ATC concept – are known as Airport Collaborative Decision Making (CDM) (Eurocontrol, 2006):
“[CDM] is a concept which aims at improving Air Traffic Flow and Capacity Management (ATFCM) at airports by reducing delays, improving the predictability of events and optimising the utilisation of resources. Airport CDM allows a[n] Airport CDM Partner to make the right decisions in collaboration with other Airport CDM Partners, knowing their preferences and constraints and the actual and predicted situation. The decision making by the Airport CDM Partners is facilitated by the sharing of accurate and timely information and by adapted procedures, mechanisms and tools.”
There would still be a need for CDM – no doubt enhanced – in a SESAR/NextGen system.
6.2. Could some kind of different queuing discipline improve Priority 2 delays?
The model uses a simple two-priority FCFS system. Is it possible to reduce the delays for the Priority 2 aircraft? The answer is Yes – but it raises another question: “Why would the ATM system operators want to do this?” There is a very good ATM system reason for giving priority to the 4D-aircraft. The reason is associated with their lower service times using the runway. A scheduling principle of shortest job first – SJF – is widely used by data processing system designers. SJF produces the minimum average waiting time for M/G/1 queues. This SJF expected delay is smaller than the expected delay in the corresponding M/G/1 system without priorities. Thus, if there is a class of arrivals with a consistently smaller expected service time then it should be made the higher priority class. The only exception would be if a group of longer service arrivals were somehow more valuable customers – for which a cost optimization analysis might change priorities around. In this case, the assumption is that all arrivals have equal value.
There is also an algebraic relationship between the waiting times of different classes of arrivals in a queue. Kleinrock (Reference Kleinrock1965), as part of research work into internet technology, discovered a work conservation law for multi-class M/G/1 queues. This states that a weighted linear sum of these mean waiting times of priority classes is independent of the processing discipline. This implies that an improvement in the average waiting of one class by a change in the processing discipline will always degrade the average waiting time of another class. Thus, for the two-class example here, if the Priority 2 average delay is reduced by some kind of clever scheduling, this would be at the expense of an increase in the average delay for Priority 1 arrivals.
6.3. Is there a better analytical queuing model?
The focus here must be on the M part in the M/G/1 categorisation. The 1 at the end means a single runway – but the future system must function adequately for single runways as well as more complex layouts, so this is a necessary test to pass. The G for the service time distributions stands for General, so this is not a restriction. In the calculations here, a deterministic service time has been used, but the queuing equations in Table 3 cover other kinds of distribution form.
The M in the categorisation stands for memory-less. In practice, this means that arrival times are assumed to be described by a Poisson distribution, which implies an exponential distribution of inter-arrival times. The great advantage of the Poisson distribution is that it lends itself to simple analytical expressions (Table 3). There is recent USA evidence (quoted in Guadagni et al, Reference Guadagni, Ndreca and Scoppola2008) that actual arrivals are slightly less random than Poisson predictions, but the difference between the two processes was quite small for all the airports observed.
As discussed earlier, airport timetables are currently rather soft, with pre-departure scheduling being overlaid by a variety of additional variabilities, which is thought to explain why a Poisson hypothesis is a reasonable first approximation. However, the context here is a future system in which the intrinsic variability of actual arrival times is reduced markedly because of various technical improvements in navigational performance and the move to time-based aircraft trajectory contracts. This suggests a stochastic model in which arrivals are scheduled as approximately minimally separated in advance, but then this schedule is subject to an additional, but comparatively small, randomness, resulting in what might be termed an S-distribution.
There do not appear to be many attempts to solve an S/G/1 queuing system. Guadagni et al (Reference Guadagni, Ndreca and Scoppola2008) have attempted to do this. However, they have not incorporated priority classes. Their results show queuing behaviour that is generally similar to a Poisson process, but which for high traffic demand are markedly different from M/G/1.
Other models could be used to explore a two-class queue in which one class does not exhibit Poisson arrivals. An example is a Batch queue. This is a well-known model used in understanding the scheduling of computer central processing units. A M X/G/1 batch priority queue has P classes of arrivals, which occur in groups whose sizes are generally distributed according to a probability function Gp. Class p arrivals follow a Poisson process. Arrivals of class p have priority over those of class q if p<q. The service times for each class are independent and identically distributed. Service is non-pre-emptive, so the first batch in the highest (non-empty) priority queue is served; and then the server deals with the first batch in the next highest (non-empty) priority queue. There are analytical models for this kind of priority batch queue. Takagi and Takahashi (Reference Takagi and Takahashi1991) present some examples, with references to earlier work. Their equations for average waiting times are similar in form to, but more complicated than, those displayed in Table 3.
For the problem here, there are two classes. The number in a Priority 1 batch is distributed with a probability function G1. However, the number in a Priority 2 batch is distributed with a special probability function G2, which is unity for size 1 batches and zero for any other size of batch. Thus, batches of Priority 1 aircraft arrive, presumably with minimal separation – i.e. matching the service rate – between each aircraft, and the individual Priority 2 arrivals have to slot in the gaps between the Priority 1 batches. The Priority 1 batches are not necessarily constructed as batches, but the ATM systems' aim of meeting demand means that the Priority 1 flights will tend to be in minimally separated sequences.
Figure 10 illustrates the problem as seen by Priority 2 customers. The boxes represent Priority 1 arrivals. If a Priority 2 customer service time is longer than a Priority 1 customer's, then in this Figure there is only one space between Priority 1 batches into which the Priority 2 arrival could potentially be slotted without affecting the Priority 1 arrivals. If the Priority 2 aircraft were to use one of the smaller intervals, then it would delay all the – already airborne – Priority 1 aircraft in the next batch. For high intensity demand, batches can be very long. For a naïve equally-spaced slot model, if the Priority 1 traffic intensity is 0·8 then batches of four or more arrivals are more than 50% probable, and very long batches of ten or more aircraft would probably occur every day. This is simply because the length of a batch has a roughly geometrical distribution.
Figure 10. Illustration of Priority 1 slots and batches.
To reduce the delays to Priority 2 aircraft, one simple technique would be to limit the maximum number of aircraft permitted in any Priority 1 batch, e.g. to five arrivals, and to add a long notional slot at their end, sufficient for the service of a waiting Priority 2 aircraft at the end on the batch. But batch queues are similarly subject to Kleinrock's work conservation law, which would mean that this tactic would increase Priority 1 aircraft delays somewhat.
7. CONCLUSIONS
New-paradigm ATM systems such as SESAR and NextGen will make dramatic changes to the nature of airport operations. Aircraft will be able to land at their destination airports with very precise timings. This would lead to a marked increase in runway capacity, because it will be possible to reduce the large separations currently required between successive aircraft movements. A significant part of these separations is required because aircraft are separated by distance, whereas with new-paradigm systems the separation will be precision time-based.
The analysis here uses some simple queuing theory models to establish rough quantitative estimates of the impact of the transition to a time-based – four-dimensional (4D) – navigational and ATM system. Such models are approximate, but they do offer insight into the broad implications of system change and its significant features. The main operational feature examined here is the time spent queuing for runway availability by non-4D-equipped arrivals.
It is unlikely that the system transition to a 100% 4D-equipped aircraft population would be a big bang, i.e. with the new paradigm not being brought into use until all aircraft were equipped. This is because there would be business incentives for early aircraft fitment, it is generally safer to introduce new technologies gradually, and researchers are already proposing potential transition steps to the new system.
4D-equipped aircraft in essence have a contract with the airport runway – they would be required to turn up at a very precise time. In return, they would get priority over any other aircraft waiting for use of the runway. In queuing jargon, they would be Priority 1 arrivals, with non-pre-emptive priority over those Priority 2 aircraft – non-pre-emptive simply meaning that Priority 2 aircraft currently in the process of landing would carry on doing so.
Using the simple M/G/1 queuing model – Poisson arrivals, general service time distribution for dealing with landing aircraft – it is easy to produce estimates of average delays for the two kinds of priority arrivals. 4D-equipped aircraft get a very good service, even at high rates of arrival, with delays generally being taken much earlier in the flight, e.g. at the originating airport. Non-4D aircraft get a reasonable service if the proportion of 4D-equipped aircraft is low, but this can deteriorate for high proportions. The onset and degree of deterioration depends on the extent to which the shorter handling time for 4D-equipped aircraft is used to add extra hourly movements to the airport schedule. If too many extra movements are added, then the delays to the non-4D-equipped flights becomes increasingly serious for high proportions of the 4D-equipped arrivals. Eventually, the non-4D-equipped operation would not be viable – delays would be so large and erratic that a sensible operational schedule could not be maintained. Obviously, this would not be allowed to happen, so preventative mechanisms would be needed, either limiting the additional growth of 4D-equipped flights and/or modifying their contracts to provide sufficient space for the non-4D-equipped flights to operate without excessive delays.
Textbook queuing theory provides useful answers to these kinds of problems. There is also the potential for non-Poisson models, for which there is little in the literature, and for more complex M/G/1 models, e.g. grouping a succession of 4D-equipped aircraft as a batch of arrivals. Queuing theory also indicates the extent to which tradeoffs are necessary, e.g. ensuring that the non-4D-equipped flights can operate without crippling delays implies the need for some increase in delays to the 4D-equipped flights.
