1. David Watling, Richard Connors, Agachai Sumalee ITS, University of Leeds Acknowledgement: DfT “New Horizons” Dynamic Traffic Assignment Workshop, Queen’s University, Belfast 15 th September 2004 Encapsulating between day variability in demand in analytical, within-day dynamic, link travel time functions

2. Aims  Dynamic modelling of network links subject to variable in-flows comprising:  Within-day variation described by inflow, outflow and travel time profiles  Between-day variation = random variation in these profiles  Thus identify mean travel times under doubly dynamic variation in flows

3. UK’s Department for Transport Work  Reliability impacts on travel decisions through generalised cost

4. Dynamic Models Cellular Automata Microsimulation Analytical ‘whole-link’ models  Many shown to fail plausibility tests (FIFO) e.g.  = f [x(t)], with x(t) = number cars on link  Carey et al. “improved” whole-link models guarantee FIFO and agree with LWR behaviour.

5. Modelling Within-Day Variation: Whole-link model (Carey et al, 2003) travel time for vehicle entering at time t in-flow at entry time out-flow at exit time Flow conservation (Astarita, 1995)

6. Whole-link Model  Combining gives a first-order differential equation:  No analytic solution for most functions h(.), u(.).  Can solve using backward differencing, applied in forward time (to avoid FIFO violations).

7. Flow Capacity  Should the link travel-time function h(w) inherently define max (valid) w and hence capacity, c?  Out-flow can exceed capacity in computation so long as inflow ‘compensates’ such that w=βu(t)+(1-β)v(t+τ(t))< c  Can ensure outflows respect flow capacity by adapting the numerical scheme. τ0τ0 τ w c Scenarios for h(w) with finite capacity c Desired meaning of capacity requires careful definition of h(w)

8. Day-to-day variation  Introduce day-to-day variation of inflow  Derive expected travel time profile in terms of mean, variances, co-variances of day-to-day varying in-flows

9. Mean travel time under between-day varying inflows Travel time at mean inflow Day-to-day variation Inflation term for between- day variation. Comprising: Variance-Covariance matrix of inflow variability and Hessian matrix “sensitivity of travel time to inflows” Not a constant!

10. Day-to-day parameterisation Practically unrestrictive: discretised case with N time slices Univariate Case General Case u(t) = u(t,  ) each day has different value of (vector)  u(t) =  = [θ 1, θ 2,…, θ N ]

11. Methodology Monte Carlo simulations of day-to-day inflows   drawn from a normal distribution gives many u(t,  i ) Whole-link model gives travel time  i (t)=  (u(t,  i )) Calculate mean of all the Monte Carlo days travel times. This is the experienced mean travel time. Calculate travel time at mean inflow, using whole-link model with inflow E[u(t,  )] Calculate the “Inflation” Term: combination of the Hessian and Covariance matrix Compare inflation term with

12. Numerical Example  BPR-type link travel time function ff = 10mins c = 2000 pcus/hour (‘capacity’)  In-flow profile with random day-to-day peak

13. Solving Carey’s model with  = 1, so that  = h[u(t)] No dependence on outflows.

14. Std dev of inflows Travel time calculated for the mean inflow Mean travel time over the days (with c.i.s) Mean inflow over the days  )(uE  Numerical difference from plot above Inflation term by calculation

15. Example:  =0.1  Asymptotic link travel time function ff = 10mins c = 7000 pcus/hour (‘capacity’)  In-flow profile with random day-to-day peak

16. Compare Two Link Travel Time Functions w τ=h(w)

19. Example:  =0.5  Asymptotic link travel time function ff = 10mins c = 7000 pcus/hour (‘capacity’)  In-flow profile with random day-to-day peak

22. Example:  =varying  Asymptotic link travel time function ff = 10mins c = 7000 pcus/hour (‘capacity’)  In-flow profile with random day-to-day peak

25. Further Work  Analytic derivation of the correction term?  Modify whole-link model to limit outflows  Augment with dynamic queuing model?  Conditions for FIFO?  Follow this approach on the links of a network to investigate its reliability under day-to-day varying demand.

26. Questions/Comments?