Transcript Analytical Methods in Airport Queueing Modeling

Airport Taxi Operations Modeling:
GreenSim
John Shortle, Rajesh Ganesan, Liya Wang,
Lance Sherry,Terry Thompson, C.H. Chen
September 28, 2007
CENTER FOR AIR TRANSPORTATION SYSTEMS RESEARCH
Outline
• Queueing 101
• GreenSim: Modeling and Analysis
• Tool Demonstration & Case Study
2
CATSR
Motivation
• GreenSim
• Airport as a “black-box”
– 5 stage queueing model
• Schedule and configuration dependent
– Queueing models configured based on historic data
• Stochastic behavior (i.e. Monte Carlo)
– Behavior determined by distributions
• Comparison of procedures (e.g. RNP procedures) and
technologies (e.g. surface management)
– Adjust distributions to reflect changes
• Rapid (< 1 week)
– Fast set-up and run
CATSR
Typical Queueing Process
Customers
Arrive
Common Notation
l:
Arrival Rate (e.g., customer arrivals per hour)
m:
Service Rate (e.g., service completions per hour)
1/m: Expected time to complete service for one customer
r:
Utilization: r = l / m
CATSR
A Simple Deterministic Queue
l
CATSR
m
• Customers arrive at 1 min, 2 min, 3 min, etc.
• Service times are exactly 1 minute.
• What happens?
Customers
in system
1
Arrival
Departure
2
3
4
Time (min)
5
6
7
A Stochastic Queue
•
•
•
•
•
CATSR
Times between arrivals are ½ min. or 1½ min. (50% each)
Service times are ½ min. or 1½ min. (50% each)
Average inter-arrival time = 1 minute
Average service time
= 1 minute
What happens?
Arrival
Departure
Customers
in system
1
2
3
4
Time (min)
Service
Times
5
6
7
Stochastic Queue in the Limit
100
90
80
Wait in
Queue
(min)
70
60
50
40
30
20
10
0
0
500
1000
1500
2000
2500
3000
3500
4000
Customer #
• Two queues with same average arrival and service rates
• Deterministic queue: zero wait in queue for every customer
• Stochastic queue: wait in queue grows without bound
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• Variance is an enemy of queueing systems
CATSR
The M/M/1 Queue
CATSR
0.018
Gamma
0.016
A single server
0.014
Normal
0.012
Inter-arrival times
follow an
exponential
distribution
(or arrival process
is Poisson)
f(x)
0.01
Service times follow an
exponential distribution
L=
Avg. # in
System
1 r
0.006
0.004
Exponential
0.002
0
40
60
80
100
120
140
x
50
45
40
35
L
r
0.008
30
25
20
15
10
5
0
0
0.2
0.4
0.6
r
0.8
1
1.2
160
180
200
220
240
The M/M/1 Queue
CATSR
• Observations
– 100% utilization is not desired
• Limitations
– Model assumes steady-state. Solution does not exist when r > 1 (arrival
rate exceed service rate).
– Poisson arrivals can be a reasonable assumption
– Exponential service distribution is usually a bad assumption.
50
45
40
L=
r
1 r
35
30
L
25
20
15
10
5
0
0
0.2
0.4
0.6
r
0.8
1
1.2
The M/G/1 Queue
CATSR
Service times follow a
general distribution
Required inputs:
• l:
arrival rate
• 1/m: expected service time
• :
std. dev. of service time
35
30
L
25
20
15
10
5
0
0
r l 
L=r
2(1  r )
2
Avg. # in
System
2
2
0.2
0.4
0.6
r
0.8
1
1.2
m = 1,  = 0.5
M/G/1: Effect of Variance
CATSR
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16
14
12
L
Deterministic
Service
Exponential
Service
10
8
Arrival Rate
Service Rate
Held
Constant
6
4
2
0
0
r 2  l2 2
L=r
2(1  r )
0.5
1
1.5

2
2.5
l = 0.8, m = 1 (r = 0.8)
3
Other Queues
CATSR
• G/G/1
– No simple analytical formulas
– Approximations exist
• G/G/∞
– Infinite number of servers – no wait in queue
– Time in system = time in service
• M(t)/M(t)/1
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– Arrival rate and service rate vary in time
– Arrival rate can be temporarily bigger than service
rate
Queueing Theory Summary
CATSR
• Strengths
– Demonstrates basic relationships between delay and
statistical properties of arrival and service processes
– Quantifies cost of variability in the process
– Analytical models easy to compute
• Potential abuses
–
–
–
–
Only simple models are analytically tractable
Analytical formulas generally assume steady-state
Theoretical models can predict exceptionally high delays
Correlation in arrival process often ignored
• Simulation can be used to overcome limitations
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GreenSim
14
CATSR
GreenSim Input/Output Model
User adjustable input:
GreenSim Airport
· Flights Schedule
Operations
· Aircraft type
Simulation
· Capacity (AAR, ADR)
15
CATSR
Outputs:
· Delays (taxi-in, taxi-out)
· ADOC
· Emission
· Departure Runway
Queue Size
· Gate Utilization
GreenSim Architecture
Data Analysis
Simulation
Actual taxi-in and
taxi-out times
Error
report
-
CATSR
Airport Performance
Analysis
Simulation taxi-in
and taxi-out times
ASPM
Databases
Individual
Airport
Service Times Settings
Data
processing
Data
processing
AAR, ADR
Capacity Settings
Arrival and Departure
Schedules
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Airport
Settings
External
demands
Airport
simulation
Queueing
model
Taxi-in, taxi-out
times
Emission
Database
Airport
performance
enhancement
model
Environmental
analysis
model
Data Analysis Process
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CATSR
Queueing Simulation Model
CATSR
NAS
lA
Runway
G / G /  /  / FCFS Runway
m
m DR
AR
G / G /  /  / FCFS
Taxiway Taxi-in
m AT times
Turn
around
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G / G / 1 /  / FCFS
Taxi-out
Taxiway G / G /  /  / FCFS
times
m DT
Ready to
depart
reservoir
lD
Service Times Settings
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CATSR
Segment Name
Settings
Arrival Runway
S1~Exponential(1/AAR)
Arrival Taxiway
S2~NOMTI + DLATI- S1
DLATI~Normal (u1 ,1 )
Departure
Taxiway
Departure
Runway
S3~NOMTO + DLATO- S4
DLATO~Normal (u2 , 2 )
S4~Exponential(1/ADR)
Notation
Performance Analysis
CATSR
• Delays (individual, quarterly average, hourly
average, daily average)
• Fuel Fuel =  (TIM j )  ( FFj /1000)  ( NEj )
j
• Emission (HC, CO, NOx, SOx)
Emissioni =  (TIM j )  ( FFj /1000)  ( NEj )  EIij
j
TIMj
FFj
NEj
EIij
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= taxi time for type-j aircraft
= fuel flow per time per engine for type-j aircraft
= number of engines used for type-j aircraft
= emissions of pollutant i per unit fuel consumed
for type-j aircraft
EWR Hourly Average Delays
21
CATSR
CATSR
EWR Quarterly-Hour Average Delays
23
CATSR
EWR Daily Average Delays
24
CATSR
CATSR

You cannot always replace a random variable with
its average value.
25
The M/M/1 Queue
A single server
Service times follow an
exponential distribution
0.018
Gamma
0.016
0.014
0.012
Normal
0.01
f(x)
Inter-arrival times
follow an
exponential
distribution
(or arrival process
is Poisson)
CATSR
0.008
0.006
0.004
Exponential
0.002
0
40
60
80
100
120
140
x
160
180
200
220
240
Poisson Process
CATSR
• For many queueing systems, the arrival process is
assumed to be a Poisson process.
• There are good reasons for assuming a Poisson process
– Roughly, the superposition of a large number of independent
(and stationary) processes is a Poisson process.
• However, the assumption is over-used.
• For a Poisson process, inter-arrival times follow an
exponential distribution.
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Notation for Queues
CATSR
A/B/C/D/E
A=interarrival time distribution
B=service time distribution
C= Number of servers
M for Markovian
(exponential) distribution
G for General
(arbitrary) distribution
D=Queueing Size Limit
E=Service Discipline(FCFS, LCFS, Priority, etc.)
.
Examples
M/M/1
M/G/1/K
M/G/1
M/G/1/Infinity/Priority
Poisson Distribution
CATSR
• Ladislaus Bortkiewicz, 1898
• Data
– 10 Prussian army corps units observed over 20 years (200 data points)
– A count of men killed by a horse kick each year by unit
– Total observed deaths: 122
• Number of deaths (per unit per year) is a Poisson RV with
mean 122 / 200 = 0.61.
120
Occurrences
100
80
Theoretical
60
Observed
40
20
0
0
1
2
3
4
5
Deaths per Unit per Year
6
Number
0
1
2
3
4
5
6
Theoretical
108.67
66.29
20.22
4.11
0.63
0.08
0.01
Observed
109
65
22
3
1
0
0
Performance Analysis
CATSR
• Delays (individual, quarterly average, hourly
average, daily average)
• Fuel Fuel = (TIM )  (FF /1000)  ( NE )
• Emission (HC, CO, NOx, SOx)
j
j
j
j
Emissioni =  (TIM j )  ( FFj /1000)  ( NEj )  EIij
j
• Cost (Directing Operation Cost and Delay
Cost)
Total DOC = Fuel Cost  Operational Cost
= Fuel Cost  Taxi - in Operation Cost  Taxi - out Operation Cost
= Fuel  Fuel Price  (Taxi - in  Taxi - out times)  DOCPrice
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Where
DOCPrice = (Labor Maintenance  Owership Other) = 38.24$/min
Some Common Distributions
Form of Probability
Density Function (PDF)
0.018
Gamma
0.016
CATSR
Normal
0.014
f ( x) =
0.012
Normal
0.01
2
2
1
e ( x  m ) /(2 )
 2
f(x)
Gamma
f ( x) =
0.008
0.006
1
  x 1e x / 
( )
Exponential
0.004
f ( x) = lel x
Exponential
0.002
0
40
60
80
100
120
140
x
Mean = 106
Std. Dev. = 26.9 (normal and gamma)
160
180
200
220
240
Some Common Distributions
CATSR
Form of Probability
Density Function (PDF)
Normal
f ( x) =
0.018
Gamma
0.016
2
2
1
e ( x  m ) /(2 )
 2
0.014
Normal
0.012
Gamma
f(x)
0.01
0.008
f ( x) =
0.006
0.004
Exponential
0.002
1
  x 1e x / 
( )
0
40
60
80
100
120
140
160
180
200
220
240
x
Exponential
f ( x) = lel x
Mean = 106
Std. Dev. = 26.9 (normal and gamma)
Queueing Theory
CATSR
Queueing Theory: The theoretical study of
waiting lines, expressed in mathematical terms
• How long does a customer wait in line?
• How many customers are typically waiting in
line?
• What is the probability of waiting longer than x
seconds in line?
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Queueing 101
Queue
Customers
CATSR
Teller
Need vs. Capability
• Mismatch between demand for analysis and
capability of existing tools
• Analysis Capability:
• Airport as Single Queue (Odoni)
– Not enough resolution
• Airport as discrete event simulation (TAAM, Simmod)
– Too much resolution requires:
• Time-consuming detailed rules for individual flight behavior
• Long run times
CATSR
The M/G/1 Queue
CATSR
Service times follow a
general distribution
Required inputs:
• l:
arrival rate
• 1/m: expected service time
• :
std. dev. of service time
35
30
L
25
20
15
10
5
0
0
r l 
L=r
2(1  r )
2
Avg. # in
System
2
2
0.2
0.4
0.6
r
0.8
1
1.2
m = 1,  = 0.5
M/G/1: Effect of Variance
CATSR
18
16
14
12
L
Deterministic
Service
Exponential
Service
10
8
Arrival Rate
Service Rate
Held
Constant
6
4
2
0
0
r 2  l2 2
L=r
2(1  r )
0.5
1
1.5

2
2.5
l = 0.8, m = 1 (r = 0.8)
3