^f/
HD28
0i3
ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
Decompostion Algorithms
Transient
Phenomena
Networks
in
for
Analyzing
Multi-class
Queuing
in Air Transportation
Dimitris Bertsimas,
Amedeo Odoni
and
Michael D. Peterson
WP#
-
3540-93
MSA
January, 1993
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
\
M.LT.
UBRARIES
Decompostion Algorithms
Transient
Phenomena
Networks
in
for
Analyzing
in Multi-class
Queuing
Air Transportation
Dimitris Bertsimas,
Amedeo Odoni
and
Michael D. Peterson
WP# - 3540-93 MSA
January, 1993
ft^AR2.5
1993
Decomposition Algorithms
Phenomena
for
Analyzing Transient
Queuing Networks
in Multi-class
in Air
Transportation
Dimitris
Michael D. Peterson*
J.
January
Bertsimas^
8,
Amedeo
R. Odoni^
1993
Abstract
In a previous paper (Peterson, Bertsimas,
and Odoni 1992), we studied the phe-
nomenon
of transient congestion in landings at a
approach
for
computing moments of queue lengths and waiting times. In
we extend our approach
method used
hub airport and developed a recursive
for
to a network, developing
the single hub.
We
is
present computational results for a simple 2-hub
Although our motivation
applicable to
may be modeled
approach
for
all
is
drawn from
in
analyzing the interaction be-
air
transportation, our
method
multi-class queuing networks where service capacity at a station
as a
Markov
or
semi-Markov process. Our method represents a new
analyzing transient congestion
Airi^ort congestion
paper
two approximations based on the
network and indicate the usefulness of the approach
tween hubs.
this
and delay
ground delays at domestic
air])orts
iiavc
grown
phenomena
in
such networks.
significantly over the last decade.
By
1986
averaged 2000 iiours per day, the equivalent of grounding
the entire fleet of Delta Airlines at that time (2oO aircraft) for one day (Donoghue 1986).
In 1990, 21 airports in the U.S.
exceeded 20,000 hours of delay, with 12 more projected to
exceed this total by 1997 (National Transi)orlation Fiesearch Board 1991). This amounts to
'School of Public and Environmental Affairs, Indiana University, Blooniington, Indiana
^.Sloan .School of
Management, Massachusetts Institute of Technology, Cambridge, Massachusetts
and Astronautics. Mas.sachusetls Institute of Technology, Cambridge,
'[)epartiiient of Aeronautics
Massachusetts
multi-class queuing networks.
Our model
provides important qualitative insight on the interaction between hubs and
improves our understanding of hub-and-spoke syste:ns. With our model we can address a
number
•
of interesting questions, such as:
To what degree do network
effects alter the results
obtained from the study of a single
hub?
•
Wliat network effects are produced by delay propagation?
•
What
effect
•
Wiat
are the effects of hub isolation strategies
does the degree of connectivity between hubs have on system congestion?
reduced
tivity to others is
The paper
is
— strategies in which a hub's connec-
— on schedule reliability?
organized as follows,
in Section
1
we review
briefly the
methodology of
the algorithm for a single queue and describe the queuing network context of the present
problem. In Sections 2 and 3 we outline two decomposition approaches which exploit this
algorithm.
Section 2 describes a relatively simple
method
in
are adjusted according to expected upstream waiting times.
involved approach which uses second
description of
downstream
moment
arrival rates
which downstream arrivals
Section 3 describes a
infonnation about delays to give a stochastic
In Section 4
we employ these approximation meth-
ods together with a simple simulation jirocedure on a 2-hub network.
moderate
traffic conditions,
Under heavy
traffic
demand
times act to smooth the
We
the interactions between hubs do not alter
significantly, so that the j^redictions of waiting times i^roduced
are very close.
more
find that
demand
under
patterns
by the different approaches
conditions with closely spaced banks, higher waiting
]>attern significantly over the day. In this latter situation,
the two recursive algorithms deviate further from simulation than in the fonner case.
also find that isolation of a ]iroi)lein
hub
]irolects other
We
hubs from schedule disruption at
the cost of further disruption at the source of the delays.
We
provide concluding remfirks
in Section 5.
1
The Basic Model
Incoming
and
aircraft at
an
airiiort require service at three stations:
a dejjarlure runway.
The landing operation
in particular is
a landing runway, a gate,
subject to wide variations
.
p„{m)=
We
For
Pr{(t,m)^(i,m+1)) =
Pr
>
[T.
m+1
|
T.
> m|
(1)
next define the following random variables:
Qk
=
Queue length
Wk
=
Waiting time at end of interval
Ck
=
Capacity state at end of interval
Ak
=
Age of current capacity
T,
=
Random
mean queue
length
I,
A-
qmax(A"i
i) is
the
m,
= maXj
fc,
k,
state at end of interval k,
lifetime of capacity state
=:
,
1
,
.
.
.
maximum
is
A',
i.
(jmaxC^,
i
=
I,
.
.
.
m=
,S,
i.
1
,
.
.
.
,
(2)
A/,
This obeys the recursion
=
|<?max(''"-l)
+
andx^ — max(x,0).
«)
= m]
Ai
i,
attainable queue length at the end of period k, given that
9max(A".0
qmax(A")
interval k,
= E \Qk \Qi=q,Ci =
q)
at that time the capacity state
where
end of
we introduce the notation
Qk(l,
where
at
>^k
-
(3)
Mt]^
Similarly, for waiting times
we employ
the notation
WkiL
We
write the second
2,
m,
moment
q)
= E\Wk
I
=
Q,
C,
9,
=
z,
Ai
=
m\.
(4)
analogs of (2) and (4) as Ql{l,i,in,q) and Wl{l,i,m,q),
resjieotiveiy.
Let (x
A
y)
denote
and W^(/,
I.
m,
q)
niiii(x,y).
The
quantities Qk{Li,m,q), Q^(l,i,rn,q), Wk(l,i,m,q),
can be calculated recursively,
(
Peterson, Bertsimas,
and Odoni
1992).
We
rejjeat here the basic equations:
Qk{l.t.m,q)
=
^p,,(»»)Q,
(/+!, J. !,((?+ A,+
P„{m)Qk (/+l.!,m+l,(q+
Ql{l,i,m,q)
=
^P.jt".)Qn'+l,J.
l.('7
+
A, +
,
i
A;^,
+
-//,)+)
-
(i,)'^)
-/i,) + )
(5)
,
+
where
i^
=
mth stop on
t"^
=
scheduled arrival time at
s^
=
siaclc
time between stops
m—
Aircraft slack between stops
at stop rn
-
1
itinerary of aircraft v,
and
1
m
mth
stop for aircraft
m
1
the
is
—
and
»n for aircraft v.
amount of time
beyond the minimal time necessary
v,
available to the aircraft
to tuni the aircraft around.
In the
network schedules are no longer exogenous and deterministic, as delays at one airport
the schedules at others.
In the
terminology of queuing theory, the system
a multi-class
is
queuing network, with the classes being the different aircraft with their individual
Service capacity at each airport
Markov process
or
Markov
affect
itineraries.
an autocorrelated stocliastic process described by a semi-
is
Thus our task
chain.
to describe the transient behavior of
is
a multi-class queuing network with autocorrelated service rates at each node.
This high
degree of complexity suggests that approximation methods are necessary.
A
2
A
first
Simple Decomposition Approach
ap]5roximation approach
of the day, one
based on the following idea. Suppose that at the start
is
knows the schedules
for all aircraft
operating
in
assumption that delays are zero at the outset of the day, the schedule
of the day
is
fixed.
Hence the
first ])eriod
demands
waiting times for each airjiort during this jieriod
(5)
-
The
(14) to each air]iort.
the delay encountered by
all
may
for
be determined by applying equations
aircraft scheduled to land in this j^eriod.
demand
in their .sclie<kile,s
for
the next
i>erio(l,
the initial period
and mean queue lengths and
resulting exjiected waiting times for period
the slack wliich these aircraft have
accordingly, one then fixes
are fixed,
Under the
the network.
1
are estimates of
Taking into account
and ujidating future
arrival
calculates the resulting
streams
new expected
waiting times, and so forth.
More
formally, let d^
aircraft r jjroceeds
rei>resent the current
through
its itinerary,
ff
is
for aircraft
v
—
amount by which
it
is
cumulative delay
the current
i.e.
behind
schedule. Further define the terms
Ain.k)
=
set of aircraft scheduletl to land at
E |W7'|
=
mean
waitnig time
for
r?
ni
period
an aircraft arriving at
i
k,
at
end of period
as
A:,
,
First Decomposition Algorithm for Air
Network Congestion
Initialize:
=
For k
=
For n
K
to
\
N
to
1
A(n,k) =
****
For
itinerary sto])s are deteniiinistic since not affected
first
77
=
<p
=
For V
earlier delays
*****
N
to
1
by
V
to
]
A(n,K{t\)) =A{n,K(i\))Uv
Set d'
=0
Main
loop:
For
k
=
For
\
r?
Set
V
V.
K
to
=
=
A;J
N
to
1
\A{n,k)\.
Using the recursive method at
For V € A(n,
m
Set n
=
Set
;
E
[^k]
i
••
•
i
^
l^k']
k):
***** find the
Find
eacli airport, calculate
the itinerary corresjionding to this stop *****
])art of
(n^^J^^, s'^) € I(v) and K{t^^
rim,
'
=
'm
+
d", S
=
Sm,
ri'
+
<f)
= n^+i,
=
''
A"
=
«'
tm+l,
=
Sm+l-
a = k(0 -t/{At).
***** calculate propagated delay *****
Set d;„+
d'
+ aE
+
(1
-a)£
1
+
M'",,,
"(<)
***** detemiine next arrival period and ui>date data structure *****
Set A{n',K{t'
+
d''))
= A(n\K{l' + d"))Uv.
END.
Figure
1:
Decomposition algorithm
for
network based on detenninistic updating scheme
U
where
hub recursive algorithm for waiting time moments
the complexity of the single
is
with deterministic input.
Proof:
Tlie choice of data structure
0{V)
dure) requires only
calls to a
means that
time.
Hence the bottleneck of the algorithm consists of repeated
subroutine for computing expected waiting times. Because for each time period
k the algorithm must recalculate
plexity
is
specified with
The presence
was shown that
it
5
EI^Vl
.
.
Markov model
at each
in
new
Thus
]>eriod.
the second
it
hub algorithm the end conditions
However, even
for the
l^robabilities for
queue length and
Odoni 1992), there
some other
A more
where
m
Under
this
algorithm
the
number
this
is
0(NS^K^Qmi^^).
in
finds E|IV/|
the
.
.
.
first
global iteration the algorithm
E\Wl^] and E\Wj]
if it
.
.
.
E\nfl
and
were possible to store within the
of iteration k as initial conditions for iteration k+l.
mean
storing the joint
Since comi^uting these probabilities requires
cai^acity.
and
reason.
the
imiirovement
minimum number
run
for
the
first
is
is
to have the recursion restart only every
of jieriods any aircraft has
run
2in jienods.
for
the
first
and so
on.
m
new scheme
m
-^2
4-
...+
:\m
-\-
Cm
/\'
Whereas
in
the original implementation,
is
- K{K+\)/2,
+ A" =
11
periods,
periods, arrivals are updated, then the
it is
+ 2m +
m
between scheduled stops.
of iterations iierformed witlnn the recursive algorithm
1
under
1
effort as for the exjiectatioii alone (see Peterson, Bertsimas,
scheme, the aigorithni
is
Markov capacity model
no benefit to doing so unless the probabilities themselves are desired
is
jjractical
is
the
simpler Markov cajmcity model, this would
0((5max) times as much
for
if
K arises from the fact that the recursion at each hub
so forth. This duplication of effort could be avoided
single
of capacity, the complexity of the
Thus
OiS"^ K'^Q^ax)
is
of the additional factor
E\W^^],
.
hub
for a
In Peterson, Bertsimas,
capacity states, overall complexity for Algorithm
restarted from time
finds
of the preceding expected waiting times, overall com-
D
recursive algoritlim for a single
is
all
0{KNII).
and Odoni (1992)
is
inner updating loop (the disaggregation proce-
tlie
C.(C,
+
1
)7n/2
+ A"
n.k<2
Fig^ure 3;
The
traffic splitting
delay encountered.
The numbers
In order to comjjlete the
tic infoniiation
phenomenon: alternative future
aircraft
paths depend upon
{p,;} indicate probabilities.
updating scheme, the algorithm must translate the probabilis-
on individual
aircraft into information
on future
arrival rates.
Define the
stochastic arrival quantities
A(n,k)
The
- number
of arrivals at airjjort
goal of this stejj of the procedure
random
variables.
.
.
.
,
random
in
period
k.
to specify an api)roximate probability law for these
R
For some user-sjiecified number
values taken by the
and A"(l),
is
n
(representing the
variables) the algorithm estimates
number
numbers 7"(1),
of possible
.
.
.
,7^(fl)
A"(ft) which obey the relationships
\>r{Mn.k) = Xl(\)}
=
-rrO).
A'/(2)}
=
^."(S).
\>r{Mv.k) = y^{R))
=
-yr(ff).
F'r{A(7,,A-)
=
(18)
where
^7r(o
This simplified description of
variaijiiiiy in
=
(19)
i
the arrival rales
is
easily incorporated into the
2.
Translatio!) of these density functions into probabilistic descriptions of future arrival
periods for each aircraft, as given in the parameters p„(0),
3.
Translation of the individual aircraft parameters p„(0),
random
into simple discrete distributions for the
4.
The
Updating of
aircraft itineraries
and airport
what
follows,
.
.
.
,Pv{C) and/c„(0),
,PviC) and
.
A:„(0),
.
.
.
.
.,
.,
/c„(C).
kv(C)
variables A(n,/c).
arrival lists.
fourth of these procedures was described in Section
further detail in
.
.
2.
The
and a sununary of the algorithm
first
three are described in
given in Figure
is
8.
Obtaining waiting time densities
3.1
Estimation of the densities f{w) cannot be done on the basis of the recursive algorithm
alone, since this procedure gives only the first
of the third
moment would
two moments of the distribution. Knowledge
enough information to determine a unique 2-point discrete
give
distribution by solving the nonlinear system
P\W]+P2W2
=
E\W]
p,u-^+p2u>2
=
E[W^]
p,u,^,+P2wl
-
E[W^]
1
+
P2
=
u'l,
tti2
>
Pi
Pi,p2,
for
the values pi, p2,
u'l.
and
w-j-
However,
(22)
0.
this
system
is
not guaranteed to have any
solution because of the ])ositivity requirement.
An
alternative metiiod
Handscomb,
1964)..
period by jieriod,
Markov
process.
where W"'
is
is
is
suggested
liy
Monte
C'arlo
Consider a simjile simulation
determined
Prom
in
Monte Carlo
for
methods
(see e.g.
Hammersley and
a single airport in which capacity,
fashion from the
Markov chain
or semi-
the simulation we obtain the matrix of observations
the waiting time at the end of jieriod k for the
mth
simulation.
Ordering
the observations, we obtain histograms for the waiting times for each period, like the one
illustrated in Figure 4 for a constant arrival rate (p
15
%
0.85,
A
=
60 per hour).
Note the
1.0
Exponential Plot for Positive Observations of
Wait Time at Period 50 (12.5 Hours)
T
lOOORealiulioas
0.8 ••
6 ••
0.4
0.2.
.
J
Figure
//
Test
5:
for
450
300
150
60(1
= (ordered) observation number
exponential distribution of positive waiting time realizations
must be determined by solving the pair
of equations (omitting subscripts)
/•CXI
^tL'n„n
6 {w^,r,f
+
+
(
{]
-
1
-
f
6)
*^U
111
terms of the mean
w
wiyc-'^"'-'^'-"Uw
=
E\W\
t/-2,/f-"(''-"'-") du,
=
£[iy2]
/
(!))
and variance
(24)
mm
a-^
we obtain the solution (omitting subscripts)
g^
- (»^-
Wmin)^
(25)
2{w - Wm,n)
(26)
Note that
6 is
always
less
than
1
and
will
be jionnegative provided that
>
111
the tj^iicai case where Wm,n
^^
zero, this
of variation for waiting times exceeds
was
1
1.
equivalent to the condition that the coefficient
is
Only
in
rare instances of the tests presented shortly
this condition
found not to hold.
In tiiose c.uses, the
entire distribution
was assumed to be
exi)onential.
17
jmrameter 6 was set to
and the
For practical reasons,
it is
necessary to choose some upper bound
C
on the number of
periods of delay to allow. Hence
v(C-i)
Together with the numbers
{/.-^(c)},
the probabilities {pv(c)} then constitute a probabilistic
description of the next period in which aircraft v will
3.3
demand
to land.
Characterizing arrivals
In order to translate the
demand
numbers
rules A{n.k), define the
random
if
1
Xn'l.nkiv)
=
into a probabilistic description of the future
{p^.ic)]
variable
e A{n',
r
next
is
1}
delayed such that
its
be n at period k
sto]) will
otherwise
This random variable denotes the "contribution" of an
arrival rate at a future place
and time. Note that
demand
i'
£ A(n'J), the
arrival rates.
jirobability that
n, then
will
it
contribute to the landing
variables X„'i.nk('^') l>rovide the necessary connection between aircraft
and
is its
next scheduled stop)
is
Then
^^^A-„.,,„,(r).
In words, this says that the arrival rate at
random
I) is
— l).
n'=
in
,
pv{k
A(rK/.-)=
points
the next stop of v € A{n'
at airjiort n during jieriod k (as,suming that n
The random
one place and time to the
\}=p,(k-l).
Pr{A-„.,.„,(r)=
In words, for aircraft
if
arrival at
the itineraries (see Figure
fi).
/<;,
I
(ri,
k)
1
=
is
(30)
1
the
sum
Thus the random
of
all
contributions from previous
variables {A} are
sums of Bernoulli
variables. Defining
NL{v.k) = next destination Of
the expectation
is
easily obtained
E\Mn.k)]
aircraft
i'
after period
A-
ius
=
^^^E|A',..,,„,(")111'=
=
I
(<*. t.=
1
EE E
n'=l l<k v:NL(v.l) = n
19
^^'(^--'^
(31)
Histogram ofA
200
(1>22)
+
(1000 Realizations)
1
50
o
o
100
••
E
50
n
10
Arrivals in Period 22 at
7:
a^
one may occur
still
Hub
I
Histo^aiii of A( 1,22) obtained from simulation. Unusual skewness patterns such
Fiffure
tills
25
20
15
in
the early part of the day
when
tiie
contributing prior arrivals are
largely deterministic.
considerable degree of insensitivity to the
assumption while acknowledging
its
demand
rate distribution.
We retain
the normality
im])erfections.
Although Algorithm 2 involves considerably more modeling work than Algorithm
comjiutational complexity
is
not significantly higher, as our next result indicates.
0(RK NU),
Theorem
2 The coviplexity of Algontlmt 2
number of
values used ni the approximate distributtoit for the arrival rates
is
where
complexity of the single hub recursive algorithm for waiting time
input.
If tlie
Markov
1, its
capacity model
is specified until
S
R
is
the user- specified
moments with
and
U
is
the
deterministic
capacity states, overall complexity is
Proof:
Within the principal
time moments
in
ioo]),
the bottleneck ojieration remains that of calculating the waiting
the recursion.
Because the
21
arrival
stream
is
specified probabilistically
rather than deterministically, there
is
an additional factor
specified for each arrival rate distribution.
Both Algorithms
1
and 2 are suitable
The
for
R
equal to the number of values
D
result follows.
any kind of network. Without the streamlining
running times are somewhat high.
suggested at the end of Section
2,
simple 2-airport network with A'
= 80
For example, on a
periods at each airport, Algorithm
1
takes about one
hour on a DEC-3100 workstation while Algorithm 2 takes about three hours (with
With the reduction
is
network of a large
the problem
is
= 3).
the recursion achieved by the streamlining procedure, there
in calls to
roughly tenfold improvement
full-size
/?
in
{400+ nodes)
airline
Even with
these figures.
well suited to parallel
is
improvement, modeling a
this
a daunting problem.
On
the other hand,
computation, with different processors handling the
individual nodes and a central processor controlling the bookkeeping of aggregation and
disaggregation
In the ])resent context, further siin])lification
to understand congestion in
delays at
its
its
spokes.
hubs have
its
That
hubs as
negligible.
setting aircraft itineraries
now each
own hul>and-spoke network. Prom
and
treat
all
congestion delays other than those emanating
In the resulting network,
As
im refers to a hub airjiort and each
sjjoke.
External aircraft add to
visits to
schedules are considered fixed
place between hubs, but
have intermediate si^oke
fiiglil
we
incorjiorate spoke information in
before, these consist of ordered triples {(irm ^mi «„)}, but
between successive
reflects the total slack available to
mternal
time,s vary
congestion
in
the system, but their arrival
flights in the collaj>sed
network appear to take
widely to reflect the fact that in
reality,
the aircraft
stoi)s.
size of a large airline's
changes reduce the model's realism.
Since one of the main goals
is
network from 400+ nodes to perhaps 5 or
Init tiie
6.
These
reduced model should capture essential behavior.
to imjirove our understanding of interactions
between hubs
the issue of isolating C'hicago), this sim|iliHcalion seenus to be furtiier justified.
testing
an
hubs, including the slack available at an intervening
demand and
All
.s„,
ignoring congestion at the sjioke.s of the system and concentrating only on the hubs,
we can reduce the
(e.g.
schedule than delays at
keep track only of aircraft belonging to the hub carrier,
is,
aircraft
By
this carrier's perspective,
This observation suggests a simplification: reduce the hub-and-spoke network
treat other arrivals as fixed,
tiie
possible. Consider a single carrier trying
far greater implications for disru])tion of its
to a network of hubs.
from
is
and analysis jjresented
in tiie
next section
23
is
The
conducted on a simple 2-hub network
case
#
isolation,
(p
= 0,
(p=
is
by considering an instance
two hubs have no
in whicli the
the disconnected case) and an instance in whicli they have
1, tiie
fully
connected case). In case 5 we examine four instances
all
in
aircraft in
common
aircraft in
common
which aircraft slack
varied.
Results and Discussion
4.2
Considering cases 1-5 together, we note that model parameters should have a noticeable
effect
on the mechanics of the network. For example,
are of the
same order
and there
is
DFW
case (#1), waiting times
substantial separation between major
For these reasons, we expect delay propagation to be relatively low and have
traffic ])eaks.
a
as aircraft slack,
in the
less disru]itive effect
on the schedule. In case
closer together, traffic intensity
is
2,
on the other hand, major peaks are much
shar]>ly higher,
and delay propagation should be more
important.
Using a DEC-3100 workstation we ])erfonned computations
for test cases 1-5, using
both
of our recursion-based approximations as well the the simple simulation procedure discussed
above.
Our
investigation
is
primarily motivated by the desire to develop qualitative insight
into the transient ])heiiomena of the network.
will
Accordingly, the following set of questions
guide our discussion of the results:
•
To what degree do network
effects alter the results
obtained from the study of a single
hub-?
•
What
are the network effects produced by delay i^ropagation
and under what circum-
stances do they become important?
•
How
closely
do they
•
What
•
How
•
What
results
match
tiiose of
simulation?
Where
differ?
is
is
do the network a]>]iroximation
the effect of congestion at one hub on
this effect altered
is
demand and
by the amount of slack
the effect of isolating a congested
with the other hub?
27
in aircraft
congestion at the other?
schedules?
hub by not allowing
its flights
to connect
preservation of
tlie
peaked delay pattern,
by delay propagation (the "network
botli of
effects"
which indicate that the
are relatively minor. Because there
)
induced
effects
is
ample
space between major banks and slack values are close to the mean waiting times, the general
])eaked pattern
to
is
preserved.
ensure a moderate
The
result suggests that
traffic intensity
when space between banks
and when mean waiting times are not
adequate
is
substantially
greater than aircraft slack, network effects are outweighed by the "deterministic" congestion
effects resulting
from
tlie
banked structure of arrivals at hubs.
Behavior Under Heavier Traffic and Closer Spacing
The
weakness of the network
relative
effect in the
preceding example obscures differences
between the network approximations and the simulation.
vided by case 2a (Figure
12).
and there
is
in the arrival
is
fit
stream become significant.
overestimate delay during the middle
same
pro-
between the simulation and the
a noticeable disparity in the middle part of the day,
certain periods. This
is
only a 15-minute gap between successive
While the early part of the day shows a close
algorithms, there
revealing picture
Here expected waiting times (30-40 minutes) are quite high
relative to aircraft slack (5 minutes),
banks.
A more
effect
is
when
alterations
Relative to simulation, both algorithms tend to
]iart of
the day, with the difference as high as
jireseiit in
case
to a
1
much
30%
for
lesser degree.
For a given hub and algorithm we define a standard error in the predictions relative to
simulation. Let
A'/^
denote the waiting time value predicted by algorithm
Yh denote the corresponding value for the simulation.
This ]jrovides a measure of how
Algorithm
numbers
1.
for
with worse
Case
far ajiart the
these values are 4.5 and 2
fi
Then
for
period k and
the standard error s
is
given by
simulation and algorithm results are.
For
minutes at the two hubs, while the corresponding
Algorithm 2 are 4.o and 2.3 The numbers rei^resent an average error of 10-20%,
fits in
3, in
the middle ])art of the day.
which demand
is
allowed to he continuous over the day (no banks), also shows
a discrepancy between the a])i)roximations and the simulation during the middle part of the
day, as Figure 13 indicates.
than case
2.
The
The
traffic intensity for this
difference between the algorithms
part of the day at hub
2,
case
is
higher than case
and simulation exceeds 20%
1
but lower
for a large
and the standard errors are approximately 15% of the delays:
29
2.2
minutes at hub
hub
2) at
2.
(for
1
Thus
both
algoritiinis)
in all cases,
and
2.4
and
2.6
minutes (Algorithm
1
and Algorithm
the simulation produces lower waiting time estimates during
the middle part of the day than both of the approximation algorithms.
We
next consider
the likely explanation for the discrepancy.
The Network
Demand Smoothing
Effect:
Consider the waiting time profiles
time
much smoother
are
jjrofiles
sejiaration between banks,
to
tiie
for cases
in
this effect
profiles at
is
is
relatively high waiting times
Hub #1
see that
the
to]) half
of Figure 14, where
The
where
original schedule
sharj)
waiting
With only a 15-minute
traffic intensity is
aircraft slack
very high and
down. Further evidence of
we have plotted the
aircraft slack
is
labeled "slack
is
large
original
demand
=
500", corresponding to the
enough to eliminate propagation completely.
see by comparison with the situation "slack
demand
when
12). Evidently,
together with those which are produced as a result of delayed arrivals
2a.
artificial situation
We
in
and
combine with low
low, the order of the network's schedule breaks
given
under scenario
11
the latter than in the former.
overwhelm the bank structure. Thus we
aircraft slack
and 2a (Figures
1
=
5" that
propagated delays smooth the
The
pattern substantially, with large numbers of aircraft shifted to late periods.
peak structure of the original demand
is
considerably altered.
This smoothing phenomenon exi>lanis why Algorithms
sistently in the
given jieriod
middle part of the day
may
1
and 2 overestimate delays con-
In the actual jirocess,
an
aircraft scheduled at a
exjierience a delay ranging from zero ui> to 3 hours or more.
In cases
of high waiting times, the aircraft's next arrival time will be considerably later than
scheduled, and
]>eriods.
its
contribution to later
Thus over
a large
number
is
more than
is
])ushed back by a significant
number
))arl
a(le(|uatc
of the day.
liieii,
when there
is
no scheduled
the result of this traffic shift
is
traffic.
to reduce
overall waiting times. Ideally, the <()iii|)uiatioiia! algorithms should reflect this shifting
smoothing of demand
have to
However.
kee]i track of the
;us
w;ls reiiKirke<l earlier, to
size,
waiting time.
both algorithms
The
result
is
u])tlatc ainraft srlirduli-s
that
i)otli
To
may
follow as a result
limit the state space to
man-
according to one number, expected
algorithms tend not to
31
and
do a thorough job they would
thousands of potential ))alhs which aircraft
of delay, a seemingly im]>ossii)le coiMimtaiional burden
ageable
of
of simulations with heavy traffic, a noticeable fraction
of arrivals are ])ushed back to the later
Because cajiacity
demand
was
shift aircraft to
the very late
part of the day sufficiently but ratlier to concentrate
in hiffher
The
demand more
in tlie middle, resulting
predicted waits.
Effect of
Demand Smoothing on Waiting Times
The phenomenon
of
demand
seem counter-intuitive
at
shifting
and smoothing explains certain observations which
An example
first.
of such a result
is
the fact that higher aircraft
slack can inc.remte expected queuing tivies at the hubs. Cases 2a
have heavy
organized into narrowly separated banks.
traffic
2b. artificially high slack prevents the
allows the
demand
to
network
effect of
become smoother over time
and 2b
The
illustrate this.
difference
that in case
is
demand smoothing, whereas
we
a closer
is
means that
slack
half of Figure 14.
peaked pattern
also preserves the
case 2b there
bottom
see in the
fit
case 2a
end of
as aircraft are pushed back to the
the day. in the high slack case, the higher concentration of demand produces higher
delays, as
Both
qnemng
Because slack preserves the schedule,
it
Note that
in
in that schedule,
which produces queues.
between the algorithms and the simulation, because the high
the schedule becomes essentially deterministic.
Assessing the Network Approximations
The
under which network
results of cases 1-3 suggest the circumstances
effects
become im-
portant, and they also indicate that under these circumstances, the network approximations
developed
day.
Case
in this
1
suggests that for networks of air])orts
jjrobably not high
]>art of
paper tend to overestimate waiting times during the busy period of the
enough
the schedule
trate, the situation
is
(i.e.
on.
traffic,
it
waiting times on average are
average to create significant network effects: the deterministic
changas when
may
traffic
becomes heavier and
s])acing
and
3 illus-
between major banks
only describe a few airi>orts at present in this country
rejjresents a future scenario
which
is
(e.g.
quite possible, hi the cases of heavier
lower slack, and less sejmration, the ])erformance of the algorithms worsens as they
tend not to capture the true sj^reading of
Slack, Connectivity,
We
DFW,
the bank structure) j^redominates. However, as cases 2
decreased. This situation
O'Hare), but
like
demand which
is
the major network effect.
and Hub Isolation
consider next the effect of network coiinectimty and aircraft slack on cumulative aircraft
delay.
We
distinguish between
tliis
latter meiisure
33
and that of
tiie
waiting times at the hubs.
Tlie latter correspond to the waiting times of aircraft at the various stations in the network,
while the former
We
is
sum
really the
measure network connectivity
hubs
at both
fully
in
of such waiting times with aircraft slack subtracted.
tenns of the percentage of
in
Case 4
the network.
having operations
two opposing extremes of connectivity: a
illustrates
disconnected network (case 4a), where each hub has
connected network (case 4b), where
flights
its
own
set of aircraft;
and a
fully
alternate between the two hubs in between
all flights
visits to spokes.
Case 4a models the idea of hub isolation referred to
= 0),
com]>letely disconnected (p
Because the network
earlier.
is
scheduled bank times at one hub cannot be disrupted by
from the other. In contrast, case 4b ensures that aircraft encountering delays at
late arrivals
maximum
one hub have the
chance to disrupt the schedule at the second, since that
is
their
next destination after the intervening s))oke. Case 4's scenario thus allows us to explore the
network
effects of low cajjacity at a single location.
we take the
initial
state of the
to be 3 (highest capacity).
at the first to the
Our
first
hub
to be
The phenomenon
The
aircraft.
early banks
#
1
whe7i
of interest
it is
delay, while later
The
further examination, these results
But we
also see that
opijosite direction
#1 more
closely,
the propagation of delays created
—
banks
figure indicates a degradation in
make
it is
#
1
at the exjjense of
performance
intuitive sense. Clearly
c;ise
in
the connected case
by about 2 banks (2 hours)
iiours (4 banks), while in the disconnected case
produced by the congestion
at
case, jiroducing the 2-liour lag.
hub ^
!
are
However,
in tlie toj) half
felt
This
seem to
is
it
is
only
2.
in
the
does not
of Figure
\^>.
lag behind the
no coincidence:
visits to
the
in
the
same hub
Thus the schedule delays
2 hours later at that
this lag
35
moves
it
connected. Examining the situation at hub
connected case, the inininium time between any aircraft's successive
heights of the two curves
we expect hub #2's
disconnected from the disrui^tions produced by
fully
we notice that the delays
hub #2.
#2.
hub # Is schedule i^erfonnance im])roves when
from disoomiected to
delays ni the disconnected
is 4
reflect delay carried
isolated, as well as tiie rorres)ionding benefits of isolation at
schedule to become more reliable when
#1.
is
hub
which plots average cumulative delay per
1.5,
Conversely, the fully connected case benefits hub
Upon
both case 4a and case 4b
(lowest capacity) and that of the second
1
show zero
over from ])revious points in the itnierary.
at hui)
in
tliis,
banks of the second.
results are sunnnarized in Figure
arnmng
To do
hub
in the
connected
fully ex]>lain the difference in tlie
In the
comiected case,
late aircraft
leaving
hub #1 have the opportunity
of recovering
next stop (uncongested hub #2). This oi)portunity
since the next stop
is
#
1
,
is
of the delay tlirough slack at their
not available in the disconnected case,
a fact which explains
why
the corresponding delay
hub
isolation. In the case of
higher even after we take account of the lag.
These
a
(congested) hub
is
some
results have interesting implications for a strategy of
hub which
is
indeed protect other hubs
On
the problem hub.
in
5a, Hb, be,
and bd
that higher slack preserves
the hubs.
On
of congestion, such a strategy will
the system from the uncertainties and disruptions produced by
the other hand, disruption at that hub itself
of its later arrivals will have
Cases
amount
believed to be the source of a large
had an
earlier
since
jieaking and
may
We noted earlier
on aircraft lateness.
actually increase queuing delays at
other hand, slack reduces each aircraft's cumulative delay.
tlie
illustrates that this
second
effect
predominates
in this relatively
hght
traffic.
Figure 16
For varying
slack values, the figure plots the average cumulative delay per aircraft arriving at each
of the day, not including any waiting at the current stop.
contain any surprises.
models are
Finally,
We
include
we note
tliat in
which
Certainly the figure does not
order to illustrate the kind of planning for which the
])henomena of
witli
a situation of major capacity shortfalls, airlines
disrujjt the schedule.
by canceling and rerouting
connection
bank
well suited.
acce])t long delays
into the
it in
many
scheduled stop there already.
illustrate the effect of slack
demand
may worsen
flights.
interest
The
and
do not
passively
Instead, schedulers respond in "real time"
jireceding discussion
is
intended to provide insight
to the strategic issues that airlines
must plan
for in
schedule disrujJtions due to congestion at their hubs. At the tactical and
oi>erational level, airline behavior
is
in actuality
more dynamic.
Conclusion
5
In tins i)a])er
l^robleni of
we
iiave develojied
modeling transient queiiiim behavior
summarize our major findings as
1.
two related ai)]iroximation a])])roaches to the
is
network.
We
would
follows:
Iviporiance of traffic fpltttivq plininmnimi.
countered by aircraft
in a liul)-and-s))oke
difficult
High uncertainty
in levels of
a iiroiniiieiil feature of the network i^roblem.
37
delay en-
Unfortunately,
accuracy
in
keeping track of aircraft amid this uncertainty
is
limited by high
compu-
tational complexity.
Importance of deterministic
2.
The peaked pattern
effect.
remains a strongly determining factor
of
demand
at
hub
aiirports
waiting times, particularly
in predicting
when
major banks are separated by ample lengths of time.
Delay and smoothing.
3.
On
the other hand, in cases where banks are narrowly spaced,
demand and
delay propagation exerts a strong smoothing effect on the
waiting time
profiles.
Effects of
4.
hub
isolation.
A
policy of isolating a congestion-prone
On
the effect of improving perfonnance at others.
with some remarks about running times. As
times for Algorithms
one hour
for
1
Algorithm
and 2 are
1
higii
even
and three hours
for the small
clearly does have
the other hand, under this policy
the isolated hub produces congestion delays which disrupt
We conclude
hub
its
own
we reported
2-hub
future schedule.
earlier,
test network:
the running
approximately
Algorithm 2 on a DEC-3100 workstation. These
for
times are particularly poor considering that the running time for the simulation program
(5000 sami)les)
ments
in
is
significantly shorteT
—
about 10 minutes.
the algorithms, this observation favors simulation.
of Algorithms
I
and
2 used in our tests
is
In the absence of improve-
However, the implementation
a rather inefficient one. Incorporating the earlier
suggestion that the recursion be restarted every
m
than at every period would
jieriods rather
reduce running times by at least a factor of
A-
+
1
(h/m)+
A
m=
value
10 (2 1/2 hours), which
would have between successive
A'
= 80
periods). This
into the
same range
vi.sjts
is
~
api>roxinialely the
to hub.s,
minimum time
a typical aircraft
would reduce running times by a factor of 9
improvement alone would
as simulation.
in
I
iiring the
The reduction
is
running times of the algorithms
nni^ortant for the general problem
because the number of simulations necessary cannot be known
least in this test case, the smiulation jirocedures themselves,
original
Markov and semi-Markov cajwcity models,
network
effects.
39
(for
in
advance.
However, at
based on the same ideas of the
offer a third
approach to understanding
Roth, Emily.
1981.
An
Investigation of the Transient Behavior of Stationary
Queuing
Systems. Ph.D. dissertation. Operations Research Center, Massacusetts Institute of
Technology, Cambridge,
Whitt, W.
1983.
MA.
The Queuing Network
Analyzer. Bell System Technical
JoumaZ 62:9,
2779-2815.
National Transportation Research Board.
1991.
Winds
of Change: Domestic
Air Transport Since Deregulation. Transportation Research Board National Research
Council Special Report 230. Washington, D.C.
41
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