What are traffic models for?

Transcription

1 Wht re trffic models for? Benjmin Heydecker Centre for Trnsort Studies

2 Dynmic Trffic Models: From ssignments to flows nd trvel times Assignments secify route inflows Clculte: link inflows, outflows nd trvel times Deduce route trvel times Dynmic link models Dynmic network loding Trvel times Assignment Trffic model Flows Centre for Trnsort Studies

3 Comrison of Sttic nd Dynmic Assignment Sttic ssignment Constnt flow long routes Cost functions: Continuous Incresing Serble? Ccity? Congestion Dynmic ssignment Time-vrying trvel times Reresents congestion exlicitly Consistency between Flows Trvel times Centre for Trnsort Studies

4 Dynmic Network Loding Modelling trffic in network: rescribed route inflows f(s) Clculte: rrivl times τ(s), rrivl flows g(t). Centre for Trnsort Studies

5 Dynmic Link Trffic Model Inflow e (t) Link x (t) Outflow g (t) Link inflow e (s) link exit time τ (s), link outflow g [τ (s)]. Centre for Trnsort Studies

6 Dynmic Trffic Flows First-In First Out: e E [ ] ( s) = G τ ( s) Flow rogtion [ ] & τ ( s) ( s) = g τ ( s) Trffic A Accumulted flow A = E(t) A = G(t) 0 s τ(s) Time t Centre for Trnsort Studies

7 Dynmic Link Trffic Model Link inflow e (s) Link exit time τ (s), Link outflow g [τ (s)], Trffic on link x (t) = E (t) G (t) (control vrible) (stte vrible) (stte vrible) (stte vrible) dx dt = e ( t) g ( t) (stte eqution) Centre for Trnsort Studies

8 Requirements on Dynmic Trffic Models 1. Positivity: e (.) > 0 g (.) > 0, x (.) > 0 2. Conservtion: 3. Flow rogtion: 4. First In First Out: x E τ& ( t) = E ( t) G ( t) ( s) = G τ ( s) ( s) 0 [ ] Distnce x Vehicle Instnt 0 s τ(s) Time t 5. Cuslity: τ (s) indeendent of e (t), t > s Centre for Trnsort Studies

9 Cuslity Vehicle-following: Informtion trvels more slowly thn does trffic Outflow t time of exit τ(s) is determined by inflows before corresonding time of entry s Distnce x Time - distnce Wve Vehicle 0 s τ(s) Time t Centre for Trnsort Studies

10 Condition for deterministic equilibrium Assignment roortions µ (s) stisfy µ ( s) g = q P od [ τ ( s) ] τ ( s) g q [ ] od P Heydecker nd Addison (1996) Assignment is roortionl to outflow Centre for Trnsort Studies

11 Anlysis of stochstic equilibrium Logit: Assigned flows e (s) stisfy ex( θ rc ( s )) od e ( s) = e ( s) Pod s ex( θ c ( s)) q P od r q e (s) is continuous in th costs c (s) c (s) is continuous in stte x (s) for finite inflows, x (s) is continuous in time s e (s) is continuous in time s Centre for Trnsort Studies

12 Links to Networks: Outflow g Link outflow is rogted inflow: g [ τ ( s) ] = e ( s) & τ ( s) Prtil outflows: g (Pgeorgiou, 1990) od [ τ ( s) ] = e ( s) τ& ( s) Sme trvel time for ll rts of flow: Multi-commodity FIFO od Distnce x Vehicle Instnt 0 s τ(s) Time t Centre for Trnsort Studies

13 Dynmic Network Loding Modelling trffic in network with rescribed route inflows f(s) Clculte: rrivl rtes g(t), rrivl times τ(s). Inflow Arrivl time Outflow Route f(s) τ (s) g[τ (s)] Link e(s) τ(s) g[τ(s)] Centre for Trnsort Studies

14 Deendence of Trvel Time Suose tht τ ( s ) s = F[ e( s), x( s), g( s) ] Dgnzo showed tht deendence on e(s) nd g(s) is null Mun showed tht F (x) = φ + x / Q (FIFO) Affine trvel times: free-flow trvel time φ ccity Q Centre for Trnsort Studies

15 A Clss of Trvel Time Models Subdivide link: Free-flow rt φ - α Congestible rt α Exit time: τ ( s) = s + φ + x( s + φ α) Q Link Free-flow φ - α α x (t) Cse Model α = 0 Verticl queue α = φ Friesz et l liner α = t Mun divided liner Centre for Trnsort Studies

16 Outflow Functions g (x ) (Merchnt nd Nemhuser) Link outflow is function of stte: But g x ( t) g [ x ( t) ] = Deends on flows entering fter σ (t) [ ] ( t) = E ( t) E σ ( t) Distnce x Vehicle Instnt 0 σ(t) t Time t Cuslity? Centre for Trnsort Studies

17 Comuttion with wve model Heydecker nd Addison (1996) Wves determine flow: Position (metres) 600 g s + [ ( s) ] w k l e = e( s) Deends on k e (s) Time (seconds) Centre for Trnsort Studies

18 Dynmic equilibrium with wve model Heydecker nd Addison (1996) Equilibrium ssignments Equilibrium trvel times 0.12 Assignment roortion (route 2) 55 Trvel time (seconds) o Route 1 d Route 1 Route Route Time of entry s (seconds) Time of entry (seconds) Centre for Trnsort Studies

19 Dynmic equilibrium with wve model Heydecker nd Addison (1996) Equilibrium ssignments Equilibrium trvel times Flow rte (vehicles er second) 60 Trvel time (seconds) 0.6 Route Demnd Route 1 Route 2 50 Route Time of entry (seconds) Time of entry (seconds) Centre for Trnsort Studies

20 Conclusions Good trffic models re crucil for dynmic ssignment Underinning for network loding Limited choice of trffic models: Required roerties cf Stedy-stte counterrts Centre for Trnsort Studies