The morning commute in urban areas with heterogeneous trip lengths. ISTTT22, Northwestern University Raphaël Lamotte, Nikolas Geroliminis

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1 The morning commute in urban areas with heterogeneous trip lengths ISTTT22, Northwestern University Raphaël Lamotte, Nikolas Geroliminis

2 In a nutshell Vickrey (1969): the morning commute with a local bottleneck. How to extend it to a city? Probability density 2

3 In a nutshell The speed MFD or production MFD (NFD) [Geroliminis and Daganzo, 2008] Should not be confused with the outflow MFD (or NEF) O(n)= nv(n)/l 3

4 In a nutshell The speed MFD or production MFD (NFD) [Geroliminis and Daganzo, 2008] Dynamics imposed by the fundamental equation: t d t a v(n(t))dt =l 4

5 LITERATURE REVIEW 5

6 Review of literature with MFD-like dynamics Homogeneous users Small and Chu (2003) Geroliminis and Levinson (2009) (Stochas4c iden4cal trip length) Fosgerau and Small (2013) Heterogeneous trip length Fosgerau (2015) (2013) Kokoza and Naji (2016) Daganzo and Lehe (2015) 6

7 Review of literature with MFD-like dynamics Homogeneous users Small τ(t)= L/v(t) and Chu (2003) Geroliminis and Levinson (2009) Fosgerau and Small (2013) Heterogeneous trip length (Stochas4c iden4cal trip length) (2013) n Fosgerau (2015) Arno@, Kokoza and Naji (2016) Daganzo and Lehe (2015) 7

8 Scheduling preferences Based on marginal utility rates: U= 0 t d h(t)dt + t d t a 0 dt + t a 0 w(t)dt Marginal ullity rate Time 8

9 Scheduling preferences Based on marginal utility rates No strong consensus on the shape despite several RP and SP studies [Small, 1982 ; Hendrickson and Planck, 1984 ; Tseng and Verhoef, 2008 ; Hjorth et al., 2015] Several functional forms proposed Marginal ullity rate β α γ Time 9

10 Scheduling preferences Based on marginal utility rates No strong consensus on the shape despite several RP and SP studies [Small, 1982 ; Hendrickson and Planck, 1984 ; Tseng and Verhoef, 2008 ; Hjorth et al., 2015] Several functional forms proposed 10

11 Literature review (continued) If Users differ only by trip length (l), d/dt ( h/v )(t)<0 and d/dt ( w/v )(t)>0 for all times t, t d (l) and t a (l) continuous and differentiable, [some mild technical conditions], Then UE exhibits the regular sorting property [Fosgerau, 2015]: 11

12 ANALYTICAL RESULTS 12

13 Continuity (proposition) A1 Assumptions: Continuum of users At home: h(t,θ) [positive and continuous w.r.t. t] At work: w(t,θ, T ) [continuous w.r.t. t, except possibly at t= T ] l, T and l T are continuously distributed n(t) and v(t) continuous Attention Continuity result valid for a large set of h and w functions, including α β γ preferences. - The next results require the same assumptions, plus α β γ preferences. 13

14 Fundamental relationships u( t a, t,l)= ατ( t a,l) sp( t a, t,l) 1 st order condition of User Equilibrium: For early users: For late users: For on-lme users: 14

15 Fundamental relationships 2 nd order condition: if t a t and v(t) is differentiable 1, then at equilibrium Or, equivalently: v ( t d )/v ( t d ) 2 v ( t a )/v ( t a ) 2 1 The exogenous differenlability assumplon is avoided in the paper. 15

16 Intuition in the ( x,ln (V) ) space Bijection: x=f(t)= 0 t v(u)du V(x)=v( f 1 (x)) 1 st order condition (FOC) for early users: ln (V( x d )) ln (V( x a )) = ln ( α/α β ) 2 nd order condition (SOC) for early users: d ln V /d x ( x a ) d ln V /d x ( x d ) 16

17 Intuition in the ( x,ln (V) ) space ln V ln ( α/α β ) SOC imposes a point-to-point concavity to ln (V) for early users L x Locally FIFO Trip-length serves as sorlng criterion 17

18 Analytical results Single peak assumption (A2): n(t) strictly increasing before t p, strictly decreasing after. A2+(α β γ) => Early users travel before t p, late users after (Corollary of FOC + SOC). Proposition: A1+A2+ (α β γ) => FIFO sorting among early (resp. Late) users having identical β/α (resp. γ/α). Longer trips start and finish earlier (resp. later). See also: Alternative approach based on exogenous assumptions only. 18

19 Adjustment mechanism SIMULATIONS SimulaLon 19

20 Simulation description Agent-based event-based dynamics 4000 agents, 10 families of α β γ preferences l, t uniformly distributed. MSA: full knowledge of last iteration, 2% update their decision at every iteration. 20

21 Lessons from simulations Confirms theoretical findings (continuity, FIFO, sorting). 21

22 Influence of demand-to-capacity ratio Vickrey (1969) (FIFO) Present work (FIFO) LIFO Cost per user Cost per user Cost per user Demand to capacity Delay Total cost Delay + Earliness Demand to capacity Smoothness Cost composition Instability and heavy congestion 22

23 FUTURE RESEARCH DIRECTIONS 23

24 General scheduling preferences Transition FIFO/LIFO Impact of congestion 24

25 Social optimum Applicability of enveloppe theorems? An example with hypercongestion Production (P) 1 short trip N long-distance trips N Accumulation (n) Time 25

26 Further research directions Stability: Depends on the adaptation mechanism, Easy to observe with simulations, Difficult to characterize analytically. Spatially heterogeneous settings. 26

27 The morning commute in urban areas with heterogeneous trip lengths Raphaël Lamotte, Nikolas Geroliminis THANK YOU

28 References Arnott, R., A bathtub model of downtown traffic congestion. Journal of Urban Economics 76, Arnott, R.J., Kokoza, A., Naji, M., Equilibrium traffic dynamics in a bathtub model: A special case. Economics of Transportation 78, Daganzo, C.F., Lehe, L.J., Distance-dependent congestion pricing for downtown zones. Transportation Research Part B: Methodological 75, Fosgerau, M., Small, K.A., Hypercongestion in downtown metropolis. Journal of Urban Economics 76, Fosgerau, M., Congestion in the bathtub. Economics of Transportation 4, Geroliminis, N., Daganzo, C.F., Existence of urban-scale macroscopic fundamental diagrams: Some experimental findings. Transportation Research Part B: Methodological 42, Geroliminis, N., Levinson, D., Cordon pricing consistent with the physics of overcrowding, in: Lam, W.H.K., Wong, S.C., Lo, H.K. (Eds.), Transportation and Traffic Theory 2009: Golden Jubilee. Springer US, pp Hendrickson, C., Plank, E., The flexibility of departure times for work trips. Transportation Research Part A: General 18, Hjorth, K., Börjesson, M., Engelson, L., Fosgerau, M., Estimating exponential scheduling preferences. Transportation Research Part B: Methodological 81, Part 1, Small, K.A., The scheduling of consumer activities: Work trips. The American Economic Review 72, Small, K.A., Chu, X., Hypercongestion. Journal of Transport Economics and Policy 37, Tseng, Y.Y., Verhoef, E.T., Value of time by time of day: A stated-preference study. Transportation Research Part B: Methodological 42, Vickrey, W.S., Congestion Theory and Transport Investment. American Economic Review 59,