ENRI Int. Workshop on ATM/CNS. Tokyo, Japan. (EIWAC 2010)[EN-012] Validation of En Route Capacity Model with Peak Counts
from the US National Airspace System(EIWAC 2010)+

J. Welch*, J. Andrews*, J. Post***MIT Lincoln Laboratory, Lexington, MA, USAwelch @LL.MIT.edu, jandrews@LL.MIT.edu**Federal Aviation Administration, Washington DC, USA
joseph.post@faa.gov
Abstract: Airspace capacity estimates are important for managing air traffic and predicting the effectiveness of new
airspace designs and proposed decision support tools. Because air traffic management relies on manual procedures,controller workload determines the traffic limit of most sectors. Current operational procedures for estimating capacity in
United States airspace do not account for conflict avoidance and recurring workload. This paper examines a more complete
analytical model. Each sector has a capacity determined by the workload intensity of inter-sector coordination, aircraft
separation assurance, and repetitive activities. As the total workload intensity approaches unity, the sector reaches capacity.
The resulting workload equation is quadratic in traffic count. Its solution provides a simple formula for capacity. In thispreliminary study we for the first time determined its parameters by regressing for all sectors in the 20 United States
continental centers. We regressed for individual centers and for the overall set of centers. The capacity parameters from theoverall regression estimate the inherent capacity potential of a sector. The capacity parameters from a center regression
estimate the achieved operational capacity of the sectors in that center. The inherent capacity differs significantly from theachieved capacity for most centers.
Keywords: airspace, capacity, workload, sector.
queuing workload intensity of background activities,
inter-sector coordination, aircraft separation assurance,and repetitive activities such as traffic scanning. The
workload intensity associated with each task is the
product of the task rate and the mean time required toservice the task. The four task types have different
occurrence rates and service times. As the sum of thefour workload intensity products approaches unity, thesector reaches capacity. The resulting workload equationis quadratic in traffic count, and its solution provides asimple formula for sector capacity.

1. INTRODUCTION
Airspace capacity estimates are important for managing
air traffic and optimizing air traffic procedures. Air
traffic managers need accurate estimates of the capacityof current and reconfigured sectors to minimize delay
from storms and demand peaks. Capacity estimates arealso essential for predicting the effectiveness of new
airspace designs [1-3] and the benefits of proposeddecision support tools.
Air traffic management relies largely on manual control
procedures. Therefore, controller workload determines
the operational traffic limit of most sectors [4-9]. The
current operational procedure [10] for estimating
capacity in the United States National Airspace System(NAS) is useful, but has limitations. It does not
explicitly account for workload from conflict avoidance
activities and recurring tasks, and it includes a fixed limitthat can underestimate the capacity of large sectors.

The formula for workload intensity includes a number ofvariables that can be estimated from known airspaceparameters. However, three of the four task components
have at least one unknown parameter. In this study wedetermine the unknown parameters by fitting the model
predictions to observed peak daily counts for 790 enroute sectors in the 20 United States continental centers.
This is our first attempt to fit the model to data for theentire NAS.

We have examined a more complete analytical model foren route sector capacity [11-13] that can be adapted totransit time and sector volume changes caused byweather or sector redesign. Each en route air traffic
sector has an inherent capacity determined by the

We regress for individual centers and for the overall setof centers. The capacity parameters from the overall
regression provide an estimate of the inherent capacity
potential of a given sector. The capacity parameters from

This work is sponsored by the Federal Aviation Administration under Air Force Contract #FA8721-05-C-0002. Opinions,interpretations, recommendations and conclusions are those of the author and are not necessarily endorsed by the United States
Government.

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J. Welch, J.Andrews, J. Posta center regression are more indicative of the achieved
operational capacity of the sectors in that center. We
examine how the achieved capacity varies from center tocenter to determine whether a single set of capacity
parameters will suffice to fit the capacity model tocurrent NAS operational data.

and the sector conflict rate is approximately
When all aircraft in a sector cruise at constant altitude,
the vertical miss distance that determines loss of
separation is constant. In sectors in which aircraftroutinely change altitude we increase the vertical missdistance to reflect the resulting increase in altitudeuncertainty. We increase the vertical miss distance fromits nominal value of 1000 ft to a maximum value M vmax
that is proportionally to Fca, the daily fraction of sector
aircraft that change altitude by more than 2000 ft.

Background tasks, such as routine supervisoryinteractions or equipment checks, produce a constantworkload intensity Gb without respect to the aircraft
count in the sector. We assume that the backgroundworkload is identical for all centers. This implies thatobserved center-to-center capacity differences are causedby the other workload components, and it allows us toarbitrarily assign Gb a value of 0.1.

When all aircraft in the sector change altitude, Fca = 1
and the vertical miss distance is Mvmax. When Fca < 1,
the vertical miss distance is
Mv = 1000 + Fca (Mvmax – 1000).

Transit tasks associated with inter-sector coordination
occur at a rate
(1)

The complete equation for workload intensity G is
G = Gb + tt + rr + cc,(2)Here P is the mean task recurrence period per aircraft.
Conflict resolution tasks occur when potential aircraft
separation violations arise. We calculate the conflict rate
for an aircraft by considering the rate at which otheraircraft trajectories penetrate its protected airspace [14].
We define κ as the volumetric traffic density along theaircraft’s flight path, and Mh and Mv as the horizontal andvertical miss distances that define a separation violation.
The aircraft sweeps out a volume 4Mh Mv V21 of protected
airspace per unit time, where V 21 is the mean of the pairwise closing speeds between the subject aircraft and allother closing aircraft that could pass within the defined
miss distances. If the volumetric traffic density κ isuniform, the conflict rate is

3. REGRESSION
3.1 Regression Unknowns
The capacity equation includes four unknown factors.
The first is t, the transit service time. The second
unknown factor involves recurrence parameters. Because
we do not know the magnitude of the recurrence period
P, we perform the regression for recurring workload byfitting the dimensionless product r/P, which is thefraction of total time devoted to recurring tasks for eachaircraft. The third unknown is the product of conflict
service time c and relative mean relative closing velocity
V21. We call this product d. It has dimensions of
distance and represents the mean separation lost while
resolving each conflict. The NAS regression value of
this “conflict distance” is about two nautical miles. The
fourth unknown is the maximum vertical miss distance
Mvmax.

(3)

The sector controller is responsible for all aircraft in the
sector. If there are N aircraft in the sector eachexperiencing conflicts at an average rate of a, then the
total conflict rate for all aircraft is Na. At practicaltraffic densities, very few of these conflicts involve morethan two aircraft. Thus, the rate of conflicts [15] seen by
the sector controller is approximatelyc = Na/2.

To speed the regression process we use scale factors toquantize each of these variables as integers. The
resolution of transit service time t is one second. The
resolution of the dimensionless recurring workload factor
r/P is 0.1. The resolution of the conflict distance d is 0.1
nautical mile. The resolution of Mvmax is 200 ft.

(4)

If Q is the sector volume, the density of aircraft in the
sector isκ = N/Q,(7)

where t randc are the mean service times for transit,
recurring, and conflict tasks. When total workload
intensity G exceeds 0.8, Schmidt [4] determined thatsafety begins to degrade. That intensity limit defines thesector capacity.

Recurring tasks, such as checking route compliance ofsector traffic, occur at a rate

λa = 4κ Mh Mv V21.

(7)

Regression over all 20 centers gives an Mvmax ofapproximately 1600 ft.

where N is the number of aircraft in the sector, and T is
the average transit time through the sector.

λr = N/P.

(6)

Although we do not know the magnitude of the mean
relative velocity, we can estimate it by regression.

2. CAPACITY MODEL

λt = N/T,

c = (2 N2/Q) Mh Mv V21.

(5)

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3.2 Regression Data Sources
We determine the unknowns by fitting the model’s
capacity predictions to observed peak daily counts foreach of 790 sectors in the 20 U.S. continental Air RouteTraffic Control Centers (ARTCC).
We obtained
historical summaries of peak traffic counts and sectortransit times from the Federal Aviation Administration(FAA) Performance Data Analysis Reporting System(PDARS) [16] and the FAA Sector Design and Analysis
Tool (SDAT) [17].
We obtained sector airspace
definitions and aircraft altitude information from SDAT.

3.3 Regression Objective Function
The model provides a capacity bound. Regressing to fit abound requires an asymmetric objective function. Our
objective function is a simple rule designed to maximize
the score of the capacity fit to the observed traffic peaks.
It penalizes the fit if the actual traffic count significantly
exceeds the model prediction. It does not penalize low
traffic counts that might be caused by lack of demand orother factors unrelated to workload. The objectivefunction also reduces the regression influence of extreme
outliers, that is, occasional high peak counts that result
from brief traffic overloads made possible by temporary
increases in controller workforce or other unusual
factors.

Most of the peak counts we used were from days in Julyand August of 2007. We regressed system-wide andcenter-wide for all sectors in the 20 NAS continental
centers. The system-wide regression included one peakcount for each sector in the NAS. That count came fromthe day with the highest overall operations count in thisperiod. Each center-wide regression included ten peakcounts for each of the sectors in the center. Those counts
came from the ten days with the highest overall
operations counts in this period.

Fig. 1 illustrates the objective function. It is based on
integer peak daily counts and rounded model capacity
bounds. It assigns a score of zero to any sector whose
capacity bound exceeds its peak daily count by five or
more aircraft. The score grows to +5 for a sector whose
capacity bound equals its peak daily count or exceeds its
peak daily count by one aircraft.

We use the system-wide regression to determine thesensitivity of effective miss distance to the altitudechange fraction of each sector.

5

0
-10

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

-5

Regression Score

The regression data sources also provide information thatallows us to determine the mean sector transit times thatare key determinants of transit workload. Air traffic
managers currently use historical mean sector transit
times to derive a Monitor Alert Parameter (MAP) whichwarns against operational sector overload. Althoughdecision support algorithms could dynamically predict
transit times with accuracy from flight plans andsurveillance tracks in real-time simulations andoperational situations [1,2], MAP transit times are
essentially static. To derive a nominal value for sector
capacity, the MAP algorithm uses a fixed value of transit
time for each sector. The rule used to calculate the MAP
traffic count is constrained by FAA Order 7210.3 [10] toapply uniformly to all sectors in the NAS. It includes anominal upper limit of 18 aircraft per sector. However,to account for local conditions, air traffic managers areauthorized to manually adjust nominal MAP counts byplus or minus three aircraft.

-9

-10

-15

-20

-25

Difference Between Sector Peak Count and Model Bound (N p -N b )Figure 1. Regression Objective function.

The rule penalizes sectors with peak counts that exceed
the capacity bound. The rule assigns a cost of -5 to asector with a count one aircraft above the bound. The
rule assigns a cost of -15 to a sector whose peak aircraft
count exceeds the bound by two aircraft. The ruleassigns a cost of -25 to a sector whose peak aircraft countexceeds the bound by more than two aircraft. The rule
limits the cost to -25 to reduce the regression influence of
extreme outliers.

When we regress our capacity algorithm against
historical data, we estimate transit workload for eachsector from the mean aircraft transit time of all aircraft
present in the sector at the time of the observed traffic
peak. PDARS and SDAT derive sector boundary
crossing times from inter-sector coordination hand-off
logs. Those boundary crossing times have one-second
resolution, which provides dynamic sector count data andallows us to compute the mean transit time for the sector
aircraft at the time of each peak traffic event.

3.4 Regression Algorithm
The use of integer search variables and the relatively
narrow range of operationally feasible values for thosevariables allow us to use a simple exhaustive searchalgorithm. The data sets were sufficiently large toprovide unique objective function maxima.
4. RESULTS

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J. Welch, J.Andrews, J. Post
mainly from sector transit time variations, which are
related to the length of the sector in the direction of its
principal flight routes.

4.1 Comparison of Center Peak Counts
One of the objectives of this study is to determine theminimum number of variables needed to accurately
predict the capacity of every en route sector. (Recall that
the MAP algorithm uses only a single variable, thehistorical mean transit time for each sector.)

NAS
30
NAS Model Capacity
Observed Peak Count

25

Peak daily sector traffic counts for the 20 NAS centersshow significant center-to-center differences, both intheir peak counts and in the distributions of their sector
volumes. These large differences indicate that more than
one variable will be needed to accurately fit the capacity
model to the data.

Aircraft Count

20

15

10

Centers with busy airports have small sectors with hightraffic densities, and exhibit clearly-bounded peak traffic
contours. Centers with large airspace volumes overoceanic or sparsely populated regions tend to have large
sectors with low traffic densities, and indistinct traffic
contours. Fig. 2 illustrates this by contrasting the peakcount versus sector-volume profile of the Washington
DC Center (ZDC) with that of the Miami Center (ZMA),
which has a large volume of oceanic airspace. For
clarity, the chart only includes results for sectors lessthan 80,000 cubic nautical miles in volume.

5

0
0

10000

20000

30000

40000

50000

60000

70000

80000

3

Sector Volume (nm )Figure 3. Peak counts and model capacities for NAS sectors.

Fig. 4 comprises the same data as Fig. 3, but directly
plots the model capacities against the observed peaksector traffic counts.
NAS Model Capacity vs Observed Peak Count
30

Peak Daily Traffic Counts for ZDC and ZMA
30

25

ZDC
ZMA

25
NAS Model Capacity

20

Aircraft Count

20

15

15

10

10

5

5

0
0

5

10

15
Observed Peak Count

20

25

30

0
0

10000

20000

30000
40000
50000
Sector Volume (nm3)

60000

70000

80000

Figure 4. Peak count versus model capacity for NAS sectors.

Low traffic demand and other factors cause most sectors
to peak at densities below capacity. The “frontier” trend
for maximum count is consistent with the model.

Figure 2. Comparison of peak daily en route sector counts inWashington and Miami Centers.

Although some sectors in these two centers have similar
airspace parameters and thus should have similar
intrinsic capacities, many ZDC sectors routinely peaknear capacity, while many ZMA sectors never reach theirintrinsic capacity because of low demand. As a result,capacity parameters regressed from ZDC peak traffic
differ significantly from parameters based on ZMA
traffic.

4.3 Center Regression Results
Any study based on the current operational capacity ofthe sectors in a specific center is best based on regression
parameters resulting from analysis of that center.
We evaluated the model capacity for each center using
both NAS regression parameters and center regression
parameters. The NAS regression parameters provide anestimate of the inherent capacity of the sectors in thecenter. The local center regression parameters provide anestimate of the achieved operational capacity of thesectors in the center. For most centers, the inherent andachieved regression parameters differ significantly.

4.2 System-Wide Regression results
Fig. 3 shows the capacity estimates resulting from the
NAS system-wide regression. This chart also restricts
results to sectors smaller than 80,000 cubic nautical
miles. It plots the model estimate of capacity and theobserved peak daily traffic count versus sector volume
for each NAS sector. The scatter in the capacities for
sectors with nearly identical airspace volumes results

We define the overall capacity of a center as the sum of
the model capacities of its sectors. Overall inherent

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ENRI Int. Workshop on ATM/CNS. Tokyo, Japan. (EIWAC 2010)
capacity is based on the NAS regression, and overall
achieved capacity is based on the center regression.

Most ZSE sectors have low demand relative to similar
sectors in other centers.
The achieved regression
capacities of the larger ZSE sectors are typically five or
more aircraft below their inherent capacities.

The overall inherent capacity for every one of the 20
NAS centers is larger than its overall achieved capacity.
This is to be expected because the asymmetricalobjective function fits the model to the busiest sectors in
the NAS.

4.4 Normalized Capacity Density of Centers
The notion of overall achieved capacity by itself is not
sufficient to compare the capacity characteristics of air
traffic control centers.
A meaningful capacity
comparison must normalize for differences in center sizeand sector count. The normalized capacity density of acenter is essentially its mean achieved traffic density per
sector. Specifically, it is the sum of the local achieved
model capacities for all sectors in the center, divided by
its total airspace volume (in units of 10,000 cubic
nautical miles), and divided by the number of sectors in
the center.

Overall achieved capacity is closer to overall inherent
capacity for centers with smaller sectors because small
sectors tend to be more constrained by workload thanlack of demand. Furthermore, the MAP rule does not
limit most small sectors since it usually exceeds theirobserved peak counts.
Large sectors contribute less to the regression score.
Their inherent capacity tends to be an extrapolation ofsmall sector capacity. The operational limit of the MAP
rule and lack of demand combine to limit the peakobserved counts in most large sectors. Most large sectors
could handle peak traffic that exceeds current procedural
limits, and the NAS regression provides a truer indicationof their inherent capacity.

Fig. 6 sorts the 20 centers (and the NAS as a whole) by
this measure. Normalized capacity density is lowest in
centers where large sectors serve mainly oceanic orsparse population areas. It is highest where many small
sectors serve major airports. The normalized capacity
density of the NAS as a whole is 0.15. (The NAS metricuses the same formula as the center metric, multiplied by
20 to account for its center count.)

Small sectors are more prevalent in centers with busy
airports where additional controllers are needed to handle
dense traffic. Boston, Washington, and Atlanta are theonly centers whose achieved capacities approach theirinherent capacities.
Furthermore, the Boston andWashington local capacity model parameters closely
approximate those of the NAS capacity model. The
individual capacity parameters resulting from the Atlanta
regression differ from those of the NAS regression, butthey combine to keep Atlanta’s achieved capacity close
to its inherent capacity.

Total Capacity/Sector Count/(Airspace Volume/10,000 nmi3)0.45

Capacity/Sector/Volume

0.4

The Seattle Center (ZSE) is an extreme example of thetendency of the local regression to under-estimate thecapacity of large sectors. Fig. 5 compares the individualZSE sector capacity estimates resulting from the NAS
regression with those from the local center regression. It
also shows the observed peak count Np for each ZSE
sector day.

Figure 6. Centers sorted by normalized capacity density.

The range of this metric is indicative of the large
variation in current operational requirements among the
20 NAS centers. Low normalized capacity density is not
an indicator of inferior performance. Centers with low
normalized capacity density are, by definition,
characterized by low demand and large airspace volumes.
International boundaries and center boundaries often
constrain the designs of their large sectors. Additional
constraints include surveillance coverage limitations,special use airspace, and the MAP limit itself. Centerswith high normalized capacity density tend to include
many active routes serving major hubs or multiple busy
airports. High traffic density forces them to employ
small sectors, and small sectors allow them to organize
their operations and design their airspace with more
freedom from the large-sector constraints listed above.

Aircraft Count

20

15

10

5

0
40000
50000
Sector Volume (nm3)

60000

70000

0.1

Center

25

30000

0.15

ZDC ZID ZNY ZAU ZTL ZOB ZME ZFW ZKC ZOA ZJX ZLA ZAB NAS ZDV ZBW ZSE ZMP ZLC ZHU ZMA

NAS Regression Capacity
Center Regression Capacity
Peak Daily Count (Np)

20000

0.2

0

30

10000

0.3
0.25

0.05

ZSE

0

0.35

80000

Figure 5. Peak counts, NAS regression capacity, and local center
regression capacity for Seattle Center.

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J. Welch, J.Andrews, J. Post
The uneven distribution of traffic is evidenced by the factthat the highest normalized capacity density for a center(0.41 aircraft in 10,000 cubic nautical miles for ZDC) issignificantly less than the peak traffic density of many
busy NAS sectors. For example, the peak traffic density
handled by sector ZDC32 is 20 aircraft in 10,000 cubic
nautical miles, about fifty times the normalized capacity
density of ZDC as a whole.

and conclusions are those of the authors and are notnecessarily endorsed by the United States Government.
7. REFERENCES[1] Bloem and Kopardekar, “Combining Airspace

Sectors for the Efficient Use of Air Traffic Control
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5. CONCLUSIONS[2] Zelinski, “A Comparison of Algorithm Generated

Sectorizations,” 8th USA/Eurocontrol ATM R&D
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This paper continues an on-going examination of asimple analytical model for the capacity of en route air
traffic control sectors. Each sector has a capacity
determined by the workload intensity of inter-sector
coordination, aircraft separation assurance, and repetitive
activities.

[3] Tien and Hoffman, Optimizing Airspace Sectors forvarying Demand Patterns using Multi-Controller
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This is our first application of the model to the entire
NAS. We determined the unknown parameters of themodel by regressing its capacity predictions against FAA
instantaneous count reports for 790 sectors in the 20
United States continental centers. We based the center
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Most of the peak counts were from days in July andAugust of 2007. We regressed system-wide and centerwide. The system-wide regression included a peak countfrom the day with the highest overall operations count in
the period for each sector in the NAS. Each center-wide
regression included peak counts for each of the sectors in
the center from the ten days with the highest overall
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[6] Majumdar, Ochieng, and Polak, “Estimation of

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The results are informative. The peak count data reflect
the wide range of complexity, demand, and airspacecharacteristics of the NAS. The capacity parameters
derived from the NAS-wide regression provide anestimate of the inherent capacity potential of NAS
sectors. The capacity parameters derived from the center
regressions estimate the current operational achieved
capacity of each center. The achieved capacity varies
significantly from center to center, and the achieved
capacity for a typical center is significantly less than its
inherent capacity.

[8] Song, Wanke, and Greenbaum, “Predicting Sector

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[10] FAA Order 7210.3, Facility Operation and

These large center-to-center differences indicate that asingle set of fixed capacity parameters will not suffice to
fit the capacity model to current NAS operational data.
Individual adaptation to each center will be necessary.

Administration, Sec. 7, 30 Aug 2007.
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Workload Model for Estimating En Route Sector
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6. ACKNOWLEDGMENT
This work is sponsored by the Federal Aviation
Administration under Air Force Contract # FA8721-05C-0002. Opinions, interpretations, recommendations,[12] Welch, Andrews, Martin, Shank, “Applications of a

Macroscopic Model for En Route Sector Capacity,”
AIAA GNC Conference, Honolulu, August 2008.

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Copyright Statement[15] Andrews, Welch, and Erzberger, “Safety Analysis

The authors confirm that they, and/or their company or
institution, hold copyright of all original material
included in their paper. They also confirm they haveobtained permission, from the copyright holder of any
third party material included in their paper, to publish it
as part of their paper. The authors grant full permission
for the publication and distribution of their paper as partof the EIWAC2010 proceedings or as individual off-prints
from the proceedings.

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