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BEHAVIOURAL FOUNDATIONS FOR THE TWO-‐FLUID MODEL
Vinayak V. Dixit School of Civil and Environmental Engineering
University of New South Wales
The Two-‐Fluid Model The two-‐fluid model describes the relaDonship between the running Dme per mile and travel Dme per mile of a vehicle in an urban network. This model was developed based on ‘parDcle physics’ and lacked the understanding of behavioral significance.
( )11 1
nm r
nn n
m
v v f
T T Tr + +
=
=Stop time, Ts (min/km)
The Two-‐Fluid Model Two-‐fluid model has been used to characterize:
– Traffic flow on urban networks. [Herman and Prigogine (1971), Ardekani (1984)].
– Traffic flow on urban arterials. [Jones and Wahid (2003)]
– Individual driver behavior, [Herman, Malakhoff and Ardekani (1988)]
– Safety [Dixit et al. (2009)]
MoDvaDon: Two-‐Fluid Model
(Herman et al., 1988)
Weather Effects
Time of Day Effects
Dixit, V., Gayah, V., and Radwan, E. (2012). ”Comparison of Driver Behavior by Time of Day and Wet Pavement CondiDons.” J. Transp. Eng., 138(8), 1023–1029.
CorrelaDon • NegaDve correlaDon between Tm and n • As n increases the correlaDon with
crash severity increases.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Rate of No-InjuryCrashes
Rate of Possible InjuryCrashes
Rate of Non-Incapicitating Injury
Crashes
Rate of Severe Crashes
Severity of Crash
Cor
rela
tion
with
n
Risk of Driving
Descrip4ve Psychological Theories Task difficulty Homeostasis-‐ Risk Homeostatsis
Individual Driving Behavior
• Driving can be described in the form of a state-‐dependent approach. – S = {crash, no crash}.
• The uDlity of being in state “no crash” is associated to reaching the desDnaDon as quickly as possible.
• The (dis)uDlity of being in the state “crash” is the associated to severity of the crash.
Model AssumpDons
• Perceived Probability of Crashing "↓$%&'ℎ =*(,/,↓% )↑0 – Conforms with Empirical Findings by Elvik et al. (2004)
• DisuDlity of crashing – Related to perceived severity (kineDc energy)
• UDlity of Not Crashing – Travelling faster
( ) ( )ku crash w vr= −
( ) u no crash v=
Expected UDlity For A Driver
Where
( ) ( ) EU P u no crash P u crashno crash crash= × + ×
(1 ) ( )kEU f v f w vr r rβ β
α α= − −
( ) ( ) ( )(1 ) ( )kEU v v v w v vr rβ β β β
α α− −
= − −
vfr
vr=
UDlity MaximizaDon of Driver ( ) ( ) ( )1 1 1
( ) ( ) 0EU k
v v k w v vr rvr
β β β βαβ α β
∂ − − + − −= − − =
∂ 11 1
1 ( ) 11
kk
k w k kv vr
ββ
−⎛ ⎞
⎛ ⎞ ⎜ ⎟⎛ ⎞− ⎛ ⎞−⇒ =⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠⎝ ⎠ ⎜ ⎟⎝ ⎠
Solution exists if k β>
PcrashP v vr rcrash
β∂= −
∂
( )( )u crash k
u crash v vr r
∂=
∂
The marginal rate of change for the perceived disutility
is larger than
the marginal rate of change for the perceived probability to crash.
Gulf Coast Center for Evacuation and Transportation Resiliency
UDlity MaximizaDon of Driver
1(1 )
ndT w n n wmdn n n
β
β β
−− + −
⇒ =⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
11( 1 ) 1
nn nn n w nT Tr nββ
⎛ ⎞ +⎛ ⎞+ − +⎜ ⎟= ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
(1 )n
n n wTm
n
β
β
+ −⇒ =
⎛ ⎞⎜ ⎟⎝ ⎠
In order to get the form of the traditional two-fluid model. To ensure k>1 Substitute k=(n+1)/n
11 1
nn nT T Tr m+ +=
Comparing:
Tm and n are negatively correlated
RelaDonship Between Tm and n
• Using Data from 1983, 1990 and 1991 from the ciDes of Dallas, Forth Worth, Arlington, AusDn, Lubbock, Houston, San Antonio, Albuquerque, Mexico City and Matamoros. (Ardekani, 1981)
• Tm and n have a nega4ve correla4on of -‐0.47
• Test validity of ( )1 1 1n w
T wm nβ β= + −
⎛ ⎞⎜ ⎟⎝ ⎠
Empirical ValidaDon Urban Network Data
0
0.5
1
1.5
2
2.5
3
3.5
4
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
(Tm
)(1/n
)
1/n
Effect of Network Features
k=0.75-‐1.169 (fraction of one-‐way)+0.147 (#Lanes) +0.005 (Intersection Density)+0.502 (density of actuated signals) R2=0.58
severity factor (k)
crash likelihood factor (0) β=1.075-‐0.295 (fraction of one-‐way) @↓A↑ 1/B =( 2.081/1.075−0.295 C↓2 )( B+1/B )−2.081 R2=0.89
Two-‐Fluid Model Arterials
0
1
2
3
4
5
6
7
0 1 2 3 4 5
Ts (Stopped Time/Mile)
SR 50 West of SR434
SR 50 (SR434-SR436)
SR 50 East of SR434
SR 426 SR 551
SR 436 (South Section)
SR 434
SR 436 (North Section)
T (T
rave
l Tim
e/M
ile)
(Tm) (1/n) = 0.6276 (1/n) + 0.3566 R² = 0.92
0
0.2
0.4
0.6
0.8
1
1.2
1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9
(Tm) 1/
n
1/n
Arterial Data (Weak Evidence)
k=-‐2793.56 (total crash rate) + 5.35 R2=0.60 @↓A↑1/B =1.476(B+1/B )−1.83 R2=0.92
Physics of Traffic Flow
• Models based on physics of parDcles and fluid – Finng models from fluid dynamics and parDcle physics to explain traffic behavior
– Useful for engineering
• Models based on explicitly assuming behavior (risk antudes and UDlity Models) – Enhances understanding for safety
Example: Fundamental Diagram
Story of the hare and slugs – Hares are aggressive and maintain
shorter gaps and therefore greater flows – In congesDon, flows are constrained
Flow
Density
TRADITIONAL THEORY
Flow
Density
OBSERVED
Banks, James H.; Amin, Mohammad R.; Cassidy, Michael; Chung, Koohong “ValidaDon of Daganzo's Behavioral Theory of MulD-‐Lane Traffic Flow” California Partners for Advanced Transit and Highways (PATH), UC Berkley Final Report, 2003
Dominance of physics of traffic, with systemaDc addiDon of behavioural parameters
Conclusion • This study puts the two-‐fluid model from a behavioral
perspecDve. • The condiDon that k> β is a necessary condiDon for the two-‐
fluid model to exist. – On freeways this might not exist (the perceived probability to crash
might increase at a larger rate than the perceived uDlity to crash.) • EvaluaDon of training and educaDonal programs for new
drivers. • The two-‐fluid model can be used on corridors to evaluate
safety. • The uDlity model has the potenDal of being used to engineer
human driving behavior. (IncenDves, disincenDves and Insurance)