Departure Time and Transport Mode Choices in Cities with Road Bottlenecks and Crowding in Transit Takao Dantsuji *1 and Daisuke Fukuda 1 * Corresponding author Email: t.dantsuji@plan.cv.titech.ac.jp 1 Transport Studies Unit (TSU), Tokyo Institute of Technology, M1-11, 2-12-1, O-okayama, Meguro-ku, Tokyo 152-8552, Japan 1. Introduction Congestion during the morning rush hour remains problematic even though multiple transportation modes may be available. In the Tokyo central business district the average car speed is 16 km/h and the worst in-vehicle congestion rate in railway is 199% of their related capacity. It is well-known that the exercise of free choice may render transportation systems inefficient. Price manipulation is one of the traffic demand managements to maximise efficiency. The bottleneck model of Vickrey (1969), as further developed by Arnott et al. (1993), can be used to analyse peak traffic demand problems. This tractable model can accommodate multimodality, as initially shown by Tabuchi (1993), who explored the equilibrium conditions attained under different pricing policies when roads and railways linked the home and workplace. In an extension of the cited work, Danielis and Marcucci (2002) analysed transit fares and road tolls; Huang (2000) explored transit congestion; and Wu and Huang (2014) investigated consumer departure times and transport mode choices. However, none of the cited works evaluated hypercongestion (which develops when increased traffic demand reduces traffic flow); hypercongestion is a major problem in urban cities. Daganzo (2007) and Geroliminis and Daganzo (2008) proved both theoretically and empirically that hypercongestion at the network level affected transport flows (Fig. 1). Some authors 12)-19) explored network equilibrium by imposing hypercongestion on the bottleneck model, creating the so-called bathtub model. Gonzales and Daganzo (2012) analysed situations in which private cars and railways shared public space and Gonzales (2015) investigated the coordinated pricing of rail fares and road tolls. However, both cited works assumed that transit capacity was infinite; crowding in transit was not a factor. Here, we develop a departure time/transport mode choice model that considers both crowding costs and hypercongestion. 2. Equilibrium in bi-modal corridors Here, we model user equilibrium in a bi-modal corridor following Wu and Huang (2014). Consider
Network Flow, q (veh/sec lane) 0.15 0.125 0.1 0.075 0.05 San Francisco: 21 km/hr Nairobi: 18 km/hr Yokohama San Francisco 0.025 Nairobi a public corridor wherein two transport modes (i.e. car and rail) link the home and the workplace. N identical commuters choose departure times and transit modes. First, we introduce the departure time choice model and then measure equilibrium in the bi-modal corridor. 2.1 Car commuters departure time choice car commuters drive through a bottleneck of capacity between the home and the workplace. We consider the queue to be a point queue, and assume that the cars follow the first-in-first-out (FIFO) principle. Then, the travel time is: μ (1) where is the free-flow travel time and the number of cars waiting at the bottleneck at time. Let be the departure rate from home at time ; is then: where is the time at which the rush hour starts. Commuters choose their departure time to minimise the generalised cost: α β t α γ t μ(t ) (2) where is a travel time parameter, and are earliness and lateness parameters, respectively, and and are the time when the rush hour finishes and the departure time that allows a car to arrive at precisely the desired time, respectively: t = (3). Employing the user equilibrium definition of Wardrop (1952), no individual can reduce his/her generalised cost by changing the departure time when equilibrium is attained. The generalised cost is constant over the rush hour [ the departure rate is: 0 0 0.02 0.04 0.06 0.08 0.1 Network Density, k (veh/m lane) Fig. 1. The network-wide relationship between commuter flow and density Figure 9. Comparison of central Nairobi with the San Francisco and Yokohama M FDs on aper lane basis. (Source: Gonzales et al. 2011). Substituting equation (4) into equation (2), we obtain: ]. Thus, (4)
(5) As no queue exists before the rush hour starts or after it finishes [ ], and as all commuters finish their trips during the rush hour [i.e. ], the following times can be determined: (6a) (6b) (6c) where =. The generalised cost at equilibrium is determined by: (7) 2.2 Model of rail commuters departure time choice Consider that commuters take the railway to the workplace. Assume, for simplification, that the travel time is zero. A constant headway of during the rush hour ( [ ]) is assumed. Let be the departure rate; then, the number of commuters who take rail services ( ) is. As headway is normally very short during the rush hour, the departure rate of all trains can be assumed to be constant:. The generalised cost is: (8) where is a crowding cost parameter. As the generalised cost is the same both during the rush hour and at equilibrium, the generalised costs when the rush hour starts and ends are identical. Thus: (9) The departure rate is derived by substituting equation (9) into equation (8): (10) As all commuters finish their trips during the rush hour [ the rush hour start and end times: ], equations (8) and (9) yield
(11) Thus, the generalised cost at equilibrium is: (12) 2.3 User equilibrium in a bi-modal corridor As all commuters travel either by car or rail,. At equilibrium in a bi-modal corridor, no person can reduce his/her generalised cost by changing the departure time or mode. Thus: (13) 3. Equilibrium in bi-modal cities In reality, the trip completion rate is not constant when hypercongestion is a factor. As shown by Geroliminis and Daganzo (2008), trip completion falls, and more people arrive late, when hypercongestion develops. Thus, we now extend the corridor model to real life. 3.1 Car commuters departure time choice Let be the length of the road network and the average trip length. Then, the trip completion rate, where is the average traffic flow (Daganzo, 2007), which is in turn equivalent to the product of the traffic density and the average traffic speed : (14) Following Small and Chu (2003), we assume that the traffic speed is reflected by the Ardekani-Herman (AH) formula: (15) where is the free flow speed, is the standardised traffic density [i.e. ], and is a traffic parameter. Then, the maximum average flow and critical density are given by: (16a) (16b) Although the travel time in the bottleneck model is given by equation (3), the travel time at the network level is: (17) Note that the travel times of commuters who end their trips at time depend only on their speed at time, as also assumed by Small and Chu (2003). As the generalised cost is constant over the rush hour when equilibrium is attained, the following equation can be derived by differentiating equation
(3) with respect to : (18) Also, on differentiating equation (17), we obtain: (19) From equations (18) and (19), we derive the standardised density by initially noting: (20) where: (21) Solving equation (20) for a fixed : (22) where is the constant of integration. As no trip is ongoing when the rush hour starts and ends [ ], the standardised density can be derived from equation (22): (23) As all trips are completed during the rush hour, the following equation is satisfied: (24) Then, from tedious calculation: and equations (14) (15) and (22), we obtain equation (25) after some (25) where: (26) This equation can be numerically solved for. 3.2 User equilibrium in the bi-modal system As the generalised cost is constant over the rush hour, we obtain the following equation: (27)
Note that the departure time choice model for rail commuters is the same as that of equation (12). From equations (12), (25), and (27); and, the total number of commuters is: (28) This equation can also be numerically solved for. 4. Optimal uniform pricing We next explore optimal uniform pricing in a bi-modal system. This is the pricing that minimises the total social cost, thus essentially the GF pricing of Wu and Huang (2014): (29a) where is the price. We assume that pricing is levied at only one mode. Thus, if a toll is levied (for example), the constraints are: subject to: (29b) (29c) (29d) If a rail fare is levied, the constraints are: subject to: (29e) (29f) (29g) Note that we do not consider a fare that covers both operation and fixed costs. We will address this topic in future. Also, please note that Wu and Huang (2014) consider a negative toll to be a subsidy; however, we view this as a rail fare. 5. Numerical example persons h h h c h person h μ vehicles h icles h icles h 5.1 Comparison of user equilibria We compare the outcomes of the equilibrium conditions derived in section 2.3 (the bottleneck model)
and section 3.2 (the bathtub model). Table 1 indicates that the duration of the car rush hour of the bathtub model is 2.02 h longer than that of the bottleneck model. Also, Figure 2 shows that the trip completion rate of the bathtub model decreases after the maximum rate is attained, because of hypercongestion. This is the principal reason why the car rush hour is longer, and thus associated with a longer travel time (Fig. 3). Table 1 also shows that the rail component of the bathtub model is 9% higher than that of the bottleneck model, again attributable to hypercongestion that increases the generalised cost. Therefore, the duration of the rush hour and the rail share of the bottleneck model are underestimates of the real values in cities. 5.2 Effects of optimal uniform pricing Figures 4 and 5 show the total social cost of both models at different pricings. This cost increases monotonically as the uniform fare rises and there is a point at which the cost is minimised if a uniform toll is levied. However, the toll value of the bathtub model is higher than that of the bottleneck model. 6. Conclusions and future work We developed a joint departure time/transport mode choice model that considered both crowding in trains and network hypercongestion. People choose their departure times by considering trade-offs between travel time and cost, or crowding and cost. Each person chooses whether s/he considers a longer travel time or crowding to be more tolerable than the other. When we numerically compared our model with the bottleneck model, we found that rush hour duration, the rail share of transport, and the optimal pricing were all underestimated by the bottleneck model. In future, we will explore the uniqueness and stability of user equilibria; for this, further statistical comparisons are required. As we focused only on optimal uniform pricing in the present paper, we will also explore time-dependent pricing and different pricing objectives. Finally, user heterogeneity must be considered; poorer people are more likely to use public transport if road tolls are in place. Acknowledgements This work is supported by the Committee on Advanced Road Technology (CART), Ministry of Land, Infrastructure, Transport, and Tourism, Japan.
Table.1. Comparison of user equilibria. Bathtub model Bottleneck model Duration of rush hour Car 6.24 4.20 (h) Rail 6.67 4.68 Number of commuters Car 24,827 (83%) 27,509 (92%) Rail 5,173 (17%) 2,492 (8%) Generalised cost 20.69 14.36 Bathtub model Bathtub model Bottleneck model Bottleneck model Fig. 2. Trip completion rate. Fig. 3. Travel time. Fig. 4. Total social cost by price (bathtub model). Fig. 5. Total social cost by price (bottleneck model).
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