Structure of Competitive Transit Networks
Carlos F. Daganzo
WORKING PAPER
UCB-ITS-VWP-2009-6
August 2009
STRUCTURE OF COMPETITIVE TRANSIT NETWORKS
Carlos F. Daganzo
Institute of Transportation Studies
University of California, Berkeley, CA 94720
(August 6, 2009)
ABSTRACT
This paper describes the network shapes and operating characteristics that allow a transit
system to deliver a level of service competitive with that of the automobile. To provide
exhaustive results for service regions of different sizes and demographics, the paper
idealizes these regions as squares, and their possible networks with a broad and realistic
family that combines the grid and the hub-and-spoke concepts. The paper also shows how to
use these results to generate master plans for transit systems of real cities.
The analysis reveals which network structure and technology (Bus, BRT or Metro)
delivers the desired performance with the least cost. It is found that the more expensive the
system’s infrastructure the more it should tilt toward the hub-and-spoke concept. Both, Bus
and BRT systems outperform Metro, even for large dense cities. And BRT competes
effectively with the automobile unless a city is big and its demand low. Agency costs are
always small compared with user costs; and both decline with the demand density. In all
cases, increasing the spatial concentration of stops beyond a critical level increases both, the
user and agency costs. Too much spatial coverage is counterproductive.
1. INTRODUCTION
This paper examines the structure of urban transit systems that can deliver a level of service
comparable to that of the automobile and the character of the cities in which this can be done at
reasonable cost. These transit networks should provide good service between every pair of points in the
city throughout the day, and be easily understood by the public. Only if they do this will they encourage
auto users to leave their cars home when their daily plans include complex trip chains with impromptu
and non-routine links.
To achieve these standards, transit systems must uniformly cover the service region in space and time
with well-spaced transit stops and frequent reliable service. Good spatial coverage limits the walking time
to/from every point in the service region, and good time coverage the waiting and transfer times.
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Coverage should be dense enough to ensure that the sum of these times for any trip is comparable to the
time that auto travelers spend walking to or from their cars, looking for parking and inside garages; i.e.,
not more than about 10 minutes. The system should also have a structure that guarantees in-vehicle-travel
times comparable to those of auto trips; and just as importantly be cost-competitive and reliable. If all
these criteria are met, public transit can become a viable alternative to the automobile because most auto
trip chains could then just as well be completed by transit.
Other authors have asked structural transit questions. For example, Holroyd (1965) examined grids,
Newell (1979) a hub and spoke system where the hub was a large street, and Wirasinghe et al (1977)
corridors. These works determined the stop spacing, service frequency and, where appropriate, the line
spacing that best balances the generalized user cost with the cost of providing service. Unfortunately, the
structural families analyzed in these and other papers are too narrow to answer the type of question posed
here. A more general family is needed.
Section 2 below describes the new family as well as the basic notation and assumptions. Section 3
expresses the systems’ measures of performance in terms of key system descriptors. Then, using these
formulae Section 4 formulates the design problem as a constrained optimization problem that answers our
questions quantitatively. Section 5 discusses the results qualitatively, suggests model refinements, and
proposes a way to develop master plans for real cities.
2. THE HYBRID CONCEPT
The service region is a square of side D (km) that generates
Λ
passenger trips per hour (p/hr) during
the rush hour and an average of
λ
p/hr during the day’s hours of service. Its origins and destinations are
uniformly and independently distributed over the service region. The uniformity assumption is appealing
because it does not require any extra parameters. Although it penalizes transit, compared with more
centripetal distributions, this is a good thing because it sets a high bar for success.
Service is provided by buses or trains which are characterized by: their design capacity, C (p); their
cruising speed including stops due to traffic and pedestrian interferences v (km/hr); the time lost per stop
due to the required door operation, deceleration and acceleration,
τ
(hr/stop); and the time added per
boarding passenger
τ
’ (hr/p). We assume that the buses or trains can be operated at the cruising speed
with only moderate variability in the headways; see Daganzo (2009) for a discussion of control methods.
Transit routes must lie on streets, which form a fine square mesh parallel with the square’s sides. To
provide uniform spatial coverage the square is covered with a square lattice of stops. This lattice is also
oriented with the sides of the square, and its spacing is s (km). To provide good temporal coverage the
service headway is assumed to be H (hr) in the central part of the square where all the transfers will take
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place, but it is allowed to be higher along its fringe. (Values of our coverage variables comparable with s
= 0.5 km and H = 3 min seem reasonable if our transit system is to be competitive with the automobile.)
A focus of this paper is the system’s structure and layout. The key choice on layout is whether each
stop is covered by one or two lines. Figure 1a shows a hub and spoke pattern with trunk and branch that
provides single coverage everywhere. Only one fourth of the square is shown because the other three
wedges are obtained by rotating the picture 90°, 180° or 270°. Note each stop is associated with s km of
infrastructure if one ignores the few stops where lines branch. Figure 1b shows a grid pattern that
provides double coverage. Here, every stop is covered by two lines and is associated with 2s km of
infrastructure. Therefore, the grid system includes more kilometers of line and is more expensive to
operate. However, it also provides a higher level of service since users can travel from any origin stop to
any destination stop without detouring from their shortest paths. Thus, the grid system should be preferred
if enough people can share the rides to pay for its higher cost; i.e., if
λ
is high enough.
The choice does not have to be black or white, however. Figure 1c displays an example of a hybrid
system that provides double coverage in a central square of side d ≤ D and single coverage in its
periphery. We shall see in the next sections that by varying the value of the ratio
α
≡ d/D from
α
= s/D
(hub and spoke) to
α
= 1 (grid) we can find structures that outperform both extremes.
The hybrid system is intuitive and easy to navigate. For maximum transparency, maps could use a
coordinate system to number the lines. For example, N-S lines could be numbered from left to right with
letters (A, B, C…) and offshoots from their trunks also from left to right, Aa, Ab, … Ba, Bb, …etc.
Likewise, its E-W lines could be numbered from top to bottom (1, 2, 3…), and their offshoots 1a, 1b,…
2a, 2b, … etc.
3. ANALYSIS
Formulae for key performance metrics of the hybrid network are now presented.
3.1 Agency metrics
Important agency metrics are: the total vehicular distance traveled per hour of operation, V (veh-km/hr);
the vehicle hours traveled during the rush hour, M (veh); the infrastructure length L (km); and the peak
vehicle occupancy during the rush hour, O (p/veh). The parameters V and M correlate with the agency’s
operating cost (M is also the required fleet size in operation); L correlates with the fixed capital costs; and
O with the required vehicle size. The following approximate formulas are derived in Appendix A:
L = [D
2
/s][1+
α
2
]; (1)
V = [2D
2
/sH][3
α
−
α
2
]; (2)
3
