Transcript Landing Safety Analysis Using an M/G/1 Queuing Model

Landing Safety Analysis of An
Independent Arrival Runway
Author: Richard Yue Xie
John Shortle
Presented by: Dr. George Donohue
22/11/2004
ICRAT 2004
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CENTER FOR AIR TRANSPORTATION SYSTEMS RESEARCH
Problem Statement
• Growth of traffic demand requires more capacity
both of airports and airspace.
• Separation reduction is an effective way of
increasing capacity.
• How will safety be affected?
• What’s the current safety level?
• What are major factors that will affect safety?
• How?
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Safety-Capacity Hypothesis
More Safe
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Safety/Capacity
IT Extension
Safety
(Departures / Hull Loss)
Less Safe
Low
[Donohue et al., 2001]
[Shortle et al. 2004]
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Capacity
(Departures / Year)
High
Safety Issues Considered
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Simultaneous runway
occupancy/landing
Runway collision/landing
Incidents
Accidents
Wake-vortexencounter/approach
Loss of control/approach
Airplane i+2
Airplane i+1
Wake Vortex Encounter
Simultaneous Runway Occupancy
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Airplane i
Key Safety Metrics
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Ease of predicting
Loss of wake vortex separation
Wake vortex encounter
Simultaneous runway occupancy
Incidents
Loss of control
due to turbulence
Runway collision
Accidents
Metric relevance
This paper focuses on
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Data Samples of Landing Time Interval
Loss Safety
Loss Capacity
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Courtesy of Haynie, Doctoral dissertation, GMU, 2002
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An Observation of ATL Landing Runways
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Mar.5 and 6, 2001 ATL 26L, 364 valid data
(Haynie, 2002)
ROT
LTI
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LTI:
Landing Time Interval
ROT:
Runway Occupancy Time
SRO:
Simultaneous Runway Occupancy
Indicate a positive probability of Simultaneous
Runway Occupancy.
Analytical Model Vs. Simulation Model
• Advantages of analytical models:
•
•
•
•
Computational efficiency
Consistency
Clarity
Accuracy
• Disadvantages of analytical models:
• Limited applicability
• Over simplification
• Dependencies
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A Queuing Model for Safety Analysis
TRACON – Final Approach
Final Approach - Runway
aircraft
separation
aircraft
separation
TRACON
RWY Threshold
RWY Exit
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Simplification in the Model
• Fleet mixture is not explicitly modeled (In VFR, separation
differences for different mix are not remarkable);
• Arrival process is approximated using a Poisson process,
although not justified theoretically;
• Service time, which is the desired separation, can be
approximated using a Gaussian distribution.
• Runway occupancy time follows a Gaussian distribution
N(48,82) seconds.
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Model Validation
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•Simulation results of an M/G/1 model shows good consistency
with observations. Arrival rate 29 acft/hour, G is Gaussian(80,112)
in second.
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Analytical Evaluation of Safety
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• Prob(SRO) = Prob(LTI* < ROT) =
Prob(LTI-ROT <0)
• LTI is the inter-departure time of the M/G/1**
queuing model.
• LTI is a function of M and G.
• LTI’s distribution = ?
* LTI: Landing Time Interval
** M means the arrival process is a Poisson process; G means the service time
follows a non-exponential distribution.
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Departure Process of A Queue
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1. If server is busy, inter-departure time is the same as service time;
2. If server is idle, inter-departure = inter-arrival + service
observer
aircraft
separation
t
pd 2  [ p1  p2 ](t ) 
p
2
(h) p1 (t  h)dh
0
pd= prob(server busy)*pd1+prob(server idle)*pd2
For an M/M/1 queue:
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pd (t)=et
r<1
 ueut
r>1
Service time in M/G/1
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A Gaussian distribution can be approximated by a finite sum
of xkexp(-ux).
P   q   
P is transition matrix,
q is exit vector
 is a vector of 1’s
Define the completion rate matrix M as a
diagonal matrix with elements Mii = mi , where
mi is the rate of leaving state i.
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Service rate matrix B is
B  M (1  P)
Service time matrix V is
V = B-1
Distribution of Service Time
Define the operator [X] = pX' , p is the entrance
vector
CDF of service time:
S (t )  1  [exp(tB)]
PDF of service time:
dS (t )
s (t ) 
 [ B exp(tB)]
dt
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Inter-Departure Time in M/G/1
 0 p
Pd  

0'
P



Md   '
0
   p 
Bd  M d ( I d  Pd )   '
0
B 
Vd  Bd 1
d (t )  s(t )  (1  r )

 [( I  V )1 ] e x  [( I  V ) 1 B exp( xB)]
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
1
 
 '
0
0
M 

pV 

V 
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Inter-Departure Time in M/G/1
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d (t )  s(t )  (1  r )

 [( I  V ) ] e
1
 x
 [( I  V ) B exp( xB)]
1
If r goes to 1, d(t) = s(t).
If r goes to 0, (1-V) is close to 1, the inter-departure time
will distribute like the inter-arrival time with the density
function e-x.
For more information, please refer to
Xie and Shortle, Landing Safety Analysis of An Independent Arrival
Runway, ICRAT, 2004;
Lipsky, Queuing Theory – A Linear Algebraic Approach, 1992.
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
Landing Time Interval Distributions
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Erlang’s
Mean Std.dev
70.7
9.7
118
15.4
70
8.4
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Prob.(SRO)
Prob(SRO)=Prob(LTI<ROT)
0 


f LTI ( x  z ) f ROT ( x)dxdz
 
Parameter values:
Inter-arrival: exponential(124 sec)
Mean of desired separation: 80 sec, std.dev is 11 sec.
Mean of runway occupancy time is 48 sec., std.dev is 8 sec.
The calculated probability of SRO is 0.00312.
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Factors That Affect Safety
Arrival rate
Mean and variance of
desired separation
Mean and variance of
runway occupancy time
Other incidents, e.g.
human error, equipment
failure,…
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Landing safety
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How ROT Affects Safety
Mean and variance of
runway occupancy time
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ROT: Runway Occupancy Time
Landing safety
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How Separation Affects Safety
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Prob.(SRO)
Mean and variance of
desired separation
Std.Dev of desired
separation (sec.)
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Landing safety
Mean of desired separation (sec.)
Separation Vs. Safety
0.14
Separation
Deviation (sec.):
Prob(SRO)
0.12
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5
7
9
11
0.1
0.08
0.06
0.04
0.02
0
-0.02 40
50
60
70
80
90
Mean(Desired Separation) (sec.)
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100
Landings/SRO
Capacity-Safety
2400
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7
11
15
Separation
Deviation (sec.):
2200
2000
9
13
1800
1600
1400
1200
1000
800
Here we are!
600
400
200
0
45
46
47
48
49
50
Landings/hour
24
51
52
53
54
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Safety-Capacity Hypothesis
More Safe
CATSR
Safety/Capacity
IT Extension
Safety
(Departures / Hull Loss)
Less Safe
Low
[Donohue et al., 2001]
[Shortle et al. 2004]
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Capacity
(Departures / Year)
High
Summary
• An M/G/1 queuing model can effectively
represent a randomly,unsynchronizedly
scheduled airport’s arrival process.
• Landing safety is significantly affected by
variances of runway occupancy time and
separation
• Both average value and variance should be
considered in policy making.
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