A Scalable Methodology For Evaluating And Designing

Coordinated Air Traffic Flow Management Strategies Under

Uncertainty

Yan Wan∗ Sandip Roy∗

Abstract

As congestion in the United States National Airspace System (NAS) increases, coordina-tion of en route and terminal-area traffic flow management procedures is becoming increasinglynecessary, in order to prevent controller workload excesses without imposing excessive delayon aircraft. Here, we address the coordination of flow management procedures in the pres-ence of realistic uncertainties, by developing a family of abstractions for implementable flowrestrictions (e.g., miles-in-trail restrictions, ground delay programs, or slot-based policies). Us-ing these abstractions, we are able to to evaluate the impact of multiple restrictions on generic(uncertain) traffic flows, and hence to design practical flow management strategies. We use thedeveloped methodology to address several common design problems, including design of multiplerestrictions along a single major traffic stream and design of multiple flows entering a congestedterminal area or Sector. For instance, we find that multiple restrictions along a stream can beused to split the backlog resulting from a single restriction, and hence to reduce congestion.We conclude the discussion by posing a tractable NAS-wide flow management problem, using asimple algebraic model for a restriction.

1 Introduction

Traffic flow management (TFM) procedures have been used successfully for many years in theUnited States National Airspace System (NAS), in order to meet the capacity constraints result-ing from arrival-rate and controller-workload requirements in the face of uncertainties includingadverse weather, unexpected airborne delays, and departure delays [1–4]. Operationally, TFMis implemented using procedures such as Air Traffic Control (ATC) preferred routes, time-basedmetering, miles-in-trail (MIT), minutes-in-trail (MINIT), ground stops (GS) and ground delay pro-grams (GDPs) (see [5–7] for details of the TFM mechanisms). In recent years, there has been agrowing need to coordinate these TFM actions over multiple Centers or even NAS-wide. As anexample, the congestion at Philadephia (PHL) airport needs be alleviated by redistributing thedelays and workloads further upstream [8]. Effectively bringing multiple Centers into collabora-tion for flow management is complicated because management actions cause complex correlationsamong aircraft in uncertain flows, both upstream and downstream. This need for coordinationmotivates study of TFM from a network point of view, which could provide a better understandingof global TFM performance, and hence permit better management of aircraft traffic flows usingexisting mechanisms (or new ones). With this motivation, we develop a methodology for analyzing

∗Both authors are with the School of Electrical Engineering and Computer Science, Washington State University.The authors are grateful for the support provided by the National Aeronautics and Space Administration (underGrant NNA06CN26A) and the National Science Foundation (under Grant ECS0528882). Both authors contributedequally to this work.

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the performance of networked TFM strategies operating in the face of realistic uncertainties, and inturn design high-performance TFM strategies that can be implemented using existing mechanismsand/or new low workload procedures (e.g., the slot-based approach [9]).

A formulation of the multi-Center TFM problem was given in [10], where the technical chal-lenges associated with implementing time-based metering of traffic to Philadelphia airport wereillustrated. This work was followed by development of detailed architectures for coordination, in-cluding a distributed scheduling architecture [11] and Multi-Center Traffic Management Advisor(McTMA)-based architecture [8]. Some other work related to multi-Center TFM focuses on theabstractly modeling traffic flows (in a deterministic way), and in turn pursuing TFM design bysolving optimal scheduling problems based on the models [12–15]. Our work here builds on boththe implementation-focused studies [8, 10, 11] and the analytical approaches [12–15], in that weuse models that capture practical considerations of implementation/robustness, but in a way thatpermits analysis and design. In taking this approach, we draw on network models for air trafficflows [1, 16], which can be used to predict the transients and steady-state statistics of regionalaircraft counts, as well as queueing models for flow restrictions (e.g., [5]).

Specifically, we make two advances in network TFM: 1) we capture the specifics of existing TFMrestrictions within the network models, and 2) we study the impact of coordinated TFM strategieson generic (typical uncertain) flows (see [5] for such analysis, for a single restriction). These effortsprovide the benefit of evaluation of managed air traffic flows in advance (since uncertainties in flowsare considered), and lead to designs that are practical for implementation in the existing operationalframework (i.e. using existing procedures or new ones that require little effort from controllers suchas the slot-based methodology introduced in [9]). The reader will note that we emphasize onthe modeling and design of restrictions on aircraft flows since many currently-used restrictions(e.g., MIT, MINIT, some GDPs) have impact on aircraft flows rather than requiring scheduling ofindividual aircraft. Our goal is to model such restrictions in a scalable way, i.e. such that NAS-wideevaluation and design is permitted. We note that our focus on flow management under uncertaintyis closely aligned with the perspective given in the articles [17, 18], which consider optimizationof coordinated flow management strategies (specifically, replanning strategies) and testing of thesestrategies in an uncertain simulation environment. Our approach builds on theirs in that we designmanagement strategies that are robust to uncertainty (rather than designing based on a forecast)and obtain simple structural insights into good strategies, albeit at the cost of using much moreabstracted models for aircraft flows.

To facilitate the NAS-wide design task described above, we need to pursue two abstractiontasks: one is to represent the traffic flow between regions (network flow/routing/splitting model)under uncertainty; and the other is to model restrictions in a way that would allow design on anetwork level. The key idea here is that we view this as a tractable network controller designproblem with stochastic air traffic flows as inputs. Each traffic flow can be modeled as a stochasticprocess (e.g., Poisson Process), as justified in [1, 5]. The restriction placement can be viewed asa controller design, in the sense that the downstream flow is managed by the controller based onthe rate of flow in the upstream. In our previous work [5], we understand the essential impactof a single restriction on traffic flow: the restriction smooths out the downstream flows (reducestheir variance), at the cost of upstream backlog and delay. Unfortunately, as we note in [5], thesingle-restriction analysis does not easily scale to multi-restriction and hence multi-Center designs.The complexity of the relationship between upstream and downstream flow under restriction drivesus to seek abstract restriction models that can capture the intrinsic relationship between the flows,

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while significantly reducing the complexity of the analysis, and so permitting multi-Center TFMdesign. In particular, we suggest a saturation model for restriction, as well as linear dynamic andalgebraic approximations. As we shall show, majorization and linear system analysis are the toolsthat allow performance evaluation for the new models.

The remainder of the paper is organized as follows: in Section 2, we introduce our framework forstudying flow management restriction design and overview the family of abstractions for restrictions.In Section 3, we briefly review detailed queueing models that permit analysis of a single restrictionanalysis. In Section 4, we describe the saturation model, and show that it can be used to analyzeflow statistics in several topologies. In Section 5, we describe the linear abstraction and use it fordesign of multiple restrictions. In Section 6, we give some preliminary discussions on using a simplealgebraic model for network-wide flow control design.

2 A Family of Abstractions For Flow Management: Overview

In this section, we introduce a framework for studying coordination of flow management restrictionsand overview our abstractions for individual abstraction. Each abstraction captures the essentialeffects of restrictions on general aircraft flows, but at different levels of detail/tractability. Together,they permit evaluation and some design of multi-Center (coordinated) TFM.

Broadly, traffic in the NAS can be viewed as flows of aircraft (network flows) entering andleaving regions (Sectors/Centers/airports). Traffic flow management is implemented by placingrestrictions on boundaries of the regions to avoid congestion in downstream regions, while notcausing unsatisfactory delay. We are interested in evaluating the effects of practical restrictions ontypical flows, which we expect to vary day-to-day due to unpredictable events such as weather andtake-off time uncertainties.

Let us first consider the flow into and dynamics of each restriction. We model the arrival ofaircraft (or entering flow) at each boundary restriction in a quite-general way, as a stochastic(arrival) process. We denote the the arrival rate (the mean of number of aircraft arriving at theboundary per unit time) as λ 1. In some cases, e.g., when a flow is comprised of traffic from severaluncoordinated flows or when it models departures from a busy airport, it is modeled as a Poissonprocess ( [5]). We address both the general case and the special Poisson case here.

Let us now consider the dynamics of the restriction (control) at each boundary. To ease un-derstanding, we imagine the existence of a virtual buffer at each boundary, as shown in Figure 1.Every approaching aircraft automatically enters the buffer (at time t, the number of aircraft inbuffer i, or the buffer count, is denoted by bi(t)). However, only a portion of the aircraft in thebuffer are allowed to enter the downstream region. We denote the stochastic process describing theflow of aircraft into the downstream region by ei(t), and call it the crossing flow. The numberof aircraft being delayed by the restriction at time t (backlog) is denoted by Bi(t). Note thatBi(t) is different from bi(t) in that Bi(t) does not count the aircraft that enter the buffer and thenleave smoothly without being held by the restriction. The relationship of a boundary’s crossingflow ei(t) with respect to its buffer count bi(t) is what we aim to design. Specifically, we developseveral abstractions for implementable restrictions (e.g., MINIT, MIT, and slot-based restrictions)within this framework with the aim of achieving designs that can be put in place easily.

1As noted in [5], the arrival rate at boundaries are in fact time-varying. Here, we are concerned with flows duringthe busy part of the day (when the restrictions are needed), so we use a time-invariant model for simplicity. Many ofthe results generalize naturally to the time-varying case.

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Figure 1: Boundary restriction framework.

In some analyses, we explicitly model counts in downstream regions. We assume each aircraftin a stream takes a constant time T to cross a downstream region with a capacity of C aircraft.This assumption is often reasonable for en route airspace, where the aircraft follow a route througha region at an approximately constant speed. The number of aircraft in a downstream region (orregion count) is denoted as ri(t).

At a network level, we view streams of aircraft counting from various boundary restrictions asmerging and splitting within the regions. While restriction design for various network topologiesare of interest, a couple simple topologies are especially common, and so we often focus on thesetopologies or study them in examples. One common phenomenon is the existence of a major streampassing through a sequence of restrictions (for instance, traffic from the West Coast to Northeastairports) together with some minor flows entering and leaving the major flow. We thus oftenconsider a stream of aircraft passing through a series of boundary restrictions, and aim to set theseboundary restrictions for suitable management. A second common topology is one with a singlecapacity-constrained region with multiple flows that can be restricted before entering the region.Such a network model is appropriate, for instance, in developing restrictions around a crowdedterminal area or Sector.

Our primary goal is to develop a series of abstract restriction models that capture the keyattributes of en route and terminal area restrictions, while permitting analysis of networked (coor-dinated) restrictions to varying degrees. The models that we have studied, in order of increasingabstraction (and also greater tractability), are: 1) Detailed queueing models for en route and ter-minal area restrictions; 2) A discrete-time saturation model; 3) A dynamic stochastic linear model;4) An algebraic linear model, for design in arbitrary networks.

The evaluation of each model is discussed in detail in the following sections.

3 Detailed Queueing Models: Review

In the literature, queueing models have been used to represent various en route and terminal arearestrictions [5, 19, 20]. Since we are seeking abstractions for restrictions that permit network-levelanalysis, we first find it worthwhile to review these detailed queueing models and their analy-sis/verification.

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In [5], we modeled an en route flow restriction as a single-server queue with deterministic servicetimes. Specifically, as developed fully in [5], one can see that a MINIT/MIT restriction enforcesthat only a single aircraft can be located in a region of airspace just beyond a waypoint wherethe restriction is placed. Hence, the restriction can be represented as a single server queue, wherea single aircraft just beyond the waypoint is viewed as being served. The service time is well-modeled as deterministic, since the duration of the restriction is a fixed time (or, in the case ofMIT restrictions, a fixed distance which roughly corresponds to a fixed time for en route aircraft).

The article [5] partially characterizes the backlog/delay and downstream smoothing effectedby a single restriction using the queue model, assuming a generic (Poisson) flow entering the re-striction. In particular, we were able to use classical results on M/D/1 queueing models (Poissoninput, deterministic single server) to identify the expected backlog and aircraft delay (see [21]).Through simulations, we were also able to characterize the downstream process, e.g., find the prob-ability of violating a capacity restriction in a downstream sector. In [5], we also present validationof the M/D/1 queueing model using several canonical examples, e.g., for flow management nearthe Atlanta airport. As a whole, these modeling and analysis/simulation efforts make clear theessential tradeoff between upstream backlog/delay and downstream flow smoothing effected by aMIT/MINIT restriction.

A detailed queueing model for a time-based metering restriction has also been developedin [5]. Similarly, temporal constraints on arrival and departure aircraft streams have been rep-resented/evaluated using queueing models [19, 20]. While these models differ in their details, theyhave in common that trade off upstream backlog and downstream flow smoothing.

Unfortunately, explicit analytical results concerning the statistics of the downstream processare difficult to obtain for these various queueing models, especially in the case where the the pro-cess impinging on a restriction is not a Poisson process. Thus, while we can characterize a singlerestriction fully, this analysis does not easily scale to a sequence or network of restrictions. Theabstractions that we develop subsequently have the advantage of scalability. They also provide acommon framework for representing various restrictions, and hence permit comparison and inter-facing of restrictions.

4 Saturation Restriction Model

In order to better understand a restriction’s impact on flows and achieve design of coordinatedrestrictions, we develop a more abstract model, which we call the saturation restriction model.The saturation model is a discrete-time abstraction of the queueing model. This discrete timeabstraction allows the analysis of downstream variability for Poisson inflows, and in turn permitsqualitative study of multi-Center restriction placement.

4.1 Description of the Saturation Restriction

The saturation restriction works as follows: during each time step of length ∆T , it allows a max-imum of Nc aircraft to pass the boundary, while the remaining aircraft remain in the buffer.Formally, let us denote the the number of aircraft arriving at a boundary between times k∆T and(k + 1)∆T as x[k], the number of aircraft allowed to pass the boundary during this time as e[k],the number of aircraft in the buffer at time k∆T as b[k], and the backlog at time k∆T as B[k].

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These variables evolve as follows:

e[k] =

{b[k − 1], (b[k − 1] ≤ Nc)

Nc, (b[k − 1] ≥ Nc)(1)

b[k] = b[k − 1] + x[k] − e[k]

B[k] = b[k − 1] − e[k]

The parameter Nc of the saturation model can be matched with the specifics of actual TFMrestrictions. In particular, many restrictions enforce that aircraft are separated by τ minutes, andhence that Nc = ∆T

τaircraft are allowed to pass the boundary in ∆T minutes. For instance, using

this reasoning, Nc can be chosen as ∆Tq

to represent q-MINIT restrictions and similarly a q-MIT

restriction can be matched with Nc = ∆Tvq

assuming that each aircraft travels at velocity v. Indesigning restrictions, it is worth noticing that Nc needs to be larger than the average aircraft flowrate λ∆T ; otherwise, there would be an unlimitedly growing aircraft count in the buffer. We expectthe setting of Nc will affect the flow (e.g., downstream volume, upstream backlog and delay). Themapping from a TFM restriction to Nc provides an approach to evaluate the restriction’s impacton flow, and (conversely) a designed Nc can be matched with a restriction duration.

The saturation model can be viewed as a discrete-time counterpart of the various detailedqueueing models. Both models delay the aircraft at busy times and permit them to go throughat times with a small arriving flow. However, as we shall show, the saturation model allows us tocharacterize the downstream variance in terms of the parameter Nc and the arrival flow, and henceprovides more flexibility to evaluate a restriction’s impact on flow. This flexibility in turn permitsdesigning restrictions to satisfy certain performance requirements.

We also note that the saturation model captures rather precisely the slot-based managementstrategy proposed in [9]. In particular, a restriction where aircraft are forced into slots that areq minutes apart can be represented using a saturation model with ∆T = q and Nc = 1. Thisrepresentation captures the behavior of the slot model, with only the slight error that the initialdelay of a fraction of a slot when an aircraft first approaches a boundary is not captured. Ourapproach thus permits evaluation of the slot-based strategy for stochastic entering flows.

4.2 Model Evaluation for Poisson Input Flow Statistics

For the saturation model, the statistics of upstream and downstream flows can be explicitly com-puted when the approaching flow is a Poisson process. In this case, the numbers x[k] that entersthe buffer at different time steps are independent because of the Poisson process’ independent in-crement property [22]. Specifically, we recall that when the input process is a Poisson Process ofrate λ, the probability that x[k] = c aircraft arrive at time step k is as follows:

Pλ(c) =(λ∆T )ce−λ∆T

c!, c ≥ 0. (2)

Next, we contend that statistics of the buffer count b[k], crossing flow e[k], and backlog B[k] canbe determined from Markov chain analysis [23]. In the infinite-state Markov chain representation,each state i ∈ {0, 1, 2, ...,∞} represents a possible buffer count, and each branch weight pij denotesthe probability that the buffer count transitions from one state i to another state j during onetime step (i.e. P (b[k + 1] = j|b[k] = i). Based on the evolution equations (1) and the probabilisticdescription of x[k] (2), the transition probability pij can be calculated:

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pij =

Pλ(j), 0 ≤ i ≤ Nc, j ≥ 0

Pλ(j − i + Nc), i > Nc, j ≥ i − Nc

0, i > Nc, j < i − Nc

(3)

We are interested in the steady state probability that the buffer count b[k] equals i (we call it πi),which enables us to calculate the statistics of flow at steady state. These probabilities can be foundfrom the transition probability matrix using classical techniques for Markov chains [23]. Withknowledge of the steady-state probabilities πi, the mean backlog in steady state can be calculatedas

E(B[k]) =∞∑

i=Nc+1

(i − Nc)πi. (4)

The mean crossing flow E(e[k]) equals the inflow rate λ∆T since we require that Nc is larger thanλ: this condition guarantees that there is no growing aircraft count in the buffer, and hence theaverage crossing flow is same as that of the entering flow. Moreover, the variance in the crossingflow can be calculated as

V (e[k]) =∞∑

i=Nc+1

(N2c πi) +

Nc∑

i=1

(i2πi) − (λ∆T )2. (5)

The above calculation shows that the saturation abstraction permits the calculation of flow statis-tics (e.g., mean backlog and mean and variance of crossing flow) using a standard Markov chainrepresentation. Figure 6 illustrates the Markov chain analysis, in particular demonstrating its usein comparing restriction of various flows entering a congested region.

However, the calculation of the flow statistics does not provide us a direct insight into a restric-tion’s effect on non-Poisson flows. Also, unfortunately, the Markov chain analysis does not easilyscale to the multiple-restriction case.

4.3 Analyzing the Effect of One or More Saturation Restrictions on Arbitrary

Aircraft Flows

In this section, we use the saturation model to give some qualitative insights into a restriction’seffect on arbitrary stochastic aircraft flows. In turn, we analyze sequences of restrictions using themodel.

Before modeling networked flows and restrictions, it is important to understand a restriction’simpact on a single generic flow (i.e. one that may originate from another restriction, and hence benon-Poisson). We expect that the placement of a single restriction would smooth out the down-stream flows at the cost of increasing backlog, and that the impact of the restriction becomes morepronounced as the restriction is made stronger (i.e. Nc is decreased). We formalize this notion inResult 1.

Result 1 Consider two ergodic 2 aircraft flows with identical statistics approaching two saturationrestrictions c1 and c2 respectively, where restriction 1 is weaker than restriction 2 (Nc1 > Nc2).

2Ergodic is a term dealing with the estimation of statistics of random processes using real data. The statistics ofa ergodic process can be estimated from the time average of a single sample sequence with the length of the sequenceapproaches infinity. It is reasonable to assume x[k], B[k] and e[k] are ergodic random processes, since the processesare strict-sense stationary, and uncorrelated for large interval.

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The variances of the downstream flows V (ec1 [k]) and V (ec2 [k]) and the mean of backlogs E(Bc1 [k])and E(Bc2 [k]) are related as follows: E(Bc1 [k]) ≤ E(Bc2 [k]), V (ec1 [k]) ≥ V (ec2 [k]).

Proof: The statistics of an ergodic process can be calculated using time-averages of one of itssample sequences. We study the effect of each restriction on an arbitrary sample sequence over asufficiently long time.

In particular, the relationship of the mean backlog can be analyzed directly from the backlogat each time step, for any sample sequence of the input process. Since Nc1 > Nc2 , at the initialtime step (k = 1), if an aircraft is able to get through c2, it must be able to get through c1; orequivalently, if an aircraft is delayed by c1 at the initial time step, it must be also delayed by c2.Therefore, at the initial time step, the backlog of c1 is smaller than that of c2. At all subsequenttime steps, we have the same conclusion that if aircraft is delayed by c1, it must be delayed byc2 too, since the backlog caused by c2 is larger than that by c1 at the previous time step, andless aircraft are permitted through by c2 than c1 at the current time step. This leads to theconclusion that Bc1 [k] ≤ Bc2 [k] for any k and thus the time-average of the backlog of each sampleflow has the relationship 1

T

∑Tk=1 Bc1 [k] ≤ 1

T

∑Tk=1 Bc2 [k] for all T . From ergodicity, we conclude

E(Bc1) ≤ E(Bc2)3.

The above result easily extends to the buffer count, since for any k, buffer count and backlogonly differ by a single term x[k], which is the same for the two restrictions c1 and c2. Thus, thesame reasoning that yields the relationship Bc1 [k] ≤ Bc2 [k] leads to bc1 [k] ≤ bc2 [k] for any k andany sample sequence. Thus, we have E(bc1) ≤ E(bc2).

The comparison of the variance of flows crossing the two boundaries can be analyzed from themanipulation of the two flow sequences ec1 [k] and ec2 [k] over some time. Firstly, the time averageof ec1 [k] is not less than that of ec2 [k] because the backlogs have the relationship Bc2 [k] ≥ Bc1 [k]for all k. Secondly, the conclusion that bc1 [k] ≤ bc2 [k] for all k leads to two conclusions: 1) if ec1 [k]is larger than Nc2 , restriction 2 must be saturating and thus ec1 [k] ≥ ec2 [k] = Nc2 ; 2) if ec1 [k] isless than Nc2 , ec2 [k] must be no less than ec1 [k]. The inequality for the flow average and the tworelationships 1) and 2) above leads to the conclusion that the sum of ec1 [k] − Nc2 for those k suchthat ec1 [k] ≥ ec2 [k] ≥ Nc2 is larger than or equal to the the sum of ec2 [k] − ec1 [k] for those k suchthat ec1 [k] ≤ ec2 [k] ≤ Nc2. Thus, by reducing ec1 [k] by 1 for those ec1 [k] > Nc2 and adding it tothose ec1 [k] with ec1 [k] < Nc2 step by step as shown in Figure 3, we can reshape the sequence ofec1 to ec1 , which is same as ec2 , except possible with extra ec1 [k] > Nc2 for some k. Therefore, wehave V (ec1 [k]) ≥ V (ec2 [k]) assuming ergodicity. In the reshaping process, the second moment andhence variance of ec1 is decreased monotonically. From ergodicity, we have V (ec1 [k]) ≥ V (ec1 [k]).Thus, we see that V (ec1 [k]) ≥ V (ec2 [k]). 2

This result clearly illustrates a boundary restriction’s role in flow control in a general way: itreduces the variance of downstream flow, at the cost of increasing the mean backlog. A stronger re-striction produces smoother downstream flow, with more backlog in the upstream. Thus, we againsee that, in contrast to the detailed queueing models, the saturation model allows us to analyzedownstream variability.

The saturation model also provides us with the flexibility to characterize the performance of

3The proof of this inequality does not actually require ergodicity. The fact that Bc1[k] ≤ Bc2

[k] for any k and anysample sequence guarantees E(Bc1

) ≤ E(Bc2).

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Figure 2: A single flow passing through a chain of two restrictions.

Figure 3: Downstream flow variance analysis for two saturation restrictions with different thresh-olds.

multiple restriction, and hence to develop coordinated TFM strategies. In Result 2, we discussrestriction placement for a sequence of two regions (which can be easily extended to a sequence ofmultiple regions) with a single flow passing through them (see Figure 2). This topology of aircraftflow is common in many situations, such as the flow following major routes from the Western statesto the Northeastern states, crossing several Centers along the route. An interesting question forthis type of flow is where to place restrictions on the route, i.e. at the boundaries of downstreamregions or further upstream. Result 2 gives insight into the combined effect of the multiple restric-tions along the route, in terms of backlog and downstream congestion caused by the restrictions.As a canonical example, we assume a single flow (with no merging or splitting of other streams)that is restricted in several places; the result is useful in practice whenever there is a major aircraftflow passing through several restrictions.

Result 2 Consider an arbitrary aircraft flow approaching a two-region chain. Placing a restric-tion c1 at the first boundary with threshold Nc1 and another restriction c2 at the second boundarywith threshold Nc2 achieves the same total delay (on each aircraft)/backlog (over time) as placinga single restriction c3 at the front of region 1 with the threshold designed as the strong restriction,i.e. as Nc3 = min(Nc1 , Nc2). In fact, the crossing flow processes of the second region for the twoschemes are identical.

Proof: When Nc1 < Nc2 , we have Nc3 = Nc1 . In this case, the result is obvious since everyaircraft passing Nc1 will pass Nc2 without any delay.

When Nc1 ≥ Nc2 , we have Nc3 = Nc2 . Showing the equality of the delay of the single- and

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two-restriction cases is equivalent to showing that any aircraft crosses boundary of region 2 at thesame time step for the two cases. The proof has two sides. First, we can prove that the delay of thetwo-restriction design is not smaller than the delay of the single restriction design, i.e. if an aircraftis able to cross boundary 2 for the two-restriction case, it must has crossed the boundary by thesame time for the single restriction case. This is obvious since the delay on each aircraft would bethe same for the two cases, if Nc1 were 0. Introduction of the first restriction for the two-restrictiondesign can only delay aircraft, hence the delay is at least as large in the two-restriction case.

Second, we show the reverse is true too—i.e. the delay of the two-restriction design is not worsethan that of the single restriction design—by contradiction, as follows. At the initial time step thatthe restrictions are set, the number of aircraft in region 2 in both cases are the same. Suppose thestatement that we are going to prove does not hold, that means there exists some time step when anaircraft is coming through boundary 2 in the single restriction case, but not in the two-restrictioncase. We denote the first aircraft satisfying the above condition as a1. All the aircraft before a1in the flow are able to go through boundary 2 at the same time step in both cases. The only tworeasons that a1 can be delayed in the two-restriction case, but not in the single restriction case,are: 1) for the two-restriction case, a1 has more aircraft in the line before it than the thresholdNc2 ; and 2) a1 is not at boundary 2 yet because it was delayed at boundary 1. Neither is possible,as shown by the following argument: If 1) happens, we know that a1 is not the first aircraft whichwas delayed, since a1 has fewer aircraft before it in the buffer in the single restriction case than thetwo restrictions case. The controversy denies reason 1. For reason 2: this cannot happen, since therestriction at boundary 1 is weaker than that in the single restriction case. As we argued for Result1, if an aircraft is able to coming across a stronger restriction as in the case of single restriction, itmust be able to cross boundary 1 with a weaker restriction in the two-restriction case.

Thus, we have proved the delays are identical. The result for delays automatically implies thebacklog result. 2

For a chain of more than two regions with a single flow, Result 2 can be generalized to thefollowing: placing multiple restrictions along the boundaries of the regions achieves the same totaldelay as placing a single restriction upstream, that is equal to the strongest of the original restric-tions. The proof is very similar, noticing that if the strongest restriction is at the boundary i in achain of m regions, the restrictions placed between boundary i and boundary m do not cause anydelay.

This result is interesting in that it shows for a major stream of aircraft, multiple restrictionsplaced along the route have the effect of splitting the delay, without increasing or reducing it.Also, placing a single strong restriction further upstream has the benefit of no congestion in thedownstream regions along the route. For example, in this two-region case, using a single strongrestriction before region 1 causes no congestion in region 1; in contrast, in the two-restriction case,congestion may result from the delay caused by the restriction before region 2, as well as theincreased variance of the downstream count due to the weaker restriction before region 1.

In practice, when considering a single major flow passing a series of regions, the end regionoften has the most stringent capacity restriction since it represents the busiest region where manyflows merge. Result 2 suggests that in this case, it is better restrict flows entering the regionfurther upstream. This will reduce the downstream congestion with the same delay. It is wise toplace the restriction at a boundary where the upstream region has little congestion concern, andso can absorb the backlog. In some cases, if such a upstream region that can hold all the backlog

10

0 5 10 15 20 25 300

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Distribution of Backlogs, Upstream Restriction

Backlog

Fre

quen

cy o

f the

bac

klog

Nc1=10, Nc2=6Nc1=7, Nc2=6Nc1=6, Nc2>6

0 5 10 15 20 25 300

0.1

0.2

0.3

0.4

0.5

0.6

0.7Distribution of Backlogs, Downstream Restriction

Backlog

Fre

quen

cy o

f the

bac

klog

Nc1=10, Nc2=6Nc1=7, Nc2=6

Figure 4: Distribution of backlogs for three different combinations of restrictions. The two restric-tions can be carefully selected (Nc1 = 7, Nc2 = 6) to split the backlog between the two regions.These distributions were computed using actual arrival data from January 2006.

does not exist, we need to design multiple restrictions along the route to split up the delay and inconsequence the backlog.

Result 2 also makes clear that, in contrast to current practice, multiple restrictions actingin a stream must be designed together because the upstream restriction strongly impacts thedelay/backlog caused by the downstream one. Thus, for instance, it is unwise to first place adownstream restriction and then place another one upstream to account for the backlog caused bythe downstream restriction: the upstream restriction in this case will change the backlog caused bythe downstream restriction.

4.4 Canonical Example: Restricting Flows into PHL

Using the saturation restriction model, we have found that multiple restrictions along a streamcan be used to split delays/backlogs. This suggests that multiple restrictions may be effective inpartially pushing backlog upstream near complex terminal areas, e.g., near Philadelphia Interna-tional Airport (PHL). Here, as a canonical example, we have evaluated the impact of using multiplerestrictions on the entire flow arriving at PHL.

We use the arrival times of all aircraft coming to PHL in January 2006 as the inflow. Let’s say wewish to limit the flow into the airport to 6 planes per 20 minutes (e.g., because of arrival capacityrequirements or constraints on nearby airspace). We have three combinations of the restrictionstrengths N1 and N2 on the flow that each achieve the requirement. For each case, the meanbacklog and distribution of backlogs due to each restriction have been found (Table 1 and Figure4). As expected, the total backlog for the three cases are the same. A more stringent restrictionupstream reduces the backlog caused by the second restriction. This informs us that designing astringent restriction at a upstream region will succeed in moving the backlog further upstream. Agood choice would be to locate a upstream region with little traffic congestion concern, and placeall the backlog there. If such a region does not exist, we can split the backlog carefully among someupstream regions (the Nc1 = 7, Nc2 = 6 case).

11

Nc1 Nc2 EB1EB2

10 6 0.0542 2.7852

7 6 0.9380 1.9014

6 10 2.8394 0

Table 1: The mean backlog of the two regions for three different restriction settings.

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AAR=30/hr.SFO,

NorthboundFlow

Flow (Major)Westbound

FlowSouthbound

550 600 650 700 750 800 850 900 950 1000

0

5

10

15

Backlog in Terminal Area

Time (Minutes)

Bac

klog

(P

lane

s)

Figure 5: a) Abstract illustration of traffic flows entering SFO and arrival capacity during stratusevent. b) Backlog at SFO during stratus event assuming flows are not restricted upstream; backlogwas computed using actual arrival data from June 1, 2006.

4.5 Another Example: Which Flow Should Be Restricted?

Our analyses of the saturation abstraction also permit us to develop coordinated strategies forother flow topologies (not only multiple restrictions along a stream). Here, we illustrate through anexample that the saturation model helps us to choose which of multiple flows entering a congestedregion to restrict.

In particular, let us consider the effect of an unexpected or incorrectly-predicted stratus eventat San Francisco airport (SFO). Stratus at SFO severely limits the arrival capacity of the airport,because only one of two parallel runways can be used. In cases where stratus impacts the airportunexpectedly (or where the time at which the stratus will clear is underestimated), the rate ofalready-airborne traffic approaching the airport may exceed the the permitted traffic. As shown inFigure 5, if these approaching flows are not restricted upstream, there may be backlogs of up to10-15 aircraft in the terminal airspace (depending on the time of day of the stratus event). Giventhe complexity of the airspace around SFO, such a backlog can unacceptably increase controllerworkloads, and hence upstream management (en route, or through ground delay programs for shortflights) is needed.

Aircraft traffic approaches SFO along several jet routes, along which restrictions can be placed.Abstractly, we can roughly separate the traffic flow approaching SFO into Northbound, Southbound,and Westbound traffic4, where the Westbound flow is the major one (see Figure 5). If an unexpectedstratus event necessitates placement of a restriction upstream, the Oakland ARTCC in coordination

4The frequency of Eastbound traffic is small enough to be negligible.

12

0 1 2 3 4 5 6 70

5

10

15

20

25

30

35

40

45

50

Decrease in standard deviation of crossing flowM

ean

back

log

Comparison of Restrictions on Major and Minor Flow

Minor flow (rate=5)Major flow (rate=45)

Figure 6: Comparison of restrictions on major and minor flows using the saturation model. Wefind that restriction of the major flow is more effective in decreasing downstream flow variability,for a given backlog.

with the ATCSCC must decide which flow(s) to restrict.The Markov chain analysis of the saturation restriction indicates that restriction of the ma-

jor aircraft flow is most effective in smoothing the downstream flow with minimum backlog anddelay (see Figure 6). In Figure 7, we have compared restriction of the major (Westbound) flowwith restriction of the Northbound flow, for traffic approaching SFO on a particular day (June 1,2006). These experiments bear out that restriction of the major flow achieves sufficient decreaseof terminal-airspace backlog with lower upstream backlog and delay. Thus, from the perspectiveof minimizing average delay and preventing upstream congestion, we find that restriction of themajor flow is most effective, as predicted by the saturation restriction model.

In summary, the saturation restriction model provides insight results into the relationships be-tween upstream and downstream flows. It also allows analysis of multiple restrictions. However,explicit expressions for region-count and flow statistics cannot easily be found when multiple re-strictions are considered, and therefore the model is difficult to use for explicit design of coordinatedTFM strategies.

We notice that the saturation model can also be to used for the comparison of MIT/MINITwith time-based metering, in that we can model a region adopting the metering strategy using thesaturation model by viewing the whole region as a buffer. We do not pursue this direction anyfurther here.

5 A Linear Abstraction for Shaping Region Counts

Linear abstractions of boundary restrictions are very appealing because explicit expressions of flowstatistics and cross-statistics can be developed and, in consequence, multiple restrictions can bedesigned to shape Sector counts. In this section, we show that the linear model can capture theintrinsic characteristics of flow restrictions, and yet is suited for multi-region analysis.

13

0 500 1000 15000

1

2

3

4

5

6Backlog at Restriction

Time (minutes)

Bac

klog

(pl

anes

)

0 500 1000 15000

2

4

6

8

10

12

14

16

18Backlog at Restriction

Time (minutes)

Bac

klog

(pl

anes

)

Figure 7: Upstream backlog caused by restriction of the a) major flow and b) minor flow, for thepurpose of limiting backlog at SFO during stratus event to 9 aircraft.

Figure 8: Linear boundary restriction scheme.

5.1 Model Description

Our discrete-time linear abstraction for a boundary restriction is shown in Figure 8. At eachtime step (i.e. between times k∆T and (k + 1)∆T ), the number of aircraft allowed to cross theboundary e[k] is calculated as a fraction (denoted by a) of the aircraft in the buffer at the previoustime step plus a constant c. In Section 5.3, we will argue that such a model can be obtainedthrough a stochastic linearization of the saturation model. One advantage of the linear model isthat, when en route restrictions are modeled, statistics of aircraft counts in downstream regionscan be computed, and hence we find it convenient to explicitly model a downstream region. Forsimplicity, let us assume that each aircraft takes a fixed number of time steps say L to cross thedownstream region. (This is often quite reasonable for an en route restriction, where each aircraft inthe flow is usually traveling at roughly the same speed.) The dynamics when the linear abstractionis used are the following:

e[k] = ab[k − 1] + c (6)

b[k] = b[k − 1] + x[k] − e[k]

B[k] = b[k − 1] − e[k]

r[k] =L∑

k=1

e[k − L + 1]

14

where b[k], B[k], and e[k] are as defined before, and r[k] is the number of aircraft in downstreamregion (downstream region count) at time k.

5.2 Analysis of the Linear Abstraction with Poisson Input

We are concerned with two measures that indicate the performance of a boundary restriction,namely downstream region count and upstream backlog [5]. For a Poisson input, the dynamic andsteady state statistics of these measures can be calculated from the linear system representation,using the classical two-moment analysis of linear system driven by random processes. Here, let uspresent the steady-state mean and variances of the backlog and downstream region count togetherwith the crossing flow (which is needed for the evaluation of the model), when the input process isPoisson with rate λ. The mean and variance of backlog B[k] caused by the restriction are

EB =1

a(λ∆T − c) − λ∆T (7)

VB =(1 − a)2

1 − (1 − a)2λ∆T ; (8)

the mean(Ee) and variance (Ve) of crossing flow are

Ee = λ∆T (9)

Ve =a2

1 − (1 − a)2λ∆T ; (10)

and the mean (Er) and variance (Vr) of downstream region count are

Er = Lλ∆T (11)

Vr =

{La2

1 − (1 − a)2+

2a2

1 − (1 − a)2

L−1∑

k=1

(L − k)(1 − a)k

}λ∆T. (12)

Let us briefly discuss the results of this analysis, from the perspective of designing restrictions.First it is important to realize that λ∆T has to be smaller than C/L (where C is capacity of thedownstream region) to be able to reduce downstream congestion while not causing growing delayin the upstream. The parameters a (a ≤ 1) and c are responsible for the downstream and upstreamperformance. A decrease in a decreases the variance of the downstream region count, but increasesthe mean and variance of backlog, as shown in Figure 9. An increase in c reduces the mean of thebacklog, and does not affect the statistics of the region count. Thus, based on these observations,it is tempting to design a to make the variance of region count small enough, and then choose c tomake the mean backlog arbitrarily small. However, such a restriction is unachievable in practice,because it requires movement of more aircraft into the downstream region than are in the buffer.For a restriction to be achievable, we need c to be small, and in this case, the boundary restrictionthat we proposed also reduces the downstream congestion with the cost of upstream backlog.

Specifically to obtain a good performance while using an achievable restriction, we need tochoose parameters a and c carefully. Let us consider the following two cases:

• Large aircraft inflow. For large λ (λ∆T close to CL

), the prevention of downstream capacityviolation is the focus of the design. Suppose no restriction is placed at the boundary; the

15

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

The Dependence of Vt on Parameter a

a

Vt

0 0.2 0.4 0.6 0.8 10

10

20

30

40

50

60

The Dependence of VB on Parameter a

a

VB

Figure 9: Dependence of downstream region count’s variance (Vt) and the backlog’s variance VB onthe linear model parameter a. Again, we see a tradeoff between backlog and downstream variance.

Figure 10: Stochastic linearizations of saturation restriction.

variance of crossing flow is λ∆T , which is very likely to cause congestion. By placing arestriction with small a, we can significantly reduce the variance of region count (in anotherwords, smoothen the aircraft flow). In this case, we can have an achievable restriction evenwith a moderate c, since the number of buffered aircraft is large, and so the buffer will notbe overdrawn even with moderate c.

• Small aircraft inflow. For small λ (λ << CL∆T

), it is a rare event that downstream congestionoccurs. In this case, we can choose a to be large (close to 1) to reduce the backlog. Essentiallyin this case, we are using little control, so c must approximately be 0.

The above analysis indicates that our linear restriction can be chosen to resemble the saturationrestriction, as shown in Figure 10. With small inflow λ, we design large a and small c to resemblethe linear-increasing region of a saturation restriction; with large λ, we design small a and moderatec to resemble the constant region of the saturation restriction. Using this stochastic linearization,we can approximate the saturation restriction with a linear abstraction, and hence gain significantadvantage in its tractability.

5.3 Restriction Design

The similarity between the linear model and the saturation model provides us with an approachfor restriction design. As developed in Section 5.2, the linear model allows the explicit evaluationof downstream count and backlog. The parameters of the linear model a and c can be calculatedfrom Nc of the saturation model, so as to match the variance of the crossing flow and the mean

16

backlog for the two models (assuming a Poisson flow). We thus can use the linear model to checkthe performance of a saturation model and hence of an actual restriction (e.g MINIT or MIT).Thus, using the linear abstraction, we can design restrictions to meet downstream region-countgoals.

As an example, we study the performance of some MINIT restrictions acting on a Poissoninflow with rate 0.2 (See Table 2). Here, we assume that the discrete time interval is ∆T = 20 andthat each airplane takes 3 time steps to cross the downstream region. This example shows thatthe analysis of downstream region count variance permitted by the linear abstraction can help usdesign appropriate MINIT restrictions. Also, the values of the parameters a and c in the tableverify that the linear restriction abstractly represents the saturation restriction: a is close to 1 andc is close to 0 when the inflow λ∆T is small compared to Nc; and a is close to 0 and c is moderatewhen the inflow λ∆T is comparative to Nc.

MINIT Nc EB Ve a c Vr

2 10 0.004 3.950 0.994 0.021 11.95

4 5 1.156 1.688 0.593 0.940 8.365

5 4 14.713 0.160 0.077 2.561 3.920

Table 2: Matching between the saturation model and linear model aids in MINIT restriction designfor the purpose of reducing variance in regional aircraft counts. Here ∆T = 20, λ = 0.2, L = 3,λ∆T = 4.

For the above case 2, trajectories of the region count and upstream backlog for the saturationrestriction and corresponding linear restriction are shown in Figure 11. The plots show that the lin-ear abstraction captures the characteristic behavior of restriction: a restriction reduces downstreamvariance with the cost of upstream backlog.

The simulation also indicates some limitations of the linear abstraction. Since we are matchingthe two models only through two steady state statistical measures, details of downstream regioncount and upstream backlog trajectories have some differences. The differences result from thefact that the saturation restriction model permits all aircraft to come through at low traffic times,while for the linear restriction, only a fraction of aircraft can come through even at times of lowtraffic. However, since the two steady state statistical measures mostly capture the importantcharacteristics of flow under restriction, we believe this abstraction is often valid, which providesus with a lot of flexibility in design.

5.4 Analysis of Multiple Restrictions

When considering design of multiple restrictions, the classical analysis of linear systems withstochastic inputs allows us to easily determine the relationship between the linear restrictionsand the statistics of flows. Thus, using the equivalence between the saturation model and linearmodel, the effect of multiple restrictions on statistics of flows in multiple regions is made analyzable.Specifically, we suggest the following procedure for analyzing a network of m restrictions:

1. Transfer possible combinations of restrictions durations to corresponding saturation parame-ters.

2. Calculate the crossing flow variance and mean backlog for each Nciusing the standard Markov

chain analysis given in Section 4.2.

17

0 20 40 60 80 100−1

0

1

2

3

4

5Upstream Backlog Comparison

Time step

Ups

trea

m b

ackl

og

Saturation restrictionLinear restriction with a=0.593 and c=0.940

0 20 40 60 80 1002

4

6

8

10

12

14

16

18

20Region Count Comparison

Time step

Reg

ion

coun

t

No controlSaturation restrictionLinear restriction with a=0.593 and c=0.940

Figure 11: Upstream backlog and downstream region count upon use of saturation restrictionmodel and corresponding linear restriction model, assuming mean inflow rate λ∆T = 4, Nc = 5,and L = 3.

3. For each restriction, match the two s.s. statistical measures with that of the linear restriction,by properly choosing the values of ai, ci for each i (as described in Section 5.3).

4. Calculate flow statistics for the entire network (e.g., variances for downstream region counts)using the linear models.

5. Design the proper MINIT restrictions that satisfy the performance requirements of the flowstatistics.

The performance analysis for multiple regions using the linear model is similar to that for a singleregion. In particular, we can develop a system of difference equations for the evolution of thenetwork, and analyze system output statistics (backlog and region count statistics) with the knowl-edge of input flow statistics, using the classical 2nd-moment analysis of a linear system driven bya stochastic process. We do not give a full formulation in the interest of space, and since we willsuggest an even simpler model for preliminary network TFM design.

Instead, we use a 2-region chain (where each aircraft takes 3 steps to pass the downstreamregion) to illustrate optimal restriction design for the important 2-region case. In the process, wederive the statistics of flows with the placement of two linear restrictions, and hence illustrate theanalysis permitted by the linear model.

As discussed above, for possible combinations of MINIT restrictions at the two boundaries, wecan calculate the corresponding saturation model parameters Nc1 and Nc2 , and then map these tolinear restriction parameters a1, c1, a2 and c2 by equivalencing the statistics of flow variance andmean backlog. Thus, we obtain a model with two linear restrictions driven by a single flow, as

18

Figure 12: Linear boundary restriction model for a chain of regions.

shown in Figure 12. The dynamics of the system are the following:

e1[k] = a1b[k − 1] + c1 (13)

b1[k] = b1[k − 1] + x[k] − e1[k]

B1[k] = b1[k − 1] − e1[k]

r1[k] = e1[k] + e1[k − 1] + e1[k − 2]

e2[k] = a2b2[k − 1] + c2

b2[k] = b2[k − 1] + e1[k − 3] − e2[k]

B2[k] = b2[k − 1] − e2[k]

r2[k] = e2[k] + e2[k − 1] + e2[k − 2]

These dynamic equations lead to the following first- and second-order statistics for the secondrestriction and downstream region (recall the statistics of the first restriction region have alreadybeen found in Section 5.2):

Et2 = 3λ (14)

EB2=

λ − c2

a2− λ (15)

VB2=

{(1 − a2)

2λ( a1

a1−a2)2( 11−(1−a1)2

+ 11−(1−a2)2

− 21−(1−a1)(1−a2)), (a1 6= a2)

λa12(1 − a1)2 1+(1−a1)2

(1−(1−a1)2)3, (a1 = a2)

(16)

Vt2 =

3a22λ(

a1

a1 − a2)2(

1

1 − (1 − a1)2+

1

1 − (1 − a2)2−

2

1 − (1 − a1)(1 − a2))

+2a22λ(

a1

a1 − a2)2(

2(1 − a1) + (1 − a1)2

1 − (1 − a1)2+

2(1 − a2) + (1 − a2)2

1 − (1 − a2)2

−2(1 − a1) + 2(1 − a2) + (1 − a1)

2 + (1 − a2)2

1 − (1 − a1)(1 − a2)), (a1 6= a2)

3a14λ1 + (1 − a1)2

(1 − (1 − a1)2)3+ 4a14λ(1 − a1)

2

(1 − (1 − a1)2)3+

2a4λ(1 − a1)23 − (1 − a1)2

(1 − (1 − a1)2)3, (a1 = a2)

(17)

This performance analysis allows us to evaluate or design MINIT restrictions. This analysis is notachievable simply with queueing models or the saturation model.

Noting the ability of the linear model to evaluate the statistics of region count, we further discussthe result stated in Section 4.3 that a single restriction placed before the first region in a chain

19

achieves better performance than two restrictions placed in front of both regions. This is equivalentto showing that any performance achievable by a combination of two restrictions can be achievedby a single restriction placed at the front. Using the linear abstraction, we can revisit the result,from the perspective of downstream region counts. In order to compare the cases, we enumeratethe combinations of Nc1 and Nc2 and calculate the statistics of two downstream region countsand the upstream backlog using the linear model. This analysis shows that, for any combinationwith Nc1 ≥ Nc2 , the total backlog at the two regions is close to that with the combination ofa first restriction with Nc2 and second restriction with the threshold larger than Nc2 (or almostequivalently ∞), while the variance of number of aircraft in region 1 is much larger, because ofthe backlog caused by restriction 2. The analysis confirms that placing a single restriction at thefront achieves better downstream variances of number of aircraft. This result is significant sinceplacing a single restriction achieves the design using the smallest numbers of resources for the bestperformance.

The analysis shows that linear abstraction of boundary restriction captures the intrinsic roleof a boundary restriction on flows. At the same time, it permits us to find explicit expressionsof flow statistics—including regional aircraft counts—with respect to a restriction’s parametersand the statistics of incoming flows (whether Poisson or not). The linear model thus providesus a very valuable approach for network-level analysis and design. Another advantage of thelinear abstraction is that it permits analysis of dynamics, hence we can evaluate coordinated TFMstrategies under time-varying uncertainties including weather events with uncertain duration/scope(e.g., fog at SFO). Restriction design that explicitly takes into account time-varying weather eventswill be considered in future work.

6 Algebraic Buffer Model and Network-wide Management Prob-

lem: Brief Introduction

The final model that we develop is a highly-abstracted algebraic description of a flow-managementrestriction. This simplistic model is motivated by the need for flow-management design (not onlyanalysis, which can be achieved using the linear model) for an arbitrary network. Such designproblems are in their essence decentralized control problems (see e.g., [24] for background) whenpractical restrictions are considered, since each restriction only acts on local flows and yet mustbe coordinated to achieve network-wide control. Decentralized control problems remain extremelychallenging in a broad sense (see e.g. [25]); thus, we take the perspective here that such design isbest achieved using the simplest plausible model (with more accurate models subsequently beingused to evaluate and refine the design). With this goal in mind, we develop a simple algebraicmodel for the (steady-state) behavior of a restriction. We then pose a network-wide flow controldesign problem using this model. In the interest of space and because our focus here is on modeling,we do not give a solution of the design problem here but do give some preliminary discussions inthis direction.

6.1 The Algebraic Model for a Single Restriction

At the most abstract level, a restriction simply serves to decrease downstream variance at the costof upstream backlog. Specifically, as a restriction is made tighter, the variability of the downstreamflow decreases but the upstream backlog increases. Very crudely, we can assume a simple linear

20

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Restriction 3

Restriction 4

Restriction 1(Takeoffs)

Restriction 2

Restriction 5

Figure 13: Network-wide flow-management model

dependence of the downstream variability and upstream backlog in steady-state on the strength ofthe restriction; such a linear dependence is roughly borne out by comparison of upstream backlogand downstream variance in the saturation abstraction. Specifically, we use a single parametera ∈ [0, 1] to describe the strength of the restriction. Assuming an input process with rate λ andvariability v, we model the resulting steady-state backlog as B = γaλ, where γ is a scale parameter5.Also, we model the variability of the downstream flow as w = (1−a)v, i.e. the variability decreaseslinearly with the strength of the restriction a while the backlog increases linearly with a.

6.2 Network Model

We use the highly abstracted algebraic model for a restriction to pose a network-wide flow-management design problem. Our aim here is formulate the flow management problem in sucha way that graphical insights into the structure of good flow management designs can be developed(for instance, to answer the question of whether it is better to restrict long traffic flows or crosstraffic). Also, as throughout this work, we focus on designing flows (rather than guiding individualaircraft) using the network model, with the motivation that practical management strategies can bedesigned in this way. We propose that designs and insights obtained through this highly abstractedmodel will be verified using much more detailed simulations.

In this network model, we are concerned with the characteristics of aircraft traffic just beforeand after ”boundaries” in the airspace (where boundaries include waypoints between Sectors enroute, as well as control points for arrival and departure traffic, see Figure 13). We assume that thenetwork has n boundaries in total, labeled 1, . . . , n. Each boundary i is assumed to have a meanflow rate λi through it, which is not affected by any management elements. We use the algebraicmodel for a restriction from Section 6.1 to describe flow management at each boundary. That is,we represent the variabilities of flows before and after each boundary i (as vi and wi, respectively),as well as the backlog (Bi) caused by the boundary restriction. The variabilities are related asdescribed above, by wi = (1 − ai)vi, and the backlog is given by Bi = γaiλi. Here, the strength ofthe restriction ai is a design parameter in some cases (e.g., for a MINIT restriction), and pre-set inother cases (e.g., for a restriction representing the arrival capacity or rate in a terminal area). Letus define A as the set of parameters ai that are designable.

Thus, we have modeled the boundary restrictions. We now have left to describe how the flows

5We have abstracted away the temporal dynamics of the restriction, as well as the details of its stochastics, so wesimply view the restriction as causing a fixed backlog rather than a mean backlog at a particular time.

21

merge and split within regions (between boundaries) and how they enter the airspace. By doingso, we will model the variability vi of the aircraft flow approaching each boundary. Let us considerthe following two cases:

1) For flows entering the airspace from the ground (take-off flows), the variability vi is well-modeled as the variance of the number of arrivals in a Poisson process of rate λi during a unitinterval, i.e. the variance of a Poisson random variable with mean λi.

2) Flows approaching other boundaries are formed by splitting/merging of flows in a regionin the airspace (including flows entering the airspace from the ground, and flows that have passedthrough other restrictions). Since these various flows that combine to form the approaching flow aretypically independent processes, it is sensible that the approaching flow’s variability is a combinationof the variabilities of these flows. In other words, we contend that the variability vi can abstractlybe written in the form vi =

∑nj=1 wjgji, where the constants gji can be obtained by determining

how flows merge/split within the region. We notice that the gji specify a connection network forrestrictions: for a particular pair (i, j), gji > 0 implies that there is a flow from restriction j torestriction i.

We note that our abstract model is similar in structure to the Eulerian Traffic Flow Modeldeveloped in [26], which also captures flows across boundaries which merge and split within regions.Our model differs in that we represent variabilities in flows rather than only flow densities, with themotivation that many flow management actions impact these variabilities. Our work also buildson [26] in that we explicitly model flow restriction strengths as design parameters.

In summary, our abstract network flow model captures variabilities before and after n ”bound-aries” (called vi and wi), as well as the backlogs Bi caused by these boundary restrictions. Specif-ically, for boundaries i corresponding to flows entering the airspace, we have

wi = vi = var(Pois(λi)), (18)

Bi = 0.

For all other boundaries, we have

wi = (1 − ai)vi, (19)

vi =n∑

j=1

wjgji

Bi = γaiλi,

where γ, λi, and gji are constants, ai ∈ A can be designed while other ai are fixed, and thevariabilities vi and backlog Bi are of interest to a designer.

(18) and (19) above together constitute a set of 2n linear equations that can be solved to findthe 2n variabilities vi and wi, and in turn the backlogs Bi. The design problem of interest is toset the parameters ai ∈ A to get desirable variability in flows or regional counts, while maintainingsmall backlogs (so that total counts in upstream regions do not exceed thresholds, and aircraft arenot subject to long delays). With just a little effort, one can show that the variabilities decreasemononotonically as the restrictions are strengthened (ai are increased), while the backlogs increasemonotonically with ai. Thus, the design goal is to appropriately trade off variability with backlog,by setting ai ∈ A.

A natural aim is to choose ai to optimize a performance measure that is based on the variabilitiesand backlogs. For instance, a cost measure of interest might aim to capture the total impact of the

22

control on the aircraft counts in a couple critical regions, by combining the variabilities of flows inthe region with backlogs caused by restrictions on flows out of the region. Many such measures arewell-approximated as being quadratic in the backlogs Bi and variabilities Wi.

In the interest of space, we only summarize the approach for designing ai using this model.With a little effort, the design problem posed above can rewritten as the following linear algebraproblem: consider the system of n equations (I − ZG)w = Zλ; design a diagonal matrix Z so asto minimize a cost measure that is quadratic in the entries of (I − Z) and w, where each diagonalentry of Z is constrained to be in [0, 1], the graph matrix G can be simply computed from therouting parameters gji, and the input vector λ depends on the boundary flow rates λi.

This optimization problem can be straightforwardly be solved numerically, using e.g. a gradientdescent method (see [27]). However, noting that the model is highly abstracted, we contend thatwe require insight into the graph-theoretic properties of the optimal design (e.g., whether longor short streams should be restricted, etc). With this goal in mind, we notice that our designproblem is deeply connected with recent efforts to develop optimal controllers for decentralizedsystems that take advantage of the network topology (e.g., [25, 28]) and more generally with thefield of decentralized controller analysis and design (see for instance the seminal work of Wang andDavison [24]). Such graph-theoretic design problems remain challenging (and do not match withthe primary modeling aim of this paper), and so we relegate the design to future work.

7 Conclusion

Air traffic flow management in the Sectors of the the NAS is complicatedly interrelated. Webelieve that a good flow management strategy must be designed at a network level—by taking intoconsideration traffic in multiple Centers in the presence of uncertainty.

In order to obtain optimal network-level flow management strategies, we emphasize the fullunderstanding of restrictions’ impact on generic traffic flows, with the aim of developing restrictionabstractions for analyzable network evaluation and optimization. In this paper, we examine fourabstractions, namely, the detailed queueing model, the discrete-time saturation model, the dynamiclinear model, and the algebraic linear model. The queueing model best approximates the detailsof restrictions, and permits some analysis of the flow statistics with a Poisson inflow. We suggestthe saturation restriction model as an approximation of the queueing model with the ability ofidentifying both the upstream and downstream flow statistics of a single restriction under a Poissoninflow. Note that it can be used to study restrictions in simple topologies. The saturation model alsolacks the scalability for a full network analysis. Furthermore, we introduce a stochastic linearizationof the saturation restriction model. This linearization has the advantage of 1) finding explicitexpressions for the flow and region-count statistics; 2) only requiring the statistics of inflow ratherthan a Poisson inflow; and 3) permitting a network-level analysis and design. In the end, we developa highly-abstracted algebraic linear model, and pose the problem of network-level optimization ofrestrictions using this model.

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