Access and the choice of transit technology

Transportation Research Part A 59 (2014) 204–221

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Transportation Research Part A journal homepage: www.elsevier.com/locate/tra

Access and the choice of transit technology Karthik Sivakumaran a,⇑, Yuwei Li b, Michael Cassidy a, Samer Madanat a a b

Institute of Transportation Studies, University of California, Berkeley, 109 McLaughlin Hall, Berkeley, CA 94720, United States College of Transport and Communications, Shanghai Maritime University, 1550 Haigang Avenue, Shanghai 201306, PR China

a r t i c l e

i n f o

Article history: Received 20 June 2012 Received in revised form 5 June 2013 Accepted 19 September 2013

Keywords: Trunk Feeder Transit Transit-oriented development

a b s t r a c t An urban transit system can be made more cost-efficient by improving the access to it. Efforts in this vein often entail the provision of greater mobility, as when high-speed feeder buses are used to carry commuters to and from trunk-line stations. Other efforts have focused on the creation of more favorable land-use patterns, as occurs when households within a Transit-Oriented Development (TOD) are tightly clustered around trunk stations. The efficacy of these mobility and land-use solutions are separately examined in the present work. To this end, continuum approximation models are used to design idealized transit systems that minimize the generalized costs to both the users and the operators of those systems. The assessments unveil how the choice of transit technology for the trunk-line portion of a transit network can be influenced by its access mode. If transit is accessed solely (and slowly) on foot, then the optimal spacings between lines, and between the stations along those lines, are small. This can place capital-intensive rail systems at a competitive disadvantage with transit systems that feature buses instead. When access speeds increase, the optimal spacings between lines and stations expand. Hence, if accessed by fast-moving feeder buses, Metro-rail or bus-rapid transit can become preferred trunk-line options. By comparison, the influence of altered land use patterns brought by TODs tends to be less dramatic. We find that clustering households around Metro-rail stations justifies larger spacings between the stations. Yet, this produces only modest reductions in generalized costs because the larger spacings penalize transit users who reside outside of the TODs. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction The choice of transit technology is an important decision when planning either a new urban transit system, or extensions to an existing one. This choice for any city would be based in part on the city’s characteristics; e.g. its spatial dimensions, the demand for travel within its boundaries, and the socio-economic attributes of its citizens. When accounting for these factors, previous studies find that capital-intensive rail systems tend to be less cost-effective than bus-rapid transit (BRT) or ordinary bus systems (Cox, 2002; Daganzo, 2011; Estrada et al., 2011; Tirachini et al., 2009). However, these earlier comparisons typically assume that access to a city’s transit system would occur entirely by walking and that demand for travel would be distributed more-or-less uniformly over the city. One wonders if these assumptions unfairly place rail at a competitive disadvantage. After all, a rail system is usually designed with large spacings, both between its lines and between the stations along those lines. To design a rail system otherwise would often be prohibitively expensive. A question therefore emerges: can the use of a faster-moving access mode (e.g. feeder buses) render rail a more economically feasible option for a large, hierarchical transit system? Similarly, can ⇑ Corresponding author. Tel.: +1 510 642 3585; fax: +1 510 643 3955. E-mail address: [email protected] (K. Sivakumaran). 0965-8564/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.tra.2013.09.006

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clustering the demands for travel around its stations (e.g. so-called transit-oriented development or TOD) render rail systems more economically viable? These questions are examined in the present paper. To this end, continuum approximation models are used to estimate the generalized costs imparted to both the users and operators of certain idealized transit systems. These costs are compared across three technology options for trunk-line service: heavy rail, BRT and ordinary buses. The effects of augmenting access to rail and BRT systems via feeder buses are explored. The influence of TOD when planning rail systems is examined as well. Background information is furnished in the following section. Generalized costs are compared across the three alternatives for trunk-line technology in Section 3. Effects of designing a rail network to suit TOD are explored in Section 4. Implications of the present findings are discussed in Section 5. 2. Background This section presents: the general structure of a hypothetical trunk-line transit network to be used for baseline analysis; the cost models, objective function and input parameters that will be used in these analyzes; and preliminary explorations of the transit technologies that are optimal for our baseline network. First, we offer the reasoning behind our modeling approach. 2.1. Continuum approximation Transit systems are sometimes designed using very detailed cost models (Kuah and Perl, 1989; Martins and Pato, 1998; Uchimura et al., 2002). Required inputs in these instances tend to be both voluminous (e.g. travel demands are specified by means of possibly time-dependent origin–destination matrices) and case-specific, such that even details of a city’s topography might be included in the analysis (Shrivastava and O’Mahony, 2006). These models therefore tend to take complex forms. Solutions often come via heuristic methods, which means that the design process is necessarily limited to the exploration of few and narrowly-defined alternatives. In contrast, continuum approximation (CA) models of cost use as their inputs small numbers of continuous functions; e.g. travel demand is expressed as a density, while lines and stations are specified in terms of the spacings between them (Daganzo, 1996; Newell, 1973). As a result, optimal design solutions often take the form of simple, closed-form expressions that readily unveil the kinds of relations between inputs and outputs that are useful for high-level decision-making. Additionally, the CA modeling approach often yields a convex optimization problem, which in turn provides a globally optimal solution. The designs obtained in these ways are optimal for idealized assessments, and deploying these designs in real settings requires that lines and stations be altered to suit actual topographies and travel patterns. Happily, these altered designs still tend to produce near-optimal costs, since costs tend to depend more on factors such as total network length and service frequency rather than on the circuity of individual lines (Estrada et al., 2011). Given their advantages, CAs have been used: to compare costs across distinct service strategies for feeder systems (Chang and Schonfeld, 1991; Chang and Yu, 1996; Chien et al., 2002; Clarens and Hurdle, 1975; Diana et al., 2007; Kuah and Perl, 1988; Li et al., 2009); to explore broader design alternatives for hierarchical trunk and feeder systems where the transit technologies to be used were specified a priori (Aldaihani et al., 2004; Wirasinghe et al., 1977); and as previously noted, to compare various transit technology alternatives for trunk networks under the assumption that users access these networks on foot (Cox, 2002; Daganzo, 2011; Estrada et al., 2011; Tirachini et al., 2009). Past research has also compared the costs of various combinations of transit technologies (bus and rail) for trunk services. However, efforts of this latter type assumed that all trunk-line trips occur along a single corridor (Fisher and Viton, 1975; Keeler and Small and Associates, 1975). Moreover, previous works have not examined how the redistribution of trip-making demand via TOD can affect the choice of transit technology for a planned transit system. We will now use CAs to address the literature’s above-cited limitations, by first exploring the influence of access mode. The intent is to assess the influence of users’ access speed on the optimal choice of transit trunk-line technology (and not to evaluate the merits of all possible means of access). Thus, we will examine the impacts of accessing transit on foot and via feeder buses. In this way, we distinguish the effects of slow- and fast-moving access. We thereafter explore how TODs can influence the choice of transit technology. 2.2. Idealized network Consider the rectangular-shaped city in Fig. 1. Assume for now that the origins and destinations of trips to be made via transit occur uniformly and independently over the city’s area L  W [km2], and at total rate k [trips/hr], providing an overall k trip-making density of q ¼ LW [trips/km2-hr].1 Further assume that the trunk network of the city’s transit system takes the form of a rectangular grid, such that all trips within the network can be made with a single transfer. Its parallel lines, running 1 We will relax the assumption of uniform O–D’s in Section 4. Also recall that a transit system designed from our idealized O–D pattern and modeling approach can be adjusted to accommodate a city’s real demand patterns with relatively small deviations in cost (e.g. see Estrada et al., 2011). Further, travel demand is assumed as exogenous to the transit system’s design. We note in this regard that the system-optimal solution given in our analysis might be achieved by suitably pricing transit and the alternative travel modes (Daganzo, 2012).

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Fig. 1. Rectangular city (L  W) with a grid trunk system.

in the x- and y-directions, are spaced at distances of rW and rL, as shown in the figure. As in Daganzo (2011) and Estrada et al. (2011), we assume that these spacings are integer multiples of the spacing between stations, s, such that rL = pLs and rW = pWs. For illustration (only), pL = 2 and pW = 1 in Fig. 1. 2.3. User costs As is customary, costs to the users of our trunk network will be expressed in terms of their average times spent in the system (Daganzo, 2010; Daganzo, 2011; Estrada et al., 2011). These include the average time spent accessing the network’s designated stations, denoted A [hrs]. Half of all users will access a trunk line by traveling first in the y-direction for an average distance of 1/4 rW, while the other half will travel in the x-direction for an average distance of 1/4 rL. The average distance to access a station along a trunk line is thus 1/4 s. We will assume that users cover similar distances when traveling from their final transit stations to their destinations off the network. Hence, A ¼ ð0:5rL þ 0:5rw þ sÞ 21v a , where va is the user’s speed during her access and egress of the system. User cost further includes the time spent waiting at a station for a transit vehicle, Y [hrs]. We assume that a user waits, on average, half of a vehicle headway, H, both at her origin station and again at her transfer station; and we include a transfer time, tr, which constitutes the time required to move from one vehicle to the other, exclusive of the wait time at the transfer station. Hence, Y = H + tr. Finally, the average time spent on-board trunk vehicles, T, is the product of the user’s expected on-board travel distance, (L + W)/3, and the vehicles’ pace, v1c ¼ v1 þ ss , where v is the vehicle cruise speed and s is the time lost at each transit station. Hence, T ¼ ðL þ WÞ 31v c . 2.4. Agency costs The costs incurred by the transit agency are influenced by factors that include the total length of the system’s infrastruc    1 ture, IL ¼ LW r1w þ r1L ¼ LW þ p1 , and the total number of stations in the network, IS ¼ IsL . Of further relevance is the total s p w

L

distance collectively traveled by vehicles per hour, V, which is the product of vehicle flow, 1/H, and 2IL, since service is bi  1 directional. Hence, V ¼ 2LW þ p1L and the required fleet size of trunk vehicles, M, is the product of V and the vehicle pace sH pw defined in Section 2.3. Each of the four above factors has a monetary unit cost: $IL [$/hr/km]; $IS [$/hr/station]; $V [$/km/veh]; and $M [$/hr/ veh].2 These can be transformed into the equivalent times that an average user spends in the system by dividing each of the above factors by kl, where l is the monetary value of user time [$/hr]. This latter rate can be roughly represented by the city’s prevailing wage rate. We thus obtain the four components of operator cost, with dimensions of unit time: pIL IL, pIS IS, pV V, and pMM. 2.5. Objective function The transit system’s generalized cost per user, Z, is the sum of the user and agency costs from the previous two sections:

ZðpL ; pW ; s; HÞ ¼ pIL IL þ pIS IS þ pV V þ pM M þ A þ Y þ T

ð1Þ

where the four decision variables, pL, pW, s, and H, are embedded in (1). Additionally, the transit system is constrained by its passenger-carrying capacity, K, for lines in both the N–S and E–W directions. The expected passenger load is given by: the product of the trip-making density (q); each transit line’s catchment 2

For the present assessments, $IL and $IS will be linearly amortized over the assumed life of the system; see Appendix A.

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K. Sivakumaran et al. / Transportation Research Part A 59 (2014) 204–221 Table 1 Operating parameters for three technologies: bus, BRT, and rail.

Bus BRT Rail

Time per transfer tr (sec)

Lost time per station s (sec)

Cruising speed hr)

10 20 60

30 30 45

25 40 60

vt (km/

Passenger capacity K (pax)

Line capacity Qmax (vehicles/ hr)

80 120 1000

20 30 15

area (either WpL s or LpW s); the transit vehicle headway (H); and the factor 1/4. This latter factor is derived from the bi-directionality of each line and the lines running orthogonal to the direction in question, each halving the effective line passenger demand. Thus, the objective function is constrained by both 1/4qWpL sH 6 K and 1/4qLpW sH 6 K. Additional constraints are also applied to the decision variables pL, pW and st, such that the city’s grid contains at least two lines in both the N–S and E– W direction: pL st 6 2L and pW st 6 W2 . Each transit line is further constrained by its service capacity, Qmax [vehicles/hr], which is the maximum of the allowable vehicle flow, and which differs by transit technology. This maximum flow can depend on several factors, including: vehicle dwell times at stations; the minimum spacing allowed between vehicles; and the number of vehicle berths at each station. The flow of vehicles along each line, H1, is thus constrained such that H1 6 Qmax. Given these constraints on operation and the integer restrictions on pL and pW, we obtain a mixed-integer, non-linear mathematical program. To simplify the problem, we restrict the feasible integer set to pL, pW 2 {1, 2, . . ., 5}. Thus for any combination of pL and pW, the objective function and all constraints are convex in s and H, and the optimal solution can be obtained either numerically or by analytical search methods that exploit the objective function’s convexity (Bazaraa et al., 2006). 2.6. Parameter values Tables 1 and 2 display the operating and cost parameter values for the three transit technologies here (regular bus, BRT and rail) used in the assessments to follow. These values reflect our assumptions that: regular buses require relatively small infrastructure costs, in part because they travel with cars in regular lanes; BRT systems entail articulated buses that operate on grade-separated guideways away from regular traffic; and portions of the rail system required tunneling. Further details on all this are furnished in Appendix A.3 One may quibble over the precise values that we chose for any or all of the tabulated parameters. The point is that our choices in this regard reflect the hierarchies that exist across the three technologies in terms of cost and speed: regular buses are the least expensive and slowest-moving and rail is the most costly and speediest. We further note that the operating and infrastructure cost parameters for the agency can vary according to the city’s prevailing wage rate; and that these costs will vary by technology, since a rail system will likely have a smaller percentage of costs attributable to labor in comparison to a bus system. We thus provide operating cost and infrastructure cost parameter values for two wage groups: ‘‘Low’’ ($3/hr), as roughly occurs in a city like Beijing; and ‘‘High’’ ($20/hr), as in a city like Los Angeles.4 Results are also provided for what we call ‘‘Low-Cost’’ and ‘‘High-Cost’’ scenarios for BRT and rail infrastructure cost because costs for these modes can vary significantly, even across cities with similar prevailing wage rates.5 These Low- and High-cost values are shown in parentheses in Table 2. 2.7. Optimal technology for access by walking Using the parameters from Tables 1 and 2, we now minimize (1) and determine the optimal transit technology for various combinations of trip-making density and city size. For illustration, we assume a square city and use the city length L as a proxy for size.6 With walking assumed to be the only access mode, the shading in Fig. 2 shows that for cities with a low average wage, ordinary buses would be the optimal trunk technology for wide ranges of trip-making density and city length (size). The results for high-wage cities in Fig. 3 are different in that BRT becomes the most cost-effective technology for a wider range of larger and denser cities. This is intuitive: a higher user value-of-time makes a faster trunk technology more pref3 Passenger capacity and line capacity values were taken from the Transit Capacity and Quality of Service Manual, 2nd Edition (Kittelson & Associates, 2003). The infrastructure cost parameters used for BRT and rail are comparable to those for the Brisbane Busway and the Baltimore Metro Subway, respectively (Flyvbjerg et al., 2008). 4 These values are loosely based on recent average wage levels from a recent Prices and Earnings report published by UBS (2010). Note also from Table 2 that distance-related operating costs are assumed to be equal across low- and high-wage cities. We believe this to be reasonable, since the components of this cost (e.g. energy costs) tend to be invariant to the wage rate (GIZ, 2011; International Energy Agency, 2007). 5 Particularly for rail, a high variation in per-kilometer costs has been linked to various factors. These include land procurement, the establishment of rightof-way, the amount of underground tunneling, and topography (Flyvbjerg et al., 2008; Halcrow Fox, 2000). 6 Consideration of (1) reveals that, for rectangular-shaped cities of given area where L – W, generalized cost is rather insensitive to the ratio L/W. Hence, we limit our illustrations to square cities.

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Table 2 Cost parameters for three technologies: bus, BRT, and rail.

Bus BRT Rail

Infrastructure cost – lines $IL ($/ km-hr)

Infrastructure cost – stations $IS ($/station-hr)

Operating cost – fleet size $M [$/veh-hr]

Low wage

High wage

Low wage

High wage

Low wage

High wage

7 189 (15; 351) 690 (252; 778)

10 270 (21; 501) 990 (360; 1111)

0.49 4.9 (0.27; 18) 340 (163; 1289)

0.7 7 (0.38; 26) 490 (233; 1841)

21 28 130

63 84 200

Operating cost – distance $V ($/veh-km)

0.59 0.66 2.2

Fig. 2. Optimal technology for walk access low-wage city.

Fig. 3. Optimal technology for walk access high-wage city.

erable. Note nonetheless that rail is never a preferred technology across the ranges of trip-making density and city size examined, even for cities with a high wage.

3. Idealized feeder system We now explore how the introduction of feeder access can reduce generalized costs and affect the choice of trunk-line technology. To this end, the rectangular-shaped region in Fig. 4 inscribes a quadrant of the so-called catchment area from which a trunk station draws users. In this illustration, the trunk station resides on the upper left-hand corner of the quadrant. Consider a system of feeder-bus lines that reside within the quadrant, like the system shown in the figure. Note how: parallel feeder lines running in the x-direction are spaced at rf [km]; stations for this feeder system are placed along each line at spacing sf [km]; and upon reaching a trunk line, feeder lines alter their directions and converge to the trunk station. Users of the feeder system continue to access the trunk station in the fashion described in Section 2, but do so at the higher speed afforded by feeder-bus travel. Feeder vehicle cruising speed is denoted vf [km/hr], and the time lost by stopping

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Fig. 4. Feeder bus access to a trunk station.

to serve passengers at each station is given by sf. We assume that users access (and depart from) feeder stations by walking, and that all stations within the trunk network (refer to Fig. 1) are served by the feeder system. Regrettably, as many as three transfers may be needed within our system, and this shortcoming is addressed in Section 5. In the meantime, we note that our choice of this network structure (with its attendant need for multiple transfers) is not an attempt to advocate its use in real settings. Rather, we select this geometry because its simplicity facilitates our efforts to glean more general insights regarding the impacts of access speed. 3.1. Access costs to nearest trunk station The hourly cost to operate each feeder bus is given by Cf [$/veh-hr]. We divide this cost by kl to obtain an equivalent time expended by a passenger. Access cost of feeder service, Af, thus includes: a user’s time spent walking to and from the feeder stations; the time spent waiting for a feeder bus at the origin station; the times spent riding feeder buses to and from the trunk stations; and the equivalent time costs imparted to the operator of the feeder system, given that feeder lines run in both the x- and y-directions in order to provide access to all trunk stations:

Af ¼ b1 ðr f þ sf Þ þ hf þ b3 s þ b4

s 1 1 1 1 1 þ b5 þ þb6 sf hf r f hf sf r f

ð2Þ

where

b1 ¼

1 2v a

b3 ¼ ð0:5pL þ 0:5pw þ 1Þ

1 2v f

sf

b4 ¼ ð0:5pL þ 0:5pw þ 1Þ  4 C f LW 1 1 b5 ¼ 4þ þ pL pW klv f 4C f LW sf b6 ¼ kl

h = feeder-bus headway along a line, and all other terms have previously been defined. The time lost due to stopping at feeder bus stations is reflected by the feeder bus vehicle pace during passenger collection:   sf 1 1 1 1 v þ s ¼ v þ ðsf Þsf , and the vehicle pace along station detours (parallel to trunk transit lines) is v , since stopping is not f

f

f

f

required along this section. 3.2. A trunk-feeder system The feeder system shown in Fig. 4 can be mapped to each trunk line in the trunk system of Fig. 1. In addition to the aforementioned cost components of feeder bus access, the user time cost of the intermodal transfer between a feeder bus and a trunk vehicle is included by adding to the per-passenger transfer cost, tft, a value that is dependent on the assumed trunk mode.7 Furthermore, the spacing between feeder bus lines is assumed to equal its station spacing, such that rf = sf (i.e. we work with a square lattice). Beyond the benefits of simplicity, this design allows users to seamlessly transfer between orthogonal feeder lines. Finally, we assume that the trunk network structure follows a square lattice, pL = pW = p. The revised per-passpenger user and agency costs are described in the sub-sections that follow. 7 An intermodal transfer between feeder bus and rail may be more costly than one between feeder bus and regular bus. Thus, it is assumed that tft = 30 s for Bus, tft = 60 s for BRT, and tft = 120 s for rail.

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3.2.1. User costs Our hierarchical trunk-feeder model considers the possibility that users will travel by alternative paths. For example, rather than utilizing a feeder-trunk-feeder trip chain, feeder-only travel may be preferable for trips of shorter lengths. Trips of this kind might be preferred because they require fewer transfers. To roughly account for these feeder-only trips, we assume that users of a hierarchical trunk-feeder system must choose between a feeder-trunk-feeder trip chain and a feeder-feeder trip chain. We further assume that users will estimate the feeder bus trip time to reach their destinations, and that if the estimated time falls below some critical time tC, users will only utilize the feeder system to make their trips. This proportion of ‘‘Feeder-Only’’ users is given by PF. The remaining proportion who utilize the trunk system is PT = 1  PF. Users are assumed to estimate their feeder trip times given an optimistic feeder-bus travel speed of vf, such that a critical trip length dC is defined by dC = vf tC. Since origins and destinations are uniformly distributed, the trip length can be represented by a random variable Z, and the proportion PF of Feeder-Only users is given by the probability PF = P(Z 6 dC). The  expected travel distance d feeder aboard the feeder buses for these users is given by the conditional expectation  dfeeder ¼ E½ZjZ 6 dC . Likewise, the proportion of trunk users and their expected travel distance along the trunk line is given  by PT = P(Z P dC), with d trunk ¼ E½ZjZ P dC . These conditional expectations and probabilities were derived using the probability distribution for trip length (Fairthorne, 1965); formulas are given in Appendix B. The weighted sum of their generalized user cost components for access (AfUser), out-of-vehicle station (Y), and riding (T) gives the per-passenger user cost across these two groups. These components are derived below.   Both groups share the same costs in walking to feeder bus stations and waiting for feeder bus vehicles: v1a rf þ hf . Howps  s þ 2f sr 1 ever, Trunk users will utilize the feeder bus system as an access mode and thus incur an additional cost ðpþ1Þ f . The 2v f

weighted sum of these cost components gives

Af User ¼



 ps  ðp þ 1Þ f r f þ hf þ P T s þ PT sr1 f 2v f va 2 1

ð3Þ

Additionally, trunk users experience the costs of waiting (H) and of transferring both intermodally (2tft) and intramodally (tr). As a result,

Y 0 ¼ PT ðH þ 2t f t þ t r Þ

ð4Þ

As regards the in-vehicle travel time component T0 , Feeder-Only users incur a travel time cost while detouring to trunk  d stations in addition to their expected travel time vfeeder . Since a square lattice is assumed for trunk lines (rL = rW = ps), the numf ;c  d s ber of detours of length 2 experienced by Feeder-Only users is feeder . Denoting the travel time of Feeder-Only users as T1: ps

T1 ¼

 d feeder

v f ;c

! s 1  d feeder  þ ¼d feeder 2 vf ps

1

vf

1 þ 2pv f

!

1  þ ðd feeder sf Þr f

The in-vehicle travel time for trunk users, T2, is similarly

T2 ¼

 d trunk

vt

1  þ ðd trunk sÞs

Their weighted sum gives the per-passenger travel time component T0 ,

T 0 ¼ PF

    d d 1 feeder trunk 1 1   þ PF ðd 1þ þ PT ðd feeder sf Þr f þ P T trunk sÞs 2p vf vt

ð5Þ

The revised user cost for the trunk-feeder system is thus AfUser + Y0 + T0 . 3.2.2. Agency cost We assume that the infrastructure cost for the lines in the feeder bus network is negligible, while the infrastructure cost   for the stations is proportional to their number, and is given by the product of the feeder line-haul network length 2LWr 1 , f 8 2 and the feeder station density r 1 f . Thus, the feeder station infrastructure cost is ðpF;S 2LWÞr f . The total length of the feeder   bus network, which includes detours to trunk stations, is given by LW 4 þ 2p r 1 f . Multiplying this value by the flow of feeder   1 1 2 buses, hf , the distance-related feeder bus cost component is pF;V LW 4 þ p r 1 f hf . The time-related costs while in motion

are proportional to the network length, the feeder bus flow, and the cruising vehicle pace, v1 , yielding a cost component of f   pF;M 2 1 1 r LW 4 þ h . Finally, the time-related costs while stopping are given by the product of the assumed feeder bus dwell f f v p f

1

time sf, the number of bus stations along the feeder network 4LWr2 f , and the feeder bus flow, hf , yielding a time-related cost component of 8

1 pF;M LWð4sf Þr2 f hf . Thus, the feeder agency cost is given by

We take a single feeder bus ‘‘station’’ to consist of the two curbside stops on opposite sides of the same feeder bus corridor.

K. Sivakumaran et al. / Transportation Research Part A 59 (2014) 204–221

   2 p 1 1 Af Agency ¼ ðpF;S 2LWÞr2 pF;V þ F;M r1 h þ pF;M LWð4sf Þr 2 f þ LW 4 þ f hf p vf f f

211

ð6Þ

The trunk agency cost components are the same as those in Section 2. 3.2.3. Objective function and constraints The objective function for the grid hierarchical trunk-feeder system is given by the sum of the user and agency cost components described above and in Section 2. Additionally, the constraints given in Section 2 for the trunk system’s passengercarrying capacity, line capacity, and line and station spacing are also applied. The physical constraints of the feeder bus system are now also considered. The passenger-carrying capacity of the feeder bus system, Kf, must support the passenger load from both Feeder-Only and trunk users. The load contribution from the former user class is given by the product of: their trip-making density, PFq; each transit line’s catchment area, either Wrf or Lrf; the feeder vehicle headway hf; and the factor 1/4. The load contribution from r r trunk users is given by: the product of their demand density, 0.5q; the feeder segment catchment area, f2 L ¼ 12 psr f ; and the feeder vehicle headway hf. The feeder passenger capacity constraint is thus

 PF



qL 4

rf hf þ PT

qp sr f hf 6 K f 4

Additionally, the confluence of several feeder bus lines at a single trunk station can lead to station congestion if there is insufficient space for storing feeder buses. Thus, we also consider the feeder bus storage capacity at trunk stations: 1

sr 1 f hf

6 Q f max , where Qfmax [buses/hr] is the maximum feeder bus service rate that can be supported by the trunk sta-

9

tion. The trunk system constraints from Section 2 are also applied, but the trunk passenger load value is discounted by PF. For a given p, the aforementioned objective function and constraints collectively yield the mixed-integer nonlinear mathematical program below:

Minimize Zðr f ; hf ; s; H; pÞ ¼ Af User þ Y 0 þ T 0 þ pIL IL þ pIS IS þ pV V þ pM M þ Af Agency L subject to Design feasibility : ps 6 2 Trunk passenger-carrying capacity : ð1=4qLpP T ÞsH 6 K Trunk line vehicle capacity : H1 6 Q max Feeder bus passenger-carrying capacity : PF



 qp srf hf 6 K f r f hf þ P T 4 4

qL

1

Feeder bus storage capacity at trunk stations : sr1 f hf

6 Q f max

ð7Þ ð8Þ ð9Þ ð10Þ ð11Þ ð12Þ

For simplicity, p 2 {1, 2, . . ., 5}. Unlike Eq. (1), the objective function and constraints both contain non-convex terms, such 1 10 as sr 1 and sr 1 f hf f . Nonetheless, because all terms of the objective function and inequality constraints are posynomials, this mathematical program can be converted into a convex optimization problem that yields globally optimal solutions for given p. This technique, known as geometric programming, is described in Boyd et al. (2007). 3.3. Optimal trunk technology with feeder bus access It turns out that the trunk-feeder system described by (7) can render rail and BRT systems more cost-effective. Fig. 5 shows the optimal combinations of trunk and feeder technologies for low-wage ($3/hr) square-shaped cities of varying physical length, L, and trip-making density, q. Fig. 6 shows the same for high-wage ($20/hr) cities.11 The reason for rail’s ‘‘new-found’’ preferability is revealed in the details of case-specific analyzes. These analyzes come next. 3.4. Case studies First considered is a large, high-wage city like Paris, with: trip-making density q = 250 trips/km2-hr; city area L = W = 20 km: and wage rate l = $20/hr. Table 3 furnishes the city’s optimal configuration and resulting costs for rail, when 9 If this constraint renders the problem infeasible, the optimization is repeated with sufficient relaxation of Qfmax to ensure feasibility. The trunk station infrastructure cost pIS is increased by a proportional amount to account for this increase in feeder vehicle storage capacity. In reality, adding additional feeder bus berths to a trunk station often provides diminishing returns to capacity, depending on the bus maneuvers allowed at stations. An optimistic proportional increase in capacity coupled with a pessimistic increase in cost is nonetheless assumed here for simplicity. ð1Þ ð2Þ ðnÞ 10 A monomial can be described as a function f ðxÞ ¼ Cxa1 xa2 . . . xan where the constant C is greater than or equal to 0 and the exponential constants are any real numbers: a(j) 2 R. A posynomial is simply a sum of monomials (Boyd et al., 2007). 11 In both cases, the feeder bus cruising speed, vf, is assumed to be 20 km/hr with sf = 20 s of dwell time lost at each feeder stop. The feeder bus station infrastructure-, distance-, and time-related cost parameters are all assumed to be 80% of the corresponding values for regular buses. The critical time is assumed to be tc = 0.4 h. Thus, for trips longer than tc vf = 0.4(20) = 8 km, users will avail the trunk system (accessed by feeder bus). Users traveling shorter distances will use the feeder bus system as their sole mode of transport. The passenger-carrying capacity of feeder buses is assumed to be Kf = 80.

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Fig. 5. Optimal trunk technology for feeder access (low-wage cities).

Fig. 6. Optimal trunk technology for feeder access (high-wage cities).

Table 3 Trunk network configuration and per-user costs for walk and feeder bus access: rail.

Rail line spacing, rL = rW (km) Rail station spacing, s (km) Rail headway, H (min) Rail commercial speed, vc (km/hr) Total generalized cost (min) Total agency cost (min) Feeder-bus agency cost (min) User cost (min)

Walk access

Feeder bus access

1.4 1.4 4 39 94 27 – 67

6 3 4 48 70 11 5 59

access occurs on foot and when it occurs via feeder buses. Note from the table the increased commercial speed afforded by feeder bus access, as well as the reductions in user and agency cost.12 The reason for these favorable outcomes is unveiled by comparing the optimal network configurations in Table 3. Note how rail’s line and station spacings both increase when the network is accessed by feeder buses. With higher-speed access, greater portions of the costly rail network are replaced with a less expensive network of feeder buses. Rail can thus become a more attractive option for trunk-line service. Continuing in this vein, we next compare generalized costs (in units of minutes/

12 Given the slow speed of walking, our assumption that all users walk can be viewed as conservative. If some users instead utilize the faster access modes that are often available in real settings (e.g. bicycle, auto), the resulting increase in average access speed should produce reductions in generalized cost.

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Table 4 Generalized per-user costs [mins] for low-wage, square cities.

Table 5 Generalized per-user costs [mins] for high-wage, square cities.

user) for all five trunk-access combinations and for representative cities of the world, each with its own physical size and trip-making density. The generalized costs for these trunk-access mode combinations in various low- and high-wage cities are presented in Tables 4 and 5, respectively. The optimal choice of combination for each city is highlighted in the tables with shading. While other factors, including economic geography, network morphology, and public safety can influence the costs of a transit system and its future patronage in a real-world city (Kuby et al., 2004; Mackett and Sutcliffe, 2003), we nonetheless include the names of specific cities so as to provide a rough sense of scale for the hypothetical cities we examine. For comparison, the parenthesized value beside each of the tabulated values for BRT-Feeder and Rail-Feeder provides the cost when walking is the system’s access mode. Again, we see that the introduction of feeder bus service can reduce the design cost of Rail and BRT across cities of varying size and demand density. For additional comparisons across various cost scenarios for BRT-Feeder and Rail-Feeder systems, the ‘‘Low-cost’’ and ‘‘High-cost’’ values for these two trunk-access combinations are provided in parentheses below their respective labels. These values attempt to convey the high variability of infrastructure cost and the subsequent impact on the optimal design cost, recognizing that cities with a similar prevailing wage may differ in their proclivities towards certain technologies (e.g. low right-of-way costs, geology for tunelling, and available construction methods). Note from Table 4 that rail is never the optimum choice for low-wage cities, though it comes closer in larger, denser cities like Delhi. Similarly, Table 5 shows that rail can be very close to optimal in larger denser cities with high wages. Given the level of uncertainty in our cost and demand parameters, rail may actually be the optimal choice in cities like Yokohama and Paris where cost estimates for rail are close to those for BRT, as seen in the parenthesized Low-cost results. Moreover, rail

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Fig. 7. Trunk station catchment area with TOD zone.

systems may favorably transform cities over the long run in ways that our analysis cannot capture (Atkinson-Palombo and Kuby, 2011; Bhattacharjee and Goetz, 2012; Cervero, 1998; Curtis et al., 2009; Knowles, 1996; Knowles, 2012; Mackett and Edwards, 1998; Vuk, 2005). 4. Transit-oriented development (TOD) We now explore what can occur when zones of higher trip-making density cluster closely around rail stations. To this end, suppose that the shaded, square-shaped zone in Fig. 7 delineates a TOD. We assume that all users access the rail station by walking along a dense rectangular grid of N  S and E  W streets or sidewalks. Hence all points along the TOD’s perimeter lie a distance, dmax, from the trunk station, as measured in jxj + jyj space. For simplicity, we will assume that the rail network is a square grid, with pL = pW = 1, such that the access cost can be written as a simple function of s. 4.1. Trip-making densities The literature reports that given appropriate conditions, the deployment of a rail system does not increase aggregate demand for travel as much as it redistributes that demand to locations that lie closer to the system (Cambridge Systematics et al., 1998; Cervero and Landis, 1997; Cervero et al., 2004; Knight and Trygg, 1977). We shall therefore consider what happens when a fixed baseline demand for the catchment area, of area sxs in Fig. 7, is redistributed such that disproportionately high fractions of that baseline are thereafter contained in the shaded TOD zone. We take the trip-making density within the TOD zone, qTOD, to be some multiple of the density outside it, q0; i.e., qTOD = k 2 fq0, where f P 1. The total demand for the rail station, Dstation ¼ qs2 ¼ LW s , where q is the average (or baseline) density in the s x s catchment area. The Dstation is the sum of the demands within the catchment area that reside both inside and outside of the TOD, DTOD and D0, respectively. Hence, we have:

DTOD þ D0 ¼ Dstation   qTOD 2d2max þ q0 s2  2d2max ¼ qs2 giving us a density outside the TOD of

q0 ¼ q

s2 s2

2

þ ðf  1Þ2dmax

4.2. Access cost We can view q0 as a background density in the catchment area, and concern ourselves with the TOD’s density over and above this background, qTOD  q0. The average distance from anywhere within the catchment area to the trunk station is 12 s; and from within the TOD zone, the average distance is 23 dmax , as determined by geometric probability. Thus, we can express , as the expected access distance, a

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215

Fig. 8. Relationship of cost components to trip-making density distribution.

Fig. 9. Expected access distance for non-TOD users.

 

¼ a



q0 s2 12 s þ ðqTOD  q0 Þ2d2max 23 dmax qs2



or equivalently, 3

4 d ðf  1Þ þ 12 s3  ¼ 3 max a 2 2dmax ðf  1Þ þ s2

=v a . to underscore the role of f. The access and egress time component, ATOD, is thus 2a 4.3. Generalized costs We now explore how the redistributions in demand influence costs by replacing the access cost, A, in (1) with ATOD and then minimizing the resulting expression numerically. Illustrations are furnished in Fig. 8, which presents costs for rail as functions of f. Parameters were taken from Table 1 and these parameters are specified in the caption of Fig. 8.13 Note from the figure that the generalized cost diminishes as f increases, indicating that TOD can improve rail’s economic footing. Yet, the marginal reductions are small, and these diminish with f. Even for f = 10, the reduction in total generalized cost is only about 3%, as compared against a uniform trip-making density with f = 1. The percent reductions to the transit agency are smaller still. As a reference, we note that TOD ridership has been found to be three- to five13 The maximum acceptable walking distance has often been defined in the literature as ranging from 0.4 km to 0.8 km. Distances within this range have also been used to define TOD areas (Cervero et al., 2004). We thus choose an intermediate value of 0.5 km for dmax.

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Fig. 10. Optimal trunk technology without rail TOD (high-wage cities).

Fig. 11. Optimal trunk technology with rail TOD (high-wage cities).

times higher than regional averages in various case studies for rail systems (Cervero et al., 2004); and that the case of f = 10 corresponds to a TOD demand roughly four times higher than the baseline (or average) trip-making density in our idealized city. The marginal benefits are small because consolidating demand closer to rail stations leads to larger optimal spacings between those stations. Thus, with increasing f, the users who reside within the TOD zone are subject to an expected access distance of 23 dmax . Yet, those users residing within the catchment periphery (see again Fig. 7) incur greater access distances due to the larger station spacings. Fig. 9 shows how the expected access distance for users within the periphery increases with f. This effect negates some of the benefits of consolidating demand via TOD when planning a transit system. 4.4. Optimal technology with TOD We now explore how TOD can affect the choice of transit technology. We do so: for high-wage, square-shaped cities of length, L; by varying trip-making densities, q; and where access to transit occurs on foot. The shadings in Fig. 10 show the ranges of L and q for which bus, BRT, and rail are the optimal technologies in the absence of TOD, such that q is uniform (i.e. f = 1).14 Fig. 11 shows the optimal choices when rail (only) is accompanied by TOD and where we optimistically assume that f = 10. Comparing the two figures, we see that with TOD, rail becomes the optimal choice for wider ranges of L and q. Yet the difference is rather modest (i.e., the lightly shaded region in Fig. 11 is small), despite our optimistic assumption that TOD would redistribute demand in very favorable ways. 14

For all cases, we restrict the line spacing to be equal to the station spacing, i.e. pL = pW = 1.

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Fig. 12. A trunk-feeder system with uni-directional trunk lines and bi-directional feeder lines.

5. Conclusions We have explored how the economic footing of capital-intensive transit systems, like rail or BRT, might be improved by enhancing access to them. Continuum approximation models were applied to idealized settings, since this approach is known to furnish useful insights that are relevant to more complicated real-world systems. This approach was used to examine access in terms of both mobility (via the introduction of faster-moving feeder buses) and land use (via TOD). As regards mobility, we find that traveling to and from trunk-line stations by feeder bus, rather than by walking, can improve cost effectiveness. The higher access speeds dictate that the optimal spacings between trunk lines and stations both increase, such that a greater proportion of the transit system is served by the lower-cost feeder buses. Cost reductions occur even when short trips utilize only the subordinate feeder-bus system. Hence by integrating feeder systems early in the design process, capital-intensive transit systems can provide faster service, through higher commercial speeds, and become optimal or near-optimal alternatives. This is good news given the transformative effects that rail or BRT can have on a city. On the downside, a hierarchical network of trunk and feeder services can impose significant transfer penalties on its users (Ben-Akiva and Morikawa, 2002). Happily, these penalties may be lessened by: (i) reducing the disutility of transfers via the thoughtful design of stations, e.g. to promote same-platform transfers; and (ii) reducing the number of transfers required via alternative network configurations. As an example of (ii), consider a BRT system where the same buses both collect patrons on local streets and travel on higher-speed trunk lines. Or, consider the network in Fig. 12 where (possibly many) bi-directional, meandering feeder lines pass through trunk stations. Trips in this network require at most two transfers, as exemplified with the dotted arrows. As regards land use, we find that when walking is the only access mode, modest cost reductions come via TODs that clump travel demand close to rail stations. But the optimal spacings between these stations grow as a result, and this penalizes users who live (or work) outside of a TOD. Had our models accounted for the elasticity of travel demand on access distance, we would expect to find a reduction in the rail system’s ridership due to these penalties. Hence the benefits of planning rail systems around TODs seem limited. Acknowledgment The present work was funded by a Faculty Research Grant from the University of California Transportation Center. Appendix A. Cost parameters Tables A1 through A3 contain cost estimates for a high-wage city, defined to be a city with an average wage of roughly $20/hr. For each mode, costs are decomposed into infrastructure and operating cost components (both distance- and timerelated). We will assume the operating cost parameters for low-wage cities differ as follows. Regarding operating cost components, we assume the hourly labor cost in low-wage cities to be $6/hr, due to an assumed premium for semi-skilled labor (relative to our assumed prevailing Low wage rate of $3 per hour). Other components (fuel, maintenance, etc.) are assumed to be the same. Regarding infrastructure cost components, we assume that infrastructure costs in low-wage cities are less than those in high-wage cities by 30%. This value is loosely based on the cost differences observed between heavy rail projects in Latin America and those in Europe/U.S. from Flyvbjerg et al. (2008). We now derive the rail and BRT infrastructure costs for ‘‘Low-cost’’ and ‘‘High-cost’’ scenarios within Low- and High-wage cities. First examined are High-wage cities. Rail stations currently planned for construction in high-wage cities were found to have projected costs as low as $30 M for the above-ground South Perth Railway Station (Syme Mermion & Co, 2010) and as high as $240 M for the San Francisco’s

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Table A1 Cost parameter breakdown for rail (high-wage city). Parameter

Value

Comments

Infrastructure Costs Infrastructure Line Cost [$/km] Infrastructure Line Cost [$/km] $IL, Infrastructure Line Cost [$/km-hr]

$140,000,000 $187,000,000 $990

In 2000 $, from Halcrow Fox (2000) Infrastructure cost in 2012 $ Straight-line amortization across an assumed 30-year life, with operation 350 days/yr and 18 h/day Infrastructure cost in 1983 $, from Pickrell (1985) and Flyvbjerg et al. (2008) Infrastructure cost in 2012 $ Straight-line amortization across an assumed 30-year life, with operation 350 days/yr and 18 h/day

Infrastructure Station Cost [$/station] Infrastructure Station Cost [$/station] $IS, Infrastructure Station Cost [$/ station-hr] Operating Costs (Distance) Energy Consumption per Car [kWh/carmi] Energy Consumption per Train [kWh/ train-mi] Energy Cost per kW-hr Energy Cost per Train-Mile [$/veh-mi] $V, Cost per veh-km [$/veh-km] Operating Costs (Time) # Employees per Vehicle Average Wage [$/hr] Labor Cost per Hr [$/hr] Purchase Price of Vehicle [$] Vehicle Lifespan [yrs] Depreciation per hr [$/hr] $M, Cost per vehicle-hr [$/veh-hr]

$40,000,000 $92,000,000 $490

3.6

From Sfeir and Chow (2007) regarding BART system’s energy consumption

36

Assumed 10-car trains

$0.10 $3.60 $2.20

Average from Electric Power Monthly (2011)

5 $20 $100 $20,000,000 30 $101 $201

From Wilson (2010) and Pushkarev and Zupan (1977) Based on high-wage city from UBS (2010) Estimate of $2 M per train car from Wilson (2010), assuming 10-car trains. Assumed straight-line depreciation

Table A2 Cost parameter breakdown for BRT (high-wage city). Parameter Infrastructure Costs Infrastructure Line Cost [$/mile] Infrastructure Line Cost [$/km] $IL, Infrastructure Line Cost [$/km-hr] Infrastructure Station Cost [$/ station] Infrastructure Station Cost [$/ station] $IS, Infrastructure Station Cost [$/station-hr] Operating Costs (Distance) Maintenance Cost per VehicleMile [$/veh-mi] Fuel Price [$/gal] Fuel Efficiency [mpg] Fuel Cost per Mile [$/mile] Cost per Veh-Mile [$/veh-mi] $V, Cost per veh-km [$/vehkm] Operating Costs (Time) # Employees per Vehicle Average Wage [$/hr] Labor Cost per Hr [$/hr] Purchase Price of Vehicle [$] Vehicle Lifespan [yrs] Depreciation per Hr [$/hr] $M, Cost per veh-hr [$/veh-hr]

Value

Comments

$32,000,000 $36,108,800 $270

Infrastructure cost in 1989 $, from Kain et al. (1992) Infrastructure cost in 2012 $ Straight-line amortization across an assumed 20-year life, with operation 350 days/yr and 18 h/day

$500,000

Infrastructure cost in 2001 $

$670,000

Infrastructure cost in 2012 $

$7

Straight-line amortization across an assumed 20-year life, with operation 350 days/yr and 18 h/day

$0.30

In 2007 $, from Clark et al. (2007) for 40-ft buses; assumed that use of articulated buses increases maintenance cost from $0.20/mile to $0.30/mile

$4.00 6 $0.67 $0.97 $0.66

4 $20 80 $500,000 20 $3.81 $84

From Clark et al. (2007)

Accounting for inflation, converted to 2012 $

From Wilson (2010) and Pushkarev and Zupan (1977) Based on high-wage city from UBS (2010) From Wilson (2010) for articulated 60 ft. bus Assumed straight-line depreciation

Central Subway Chinatown Station (Central Subway Outreach Team, 2012). Since cost escalation for rail infrastructure has historically averaged roughly 45% (Flyvbjerg et al., 2003), we apply a cost overrun factor of 1.45 to both these values, obtaining Low- and High-cost station infrastructure cost values of $44 M and $348 M, respectively.

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K. Sivakumaran et al. / Transportation Research Part A 59 (2014) 204–221 Table A3 Cost parameter breakdown for bus (high-wage city). Parameter Infrastructure Costs Infrastructure Line Cost [$/ mile] Infrastructure Line Cost [$/km] $IL, Infrastructure Line Cost [$/km-hr] $IS, Infrastructure Station Cost [$/station-hr] Operating Costs (Distance) Maintenance Cost per VehicleMile [$/veh-mi] Fuel Price [$/gal] Fuel Efficiency [mpg] Fuel Cost per Mile [$/mile] Cost per Veh-Mile [$/veh-mi] $V, Cost per veh-km [$/vehkm] Operating Costs (Time) # Employees per Vehicle Average Wage [$/hr] Labor Cost per Hr [$/hr] Purchase Price of Vehicle [$] Vehicle Lifespan [yrs] Depreciation per Hr [$/hr] $M, Cost per veh-hr [$/veh-hr]

Value

Comments

$1,600,000

Infrastructure cost in 1989 $, from Kain et al. (1992)

$1,805,440 $10

Infrastructure cost in 2012 $ Straight-line amortization across an assumed 20-year life, with operation 350 days/yr and 18 h/day

$0.70

For simplicity, we assume that each station requires stops on both sides of its corridor, and that the collective cost of these two stops is one-tenth the cost of a BRT station. The per-stop cost in this case is $0.35., and the per-station cost is $0.70

$0.20

In 2007 $, from Clark et al. (2007) for 40-ft buse

$4.00 6 $0.67 $0.87 $0.59

3 $20 60 $350,000 20 $2.66 $63

From Clark et al. (2007)

Accounting for inflation, converted to 2011 $

From Wilson (2010) and Pushkarev and Zupan (1977) Based on high-wage city from UBS (2010) From Wilson (2010) for standard 40 ft. bus Assumed straight-line depreciation

Table A4 Cost parameter breakdown for low-cost and high-cost scenarios for BRT and rail. Line infrastructure cost,$IL[$/km-hr] Low-wage city

BRT Rail

Station infrastructure cost,$IS[$/station-hr]

High-wage city

Low-wage city

High-wage city

Low cost

High cost

Low cost

High cost

Low cost

High cost

Low cost

High cost

$14.70 $252

$351 $778

$21 $360

$501 $1111

$0.27 $163

$18.33 $1289

$0.38 $233

$26.19 $1841

Ignoring cost outliers, rail line infrastructure costs in 2002 US$ were as low as $70 M per kilometer for the partially tunneled Copenhagen Metro and as high as $220 M per kilometer for the Paris Metro Line 14 (Flyvbjerg et al., 2008). Adjusting for inflation, these values are equivalent to $98 M and $280 M in 2012 US$. However, both cost values include station construction costs, which typically amount to 20–25% of total construction costs (Flyvbjerg et al., 2008; Pickrell, 1985). To uncouple line and station costs, we thus multiply the previous values by a factor of 0.75 and obtain Low- and High-cost rail line infrastructure cost values of $68 M per kilometer and $210 M per kilometer. For rail line infrastructure costs in Low-wage cities, we apply our assumed 30% reduction in cost to the above (uncoupled) infrastructure cost values. Low- and High-cost Per-kilometer rail line infrastructure costs in Low-wage cities are then $47.6 M and $147 M, respectively.15 Similarly, per-station rail infrastructure costs in Low-wage cities are $31 M and $244 M. BRT infrastructure costs can also vary widely across cities with the same prevailing wage, depending on the system’s implemented components (e.g. dedicated lanes, transit signal priority, large stations). In regards to BRT stations in Highwage cities, Hess et al. (2005) notes per-station infrastructure costs in 2002 US$ as low as $38,000 (Wilshire I corridor in Los Angeles) and as high as $2.6 M (New Britain-Hartford Busway in Connecticut, under construction). Adjusting for inflation, these per-station cost values are equivalent to $48,500 and $3.3 M in 2012 US$. BRT line infrastructure costs in High-wage cities have generally varied between a Low-cost value of approximately $2.2 M per kilometer for the Eugene-Springfield BRT (Hess et al., 2005) and a High-cost value of $53.2 M per kilometer for the Silver Line in Boston (Hensher and Golob, 2008). Adjusting for inflation, these values are equivalent to $2.6 M and $63.1 M in 2012 US$, respectively. To uncouple the station costs from these total cost values, we simply maintain the Low-cost scenario’s per-

15 These costs compare reasonably well to the per-kilometer line infrastructure costs of Mexico City’s Line B ($56 M per kilometer) and Cairo’s Metro system ($156 M per kilometer).

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kilometer line infrastructure cost as $2.2 M. For the High-cost scenario, we will simply assume that BRT station costs amount to 5% of BRT station costs, yielding a High-cost BRT line infrastructure value of $59.8 M per kilometer. For infrastructure costs in Low-wage cities, we apply our assumed 30% cost reduction to the above (uncoupled) infrastructure cost values. Finally applying the assumed lifespan and operating hours from Tables A1 and A2 to the above values, we obtain Table A4, which contains Low- and High-cost values for line and station infrastructure across Low-wage and High-wage cities. Appendix B. Trip length distribution in a rectangular city Assume for a rectangular city of width W and length L that user origins and destinations are respectively represented by X–Y coordinate pairs x1, y1 and x2, y2. Random variables x1 and x2 are assumed to be uniformly distributed between 0 and L and random variable y1 and y2 are assumed uniformly distributed between 0 and W. Take Z as a random variable representing the trip distance: Z = jx1  x2j + jy1  y2j. Fairthorne (1965) gives the probability distribution of Z as

2z ð6LW  3zðL þ WÞ þ z2 Þ for 0 6 z 6 W 3L2 W 2 2 fZ ðzÞ ¼ 2 ð3L þ W  3zÞ for W 6 z 6 L 3L 2 fZ ðzÞ ¼ 2 2 ðL þ W  zÞ3 for L 6 z 6 L þ W 3L W

fZ ðzÞ ¼

Consider now the users of a hierarchical grid trunk-feeder transit system. Optimistically assuming feeder buses travel at cruising (rather than commercial) speed vf, users will choose the feeder bus system as their sole transport mode as long as their estimated feeder bus travel time falls below a critical trip time tc. A critical trip length can then be defined as dc = Rd P(Z 6 dC), and thus the proportion of ‘‘Feeder-Only’’ users is given by the probability PF, such that PF ¼ PðZ 6 dC Þ ¼ 0 C f ðtÞdt.  The expected trip length for these Feeder-Only trips, dfeeder , is given by the following conditional expectation:

E½ZjZ 6 dC  ¼

1 Pðz 6 dC Þ

Z

dC

tfZ ðtÞdt ¼

0

1 F Z ðdC Þ

Z

dC

tfZ ðtÞdt

0

where F Z ðzÞ is the cumulative distribution function of z. Evaluation of this integral yields the following expressions for, depending on the value of dC

 d feeder

8 dC ð40LW15LdC 15WdC þ4d2 Þ > E½ZjZ 6 dC  ¼ 5 12LW4Ld 4Wd þd2 c > > ð > C C cÞ > < 2Wd2c þLð6d2c W 2 ÞW 3 =54d3c ¼ E½ZjZ 6 dC  ¼ > Lð12W12dC Þ4WdC þ6L2 3W 2 þ6d2c > > > > : E½ZjZ 6 d  ¼ 4d5c þL5 þ5L4 Wþ20ðLþWÞ2 d3c þ5LW 4 þW 5 15ðLþWÞd4c 10ðLþWÞ3 d2c C 5d4 þ30ðLþWÞ2 d2 þ5L4 þ20L3 Wþ20LW 3 þ5W 4 20ðLþWÞd3 20ðLþWÞ3 d c

c

c

for 0 6 dC 6 W for W 6 dC 6 L C

for L 6 dc 6 L þ W

Now consider the proportion of users who opt to utilize the trunk system for their travel, PT. This proportion of users is  simply given by PT = 1  PF. The expected trip length for these trunk users, d trunk , is given by the following conditional expectation,

E½ZjZ > dC  ¼

1 PðZ > dC Þ

Z

LþW

tfZ ðtÞdt ¼

dC

1 1  F Z ðdC Þ

Z

LþW

tfZ ðtÞdt;

dC

 which yields the following expressions for d trunk depending on the value of dC.

 d trunk ¼

8 3 2 2 3 W 8LWd3c þ3Ld4c þ3Wd4c 4=5d5c > E½Zjz > dC  ¼ 2L W6Lþ2L > 2 2 > W 12LWd2c þ4Ld3c þ4Wd3c d4c < 10L

3

þ10L Wþ5LW 30Ld2c þW 3 10Wd2c þ20d3c 15W 2 þ20WdC 60LW30d2c þ60LdC 2

2

E½Zjz > dC  ¼ > > > : E½Zjz > dC  ¼ 15 ðL þ W þ 4dc Þ

for 0 6 dC 6 W for W 6 dC 6 L for L 6 dc 6 L þ W

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