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David Watling, Richard Connors, Agachai Sumalee ITS, University of Leeds

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

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David Watling, Richard Connors, Agachai Sumalee ITS, University of Leeds

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

  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. Flow conservation (Astarita, 1995) 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

  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. τ τ0 w c 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. 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. Day-to-day variation Mean travel time under between-day varying inflows Travel time at mean inflow 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 • u(t) = u(t, ) each day has different value of (vector)  • Practically unrestrictive: discretised case with N time slices • Univariate Case • General Case u(t) = = [θ1, θ2,…, θN]

  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 = 2000pcus/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. [ ] t E ( u ) Std dev of inflows Mean inflow over the days Mean travel time over the days (with c.i.s) Travel time calculated for the mean inflow Numerical difference from plot above Inflation term by calculation

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

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

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

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

  19. 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.

  20. Questions/Comments?

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