1. Dynamics of Traffic Flows in Combined Day-to-day and With- in Day Context Chandra Balijepalli ITS, Leeds 15-16 September 2004

2. Objectives of this Presentation To introduce the combined day-to-day and with- in day context of dynamic traffic assignment To introduce the extended method of approximation To discuss the issues in computing the parameters e.g., jacobians of travel time functions To discuss some numerical results

3. The Context Day-to-day dynamics: drivers’ learning and adjusting With-in day dynamics: delays along the route based on prevailing traffic conditions Not dealing with departure time choice

4. 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 Route flows Route Costs Average route flows, costs, outflow profile, travel time flow profile, etc Route Choice Model Dynamic Link Loading Model

5. Literature Review Cantarella, G.E. and Cascetta, E. (1995) Dynamic Processes and Equilibrium in Transportation Networks: Towards a Unifying Theory, Transportation Science 29(4), 305-329 Davis, G.A. and Nihan, N.L. (1993) Large Population Approximations of a General Stochastic Traffic Assignment Model, Operations Research 41(1), 169-178 Friesz, T.L., Bernstein, D., Smith, T.E., Tobin, R.L. and Wie,B.W. (1993) A Variational Inequality Formulation of the Dynamic Network User Equilibrium Problem, Operations Research 41(1), 179-191 Hazelton, M. and Watling, D. (2004) Computation of Equilibrium Distributions of Markov Traffic Assignment Models, Transportation Science 38(3), 331-342

6. The Extended Method of Approximation Assume the drivers are indistinguishable and rational in minimising their perceived travel cost Perceived Travel Cost Measured Travel Cost Error in Perceived Travel Cost

7. Measured travel costs are updated using m = memory length λ = memory weighting

8. The number of drivers taking each possible route on day n during time period T, conditional on the weighted average of costs, is obtained as independently for T = 1,2,… independently for T = 1,2,… q T = demand during time period t p T (.) = route choice probability vector

9. Conditional Moments Then the expectation and variance of the conditional distribution for each time period would be

10. Unconditional Moments Based on standard results, the unconditional first moment is given as Davis and Nihan (1993) proved that the equilibrium distribution is approximately normal with its mean equal to the solution of SUE in each time period, as the demand grows larger

11. Unconditional Moments Diagonalised OD Demand Flow Jacobian of Choice Probability Vector Jacobian of Travel Time Vector

12. Computing the Jacobians Computing the derivatives of choice probability vector in case of logit function i.e. matrix ‘D’ is relatively straight forward But computing the matrix ‘B’ (jacobian matrix of cost vector) is not!

13. Computing the Matrix B Assume linear dynamic travel time function, where, = travel time for vehicles entering at t = travel time for vehicles entering at t a = free flow travel time b = congestion related time x(t) = number of vehicles on the link at t Major time periods, T Minor time steps, t

14. Mean travel time in major period T is Φ(t) = entry time for vehicles exiting at t

15. Numerical Jacobians Perturbation of inflow in any time period and studying its impact on the travel times of all the time periods Operate the main program to obtain SUE flows Operate a single link model to obtain the jacobians numerically

16. Numerical Example Three link parallel route network servicing a single OD pair with linear dynamic cost functions Time Period 1 Time Period 2 Time Period 3 Time Period 4 400700620550 Origin Destination Number of drivers in each periodRouteab1120.025 290.035 380.065 Network Parameters

17. Results Jacobians by numerical method for Route 1 Jacobians by analytical/ finite difference approximation method

18. Results θ = 0.01 θ = 0.4 MeanVarianceMeanVariance Simulation131.5487.9889.31324.0 Naïve131.5988.3095.6972.8 Approximation131.5991.9995.693140 Estimates of Mean and Variance for Route 1 Time Period 1

19. Plan for Further Work Analytical expressions for the jacobians Non-linear dynamic cost functions Network with multiple ODs

20. Any questions, comments, suggestions welcome!