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							 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
                   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
UK’s Department for Transport Work

 Reliability impacts on travel decisions through
  generalised cost
    Generalised cost gc:          gc  α  β1μ  β 2 σ
                                   gc α          β2 σ 
    … or generalised time gt: gt        μ 1 
                                                       .
                                   β1 β1         β1 μ 
                                                       
    that has key components:
             the mean travel time ;
             the so-called ‘reliability ratio’ β 2        ;
                                                     β1
             the travel time coefficient of variation,           .
                                                              
             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.
 Modelling Within-Day Variation:
Whole-link model (Carey et al, 2003)
   (t )  h(u(t ), v(t   ))
  travel time for vehicle entering at time t

   (t )  hw(t )  hu(t )  (1   )v(t   (t )) 

                 in-flow at entry time     out-flow at exit time
                                                       t  t 
                   u (t )                t                       
  v(t  τ(t ))                            u s ds   vs ds 
                 1  τ(t )              0
                                                        0     
                                                                  
  Flow conservation (Astarita, 1995)
               Whole-link Model

   Combining gives a first-order differential equation:
               (1   )u (t )
 (t )  1  1                FIFO   ' (t )  1
              h ( )  u (t )

 No analytic solution for most functions h(.), u(.).
 Can solve using backward differencing, applied in
  forward time (to avoid FIFO violations).
                  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.
       τ                           Scenarios for h(w) with
                                   finite capacity c
                                   Desired meaning of capacity
                                   requires careful definition of
      τ0                           h(w)
                                     w
                            c
           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
              Day-to-day variation


          E[ (t )]   E[u (t )]  , H (t )
                                     1
                                     2
Mean travel time    Travel time      Inflation term for between-
under between-day   at mean inflow   day variation. Comprising:
varying inflows                      Variance-Covariance matrix
                                     of inflow variability and
                                     Hessian matrix
                                     “sensitivity of travel time
                                     to inflows”
                                     Not a constant!
   Day-to-day parameterisation
 u(t) = u(t, )
  each day has different value of (vector) 

 Practically unrestrictive: discretised case with N time
  slices u(t) = = [θ1, θ2,…, θN]

                                             1  2 t , 
 Univariate Case E t ,    t ,                    Var 
                                             2       2
                                                            


                       E t ,    t ,   H  t ,   Cov  
                                                 1
 General Case
                                                 2
                 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 E  t ,     t , 
              Numerical Example

 BPR-type link travel time function
              w 4         ff = 10mins
  hw  ff 1    
             c 
                            c = 2000 pcus/hour (‘capacity’)

 In-flow profile with random day-to-day peak
                           πt 
            (4000  ε) sin 120  0  t  60
                               
   U (t )  (4000  ε)              60  t  120   ε  N (0,1000 2 )
                          5  πt  120  t  240
            (4000  ε) sin  240 
                                 
Solving Carey’s model with  = 1, so that  = h[u(t)]
No dependence on outflows.
Std dev of inflows


Mean inflow over the
days E u 


Mean travel time over the
days E  (u ) (with c.i.s)
       

Travel time calculated for
the mean inflow  E[u ]

Numerical difference from
plot above

           Inflation term by
           calculation
              Example: =0.1

 Asymptotic link travel time function
     hw 
             ff
                         ff = 10mins
               w
            1
                c        c = 7000 pcus/hour (‘capacity’)

 In-flow profile with random day-to-day peak

                   740000      t   2 
   u (t ,  , )         exp                N (80 ,20 2 )
                               2 2 
Compare Two Link Travel Time Functions
                                                                        w 4 
τ=h(w)                                                      hw  10 1      
                                                                        2000  
 40

          Asymp
          BPR
                                                                                
 35




 30




 25




                                                                hw 
 20
                                                                            10
                                                                              w
 15                                                                      1
                                                                            7000
 10
      0   1000    2000   3000   4000   5000   6000   7000
                                                            w
                  Example: =0.5

 Asymptotic link travel time function
      hw 
              ff
                                ff = 10mins
                w
             1
                 c              c = 7000 pcus/hour (‘capacity’)

 In-flow profile with random day-to-day peak
                                t 
                (4000   ) sin 120  0  t  60
                                     
 U (t )  500  (4000   )               60  t  120   ε  N (0,1000 2 )
                                   t 
                (4000   ) sin 5      120  t  240
                                   240 
              Example: =varying

 Asymptotic link travel time function
      hw 
              ff
                                ff = 10mins
                w
             1
                 c              c = 7000 pcus/hour (‘capacity’)

 In-flow profile with random day-to-day peak
                                t 
                (4000   ) sin 120  0  t  60
                                     
 U (t )  500  (4000   )               60  t  120   ε  N (0,1000 2 )
                                   t 
                (4000   ) sin 5      120  t  240
                                   240 
            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.
Questions/Comments?