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A Stochastic Characterization of En Route Traffic Flow Management Strategies

Donnie C. Moreau* and Sandip Roy† Washington State University, Pullman, WA 99164

Motivated by the need for coordinated multi-Center flow management for the National Airspace System, we characterize the impact of several en route flow management actions on upstream and downstream flows. Specifically, we generically model a single-stream aircraft flow using a Poisson process representation, and analytically characterize the performance of several types of flow-management restrictions imposed on this stream. Using these characterizations of performance measures (which include capacity-violation probabilities, average delays imposed by the restriction, and backlog), we are able to compare flow management paradigms, such as use of in-trail restrictions vs. time-based metering. We also present our preliminary efforts to validate our model for flows and restrictions using historical data. Finally, we discuss the robustness of one of these flow management paradigms to deviations of aircraft from their nominal schedules, and present preliminary explorations regarding several further applications of this study.

I. Introduction

n the National Airspace System (NAS), strategies for meeting capacity constraints, which are needed e.g. for controller workload management or because of arrival rate constraints at airports, have been used successfully for

many years. Recently, however, there has been increasing interest in coordinating these strategies – collectively known as traffic flow management – over a multiple Center area or even NAS-wide. This interest in coordinated flow management strategies reflects several factors, including increasing realization that flow management actions have long-range and interactive effects, interest in optimizing flow management performance through coordination, and increase in traffic and route complexity at certain critical times.

Broadly, traffic flow management strategies aim to stretch out airport arrival demand and/or sector entry demand during “rush” periods, in order to prevent capacity violations. Operationally, flow management of arrival traffic for airports surrounded by a single Air Route Traffic Control Center (ARTCC) can be achieved using time-based metering , i.e. through the scheduling of aircraft at points along their routes1,2,3. In contrast, en route Sector capacity management is traditionally directly achieved using Miles -in-Trail (MIT) and Minutes -in-Trail (MINIT) restrictions, which are cruder but simpler-to-implement management requirements that enforce spacing (in distance and time, respectively) between aircraft in a stream4,5. In such cases, these in-trail restrictions are imposed by controllers according to procedures that have been developed through experience, known as playbooks. Recent efforts to coordinate flow management for airports near multiple ARTCCs (e.g., the Philadelphia airport), and to combine flow management for en route and arrival needs, have highlighted the complexities associated with implementing such coordinated strategies2,6.

Nevertheless, coordinated flow management strategies at several scales and levels of details have been pursued. Several studies have considered in detail the implementation of multi-Center traffic management2,6,7. In particular, the article2 identifies challenges associated with implementing multi-Center traffic management around the Philadelphia International Airport; challenges include limited infrastructure for coordination among facilities,

* Student, School of Electrical Engineering and Computer Science, P.O.Box 642752, dmoreau@eecs.wsu.edu † Assistant Professor, School of Electrical Engineering and Computer Science, P.O.Box 642752, sroy@eecs.wsu.edu

I

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uncertainties in departure times for short-duration “pop-up” flights, and controller workload stresses associated with the operational changes required to adopt the coordinated metering strategy and with the concurrent use of time-based metering and in-trail spacing. In addressing these challenges, a subsequent article has developed a distributed scheduling framework for multi-Center time-based metering. These detailed studies are highly pertinent in that they expose the many operational and implementation challenges associated with coordinating flow management, as well as the specific challenges associated with particular facilities6. They also give significant conceptual insight into the interaction of multiple different flow controls. Of importance, these detailed strategies aim to codify, and make use of, historical data and controller/operator experience in building an architecture for coordinated management.

Several recent analytical works have sought to develop higher performance flow management strategies, by phrasing flow-management problems as optimal scheduling problems and then solving these problems. A thorough analysis of flow management at two scales (in particular, for the arrival traffic at a busy airport surrounded by a single ARTCC and for aggregate routing of flows NAS-wide, respectively), and of the operational implementation of these flow restrictions by traffic controllers, can be found in the thesis [8] and associated papers [3,9,10]. Specifically, the thesis [3] uses a hybrid system model as a framework for developing optimal maneuver-assignment strategies for arrival-traffic flow management. This work also addresses the question of NAS-wide flow management through routing, by developing and using an Eulerian model for air traffic. Another high-level, but Eulerian, approach to NAS-wide (en route and arrival) flow management is developed in the article [11]. This study explicitly models capacity constraints both en route and at terminal areas, and considers use of various traffic flow management strategies (including en route controls and ground holds) to prevent capacity violations while minimizing costs imposed by the controls.

Both the operational studies of multi-Center flow management and the studies of optimal scheduling discussed above are needed for the development of practical algorithms for flow management in the NAS. It is our belief, however, that more detailed modeling of restrictions/metering, and of the traffic flow through these restrictions, can allow for better evaluation and comparison of multi-Center or NAS-wide flow management strategies and hence can foster the development of good multi-Center flow management strategies. The following are three specific improvements that we believe can be achieved through further modeling and analysis of restrictions/metering:

· The impact of en route flow management on both downstream and upstream traffic can be evaluated, for typical flows in the airspace. Such evaluations can allow for better comparison of flow management strategies, and facilitate coordination of flow management at multiple points in the airspace.

· The impact of operational complexities – including limitations in the communication capabilities of facilities (which may be present in inter-Center and even inter-Sector communications2), uncertainties associated with popup flights, and short-term variations in flows dues to weather or upstream flow management – on the performance of flow management strategies can be better characterized. This ability to characterize the impact of uncertainties on flow management, and in turn to develop strategies that are robust to these uncertainties, may be especially valuable in the context of the Free Flight paradigm12.

· Optimized time-based metering strategies can be compared with direct imposition of in-trail restrictions using playbooks, and strategies that involve both time-based metering and in-trail restrictions can be evaluated.

Using the insights gained from the points above, multi-Center flow management strategies that are robust to operational limitations/uncertainties and yet perform well (in terms of controller workload and aircraft delay) can perhaps be developed.

We feel that there are two broad approaches for developing models that capture the flow characteristics and operational complexities associated with multi-Center flow management. One meritorious approach is to enhance the currently used location-specific models for en route flow management8,11 to incorporate these details. A second viewpoint – the viewpoint taken in this article – is to embrace that flows in parts of the ATS have essential uncertainties (due to uncertain events such as popup times, limitations associated with coordinating multiple flow streams and facilities, and operational uncertainties such as those resulting from the use of playbooks). Taking this second viewpoint, we study metering/restrictions from the point of view that they operate on generic flows that are modeled as stochastic processes (in particular, Poisson processes). Thus, this paper builds on our earlier work13, in which we develop a Poisson process model for flows in the NAS. Specifically, in this article, we compare several restriction/metering strategies that act on a single stream of aircraft, using a Poisson process model for the single-stream flow and queuing models for the restriction/metering. This stochastic modeling approach to flow management allows us to evaluate whether typical management strategies (e.g., imposition of MINIT restrictions

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during expected high-traffic periods) are capable of achieving capacity management, and to identify the delays/backlogs associated with the various control strategies. Finally, we explore the robustness of the strategies to unmodeled uncertainties (i.e., we explore whether flow restrictions can be designed based on a priori known aircraft schedules), and briefly discuss the use of our analysis in multi-Center flow management.

The remainder of the article is as follows: in Section 2, we review the stochastic model for single-stream flows, and present several restriction/metering strategies. In Section 3, we use our model to both simulate and analytically evaluate the flow management strategies, and then compare the flow management strategies using these evaluations. Section 4 describes out preliminary efforts to validate the modeling approach. Finally in Section 5, we give preliminary explorations in several further directions, including robustness analysis of a promising flow management strategy, application of our analysis to multi-Center flow management, and development of a tool for evaluating the impact of MINIT restrictions on specific flows.

II. Stochastic Model of a Controlled Stream

We consider a single stream of aircraft entering a Sector with a capacity of L aircraft. This stream of aircraft, and hence the aircraft count in the downstream Sector, can be managed at a fix or waypoint at the entrance of the Sector. Since we are interested in comparing several management paradigms at this waypoint, we nominally disregard any other traffic flow in the Sector and address effects of such flows separately, in Section 4. For simplicity, we also assume that each aircraft in the single stream requires the same amount of time T to cross the Sector. (This assumption is not unreasonable when the aircraft remain on the same route through the Sector, such as in the case where most of the aircraft are destined for the same airport.)

We model the aircraft flow along the single stream stochastically, as a Poisson process 14. That is, we model the times between arrivals of successive aircraft at the waypoint as statistically-independent exponential random variables, as shown in Figure 1. The use of Poisson process models for flows has been conceptually justified and experimentally verified in our earlier work13. Briefly, a Poisson process model is reasonable because a single stream typically comprises merged traffic from several uncoordinated flows (e.g., flows from different source airports and to different destinations, as well as flows impacted by weather and take-off time uncertainties). In consequence, the flow on the stream is essentially memoryless – the position of a single aircraft in the stream does not give information about the positions of the other aircraft. This memorylessness of the flow implies an exponential distribution for the times between successive aircraft14, and hence implies a Poisson process model for the arrival times at the waypoint. Thus, we adopt a Poisson process model for the traffic flow on the single stream. In doing so, we abstract away the details of the aircrafts’ schedules and individual uncertainties, and instead take the perspective that a holistic model of the flow can give insight into the performance of management strategies. We denote the arrival rate at the waypoint, i.e. the average number of aircraft arriving at the waypoint per unit time, as λ.

It is worth stressing, specifically, that the Poisson process description of a flow incorporates both features of the flow that are known a priori (e.g., features that reflect the nominal schedule or a-priori known deviations from it) and those caused by unpredictable events (such as upstream flow management or, for short-haul flights, take-off time uncertainties). We adopt this holistic viewpoint because it allows us to characterize the impact of a flow restriction in a generic way, without consideration of schedule specifics at a particular location. In Section 4, we shall explicitly separate a priori -known and unpredictable aspects, in our study of the robustness of the flow management strategies.

Two limitations of this Poisson process representation are pertinent to our study and so require discussion: first, the model breaks down if the process is considered at a sufficiently fine time-scale, because aircraft are required to maintain separation and hence the process is not memoryless at this scale. Flow management is concerned with decisions at longer time-scales (which are then applied at the finer time-scale by the controllers), and hence we do not concern ourselves with this limitation. Second, flow management itself causes a flow to lose its Poisson characteristics, since

Figure 1. A single -stream flow impinging on a en route flow management restriction. We model the aircraft flow using a Poisson process, and consider various management strategies at the waypoint.

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aircraft spacing is uniformized and hence memorylessness is lost. This uniformization of the schedule is in fact sometimes a desired goal of the flow management scheme6. Hence, upstream flow management can break the Poisson approximation. We nevertheless choose to use a Poisson process model: in the case where management at an upstream waypoint dominates the traffic flow, we note that the management problem is easier (since the arrival times are uniformized) and hence the Poisson approximation provides a worst-case analysis. In the other case where many possibly-managed flows merge or other factors such as weather have an impact, the schedule uniformization is likely to be destroyed and a Poisson assumption is again reasonable.

We consider application of four flow management strategies at the waypoint:

· The No Control strategy is the nominal case where no flow management is applied at the waypoint. That is, aircraft are allowed to enter the downstream Sector immediately upon arrival at the waypoint, and hence no delay is incurred. We denote the number of aircraft in the sector when the No Control strategy is used by Xnc(t).

· The Strict Control strategy refers to the use of a fixed Minutes-in-Trail (MINIT) restriction (i.e., one that is imposed at all times) at the waypoint. That is, we enforce that each aircraft passes the waypoint Tsc units of time after the aircraft ahead of it, at all times. We denote the number of aircraft in the Sector at each time when the Strict Control strategy is used by Xsc(t). Although in reality a MINIT restriction would never be imposed at all times, analysis of this case is important in that it clarifies the impact of a MINIT restriction on a typical flow.

· We use the term Intelligent Control for the following strategy: a MINIT restriction is used in periods of high traffic, and aircraft are not restricted at other times. This need-based application of an en route restriction is used in the NAS. Currently, the periods of time during which restrictions are applied are determined heuristically, based on experience. Some efforts have been made to study the impact of MINIT restrictions imposed during high-traffic times on particular scheduled flows5, but (as far as we know) there is little generic understanding of the impact of such restrictions on flows. For our Intelligent Control strategy, the No Control Sector count Xnc(t) is used to determine periods of high traffic. When Xnc(t) exceeds a threshold X, a MINIT restriction of duration Tnc is imposed; at other times, no flow management mechanism is used. We note that the Intelligent Control strategy requires at least aggregate knowledge of the incoming traffic flow (i.e., of Xnc(t) ), at a sufficiently early time that the desired restriction can be enforced. In Section 4, we explore whether this strategy is in fact robust to uncertainties (in aircraft schedules, due to weather and pop-ups, etc.), and hence consider whether the application of the restriction can be determined from a nominal schedule. We denote the Sector count when the Intelligent Control strategy is used by Xic(t).

· We consider flow management using time-based Metering. Our formulation here permits comparison of metering with other flow management strategies. Specifically, in our formulation of metering, each aircraft is permitted to enter the Sector at the earliest time at which the Sector count is below the capacity L. We note that application of time-based metering requires detailed knowledge of arrival times of aircraft at the waypoint, in sufficient advance that controllers can be given directive regarding the delay to be imposed on each aircraft. We refer to the Sector count when time-based metering is used by Xm(t) .

We assume that each of the management strategies can be exactly implemented (i.e., the restriction on each aircraft can be imposed) by the controllers associated with the Sector, using either playbooks or a computer-based scheduling algorithms. The article [15] discusses potential difficulties associated with imposing in-trail restrictions, and introduces a tool that can help controllers achieve in-trail separation.

The stochastic model for the stream of traffic arriving at the waypoint, together with the control strategy at the waypoint and the count in the downstream Sector, comprise our basic model for en route flow management.

III. Evaluation of the Flow-Modeling Strategies

We evaluate the performance of the four flow-modeling strategies, with the aim of comparing them and also of better understanding the impact of these strategies on flows in the NAS. Specifically, we shall be concerned with three measures that indicate the performance of a strategy: the number of aircraft within the Sector at each time (which indicates whether the strategy achieves capacity management); the delay D imposed on each aircraft by the control (which indicates the cost of the control); and the backlog B, i.e. the number of aircraft that are being delayed by the restriction at a particular time (which indicates the effect of the control on the upstream Center). For each management strategy, we simulate trajectories of the three measures for particular schedules, and also analytically characterize the measures in general for a Poisson traffic flow. Since the arrival process statistics are assumed time-

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invariant over the duration of the experiment, our analyses give statistics for a typical or randomly-chosen aircraft (for delays) or time (for backlog and counts).

In several instances, representation of the flow management strategy as a queueing process permits this analysis. We refer the reader to [16] for more discussion of the use of queueing models in air traffic management.

In the No Control case, the sector entry times are governed by a Poisson process, and hence the probability distribution for the number of aircraft in the Sector is well known to be a Poisson distribution14. In particular, we find that

!

)())((

c

eTctXP

Tc

nc

ll -

== (1)

From (1), it is clear that the Sector count can exceed the capacity L at a particular time with some probability, regardless of the arrival rate λ. The probability that the Sector count exceeds the capacity L can be determined

analytically, as å¥

+=

=1

))((Lc

nc ctXP . This chance of exceeding capacity is demonstrated in Figure 2a, in which

Xnc(t) is shown for a particular Poisson arrival stream. However, as illustrated in Figure 2b, this probability becomes vanishingly small as the average Sector count λT becomes small compared to L. Hence, for Sectors with small enough average Sector count, the No Control strategy is sufficient; this is in fact the case for most Sectors in the NAS, for which the nominal schedule dictates that aircraft counts remain well below the capacity at all times. We notice that the delay imposed on each aircraft and the backlog are both nil for the No Control strategy.

Trajectories of the three performance measures for the Strict Control strategy are shown for a particular arrival

stream, and for several MINIT restriction durations Tsc in Figure 3. We see that the minute restriction does prevent capacity violations when it is sufficiently long, but also imposes unnecessary delay by restricting aircraft at times when no capacity violation is imminent. We note that the Strict Control strategy is guaranteed to prevent capacity

violations if the duration of the restriction is sufficient, in particular if L

TTsc ³ . For

L

TTsc £ , our simulations indicate

that Strict Control is not able to prevent capacity violations: the impact of the restriction is insufficient to prevent excess flow into the Sector during high-traffic periods (see Figures 3a and 3c). The delay D incurred on an aircraft and the backlog B can be characterized analytically, by noticing that the flow dynamics imposed by the flow management strategy are equivalent to those imposed by a single-server queue with deterministic service time Tsc.

a) b) Figure 2. In plot a), the aircraft count in the downstream Sector for the No Control strategy is shown , for a particular arrival stream. In plot b), the probability of a capacity violation is shown as a function of the rate of the traffic arriving at the waypoint. In this example, each aircraft is assumed to take 50 minutes to cross the region, and the capacity is L=15 as seen in a) .

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In other words, an aircraft that is within Tsc time units past the waypoint can be viewed as being “served” by a single server, so that arriving at the waypoint are queued (restricted) in order until the server empties (i.e., the aircraft spends more than Tsc units of time within the Sector). Since the traffic approaching the waypoint is modeled as Poisson, the control strategy can thus be represented as a FIFO (first-in, first-out) M/D/1 (memoryless-flow, deterministic service-time, single-server) queue. Based on this M/D/1 queue representation, we can find statistics of the delay and backlog. In particular, using the Pollaczek-Kinchine formula 17, we find that the mean delay is

)1(2)(

2

sc

sc

T

TDE

l

l

-= (3)

and the mean backlog is

)1(2)(

22

sc

sc

T

TBE

l

l

-= . (4)

a) b)

c)

Figure 3. The performance of the Strict Control strategy on a particular arrival stream is shown, for three different MINIT restrictions. Plot a) shows the aircraft count in the downstream Sector, plot b) shows the backlog as a function of time, and plot c) shows the distribution of delays imposed by the restriction on the aircraft.

a) b) Figure 4. The performance of the Strict Control strategy is shown. A restriction of 4 minutes is used, and the average delay (plot a) and average backlog (plot b) are computed.

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Higher moments of the delay and backlog can also be determined in closed form, while the probability distributions for these random variables can be found through a recursion17. For our purposes, it is valuable to observe that the

mean delay and backlog are large if the average spacing in time between aircraft in the impinging stream ÷ø

öçè

æ

l

1 is

close to the duration of the restriction Tsc (see Figure 4). Hence, if the downstream Sector is on average fairly busy (i.e., the expected number of aircraft in the Sector in the unmanaged-flow case is close to the capacity), application of the Strict Control strategy will result in significant delay/backlog. There is reason to believe that the delay/backlog can be improved using information about the aircraft stream, since many aircraft are restricted even when capacity management is not needed. We note that MIT restrictions can be analyzed in much the same way, assuming that aircraft in the flow are traveling at similar speeds.

Based on our observation that the Strict Control strategy unnecessarily delays aircraft during low-traffic periods, we might expect a strategy that reacts to the aircraft counts in the downstream Sector to outperform it. The Intelligent Control does react to the aircraft count in the downstream Sector, in that a MINIT restriction is imposed only during periods when many aircraft are expected (based on the nominal Sector count). In Figure 5 and Table 1, the performance of the Intelligent Control strategy is compared with the performances of the other strategies. The simulations show that, indeed, capacity violation is prevented while delay and backlog are reduced in comparison the Strict Control strategy. As expected, the Intelligent Control strategy does not delay aircraft at times when the downstream Sector count is low, and hence the total imposed delay is much smaller than for the Strict Control strategy. Analytically, several features of the Intelligent Control strategy are of note. First, we notice that there may in general be some probability of capacity violation when the Intelligent Control strategy is used, but it is easy to check that this probability can be made vanishingly small by setting the threshold slightly below the region capacity. For instance, for the example described in Figure 5, one can compute that only about 1% of the No Control trajectories with capacity violations would have violations upon use of Intelligent Control. Regarding analysis of delays and backlog, unfortunately the Intelligent Control strategy does not admit a standard queuing representation like the other strategies. One promising approach for characterizing these performance measures analytically is to first characterize the durations over which the MINIT restriction must be imposed, and in turn to characterize delays and backlogs as a function of the restriction duration. Using this approach, we have been able to come up with a crude bound for the delay/backlog. We are currently working on a closer approximation.

a) b)

c)

Figure 5. The performance of each flow management strategy is shown, for a particular arrival stream. Plot a) shows the aircraft count in the downstream Sector, plot b) shows the backlog as a function of time, and plot c) shows the distribution of delays imposed by th e restriction on the aircraft. We note the 5 -MINIT restrictions are used for both the Strict Control and Smart Control strategy, and that a MINIT restriction is imposed for the Intelligent Control strategy when the number of aircraft exceeds 13.

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The performance of time-based metering is compared with the performance of Strict Control and Intelligent Control in Figure 5, for a particular arrival stream. In this simulation, capacity is not violated, and the backlog and delay are smaller than for the other flow management strategies. Of particular importance, we note that the average delay of the Metering strategy is 0.32 time units as compared to 0.72 units for the Intelligent Control strategy. Thus, we see that (relative to the other strategies), Intelligent Control and Metering are comparable in cost, though Metering outperforms Intelligent Control by roughly a factor of 2. We note that the downstream Sector’s capacity will not ever be violated when time-based metering is used, by assumption. Also, it is easy to check that time-based metering achieves minimal average delay and backlog, among all strategies for which capacity is not exceeded. Statistics of the delay and backlog for the Metering strategy can also be computed analytically, using a queueing model. Specifically, the downstream Sector can be viewed as a group of L servers each with deterministic service time T; with this model, metering is represented in that jobs (aircraft) must wait only when the servers are all being used (the Sector is full). Hence, when metering is used, the dynamics can be represented using an FIFO M/D/c queue (i.e., a queue with memoryless arrivals, deterministic service time, and multiple servers). The average waiting time and backlog unfortunately do not have known closed-form expressions in this case, but can be found approximately or through infinite summation (see [18] for a review of relevant results). Finally, it is worth noting that time-based metering can be applied to multiple flows at once. In this case, the queuing representation for the strategy would essentially still hold, except perhaps in that each aircraft would not take the same amount of time to cross the Sector. In the literature, an Erlang -3 distribution has been proposed for the statistics of the Sector crossing times of aircraft16.

Our analysis of several flow management strategies in the context of a Poisson arrival traffic model indicates that both the Intelligent Control strategy and the Metering strategy can achieve capacity management with limited delay and backlog. As expected, the metering strategy outperforms the Intelligent Control strategy and the various other restriction-based strategies, though the difference in performance between the Intelligent Control strategy and the Metering strategy is relatively small. Although time-based metering outperforms the restriction-based strategies, we note that the Intelligent Control strategy provides some advantages for multi-Center flow management:

· Since the Intelligent Control strategy uses an in-trail restriction, it can be imposed by controllers using existing playbooks. In contrast, imposition of time-based metering at a particular waypoint may require development of scheduling algorithms, which would need to be specific to the geography of that waypoint.

· In contrast to the Metering strategy, the Intelligent Control strategy requires no information about the aircraft count in the downstream Sector, and only aggregate information about aircraft approaching the waypoint. In fact, in Section 4, we will present preliminary studies indicating that the Intelligent Control strategy is robust to uncertainties in the arrival times of aircraft at the waypoint, and hence can be determined from a nominal (a priori known) schedule. Accordingly, the times at which the restriction must be imposed are known in advance, and hence characteristics of the downstream and upstream flow during these times can be determined based on our study. This a priori structure and robustness of the strategy may be especially valuable when inter-Center or inter-Sector communications are limited: in this case, an ARTCC and/or controllers associated with a Sector would not require detailed information from other regions to predict impinging aircraft flows.

It is also worth noting one significant advantage of Metering over Intelligent Control: time-based metering can straightforwardly be adapted to multi-stream flows and would achieve the same performance in this case. In contrast, the performance of in-trail restrictions operating on multiple flows is unclear; further analysis of this case is needed. Broadly, our approach is significant in that it permits precise comparison of the time-based metering with sub-optimal but simpler in-trail restriction-based strategies.

% Violation Average Delay Average Backlog

No Control 5 % 0 min 0 aircraft

Strict Control 0 % 8 min 1.6 aircraft

Intelligent Control 0.05 % 0.7 min 0.15 aircraft

Time-based Metering 0 % 0.3 min 0.06 aircraft

Table 1. The performances of the flow management strategies are shown. The c olumn labeled ‘% Violation’ indicates the fraction of time during which the downstream Sector capacity is violated, for each strategy. We note that the average delay and average backlog for the Intelligent Control and Time -based Metering strategies are approximate.

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IV. Preliminary Model Validation

We have taken some preliminary steps toward validating the significant claim of our modeling framework, that the impact of control elements on real upstream and downstream flows can be determined by studying their impact on generic (Poisson) flows. One simple means for providing some validation is to show that the delays predicted by our model capture actual delays observed in the NAS. In this preliminary study of validation, we have compared observed delays with predicted ones, for two examples. In the first example, we compare our model’s prediction with the observed en route delay distribution for aircraft traveling between a particular origin-destination pair, in particular ATL to DFW. We have obtained the observed delay distribution, which is adjusted for wind conditions and based on MD-80 traffic between the two airports during 2002, from [19]. These en route delays in many cases originate from restriction and/or rerouting of aircraft in

response to congestion or weather. Hence, we might expect these delays to be well-modeled using a Poisson flow controlled by e.g. a MINIT restriction. In Figure 6, we compare the observed delays with those generated by our model, when a Poisson flow is subject to a MINIT restriction of appropriate duration. We note the similarity between the observed delay distribution and the predicted distribution. Figure 7 indicates the change in the delay distribution that would result from a change in the bottleneck control in the example (from a 3-MINIT restriction to a 4-MINIT restriction), and hence clarifies one use of the model. In the second example, which is drawn from [20], we consider delays imposed by a single application of a MIT restriction. Here, placement of a 20-MIT restriction for a two-hour duration is necessitated by a weather disturbance, which leads to a predicted excess of traffic along a particular jet route in ZID. The actual number of planes passing through the restriction, the average delay incurred on these aircraft, and the maximum delay among the aircraft, are known. For comparison, we model the flow impinging upon the restriction as a Poisson flow of the same intensity (i.e., containing the same number of aircraft), and determine the average delay and maximum delay predicted by our model. The results are shown in Table 2.

These two examples provide some preliminary validation of our modeling approach. We note that the Poisson model for traffic flows has been further validated in our previous work [13]. We caution that the attempts at model validation described here are of course very preliminary, and much remains to be done in identifying the scenarios where the Poisson flows and the described control elements are representative. In future work, we hope to pursue further validation of delay prediction using Enhanced Traffic Management System (ETMS) data processed by the

Figure 6. The delay distribution predicted by our model is compared with the delay distrib ution observed for one origin -destination pair (ATL to DFW).

Figure 7. Our model predicts that changing the restriction duration at the congestion point significantly changes the delay distribution.

Max Delay Average Delay 77.0 36.9

a s d d s a d a s d s d a s a d s d a

77.4 36.1 77.4 38.4 77.8 32.1 77.8 37.2 78.2 37.8 79.6 35.2 81.4 38.0

Table 2. Delays due to a MINIT restriction predicted by our model are compared with observed delay data. The italicized values indicate observed data, whil e the remaining seven values are from our model.

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Post-Operations Evaluation Tool (POET) (see [21] and [22] for details on ETMS data and POET, respectively). We also plan to validate the model by comparing predictions of downstream Sector count deviations due to restriction placement with actual observations of these deviations23.

V. Discussion and Future Directions

Using a Poisson process model for a single-stream aircraft flow, we have characterized and simulated the performance of several management strategies acting on that flow at a waypoint. We have verified that these various flow management strategies prevent capacity violation in a downstream Sector by uniformizing the stream of traffic in the Sector. This management action is costly, in that it imposes delays on the aircraft in the stream and because it increases the count (and spread in count) in the upstream Sector. A specific result of this study is that the delay costs/backlogs associated with each strategy have been characterized precisely, based on the stochastic model for the single-stream flow. This characterization verifies that the cost associated with a particular management strategy depends on the fineness of the control – i.e. its ability to react to the downstream Count, so as to prevent capacity violation without incurring any unnecessary delays. As expected, our study indicates that time-based metering provides the finest control, with time-varying in-trail restriction-based control second best. A trade-off associated with using the finer controls is the requirement of more information about the impinging flow and the downstream Sector count for management, and (in the case of time-based metering) the need for new methods for imposing the restriction. For these reasons, and because the costs associated with Metering and Intelligent Control are roughly comparable, we believe that both types of management strategies must be considered further.

We have claimed that one potential advantage of using the Intelligent Control strategy is its robustness to uncertainties in the aircraft flow and in the downstream Sector count. Here, we present some preliminary explorations on the robustness of the Intelligent Control strategy. Specifically, we consider the following question: given that actual positions of aircraft in the flow stream may deviate from those predicted a priori (e.g., because of

a) b) Figure 9. A preliminary version of a tool for designing flow management strategies is shown. The user will have the option to set parameters of the management strategy (e.g., duration over which a MINIT restriction is imposed), and observe several performance metrics (e.g., downstream Sector count, backlog, imposed delay), for one or a set of possible flows arriving at the restriction.

Figure 8. The robustness of the Intelligent Control strategy is illustrated. Specifically, the Sector counts are shown to be within capacity for five arrival streams that deviate probabilistically from the nominal schedule, even though the restriction duration is set from the nominal schedule. Our simulations indicate that the cost of the Inte lligent Control strategy (in terms of restriction delay or backlog) approximately doubles when it is made robust to variations in aircraft arrival times.

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upstream flow control or because of take-off time uncertainties for short-haul flights), can an Intelligent Control strategy based on an a priori known schedule be used? For example, let us consider the case where each aircraft may, with some probability, be delayed up to 15 units of time (e.g., due to an unknown upstream flow management action). Our simulations suggest that that the Intelligent Control strategy can indeed be implemented based on the a priori -known schedule (see Figure 8), with a slight modification that takes into account the possibility for delays. In particular, instead of using a threshold on the a priori predicted Sector count (i.e., Xnc(t)) to determine when the restriction should be imposed, we find it necessary to base the decision on the worst-case count or two standard-deviation upper bound on the count. Since these bounds can be computed based on the a priori schedule together with the statistics of the delay, our simulations suggest that the Intelligent Control strategy is indeed robust to uncertainties in the flow. While further simulations and analysis (e.g., of robustness to pop-ups) are needed, it is our belief that the Intelligent Control strategy can be applied based on nominal aircraft schedules and statistics of uncertainties (perhaps together with notifications about major routing changes due to, e.g., weather).

We believe that our study is pertinent in the development of multi-Center flow management strategies, because it characterizes the impact of a restriction or metering strategy on both downstream flows (specifically, downstream Sector counts) and upstream flows (specifically, backlogs and delays). We can take advantage of these characterizations to evaluate the performance of multi-region strategies, and perhaps in turn to design such strategies. One promising approach for the design problem is to use an Eulerian model for the NAS10,13,24, such as the quasi-linear stochastic model for region Counts developed in our previous work13. This model, which does not explicitly consider flow management, provides a statistical characterization of region counts given departure flow rates. It would be interesting to incorporate flow management in this model, using the characterizations of the impact of management on upstream and downstream counts. That is, the management strategies can be modeled as impacting the stochastic flows among regions; specifically, a flow restriction will serve to increase mean upstream aircraft counts (because of backlog), while reducing the variance (spread) in downstream counts. Using region count statistics based on this modified model, one could evaluate whether or not a particular multi-Center management strategy would prevent capacity violations. Also, in this manner, the capacity management problem can be phrased as a (possibly distributed) graph-theoretic control problem with discrete-valued control variables.

A third direction that we are currently pursuing is the development of a tool that facilitates design of flow management strategies through visualization of their impact on air traffic. Figure 9 constitutes a proof of concept for a tool that can be used for designing/optimizing an Intelligent Control strategy, i.e. for deciding the time period over which a MINIT restriction should be imposed. The user of the tool would have the option to set the parameters of the management strategy, and observe the impact of this strategy on a particular arrival stream or a stochastically-generated set of possible arrival streams.

Acknowledgments

The second author thanks Dr. B. Sridhar and Dr. G. B. Chatterji for several illuminating conversations on the topic of this paper.

References 1Swenson, H. N., T. Hoang, S. Engelland, D. Vincent, T. Sanders, B. Sanford, and K. Heere, “Design and operational

evaluation of the Traffic Management Advisor at the Fort Worth Air Route Traffic Control Center”, 1st USA/Europe Air Traffic Management R&D Seminar, Saclay, France, June 1997.

2Farley, T. C., J. D. Foster, T. Hoang, K. K. Lee, “A time-based approach to metering arrival traffic to Philadelphia”, AIAA-2001-5241, First AIAA Aircraft Technology, Integration, and Operations Forum, Los Angeles, California, October 16-18, 2001.

3Bayen, A. M., C. J. Tomlin, Y. Ye, and J. Zhang, “An approximation algorithm for scheduling aircraft with holding time”, In Proceedings of the 43rd IEEE Conference on Decision and Control, Atlantis, Bahamas, Dec. 2004.

4Green, S.M., “En route spacing tool: efficient conflict-free spacing to flow-restricted airspace”, 3rd USA/Europe Air Traffic Management R&D Seminar, Napoli, Italy, June 2000.

5Wang, P.T., C. R. Wanke, and F. P. Wieland, “Modeling time and space metering of flights in the National Airspace System”, in Proceedings of the 2004 Winter Simulation Conference , R.G. Ingalls et al, eds.

6Landry, S., T. Farley, J. Foster, S. Green, T. Hoang, and G. L. Wong, “Distributed scheduling architecture for Multi-Center time-based metering, AIAA Aircraft Technology, Integration and Operations (ATIO) Conference, Denver, CO, November 17-19, 2003.

7Hoang, T., T. Farley, J. Foster, and T. Davis, “The Multi-Center TMA system architecture and its impact on inter-facility collaboration”, AIAA Aircraft Technology, Integration and Operations (ATIO) Conference, Los Angeles, CA, October 1-3, 2002.

American Institute of Aeronautics and Astronautics

12

8A. Bayen, Computational control of networks of dynamical systems: application to the National Airspace System , Department of Aeronautics and Astronautics, Stanford University, Dec. 2003.

9A. Bayen, R. Raffard, C. Tomlin, “Adjoint-based constrained control of Eulerian transportation networks: application to air traffic control”, In Proceedings of the 2004 American Control Conference , Boston, MA, June 2004.

10A. Bayen, R. Raffard, C. Tomlin, “Eulerian network model of air traffic flow in congested areas”, In Proceedings of the 2004 American Control Conference , Boston, MA, June 2004.

11Stock, S. and D. Bertsimas, “The air traffic flow management problem with enroute capacities”, Operations Research , 46:3, 406-422.

12Bilimoria, K., H. Q. Lee, Z-H Mao, and E. Feron, “Comparison of centralized and decentralized conflict resolution strategies for multiple-aircraft problems,” AIAA Guidance, Navigation, and Control Conference, Denver, CO, Aug. 2000.

13Roy, S., B. Sridhar and G. C. Verghese, “An aggregate dynamic stochastic model for an air traffic system,” 5th USA/Europe Air Traffic Management R&D Seminar, Budapest, Hungary, June 2003.

14Gallager, R. G., Discrete Stochastic Processes , Kluwer Academic Publishers: Boston, 1996. 15Green, S., “En route spacing tool: efficient conflict-free spacing to flow-restricted airspace”, 3rd USA/Europe Air Traffic

Management R&D Seminar, Napoli, Italy, June 2000. 16Long, D. et al, Modeling Air Traffic Management Strategies with a Queueing Network Model of the National Airspace

System, NASA CR-1999-208988 17Gross, D. and C. M. Harris, Fundamentals of Queueing Theory , 3rd ed., Wiley: New York, 1998. 18Altinkemer, K., I. Bose, and R. Pal, “Average waiting time of customers in an M/D/k queue with nonpreemptive priorities”,

Computers in Operations Research, Vol. 25, No. 4, pp. 317-328, 1998. 19G. Solomos, D. Knorr, M. Blucher, J. DeArmon, G.Dorfman, H. Yi, “Do flight times change from year to year? A

comparison of 2001 and 2002,” MITRE Technical Paper, May 2003. 20C. Wanke, N. J. Taber, S. L. Miller, C. G. Ball, L. Fellman, “Human-in-the-loop evaluation of a multi-strategy traffic

management decision support capability,” in the Proceedings of the 5 th Eurocontrol/Federal Aviation Agency Air Traffic Management Research and Development Conference , Budapest, Jun. 23-27, 2003.

21Aircraft Situation Display to Industry: Functional Description and Interface Control Document (Version 4.0), Volpe Center, Automation Applications Division, Internal Technical Report, Aug. 4, 2000, Tim Grovac and Rick Oiesen, ed.

22M. Klopfenstein et al, En Route User Deviation Assessment , NASA Research and Technology Organization publication, Advanced Air Transportation Technologies #37, Nov. 1999.

23J. Krozel, D. Rosman, and T. Carniol, Analysis of En Route Sector Demand Prediction Accuracy and Quantification of Error Sources , NASA Research and Technology Organization publication, Advanced Air Transportation Technologies #66, May 2002.

24Menon, P. K., G. D. Sweriduk, and K. Bilimoria, “A new approach for modeling, analysis and control of air traffic flow,” in Proceedings of the AIAA Conference on Guidance, Navigation, and Control , Monterey, CA, Aug. 2002.