1 Trip-timing decisions with traffic incidents in the bottleneck model Mogens Fosgerau (Technical University of Denmark; CTS Sweden; ENS Cachan) Robin Lindsey (University of British Columbia) Tokyo, March 2013

2 Outline 1.Literature review 2.The model 3.No-toll user equilibrium 4.System optimum Quasi-system optimum Full optimum 5.Numerical examples 6.Conclusions/further research

3 Literature on traffic incidents Simulation studies: many Analytical static models Emmerink (1998), Emmerink and Verhoef (1998) … Analytical dynamic models (a) Flow congestion. Travel time has constant and exogenous variance. Gaver (1968), Knight (1974), Hall (1983), Noland and Small (1995), Noland (1997). (b) Bottleneck model. Travel time has constant, exogenous and independent variance over time. No incidents per se. Xin and Levinson (2007). (c) Bottleneck model with incidents Arnott et al. (1991, 1999), Lindsey (1994, 1999), Stefanie Peer and Paul Koster (2009).

4 Bottleneck model studies Scheduling utility approach Vickrey (1973), Ettema and Timmermans (2003), Fosgerau and Engelson (2010), Tseng and Verhoef (2008), Jenelius, Mattsson and Levinson (2010).

5 Outline 1.Literature review 2.The model 3.No-toll user equilibrium 4.System optimum Quasi-system optimum Full optimum 5.Numerical examples 6.Conclusions/further research

6 The model

u Scheduling utility: Zero travel time Move from H to W at t* As in Engelson and Fosgerau (2010)

u Scheduling utility: Positive travel time

9 The model (cont.)

10 The model (cont.)

11 Outline 1.Literature review 2.The model 3.No-toll user equilibrium 4.System optimum Quasi-system optimum Full optimum 5.Numerical examples 6.Conclusions/further research

u No-toll user equilibrium in deterministic model Queuing time cumulative departures cumulative arrivals

No-toll user equilibrium with major incidents u cumulative departures cumulative arrivals Driver m causes incident

14 No-toll user equilibrium properties Similar to pre-trip incidents model.

15 No-toll user equilibrium (cont.) Similar to pre-trip incidents model.

16 Outline 1.Literature review 2.The model 3.No-toll user equilibrium 4.System optimum Quasi-system optimum Full optimum 5.Numerical examples 6.Conclusions/further research

17 System optimum Deterministic SO Maximize aggregate utility. Optimal departure rate = s (design capacity) Stochastic SO Maximize aggregate expected utility. What is optimal departure rate?

18 General properties of system optimum (w)

19 System optimal approaches Quasi-system optimum (x) Departure rate Choose optimal Full optimum (w) Choose optimal and

20 Outline 1.Literature review 2.The model 3.No-toll user equilibrium 4.System optimum Quasi-system optimum (QSO) Full optimum 5.Numerical examples 6.Conclusions/further research

21 Quasi-system optimum (QSO)

22 Outline 1.Literature review 2.The model 3.No-toll user equilibrium 4.System optimum Quasi-system optimum Full optimum 5.Numerical examples 6.Conclusions/further research

23 Full system optimum (SO)

System optimum with queue persistence n cumulative departures cumulative potential arrivals Driver m causes incident

Departure rate n Interval 1 Interval 2

26 System optimum (major incidents, queue persistence)

27 System optimum (major incidents, queue persistence)

28 System optimum (major incidents, queue persistence)

29 System optimum (major incidents, queue persistence)

30 Outline 1.Literature review 2.The model 3.No-toll user equilibrium 4.System optimum Quasi-system optimum Full optimum 5.Numerical examples 6.Conclusions/further research

31 Calibration of schedule utility functions Source: Tseng et al. (2008, Figure 3)

32 Calibration of schedule utility functions (cont.) Source: Authors calculation using Tseng et al. mixed logit estimates for slopes.

33 Calibration of incident duration Mean incident duration estimates Golob et al. (1987): 60 mins. (one lane closed) Jones et al. (1991): 55 mins. Nam and Mannering (2000): mins. Select:

34 Other parameter values N = 8,000; s = 4,000 f(n)=f, fN = 0.2

35 Results: Major incidents

36 Results: Major incidents Total cost of incident in NTE

37 Results: Major incidents Total cost of incident in QSO

38 Results: Major incidents Total cost of incident in QSOTotal cost of incident in NTE

39 Results: Major incidents, individual costs NTE, no incident occurredNTE, incident occurred

40 Results: Major incidents, individual costs NTE, no incident occurred QSO, no incident occurred NTE, incident occurred QSO, incident occurred

41 Minor incidents Complication: NTE departure rate depends on lagged values of itself No closed-form analytical solution. Requires fixed-point iteration to solve. Results reported here use an approximation.

42 Results: Minor incidents Initial departure time

43 Results: Minor incidents Increase in expected travel cost due to incidents

44 Modified example with SO different from QSO s = 4,000 N = 3,000 fN = 0.4 Explanation for parameter changes: Shorter peak: Lower cost from moderating departure rate Higher incident probability and duration: Greater incentive to avoid queuing by reducing departure rate

45 Results: Modified example Departure rates for QSO and SO

46 Results: Modified example

47 6. Conclusions (partial) Properties of SO differ for endogenous-timing and pre-trip incidents models Plausible that QSO is a full SO: optimal departure rate = design capacity (same as without incidents), but with departures beginning earlier

48 Future research Theoretical 1.Further properties of NTE and QSO for minor incidents. 2.SO for minor incidents. 3.Stochastic incident duration Caveat: Analytical approach becomes difficult! Empirical 1.Probability distribution of capacity during incidents 2.Dependence of incident frequency on level of traffic flow, time of day, etc.

49 Thank you