Dynamics of Traffic Flows in Dynamics of Traffic Flows in Combined Day-to-day and With-Combined Day-to-day and With-

in Day Contextin Day Context

Chandra BalijepalliChandra BalijepalliITS, LeedsITS, Leeds

15-16 September 200415-16 September 2004

Objectives of this PresentationObjectives of this Presentation

To introduce the combined day-to-day and with-To introduce the combined day-to-day and with-in day context of dynamic traffic assignmentin day context of dynamic traffic assignment

To introduce the extended method of To introduce the extended method of approximationapproximation

To discuss the issues in computing the To discuss the issues in computing the parameters e.g., jacobians of travel time parameters e.g., jacobians of travel time functionsfunctions

To discuss some numerical resultsTo discuss some numerical results

The ContextThe Context

Day-to-day dynamics: drivers’ learning and Day-to-day dynamics: drivers’ learning and adjustingadjusting

With-in day dynamics: delays along the With-in day dynamics: delays along the route based on prevailing traffic conditionsroute based on prevailing traffic conditions

Not dealing with departure time choice Not dealing with departure time choice

Logit based route choice model coupled with MSA

Dynamic loading of the flows on the routes using a whole-link

model

Time varying demand

Initial cost vector

Rou

te

flow

s

Rou

te

Cos

tsAverage route flows, costs, outflow profile, travel time flow profile, etc

Route Choice Model

Dynamic Link Loading Model

Literature ReviewLiterature Review

Cantarella, G.E. and Cascetta, E. (1995) Dynamic Processes and Cantarella, G.E. and Cascetta, E. (1995) Dynamic Processes and Equilibrium in Transportation Networks: Towards a Unifying Theory, Equilibrium in Transportation Networks: Towards a Unifying Theory, Transportation ScienceTransportation Science 29(4)29(4), 305-329, 305-329

Davis, G.A. and Nihan, N.L. (1993) Large Population Davis, G.A. and Nihan, N.L. (1993) Large Population Approximations of a General Stochastic Traffic Assignment Model, Approximations of a General Stochastic Traffic Assignment Model, Operations ResearchOperations Research 41(1)41(1), 169-178, 169-178

Friesz, T.L., Bernstein, D., Smith, T.E., Tobin, R.L. and Wie,B.W. Friesz, T.L., Bernstein, D., Smith, T.E., Tobin, R.L. and Wie,B.W. (1993) A Variational Inequality Formulation of the Dynamic Network (1993) A Variational Inequality Formulation of the Dynamic Network User Equilibrium Problem, User Equilibrium Problem, Operations ResearchOperations Research 41(1)41(1), 179-191, 179-191

Hazelton, M. and Watling, D. (2004) Computation of Equilibrium Hazelton, M. and Watling, D. (2004) Computation of Equilibrium Distributions of Markov Traffic Assignment Models, Distributions of Markov Traffic Assignment Models, Transportation Transportation ScienceScience 38(3)38(3), 331-342, 331-342

The Extended Method of The Extended Method of ApproximationApproximation

Assume the drivers are indistinguishable and Assume the drivers are indistinguishable and rational in minimising their perceived travel costrational in minimising their perceived travel cost

t)n(r

t)1n(r

t)n(r UU t)n(

rt)1n(

r

t)n(r UU t)n(

rt)1n(

r

t)n(r UU

]UUProb[)(Up

ηUU

(n)Tk

(n)Tr

(n)Trr

(n)Tr

1)T(nr

(n)Tr

Perceived Travel Cost Measured

Travel Cost

Error in Perceived Travel Cost

Measured travel costs are updated usingMeasured travel costs are updated using

m = memory lengthm = memory lengthλλ = memory weighting = memory weighting

)1/()1()(s

1 0)()(s

mm

1j

1j

m

1j

)jn(1j1)1n(

FcU

The number of drivers taking each possible The number of drivers taking each possible route on day n during time period T, conditional route on day n during time period T, conditional on the weighted average of costs, is obtained as on the weighted average of costs, is obtained as

independently for T = 1,2,… independently for T = 1,2,…

qqTT = demand during time period t = demand during time period t

ppTT(.) = route choice probability vector (.) = route choice probability vector

))(,q(lMultinomia~ / 1)-(nTT1)-(n(n)T UpUF

Conditional MomentsConditional Moments

Then the expectation and variance of the Then the expectation and variance of the conditional distribution for each time period conditional distribution for each time period would be would be

]))()(())([diag( q )(Var

)( q E

')1n(T)1n(T)1n(TT)1n((n)T

1)-(nTT)1n(T)n(

UpUpUpUF

UpUF

Unconditional MomentsUnconditional MomentsBased on standard results, the unconditional first Based on standard results, the unconditional first moment is given as moment is given as

Davis and Nihan (1993) proved that the Davis and Nihan (1993) proved that the equilibrium distribution is approximately normal equilibrium distribution is approximately normal with its mean equal to the solution of SUE in with its mean equal to the solution of SUE in each time period, as the demand grows larger each time period, as the demand grows larger

)1n()n()n( EEE UFF

TSUET)n(E FF

Unconditional MomentsUnconditional Moments

)E(Var)(VarE)(Var )1n()n()1n()n()n( UFUFF

)]/(Var[E

)]/(Var[E

)]/(Var[E)1n(T)n(

)1n(1)n(

*)1n(n

UF00

00

00UF

ΘUF

])()([)(s])/[E(Var '*'*2)1n()n( QDMBQDMBΘQDBQDBΘUF

Diagonalised OD Demand Flow

Jacobian of Choice Probability VectorJacobian of Travel Time Vector

Computing the JacobiansComputing the Jacobians

Computing the derivatives of choice Computing the derivatives of choice probability vector in case of logit function probability vector in case of logit function i.e. matrix ‘i.e. matrix ‘DD’ is relatively straight forward’ is relatively straight forward

But computing the matrix ‘But computing the matrix ‘BB’ (jacobian ’ (jacobian matrix of cost vector) is not! matrix of cost vector) is not!

Computing the Matrix BComputing the Matrix B

Assume linear dynamic travel time function,Assume linear dynamic travel time function,

where, where,

= travel time for vehicles entering at t= travel time for vehicles entering at t

a = free flow travel timea = free flow travel time

b = congestion related timeb = congestion related time

x(t) = number of vehicles on the link at t x(t) = number of vehicles on the link at t

x(t)b a τ(t)

τ(t)

t

0

ds v(s))- (u(s) x(t)

Major time periods, T

Minor time steps, t

Mean travel time in major period T isMean travel time in major period T is

ΦΦ(t) = entry time for vehicles exiting at t(t) = entry time for vehicles exiting at t

dt τ(t))t-(t

1 τ

2

1

t

t12

T

t

)t(

T

t

tT

12T

T

ds)s(u x(t)

dt u

x(t)

) t- (t

b

u

τ 2

1

Numerical JacobiansNumerical Jacobians

Perturbation of inflow in any time period Perturbation of inflow in any time period and studying its impact on the travel times and studying its impact on the travel times of all the time periodsof all the time periods

Operate the main program to obtain SUE Operate the main program to obtain SUE flowsflows

Operate a single link model to obtain the Operate a single link model to obtain the jacobians numerically jacobians numerically

Numerical ExampleNumerical Example

Three link parallel route network servicing a Three link parallel route network servicing a single OD pair with linear dynamic cost functions single OD pair with linear dynamic cost functions

Time Time Period 1Period 1

Time Time Period 2Period 2

Time Time Period 3Period 3

Time Time Period 4Period 4

400400 700700 620620 550550

OriginDestination

Number of drivers in each period RouteRoute aa bb

11 1212 0.0250.025

22 99 0.0350.035

33 88 0.0650.065

Network Parameters

ResultsResults

Jacobians by numerical Jacobians by numerical method for Route 1method for Route 1

Jacobians by analytical/ Jacobians by analytical/ finite difference finite difference approximation methodapproximation method

0.191388 u

τ 0.214824

u

τ

0.301494 u

τ 0.200349

u

τ

1

4

1

3

1

2

1

1

0.196400 u1

1

ResultsResults

θθ = 0.01 = 0.01 θθ = 0.4 = 0.4

MeanMean VarianceVariance MeanMean VarianceVariance

SimulationSimulation 131.54131.54 87.9887.98 89.3189.31 324.0324.0

NaïveNaïve 131.59131.59 88.3088.30 95.6995.69 72.872.8

ApproximationApproximation 131.59131.59 91.9991.99 95.6995.69 31403140

Estimates of Mean and Variance for Route 1 Time Period 1

Plan for Further WorkPlan for Further Work

Analytical expressions for the jacobiansAnalytical expressions for the jacobians

Non-linear dynamic cost functionsNon-linear dynamic cost functions

Network with multiple ODs Network with multiple ODs

Any questions, comments, Any questions, comments, suggestions welcome!suggestions welcome!