MCMC algorithm for investigating variation in traffic flow - PowerPoint PPT Presentation

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

Examples in reality  Traffic counter data at inner city links  2002/January~March, weekdays, 8:00~9:00 750 1900 700 1850 650 1800 600 1750 550 1700 500 1650 450 1600 400 1550 350 Sakura-honmachi （ 3 lanes ） Yotsuya-dori （ 2 lanes ） 1500 300 Average: 554.9, s.d.: 63.3 Average: 1740.7, s.d.: 50.5 International Colloquium on Transportation 2003/11/01 Networks and Behaviour 3

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. International Colloquium on Transportation 2003/11/01 Networks and Behaviour 4

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 so the flow is stochastic. travellers, Markov chain Monte Carlo (MCMC) algorithm is used to solve the stochastic flow. International Colloquium on Transportation 2003/11/01 Networks and Behaviour 5

Objective of this study By using Hazelton’s Conditional SUE, variation of traffic flow is investigated in some cases.  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 International Colloquium on Transportation 2003/11/01 Networks and Behaviour 6

Traveller’s stochastic choice t 1 O D i t 2  Traveller’s stochastic choice depends on route travel time: Pr ( R i =r 1 | t 1 , t 2 )  Route travel time is a function of other travellers’ choice: t k = F ( r 1 , r 2 , . . , r i-1 , r i+1 , . . , r N )  Thus, traveller’s choice depends on other travellers’ choice: Pr ( R i =r 1 | R 1 , R 2 , . . , R i-1 , R i+1 , . . , R N ) International Colloquium on Transportation 2003/11/01 Networks and Behaviour 7

Stochastic traffic flow  Distribution of stochastic flow is a function of travellers’ choice: Pr ( R 1 , R 2 , . . . , R N ) .  We only know Pr ( R i =r 1 | R 1 , R 2 , . . , R i-1 , R i+1 , . . , R N ). How to find Pr ( R 1 , R 2 , . . . , R N ) from Pr ( R i =r 1 | R 1 , R 2 , . . , R i-1 , R i+1 , . . , R N )? MCMC algorithm samples state according to joint distribution using conditional distribution. International Colloquium on Transportation 2003/11/01 Networks and Behaviour 8

MCMC algorithm (1) For the initial state, assign arbitrary ( R 1 (0), … , R N (0)). (2) Re-assign R 1 ( j+ 1) probabilistically according to Pr ( R 1 ( j+ 1) | R 2 ( j ) , R 3 ( j ) , . . , R N ( j )) R 2 ( j+ 1) ~ Pr ( R 2 ( j+ 1) | R 1 ( j+ 1) , R 3 ( j ) , . . , R N ( j )) …… R N ( j+ 1) ~ Pr ( R N ( j+ 1) | R 1 ( j+ 1) , R 2 ( j+ 1) , . . , R N- 1 ( j+ 1)) (3) It is proved that the iterations of (2) reach to the probabilistic equilibrium state according to the joint distribution. International Colloquium on Transportation 2003/11/01 Networks and Behaviour 9

Functions used in this study  Modified BPR function is used for link cost.   5     x = +     ( ) 1 2 . 62 t x l     C    Multinomial logit function is for route choice. { } − θ exp ( | ) t r R = = − { } k i Pr( | ) R r R ∑ − − θ i k i exp ( | ) t r R − j i j International Colloquium on Transportation 2003/11/01 Networks and Behaviour 10

Example of simulation 4100  demand = 8000, 4080 θ = 0.5, l = 5km, 4060 C = 4000 4040 4020 Link volume O D 4000 3980 3960 Link volume 3940 3920 Average Variance 3900 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 4000 58.8 Iteration（×8000） Iterations until convergence are discarded. International Colloquium on Transportation 2003/11/01 Networks and Behaviour 11

Interdependency of travellers’ choice Simple 1 OD 2 links  θ = 0.5, l = 5km, demand = 8000, C = 4000 O D Comparison between  Pr ( R i =r 1 | R 1 , R 2 , . . , R i-1 , R i+1 , . . , R N ), and  Independent choice: Pr ( R i =r 1 | E ( t 1 ) , E ( t 2 )) International Colloquium on Transportation 2003/11/01 Networks and Behaviour 12

Interdependency of travellers’ choice Inter-dependent choice Independent choice 4150 4150 4100 4100 4050 4050 Link volume Link volume 4000 4000 3950 3950 3900 3900 3850 3850 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Iteration（×8000） Iteration（×8000）  Over-estimate of variation if independence is assumed. International Colloquium on Transportation 2003/11/01 Networks and Behaviour 13

Case 1: Effects of demand, capacity, and scale parameter O D Lower Base Upper bound case bound Demand 4000 ～ 8000 ～ 16000 C 2000 ～ 4000 ～ 8000 l （ km ） 5 θ （ 1/min ） 0.1 ～ 0.5 ～ 1 International Colloquium on Transportation 2003/11/01 Networks and Behaviour 14

Effect of demand s.d. of link volume s.d. of travel time 20 0.9 Standard deviation of travel time Standard deviation of link volume 18 0.8 16 0.7 14 0.6 12 0.5 10 0.4 8 0.3 6 0.2 4 0.1 2 0 0 0 5000 10000 15000 20000 0 5000 10000 15000 20000 Demand Demand  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. International Colloquium on Transportation 2003/11/01 Networks and Behaviour 15

Effects of demand and capacity s.d. of link volume by demand s.d. of link volume by capacity 20 30 Standard deviation of link volume Standard deviation of link volume 18 25 16 14 20 12 10 15 8 10 6 4 5 2 0 0 0 5000 10000 15000 20000 0 2000 4000 6000 8000 10000 Demand Capacity  Effect of capacity is opposite to that of demand as expected. International Colloquium on Transportation 2003/11/01 Networks and Behaviour 16

Effect of scale parameter s.d. of link volume s.d. of travel time 0.4 14 Standard deviation of travel time Standard deviation of link volume 12 0.3 10 8 0.2 6 4 0.1 2 0 0 0 0.5 1 0 0.5 1 Scale parameter:θ（1/min） Scale parameter: θ（1/min）  Both fluctuations of link flow and travel time decrease according to the scale parameter. International Colloquium on Transportation 2003/11/01 Networks and Behaviour 17

Case 2: 1 OD 3 routes with overlapping  θ = 0.1, OD length = 5km, demand = 8000, C = 4000 link2 link1 link3 O D link4  Examine the effect of the length of link 1 International Colloquium on Transportation 2003/11/01 Networks and Behaviour 18

Consistency between CSUE and SUE Average link volume of CSUE SUE 5000 5000 4000 4000 Average link volume Average link volume 3000 3000 Link 1 Link 1 Link 2 Link 2 2000 2000 Link 3 Link 3 Link 4 Link 4 1000 1000 0 0 0 1 2 3 4 5 0 1 2 3 4 5 Length of link 1 (km) Length of link 1 (km)  Average link flow of CSUE is consistent with link flow of SUE. International Colloquium on Transportation 2003/11/01 Networks and Behaviour 19

Effect on link volume Average link volume s.d. of link volume 40 5000 35 30 4000 Average link volume 25 s.d. 3000 20 Link 1 Link 1 15 Link 2 2000 Link 2 Link 3 10 Link 3 Link 4 1000 5 Link 4 0 0 0 1 2 3 4 5 0 1 2 3 4 5 Length of link 1（km） Length of link 1 (km)  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. International Colloquium on Transportation 2003/11/01 Networks and Behaviour 20

Effect on speed Average speed s.d. of speed 60 0.7 Link 1 Link 2 Average speed（km/h） 0.6 50 Link 1 Link 3 0.5 s.d. of speed Link 2 40 Link 4 0.4 Link 3 30 Link 4 0.3 20 0.2 10 0.1 0 0 0 1 2 3 4 5 0 1 2 3 4 5 Length of link 1 （km） Length of link 1 （km） International Colloquium on Transportation 2003/11/01 Networks and Behaviour 21

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 Highway (¥450) O D Arterial   5     450 x = × + +     ( ) 5 ( 60 / 100 ) 1 2 . 62 t x ( ) ( / ) highway km m km     4000 VOT   International Colloquium on Transportation 2003/11/01 Networks and Behaviour 22