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MCMC algorithm for investigating variation in traffic flow Toshiyuki Yamamoto Nagoya University Introduction Needs for complex traffic control and emerging driver information systems have led to increasing interests in: Stochastic

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MCMC algorithm for investigating variation in traffic flow

Toshiyuki Yamamoto Nagoya University

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2003/11/01 International Colloquium on Transportation Networks and Behaviour 2

Introduction

 Needs for complex traffic control and

emerging driver information systems have led to increasing interests in:

 Stochastic elements to account for errors in

drivers’ perceptions, and

 Day-to-day variation in behaviour.

 Stochastic user equilibrium, despite the name,

forms a fixed flow pattern, thus unable to represent variations in traffic flow.

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Examples in reality

 Traffic counter data at inner city links

 2002/January~March, weekdays, 8:00~9:00

1500 1550 1600 1650 1700 1750 1800 1850 1900 300 350 400 450 500 550 600 650 700 750

Yotsuya-dori（2 lanes） Sakura-honmachi（3 lanes） Average: 554.9, s.d.: 63.3 Average: 1740.7, s.d.: 50.5

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Stochastic user equilibrium

 A traveller selects the route which he/she

perceives to have minimum cost, including errors.

 The traveller chooses the route stochastically.  Traffic flow results from the choices of the

travellers, so the flow should be stochastic. By The Weak Law of Large Numbers, the flow gets closer to a fixed pattern.

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Conditional SUE by Hazelton (1996)

 A traveller selects the route which he/she

perceives to have minimum cost, including errors.

 The traveller chooses the route stochastically.  Traffic flow results from the choices of the

travellers, so the flow is stochastic. Markov chain Monte Carlo (MCMC) algorithm is used to solve the stochastic flow.

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Objective of this study

 Variation of traffic flow:

 Link flow  Link travel time  Link speed

 Cases:

 Different demand, capacity, and scale parameter  1 OD 3 routes with overlapping  Heterogeneous value of time

By using Hazelton’s Conditional SUE, variation

f traffic flow is investigated in some cases.

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Traveller’s stochastic choice

 Traveller’s stochastic choice depends on route

travel time: Pr(Ri=r1| t1 , t2)

 Route travel time is a function of other

travellers’ choice: tk= F(r1, r2, . . , ri-1, ri+1, . . , rN)

 Thus, traveller’s choice depends on other

travellers’ choice: Pr(Ri=r1| R1, R2, . . , Ri-1, Ri+1, . . , RN)

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t2 t1

i

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Stochastic traffic flow

 Distribution of stochastic flow is a function of

travellers’ choice: Pr(R1, R2, . . . , RN).

 We only know Pr(Ri=r1| R1, R2, . . , Ri-1,

Ri+1, . . , RN). How to find Pr(R1, R2, . . . , RN) from Pr(Ri=r1| R1, R2, . . , Ri-1, Ri+1, . . , RN)? MCMC algorithm samples state according to joint distribution using conditional distribution.

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MCMC algorithm

(1) For the initial state, assign arbitrary (R1(0), … , RN(0)). (2) Re-assign R1(j+1) probabilistically according to Pr(R1(j+1)| R2(j), R3(j), . . , RN(j)) R2(j+1) ~ Pr(R2(j+1)| R1(j+1), R3(j), . . , RN(j)) …… RN(j+1) ~ Pr(RN(j+1)| R1(j+1), R2(j+1), . . , RN-1(j+1)) (3) It is proved that the iterations of (2) reach to the probabilistic equilibrium state according to the joint distribution.

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Functions used in this study

 Modified BPR function is used for link cost.  Multinomial logit function is for route choice.

                + =

62 . 2 1 ) ( C x l x t

{ }

∑

− − −

− − = =

j i j i k i k i

R r t R r t R r R ) | ( exp ) | ( exp ) | Pr( θ θ

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Example of simulation

 demand = 8000,

θ = 0.5, l = 5km, C = 4000

3900 3920 3940 3960 3980 4000 4020 4040 4060 4080 4100 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Iteration（×8000） Link volume

O D

Average Variance 4000 58.8 Iterations until convergence are discarded. Link volume

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Interdependency of travellers’ choice

Simple 1 OD 2 links

 θ = 0.5, l = 5km, demand = 8000, C = 4000

Comparison between

 Pr(Ri=r1| R1, R2, . . , Ri-1, Ri+1, . . , RN), and  Independent choice: Pr(Ri=r1| E(t1), E(t2))

O D

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Interdependency of travellers’ choice

 Over-estimate of variation if independence is assumed.

3850 3900 3950 4000 4050 4100 4150 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Iteration（×8000） Link volume

Inter-dependent choice Independent choice

3850 3900 3950 4000 4050 4100 4150 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Iteration（×8000） Link volume

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Case 1: Effects of demand, capacity, and scale parameter

Lower bound Base case Upper bound Demand 4000 ～ 8000 ～ 16000 C 2000 ～ 4000 ～ 8000 l （km） 5 θ （1/min） 0.1 ～ 0.5 ～ 1

O D

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Effect of demand

2 4 6 8 10 12 14 16 18 20 5000 10000 15000 20000 Demand Standard deviation of link volume

 Congestion negates the fluctuation of link volume, but

increases variation of travel time because even the small change in volume causes large change in travel time. s.d. of link volume s.d. of travel time

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 5000 10000 15000 20000 Demand Standard deviation of travel time

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Effects of demand and capacity

2 4 6 8 10 12 14 16 18 20 5000 10000 15000 20000 Demand Standard deviation of link volume

 Effect of capacity is opposite to that of demand as

expected. s.d. of link volume by demand s.d. of link volume by capacity

5 10 15 20 25 30 2000 4000 6000 8000 10000 Capacity Standard deviation of link volume

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Effect of scale parameter

2 4 6 8 10 12 14 0.5 1 Scale parameter:θ（1/min） Standard deviation of link volume 0.1 0.2 0.3 0.4 0.5 1 Scale parameter: θ（1/min） Standard deviation of travel time

 Both fluctuations of link flow and travel time decrease

according to the scale parameter. s.d. of link volume s.d. of travel time

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Case 2: 1 OD 3 routes with overlapping

 θ = 0.1, OD length = 5km, demand = 8000, C

= 4000

 Examine the effect of the length of link 1

O D

link1 link2 link3 link4

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Consistency between CSUE and SUE

1000 2000 3000 4000 5000 1 2 3 4 5 Length of link 1 (km) Average link volume Link 1 Link 2 Link 3 Link 4

 Average link flow of CSUE is consistent with link flow of

SUE. Average link volume of CSUE SUE

1000 2000 3000 4000 5000 1 2 3 4 5 Length of link 1 (km) Average link volume Link 1 Link 2 Link 3 Link 4

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Effect on link volume

5 10 15 20 25 30 35 40 1 2 3 4 5 Length of link 1（km） s.d. Link 1 Link 2 Link 3 Link 4

 Average link volumes of link 1 and 4 are different, but s.d.

are identical because of the negative perfect correlation between the link volumes of the two links. Average link volume s.d. of link volume

1000 2000 3000 4000 5000 1 2 3 4 5 Length of link 1 (km) Average link volume Link 1 Link 2 Link 3 Link 4

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Effect on speed

0.1 0.2 0.3 0.4 0.5 0.6 0.7 1 2 3 4 5 Length of link 1 （km） s.d. of speed Link 1 Link 2 Link 3 Link 4

s.d. of speed

10 20 30 40 50 60 1 2 3 4 5 Length of link 1 （km） Average speed（km/h） Link 1 Link 2 Link 3 Link 4

Average speed

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Case 3: Heterogeneous value of time

 Heterogeneity in value of time across

travellers causes no problem in CSUE.

 θ = 0.1, l = 5km, demand = 8000, C = 4000

O D

Highway (¥450) Arterial

VOT x x t

km m km highway

450 4000 62 . 2 1 ) 100 / 60 ( 5 ) (

5 ) / ( ) (

+                 + × =

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Distribution of value of time

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 35 40 45 50 55 60 65 70 75 80 85 90 95 Value of time (average = 62.5) (\/min) Percent s.d. = 0.1 s.d. = 2.0 s.d. = 4.0 s.d. = 6.0 s.d. = 8.0 s.d. = 9.9 s.d. = 10

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Effect on link volume

 Standard deviation of link volume looks fluctuated across

standard deviation of VOT, but fluctuation is small. s.d. of link volume Average link volume

3992 3994 3996 3998 4000 4002 4004 4006 4008 2 4 6 8 10 s.d. of value of time Average link volume

Arterial Highway 17.7 17.75 17.8 17.85 17.9 17.95 18 18.05 18.1 2 4 6 8 10 s.d. of value of time s.d. of link volume Arterial Highway

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Effect on generalized cost (min.)

 Something wrong with the program? Need verification?

s.d. of generalized cost Average generalized cost

17.95 18 18.05 18.1 18.15 18.2 18.25 2 4 6 8 10 s.d. of value of time Generalized cost (min) Arterial Highway

0.05 0.1 0.15 0.2 0.25 0.3 0.35 2 4 6 8 10 s.d. of value of time s.d. of travel time

Arterial Highway

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Future research

 Link nested logit model to investigate

correlation of traffic flows of overlapping routes

 Extension to elastic demand (integrated

mode choice and assignment)

 Variation of traffic flow caused by both mode

choice and route choice

 Extension to dynamic user equilibrium

 No problem with uniqueness of the equilibrium

 Application to real networks

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References

 Hazelton, M.L. , S. Lee and J.W. Polak

(1996) Stationary states in stochastic process models of traffic assignment : a Markov chain Monte Carlo approach，Proceedings of the 13th ISTTT，341-357.

 Hazelton, M.L. (1998) Some remarks on

MCMC algorithm for investigating variation in traffic flow

Toshiyuki Yamamoto Nagoya University

Introduction

emerging driver information systems have led to increasing interests in:

drivers’ perceptions, and

forms a fixed flow pattern, thus unable to represent variations in traffic flow.

Examples in reality

Stochastic user equilibrium

perceives to have minimum cost, including errors.

travellers, so the flow should be stochastic. By The Weak Law of Large Numbers, the flow gets closer to a fixed pattern.

Conditional SUE by Hazelton (1996)

perceives to have minimum cost, including errors.

travellers, so the flow is stochastic. Markov chain Monte Carlo (MCMC) algorithm is used to solve the stochastic flow.

Objective of this study

By using Hazelton’s Conditional SUE, variation

Traveller’s stochastic choice

travel time: Pr(Ri=r1| t1 , t2)

travellers’ choice: tk= F(r1, r2, . . , ri-1, ri+1, . . , rN)

travellers’ choice: Pr(Ri=r1| R1, R2, . . , Ri-1, Ri+1, . . , RN)

O D

i

Stochastic traffic flow

travellers’ choice: Pr(R1, R2, . . . , RN).

Ri+1, . . , RN). How to find Pr(R1, R2, . . . , RN) from Pr(Ri=r1| R1, R2, . . , Ri-1, Ri+1, . . , RN)? MCMC algorithm samples state according to joint distribution using conditional distribution.

MCMC algorithm

Functions used in this study

                + =

62 . 2 1 ) ( C x l x t

{ }

{ }

∑

− − = =

R r t R r t R r R ) | ( exp ) | ( exp ) | Pr( θ θ

Example of simulation

θ = 0.5, l = 5km, C = 4000

O D

Interdependency of travellers’ choice

Simple 1 OD 2 links

Comparison between

O D

Interdependency of travellers’ choice

Case 1: Effects of demand, capacity, and scale parameter

O D

Effect of demand

Effects of demand and capacity

Effect of scale parameter

Case 2: 1 OD 3 routes with overlapping

= 4000

O D

Consistency between CSUE and SUE

Effect on link volume

Effect on speed

Case 3: Heterogeneous value of time

travellers causes no problem in CSUE.

O D

VOT x x t

450 4000 62 . 2 1 ) 100 / 60 ( 5 ) (

+                 + × =

Distribution of value of time

Effect on link volume

Effect on generalized cost (min.)

Future research

correlation of traffic flows of overlapping routes

mode choice and assignment)

choice and route choice

References

(1996) Stationary states in stochastic process models of traffic assignment : a Markov chain Monte Carlo approach，Proceedings of the 13th ISTTT，341-357.

stochastic user equilibrium, Transportation Research Part B, Vol. 32, 101-108.