ENRI Int. Workshop on ATM/CNS. Tokyo, Japan. (EIWAC 2010)!"#$%&%'()(*+,+,-./(012,3(4-56(786,2+3,2()99-:;3<(+.2,9(=>(?9;@,8519A$B;<,2(CD,9;5-1.<(!"#$%&'()*)+'+

T. Nikoleris*, M. Hansen**Civil and Environmental Engineering Department
University of California
Berkeley, USA[nikoleris | mhansen]@berkeley.edu
Abstract This paper develops a queueing model for aircraft arrivals at a single server under trajectory-based flight
operations, which are expected to prevail in the Next Generation Air Transportation System. Aircraft are assigned scheduled
times of arrival at a server, which they meet with some normally distributed stochastic error. A recursive queueing model isformulated, and the Clark approximation method is employed. Exact results are derived for a special case with evenly
spaced scheduled times of arrival, and the impact of buffers in the arrival stream is explored.
Keywords aircraft, 4D trajectories, queueing model
However, even with the deployment of the very best 4D
trajectory precision and navigation tools, adherence to 4D
trajectories — in particular to scheduled arrival times at
the fixes — will not be perfect. Sources of imprecision
include airframe-to-airframe variation in aerodynamic
performance, limitations in wind prediction capability,
variations in flight crew technique, and varying degrees ofexactitude in navigational performance [1]. As the NAS
evolves from its current state to a future condition where
location precision is maximized, trajectory precision willvary in response to these factors. It will range from low
precision, corresponding to today's operations in the NAS,to high precision, enabled by full deployment andutilization of precision navigation and 4DT trajectoryawareness tools. Operations at the two ends of thisspectrum can be modeled using stochastic anddeterministic queuing models; see [2]. While the models
for such cases are well established, it is far morechallenging to consider intermediate levels of stochasticity.
Such cases must, however, be considered when modeling
NextGen since trajectory adherence will be imperfect, but
sufficient to invalidate the assumptions of existing
stochastic models. Thus, the objective of this paper is tomodel aircraft operations in NextGen, in a way that
accounts explicitly for varying levels of trajectory
uncertainty.

1. INTRODUCTION
The nation's air transportation system (NAS) will incur
major transformations in the coming years, developing
towards the so-called Next Generation Air Transportation
System (NextGen). NextGen features a shift from thecurrent static system of routes and sectors to one that is
adaptive to weather, traffic, and user preferences. NextGen
planners envision a system in which users will exchange
trajectory information and supply the Air Navigation
Service Provider with greater amounts of information
about future traffic demand. This will be used to anticipate
and resolve conflicts well in advance, reducing the needfor tactical air traffic control. It will also allow controlled
times of arrival into busy terminals, weather-impacted
airspace, and other bottlenecks. This transformation is
expected to greatly reduce human operator workload andsignificantly increase airport and airspace capacity.
The motivation for this research is the fact that the ability
to control and predict 4D aircraft trajectories (4DT) withhigh precision is a cornerstone of NextGen. 4DT
capability, with time being the fourth dimension, isdefined as the ability to precisely fly an assigned 3D
trajectory while meeting specified times of arrival atcertain waypoints [1]. This will allow high density flowsthat rely on controlled times of arrival for criticalresources, including entry and exit to/from airspace
regions, taxiways, and runways [1]. Thus, in this research
we assume that an aircraft’s flight path includes a series of
waypoints (that can be either points in the airspace or the
runway’s threshold) that the aircraft has to cross at ascheduled time. In other words, we assume that under 4DT
operations aircraft will be metered at fixes.

Existing stochastic queueing models typically assume thatthe aircraft arrival process at an airport’s terminal airspacearea is a non-homogeneous Poisson process [3]. However,for trajectory-based operations in NextGen, the Poissonarrivals assumption does not capture the concept ofmetered aircraft operations. Thus, in this paper, we
propose a queueing model that can analyze flight delays in

1133T. Nikoleris, M. Hansen
a high-precision trajectory-based operational environment,
as is envisioned under NextGen.

stochastic deviation from its scheduled arrival time. The
sources of imprecision might include airframe-to-airframe
variation in aerodynamic performance, limitations in windprediction capability, variations in flight crew technique,and varying degrees of exactitude in navigational
performance [1]. In addition, consecutive aircraft must
maintain a minimum headway h, which can vary overpairs of arriving aircraft, for safety reasons. Since air
traffic controllers, with guidance from separation rules,
decide values for h, we consider it as a deterministic
variable in our model. Moreover, we assume that h is thebinding constraint among all factors that may affect theminimum required separation between consecutive
aircraft.

Within transportation engineering context, queueing
models with scheduled arrivals have been proposed to
study port operations. Sabria and Daganzo [4] examine
single server queueing systems where customers must be
served in an order that is specified by a timetable, i.e. in a
First-Scheduled-First-Served (FSFS) order. Each customer
has a scheduled time of arrival at the server, where they
actually arrive with some stochastic lateness (positive or
negative). Exact transient solutions are obtained for thecase when the lateness distribution is Gumbel, and service
times are deterministic. In the present paper though,stochastic deviations from scheduled times of arrival are
assumed to follow a Normal distribution, while the rule of
FSFS service is maintained.

Following Sabria and Daganzo’s approach, each airplane i
has an arrival time at the server Ai that consists of adeterministic and a stochastic portion. The deterministic
component ai is the scheduled arrival time at the fix, while
the stochastic component is denoted as A! i and represents
the lateness (positive or negative) with which the aircraft
arrives at the fix, due to imprecision in trajectory
adherence. Therefore, we have Ai = ai + A! i .

In addition to developing a queueing model, we examine
the effect of buffer time between scheduled arrivals on
system delay. Although any delays due to unpunctual
aircraft arrivals can be absorbed in the inserted buffer, thesystem incurs losses in throughput performance. Therefore
two types of inefficiencies can be defined: those due toreduced throughput and those to unpunctual arrivals.
While the former can be planned ahead, the latter are
unpredictable. Thus, a different cost coefficient can be
assigned to each type of inefficiency and we seek to findthe optimal amount of buffer time that minimizes the total
cost of delay.

We assume that deviations A! i 's are small enough that
serving aircraft on a First-Scheduled-First-Served (FSFS)
order will not significantly increase delays. As an order ofmagnitude, NextGen planners foresee accuracies of ±10
seconds in aircraft meeting scheduled times of arrival [5].
Under a FSFS queue discipline, the actual time airplane ideparts from the server, Di, would be Ai if there were noqueue at the server by the time it arrived, or the time theprevious scheduled aircraft i-1 crossed the fix plus aminimum required separation headway hi-1,i between the
two aircraft. The actual times that aircraft cross the fix
under study would then be:

The rest of the paper is organized as follows: Section 2
presents the general form of our model and its
approximate solution, and report on experiments
conducted to assess the accuracy of the model against
simulation. In Section 3 we consider a special case, inwhich aircraft are metered at a constant rate and theseparation requirements as well as 4DT precision levels
are same among all aircraft. Working with a queueing
model in this simplified form enables us to shed light in
the issue of finding an optimal metering rate that attains
high throughput while keeping delays due to imprecise
adherence to 4DT’s small. Finally, Section 4 summarizes
our main findings and discusses several directions for
future research.

D1 = A1

Di = max ( Ai , Di!1 + hi!1,i ) , "i # 2

If there were no stochasticity in the system, thedeterministic time of departure from the server would be:

di = max ( ai , di!1 + hi!1,i ) , "i # 2
Accounting for stochasticity, the departure time from theserver of airplane i is:

2. THE MODEL AND AN APPROXIMATE
SOLUTION

Di = di + D! i
The distribution of the stochastic component D! i clearly

2.1 Model Formulation
Our queueing system consists of a single server, which is afix (either a point in the airspace or a runway’s threshold),and of airplanes that must cross this fix. Aircraft areassigned scheduled times of arrival at the fix, and they fly
4D trajectories to arrive at the fix at their scheduled times.
However, due to imprecision in trajectory adherence,
aircraft’s actual time of arrival at the fix has somedepends on, which captures all stochastic effects thatcause flight i to arrive at a time other than its scheduled
one ai :D! 1 = A!1

2

134

(1a)

ENRI Int. Workshop on ATM/CNS. Tokyo, Japan. (EIWAC 2010)()

D! i = max ai + A! i , di!1 + D! i!1 + hi!1,i ! di , "i # 2! ! (" X2 + " Y2 # 2 $" X " Y )

1/2

(1b)

The second pivotal assumption is that the vector of
stochastic errors A! i follows a multivariate normal
distribution with zero means (without loss of generality),
standard deviations ! i , and a covariance structure !:%!

µ X # µY!
( x2 +
1
exp * # ) 2,2'

& ( x) !

! " Normal (0, !) . The normality assumption stems fromA
the observation that the probability distribution for A! i isy

. ( y ) ! 0 & ( x ) dx#/

generated by convolving the individual distributions of
low-correlated stochastic factors. It should be emphasizedthough, that A! i 's do not represent factors such as,
departure delays, traffic management initiatives, severe
weather, or en-route congestion that cause significant
amounts of delays; lateness effects due to such factors
have already been incorporated in the estimation ofscheduled arrival times ai.

The coefficient of linear correlation between Z
and a third normal random variable W can also be
estimated, given that we know the coefficients of linear
correlation between X and W ( ! X ,W ) , and between Y andIn practice, values for schedule deviation ! i could be
aggregated to represent classes of aircraft that have similarcapabilities of adherence to 4D trajectories. For example,
one could assume two different values for the standard
deviation, ! A and ! B , in order to roughly represent
aircraft with and without Area Navigation (RNAV) andRequired Navigation Performance (RNP) capabilities.
Alternatively, one could also differentiate aircraft’s ability
to precisely fly 4D trajectories according to the en-routeweather conditions they encounter.

The above formulas give the exact mean and variance of
Z. The approximation is introduced by assuming that Z
follows a normal distribution with mean µ Z and variance

2.2 Solution with the Clark Approximation Method
In equation (1), for i=2 both terms of the max ( i ) operator
are normally distributed. The max operation on normal
random variables, in contrast to the add operation, does
not yield a normal random variable. A well-known result
due to Clark [6] derives analytical formulas for the mean
and variance of the maximum of two normally distributed
random variables. Let X and Y be normally distributed
random variables, X ~ N( µ X , ! X ) and Y ~ N( µY , ! Y ) , !
represent the correlation coefficient between X and Y, andZ be the maximum of X and Y, Z ! max(X,Y ) . The mean

E(Di ) = ai !(" i ) + $% E(Di#1 ) + hi#1,i &' !(#" i ) + ( i) (" i ) (2)W ( !Y ,W ) :r [W , Z ] = (! X " X ,W # ($ ) + ! Y "Y ,W # ( %$ )) / ! Z! Z2 .
In the context of our problem with scheduled aircraft
arrivals, the Clark approximation method can be used for
all i ! 2 to approximate Di 's as normal random variables,and estimate their mean E(Di ) and variance Var(Di ) in arecursive manner:()

Var(Di ) = ! i2 + ai2 "(# i ) ++ %Var ( Di$1 ) + %& E ( Di$1 ) + hi$1,i '( ' "($# i ) + (3)&(2+ %& ai + E ( Di$1 ) + hi$1,i '( ) i* (# i ) $ %& E ( Di ) '(r !" Ai+1, Di #$ = [% i & '1( () i ) ++ Var ( Di*1 ) & '2 & ( ( *) i )] / Var ( Di )µ Z and variance ! Z of Z are then:
2

2

(4)

where

(

! i = " i2 + Var ( Di#1 ) # 2 $ % $ " i $ Var ( Di#1 )µ Z = µ X !(" ) + µY !(#" ) + $% (" )& Z2 = (& X2 + µ X2 ) !(" ) + (& Y2 + µY2 ) !(#" )+ ( µ X + µY ) $% (" ) # µ Z2

!i =

where

ai " E ( Di"1 ) " hi"1,i
#i

)

1/2

(5)

(6)

and at each step i! = r [ Ai , Di"1 ] , !1 = r [ Ai+1 , Ai ] , !2 = r [ Ai+1 , Di"1 ] .
Note that r [ Ai , Di!1 ] and r [ Ai+1 , Di!1 ] are obtained
through equation (4) in previous iterations. Effectively, themethod is implemented by estimating at each step k3135T. Nikoleris, M. Hansenr [ Ai , Dk ] for

all i > k .

Moreover, r [ Ai+1 , Ai ] isset to hi!1,i = 30, 60, or 90 seconds. Each aircraft is
assigned a scheduled time of arrival at the server
ai = ai!1 + hi!1,i + b , where b denotes a buffer time

considered as input from covariance matrix !. Equations(2)-(6) are easy to program and they are computationallyefficient. Finally, for a stream of N flights scheduled for toarrive at a fix, the total expected delay is defined as:

inserted. Aircraft arrive at the server with some
imprecision that follows a normal distribution and has astandard deviation !. Zero covariance was assumed acrossthe aircraft arrival times at the server Ai. A total of 90
scenarios were examined:#N&
E [WN ] ! % " E ( Di ) ! ai ($ i=1'This completes the formulation of our queueing model. In
summary, the model requires as inputs a schedule ofarrival times ai, a capacity profile expressed in terms oftime separation headways hi-1,i, and a covariance matrix oftrajectory adherence errors ! i, j . These, coupled with the•••assumption that ! i 's are small enough to allow a firstscheduled, first-served policy, enable the estimation of
expected flight delays through Clark's approximation
method.

Three metrics for the approximation method accuracy
were considered:

2.3 Approximation Error
Although the maximum Z of two normal random variables
X and Y is not normally distributed, our model is based on
approximating Z with a normal random variable. In
particular, in estimating Di = max Ai , Di!1 + hi!1,i it is(

10 different sequences of hi-1,i (each sequence has
an equal mix of 30, 60, and 90 seconds)
b = 0, 10, and 20 seconds (held constant withineach sequence)! = 10 seconds (uniform across all aircraft), 30
seconds (uniform across all aircraft), and an equalmix of both.

1.

Percentage Error
in total Delay % (PE):apprsim
E [WN ] ! E [WN ]"100sim
E [WN ]

2.

Absolute

Error

E [WN ]

! E [WN ])appr

assumed that Di-1 is normally distributed. That enables theestimation of the mean and variance of Di, which is thenalso approximated as a normal random variable. However,each pair-wise operation introduces some error that is
propagated and might affect the accuracy of our estimates.
For a thorough analysis on this topic, see Sinha et al. [7]and Horowitz et al. [8].

3.

in

Total

Delay

(AE):

sim

Flight Departure Time Mean Absolute Deviation
N

"D

appr
i

(MAD): i=1! DisimN

The first two metrics evaluate the accuracy of theapproximation method in estimating the expected total
aircraft delay against assigned scheduled times of arrival.
The third metric provides with a measure of the error in
predicted outcomes for individual flights.

To test the accuracy of the Clark Approximation Methodin the context of our analysis, several operational scenarios
were considered. The estimates from the analyticalqueueing model were then compared against the averageof 104 Monte Carlo simulation runs, which is considered
as ground truth.

The results are presented in Table 1. Each entry in thetable represents the average value across the ten scenariosof different hi!1,i sequences. In all cases, the Total Delay
PE metric indicates that the approximation method is
within -8% accuracy in estimating the total delay in thesystem, as compared to simulation. Moreover, the absolute

Each operational scenario was formulated as follows: a
total of 120 aircraft must cross a fix, and the minimum
required separation between any two successive aircraft is

Table 1 Results of approximation accuracy tests

Buffer = 0 (sec)

Buffer = 10 (sec)

Buffer = 20 (sec)PE

AE (sec)

MAD (sec)PE

AE (sec)

MAD (sec)PE

AE (sec)

MAD (sec)" = 10-0.62%

13.78

0.14

-3.26%

9.17

0.09-3.93%

2.97

0.08" = 30-0.49%

36.50

0.35

-1.69%

40.92

0.35

-2.41%

31.17

0.31

Mixed

-1.52%

97.26

0.89

-5.74%

79.53

0.65

-7.70%

54.07

0.44

4

136

ENRI Int. Workshop on ATM/CNS. Tokyo, Japan. (EIWAC 2010)
error in estimating total delay does never exceed 1.3
minutes. Also, the MAD metric indicates that theapproximation method estimates the expected delay ofeach aircraft with accuracy better than 1 second, on
average. The accuracy of the method slightly decreases
when the fleet contains aircraft with different navigation
capabilities. This must be due to heterogeneity in thevariance of the normal distributions for Ai that enters in themax operator in each step of the recursion.

Since a ! h , the deterministic component of departure
times from the server would be di = ia , and thus Eq. (1)becomes:

In summary, these experimental results indicate that our
proposed model accurately predicts operational
consequences of metered operations with good but
imperfect 4DT adherence, as might be expected in
NextGen.

We are interested in deriving an analytical expression for
E D! i , but this is a rather difficult task. Instead, we

3. SIMPLIFIED MODEL

Proposition 1. For a given level of the relative buffer! ! ( a " h ) / # , the sequence of random variables D! i is
proportional to the sequence of random variables Z! by

D! 1 = A!1

(

•

(7b)

where A! i ~ N ( 0, ! ) , "i # 1 .

( )(define as Z! i " max A! i* , Z! i!1 ! ") where A! # N ( 0,1) , and*iresort to dimensional analysis by employing the following
proposition:i! standard deviations:
D! i = ! " Z! i

(8)

Proof. Assume that for some i # 2, D! i!1 = " # Z! i!1 . Then
we have:(

D! i = max A! i , D! i!1 ! ( a ! h )

(

)

)

= max " # A! i* , " # Z! i!1 ! " # $ = " # Z! i
But note that for i = 2:()
= max (" # A! , " # A! ! " # $ ) = " # Z!

D! 2 = max A! 2 , D! 1 ! ( a ! h )
*
2

*
1

Hence, for i ! 3 the result follows by induction.

( )

2

!

Therefore, the problem of estimating E D! n is reduced to( )estimating E Z! n . To estimate the mean of Z! n we first
derive its cumulative distribution function:

We focus on a single fix, and make the following
assumptions:•)

D! i = max A! i , D! i!1 + h ! a , "i # 2

3.1 Formulation
We consider the case where average demand for service
exceeds capacity over a considerable period of time, andaircraft are metered at the server (e.g. a point in theairspace, or a runway’s threshold) at a constant meteredheadway of a time units. To maximize throughput andminimize expected delay, a should be set to the minimum
headway, h. It may, however, better to set a to be slightly
greater than h, for at least two reasons. First, while
increasing a increases delay, it also make the delay morepredictable. Airlines value such predictability because it
enables them to better plan terminal operations. Second,the delays that result from setting a above h can beabsorbed in a more efficient manner, by increasing time athigher altitudes and reducing speed slightly over a longer
distance rather than dramatically as the aircraft approaches
the metering fix. For these reasons, aircraft operators may
be willing to accept an increase in overall delay inexchange for a reduction in the amount of stochastic delaythat results from imperfect trajectory adherence. In thissection we employ our model to quantify this trade-off.

•

(7a)

FZ! n ( x ) = ! ( x ) " ! ( x + # ) "…" ! ( x + ( n $ 1) # )The required minimum time headway h between
successive aircraft is constant and deterministic.
Flight arrivals are uniformly scheduled during theanalysis period, with metering headway a.

The probability distribution function of Z! n is therefore:
n

n

($ )

fZ! n ( x ) = ' & ! ( x + ( k " 1) # ) k ,Stochastic errors
are i.i.d. random variables
with zero mean and standard deviation .

i=1 k=1$k = 1for i = k

$ k = 0 for i % kwhere the term (! k ) indicates a derivative. The mean of

In a fully deterministic environment, there will be nodelays if the separation headway a is greater than theminimum headway h. However, if an airplane arrives later
than its scheduled arrival time, then it might cause delay to
airplanes upstream of it. We are interested in quantifying
the expected amount of delay that each airplane in thestream incurs.

Z! n is therefore:

5137

T. Nikoleris, M. Hansen

3!=0

E x p e c t e d Va l u e o f Z i

2.5

2

1.5
! = 0.1
1! = 0.2! = 0.4! = 0.6! = 0.8

0.5! = 1.6
0
0

10

20

30

40

50

60

70

80

90

100

Index of Flight Arrival i

Figure 1 Expected delay E

+,

n

',

i=1 k=1( Z! ) for several values of "i

Therefore, the expected total delay of the system is
proportional to the standard deviation ! of the stochasticn() )E !" Z! n #$ = - x % + * & ( x + ( k ' 1) ( ) k dx,)k = 1error A! i . We can thus plot the expected total delay

(9)

where

E [WN ] against the number of flights N in the arrival

for i = kstream, for the case when ! = 1 (see Fig. 2). Observe thatfor lower levels of the relative buffer ! , the system incurs
higher stochastic delay due to its reduced capacity for
absorbing stochastic deviations from schedule.

) k = 0 for i . k
Therefore, it suffices to consider cases for ! and compute
E Z! i through numerical integration; Fig. 1 depicts

( )

( )
a and h, values for E ( D! ) can be computed by selecting
the appropriate curve from Fig. 1 and multiplying E ( Z! )several such curves for E Z! i . Thus, for any value of ! ,3.2 Trade-offs between deterministic and stochastic
delay
After developing methods for estimating each flight’s
expected stochastic delay, we proceed to analyzing thetrade-off between stochastic delay and throughput. In thecontext of our analysis, the maximum departure rate fromthe fix that can be attained is constrained by aircraft
minimum separation requirements and is equal to 1/h.
Thus, in order to achieve maximum throughput from theserver, one would attempt to meter aircraft at the fix at aheadway a = h. However, that will result in maximum
stochastic delay, as can be inferred from Fig. 1 for ! = 0 .
By metering at a headway a > h we decrease the expectedamount of stochastic delay, since ! increases with a, but
at the same time we incur increases in deterministic
queueing delay. A graphical illustration through aqueueing diagram is presented in Fig. 3.

iiby ! . In this way, instead of an analytical formula, values
for E D! can be estimated through a lookup table and( )iwith the same level of accuracy. Note that the system
eventually reaches a steady state (except for the case
! = 0 ), after which each flight incurs the same amount of
expected delay E Z! i ! .

( )Finally, for a stream of N flights scheduled for landing anda given value a–h the total expected delay is equal to:

# n
& # n
&
E [WN ] = % " E ( Di ! ia ) ( = % " E Z! i ( )$ i=1' $ i=1
'

( )

(10)

6

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ENRI Int. Workshop on ATM/CNS. Tokyo, Japan. (EIWAC 2010)E x p e c t e d To t a l D e l a y W n f o r ! = 1

250

!=0
200

150
! = 0.1
100! = 0.2! = 0.450! = 0.6! = 0.8! = 1.6

0
0

10

20

30

40

50

60

70

80

90

100

Total Number of Flights N in Arrival Stream[ ] for several values of !

Figure 2 Expected Total Delay E WN

The coefficient ! in the above relationship is the relative
cost of stochastic delay over the delay due to reduced
throughput. As noted earlier, delays due to reduced
metering rate can be planned for well in advance, taken ata higher altitude, in less busy airspace, and with relatively
small speed reductions. As an example, we consider twovalues for the relative cost of stochastic over deterministic
delay, namely ! = 3 and ! = 10 , as well as two surges ofaircraft arrivals, N = 50 and N = 100 . For each case, theFor a surge of N aircraft arrivals, the expected loss fromthose two types of delay can be expressed as:N! [ L ] = 1 / 2 " N " ( N " a # N " h ) + $ " * ! %& Z! i '( " )i=1*! [ L ] = , 1 / 2 " N 2 " # + $ " ) ! %& Z! i '(/ " 0
+
.
i=1
N

(15)

'$!"

!"#"$%&'()*"#+(,)-.)/$01234)

or

'#!"
'!!"
N
&!"

=3034>/5/?@."=39,1"

%!"

1/a

1/h

A0<.;,?@."=39,1"$!"+,-,./01"
23034/56"#!"

7.08,9":;4<86;-80"

!"
!"

'!!!"

#!!!"

(!!!"

$!!!"

)!!!"

%!!!"
50#()

*!!!"

Figure 3 Queueing diagram with deterministic and stochastic delays

7

139

&!!!"

T. Nikoleris, M. Hansen
delays. The accuracy of the approximated method wasvalidated through simulation experiments, which indicatedsufficient accuracy of the Clark method in estimating total
system delays.

normalized (after setting ! = 1 ) expected loss E !" L* #$ as afunction of ! is plotted in Fig. 4.

The values of relative buffer that minimize E !" L* #$ , ! * ,Next, a simplified situation, with homoskedastic model
parameters, was examined in order to provide insights into
the queueing system. The impact of buffers in the arrival
stream was investigated, and was found that minimum
losses from deterministic and stochastic delays are attained
when the added buffer is small as compared to the level ofprecision ! .

range between 0.01 and 0.16. Even a value of ! * = 0.16
translates to a buffer of 3 seconds between consecutive
aircraft for a scenario where ! = 20 seconds. Also,
observe that for constant N, ! * increases with ! .
Therefore, as the unit cost of stochastic delay increases, alarger buffer is required to achieve minimum losses. On
the other hand, for constant ! , ! * decreases with N,
indicating that the loss from stochastic delays increases ata lower rate than the loss from deterministic delays, as thesurge of aircraft becomes larger. That is expected since
deterministic delays increase with N2 (see Equation 15),
while stochastic delays increase linearly with N as it canbe observed in Fig. 2.

Extending this research will relax the FSFS rule of service
and allow aircraft to re-sequence themselves as they
approach the server. It will also model explicitly the
Runway Occupancy Time of an aircraft as a separate
random variable, which is now embedded in thedeterministic minimum required headway h. Moreover, a
distribution other than normal will be considered for theunimpeded arrival times Ai. Finally, an extension to thesingle server case presented in this paper could be a
scenario with a network of servers.

4. CONCLUSIONSIn this paper a queueing system with a single server under
4D trajectory-based aircraft operations is examined.
Aircraft are assigned scheduled times of arrival at a fix,
which they meet with some stochastic error. A normal
distribution was assumed for the error, and aircraft enter
service according to a First-Scheduled-First-Served queue
discipline. A recursive queueing model was formulated,
and the Clark approximation method was employed toanalytically estimate the mean and variance of aircraft

5. ACKNOWLEDGMENTS
This research effort was sponsored by NASA under Award# NNX07AP16A.



N=100 & β=10
N=100 & β=3

Expected Loss E[L*]









N=50 & β=10
N=50 & β=3
























Δ = (a - h) / σ
Figure 4 Expected loss from deterministic and stochastic delays as a function of 

8

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ENRI Int. Workshop on ATM/CNS. Tokyo, Japan. (EIWAC 2010)
6. REFERENCES[7] Sinha, D., Zhou, H., and Shenoy, N., “Advances in[1] Joint Planning and Development Office, “Concept of

the Computation of the Maximum of a Set of Random
Variables,” 7th International Symposium on Quality
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