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Impact of Buses on the Macroscopic Fundamental1
Diagram of Homogeneous Arterial Corridors2
Felipe Castrillona, Jorge Lavala,∗3
aSchool of Civil and Environmental Engineering, Georgia Institute of Technology4
Abstract5
This paper proposes a modification of the method of cuts to estimate the6
effect of bus operations on the Macroscopic Fundamental Diagram of long ar-7
terial corridors with uniform signal parameters and block lengths. It is found8
that the impact of buses can be adequately captured by the theory of moving9
bottlenecks using only the bus average free-flow speed. This speed considers10
the effects of signals and stops but not traffic conditions and therefore can11
be calculated endogenously. A formulation is developed to approximate the12
bus average free-flow speed as a function of bus operational parameters. We13
also found that buses produce an additional capacity restriction that is sim-14
ilar to the ”short block” on networks without buses and that can severely15
reduce corridor capacity. The proposed method is validated with numerical16
approximations of the corresponding kinematic wave problem.17
Keywords: bus operations, moving bottleneck, macroscopic fundamental18
diagram19
1. Introduction20
The Macroscopic Fundamental Diagram (MFD) can be a valuable tool21
to monitor and control congestion on urban networks. First proposed by22
Daganzo (2007), the MFD relates the average flow and density of a net-23
work as an invariant property of the system. This defined relationship makes24
it possible to obtain basic traffic performance measures by simply knowing25
the number of vehicles in the network. Currently, there are three main ap-26
proaches to estimate the MFD. The first one is micro-simulation method27
which was first verified by (Geroliminis and Daganzo, 2008) on a large net-28
work in Downtown San Francisco. Also, there is the variational theory (VT)29
∗Corresponding author. Tel. : +1 (404) 894-2360Email address: [email protected] (Felipe Castrillon)Preprint submitted to Transportmetrica B January 31, 2017
network approach, which is numerical (Daganzo, 2005a,b). Finally, we have30
the analytical approach, or the ”method of cuts” (MOC) which has been de-31
veloped for homogeneous corridors (Daganzo and Geroliminis, 2008) and later32
extended to heterogenous corridors (Laval and Castrillon, 2015). Geroliminis33
and Boyaci (2012) use the analytical approach to perform sensitivity analysis34
of topological features (block lengths, signal parameters) on the MFD.35
The approaches above focus on a homogeneous stream of cars but do36
not include the effects of other modes of transportation. To address this,37
a few research studies have extended the models to incorporate a mixed38
stream of car and bus operations. From a micro-simulation perspective,39
Geroliminis et al. (2014) demonstrate the existence of a bi-modal MFD of40
Downtown San Francisco with real-life bus schedule inputs. Other micro-41
simulated approaches of mixed traffic include Castrillon and Laval (2013);42
Xie et al. (2013).43
Boyaci and Geroliminis (2011) extend VT networks to include bus stops as44
temporary bottlenecks blocking one lane. The authors explore a few corridor45
configurations and find that bus capacity varies greatly with changing input46
parameters. Xie et al. (2013) extend this work to incorporate the moving47
bottleneck effect of buses (Gazis and Herman, 1992; Newell, 1993; Munoz48
and Daganzo, 2002) by reducing capacity by one lane for a particular block49
during the time that the bus is active on that block similar to Daganzo50
and Laval (2005). These numerical methods are exact and useful tools to51
perform simulations but provide limited insight into how input parameters52
(bus operation parameters, signal timings, block length) affect the output53
capacity of a system. Eichler and Daganzo (2006) and (Xie et al., 2013)54
also focus on the analytical methodology. The authors incorporate buses to55
the MOC by introducing a moving bottleneck cut to the MFD, under the56
assumption that the average bus operating speed is known.57
Other multi-modal approaches focus on passenger flow (Geroliminis et al.,58
2014; Chiabaut, 2015a) and/or dedicated bus lanes (Eichler and Daganzo,59
2006; Chiabaut et al., 2014). Chiabaut (2015b) investigate the effects of60
double-parked trucks, which provide similar but longer lasting effects than61
bus stops. Zheng and Geroliminis (2013) develop a multi-region macroscopic62
approach to allocate road space between cars and buses using optimization63
methods. Gonzales and Daganzo (2012) extends the morning commute prob-64
lem to include multi-modal operations. Zheng et al. (2013) investigates a 3D-65
MFD for bi-modal urban traffic using a commercial micro-simulation model.66
Gayah et al. (2016) analyses the impact of obstructions such as bus stops on67
2
the capacity of nearby signalised intersections using variational theory.68
A common feature of the above references seeking analytical insight is the69
assumption that the bus operating speed is known exogenously. In this paper70
we propose a method to overcome this limitation. It turns out that a good71
approximation of the impact of buses is based on the bus average free-flow72
speed. This speed considers the effects of signals and stops but not traffic73
conditions and therefore can be calculated endogenously. A formulation is74
developed to approximate the bus average free-flow speed as a function of bus75
operational parameters. This realization allowed us to propose a modification76
of the method of cuts (mMOC), which is simple, parsimonious and uses only77
four parameters. Compared to the analytical component in (Xie et al., 2013),78
our method (i) approximates the average bus operating speed endogenously79
using renewal theory, and (ii) captures what we call the “bus short-block80
effect”, which is a result of the interactions between buses and traffic signals.81
The remainder of this paper is organized as follows. Following a brief82
background in section 2, the problem is defined in section 3. Section 4 uses83
simulation sensitivity analysis to understand the effect of key parameters84
on the MFD and to provide key insight to formulate the proposed mMOC85
method in section 5. Section 6 gives a discussion and outlook.86
2. Background87
In this section we will review the basic building blocks for the proposed88
methods: the theory of moving bottlenecks and the method of cuts.89
Hereafter, a lane will be assumed to follow a triangular fundamental dia-90
gram with capacity Q, jam density κ, free-flow speed of cars u and congestion91
wave speed −w, and critical density kc; notice that only three parameters92
are required while the rest can be derived.93
2.1. Moving bottlenecks94
The moving bottleneck model was first reported in Gazis and Herman95
(1992), but it was Newell (1998) who puts it in the context of kinematic wave96
theory. The theory is particularly useful when the fundamental diagram on97
the roadway is assumed to be triangular in shape. The empirical validation98
can be found in Munoz and Daganzo (2002), while numerical methods for99
incorporation into simulation models are presented in Daganzo and Laval100
(2005); Laval and Daganzo (2006).101
3
Dv
UD
U
A
C
v
A
-w
u
QD
vU
QU
density
flow sp
ace
time
(a) (b)
x0
v
ER
s
s
Figure 1: Moving bottleneck effect. (a) Fundamental diagram with moving bottleneck”cut”. (b) Time-space diagram showing the effects of a moving bottleneck trajectory onupstream and downstream traffic states
A moving bottleneck creates a free-flow traffic state downstream of its102
trajectory denoted by D and a congested traffic state upstream given by U .103
These traffic states are plotted on the fundamental diagram in Fig. 1(a).104
The downstream flow can be assumed to be the capacity of the remaining105
unblocked lanes QD = Q(n − 1) where n is the number of lanes. A bus106
traveling at speed v is represented by the line between traffic states D and U107
in the figure, which acts as an upper bound to the fundamental diagram as108
vehicles cannot pass the bus on one of the lanes. An example of the moving109
bottleneck effect on the time-space diagram is shown in Fig. 1(b), where110
a traffic state A is present before and after the bus enters the segment at111
location x0.112
2.2. Method of Cuts113
According to VT, the solution to a kinematic wave problem can be ob-114
tained by solving:115
N(P ) = infBεβP{N(B) + ∆BP}, (1)
where N(P ) is the cumulative count of vehicles at location x by time t,116
P = (t, x) is a generic point, βP is the boundary data in the domain of117
dependence of P , ∆BP is the maximum number of vehicles that can cross118
the minimum path connecting boundary point B = (tB, xB) and point P .119
Daganzo and Geroliminis (2008) express (1) in terms of the steady state120
flow, i.e. q = N(P )/t, t→∞, which gives the method of cuts:121
q = mins{sk +R(s)}, (2)
4
B P
u
t
x
t =B 0
BxB
constant density
-w
time
P = ( , )t xO
s
q
k
R s( )
qs
(a) (b)
Figure 2: (a) Initial value variational theory problem with constant density on the bound-ary. For each boundary point B, the minimum path is obtained from all valid paths. (b)Each minimum path corresponds to a upper boundary ”cut” to the MFD
where k is the initial density, s is the average speed of a valid path and122
R(s) = ∆BP/t is the maximum passing rate of the minimum path. Therefore,123
the MFD is the lower envelope of the family of “cuts” qs(k) = sk +R(s), as124
shown on Fig. 2b.125
3. Problem definition126
Consider an arterial corridor consisting of a large sequence of traffic sig-127
nals with common green time g, red time r, cycle time c = r + g, and offset128
between two neighboring signals, o. The distance between two traffic signals,129
or simply block lengths, is l, also constant. Buses travel the entire corridor130
with inter-arrival times obtained from a Poisson process with mean headway131
h.132
Buses travel at a maximum speed v ≤ u when unobstructed by down-133
stream traffic, red lights, or bus stops. These stops are located at each block134
and the bus will stop in each one with a probability p, for a (deterministic)135
dwell time d.136
A key variable will be the bus average free-flow speed, v, where “free-137
flow” means that the bus is constrained only by red lights and bus stops, not138
by downstream traffic.139
5
Variable Symbol Default value
cell size Δx 6.67 m
time step Δt 1.2 s
Car free-flow speed u 80 km/hr
Congestion wave speed -w -20 km/hr
Jam density κ 150 veh/km
Optimal density k c 30 veh/km
Maximum lane flow Q 2400 veh/h
Number of blocks b 75
Number of lanes n 1 or 2
Length of each block l 160 m
Cycle length c 72 seconds
Green time g 36 seconds
MFD time aggregation T 15 minutes
Bus free-flow speed v 60 km/h
Probability of stopping p s 1/3
Dwell time d 30 seconds
Headway h varies
Network parameters
Bus operation parameters
Table 1: Variables, notations, and default values for experiments
The default parameter values are given in Table 1; they are used through-140
out the paper unless otherwise stated.141
4. Simulation experiments142
In this section we run three simulation experiments to see how the differ-143
ent parameters affect the shape of the MFD. The simulation tool is described144
in Appendix A, and gives the exact numerical solution of the problem in the145
previous section.146
Each flow-density data point obtained in this section corresponds to an147
individual simulation, and the flow-density measure is taken after reaching148
steady-state. Different traffic states in the free-flow branch are obtained by149
running the simulation with initial and boundary conditions corresponding150
to a density ranging from zero to the critical density. To obtain the congested151
branch the initial and boundary conditions are maintained at capacity while152
limiting the exit rate of the last intersection on the corridor. In this case153
the entire corridor will be homogeneously congested in the steady-state, as154
required by the theory, because the queue generated at the downstream exit155
6
will have spilled all the way back to the entrance of the corridor.156
Output results are normalized to give more generality in our results since157
all parameters become dimensionless. Following Laval and Castrillon (2015)158
we set the maximum flow Q = 1 and jam density κ = 1, and define the159
dimensionless parameters as:160
λ = l/l∗, ρ = r/g τ = o/c γ = h/g, α = v/u (3)
where λ represents block length measured in units of the critical block length,161
l∗ = g(1/u + 1/w) and ρ is the red to green ratio. It is worth mentioning162
that the critical block length is the height of the “influence triangle” to be163
described momentarily on Fig. 9 to explain the short-block effect; see Laval164
and Castrillon (2015) for more details. Notice that τ and α will vary between165
0 and 1, but the remaining parameters are simply positive real numbers.166
Experiment 1- The effect of buses and topological parameters167
The purpose of this experiment is to examine the effect of bus operations168
on homogeneous corridors by varying block lengths, and signal parameters.169
Twenty-seven roadway cases are considered, which are obtained by com-170
bining three different λ values (0.5, 1, 1.5), ρ values (0.5,1.0,1.5), and τ values171
(0,0.33,0.66). Only nine cases are shown on each diagram on Figure 3 for172
clarity, but the following findings are general:173
F(a) The MFD for the ”NO BUS” case is an upper bound to the MFDs174
when buses are introduced. This upper bound is tight during regions175
of low flow (but usually not tight during regions of moderate to high176
flow), which means that buses no longer constrain traffic.177
F(b) The MFD flow is monotonically decreasing as bus headway increases178
for all experiments, as expected.179
F(c) The effect of bus headway on the MFD varies greatly depending on180
corridor configuration. As ρ increases, the flow decreases for all bus181
headways, as expected. As λ and τ changes, the MFD shape changes182
drastically.183
Notice that although the bus maximum speed v is kept constant in all the184
experiments, one can see from the figure that the MFD free-flow speed in the185
figure seems different in each case, and that it explains most of the differences186
with respect to the no-bus MFD. As we will see in the next section, this187
corresponds to the the average bus free-flow speed, v.188
7
]
Figure 3: Experiment 1: Each diagram represents a different corridor configuration byvarying corridor parameters λ, ρ, and τ . For every diagram, there are 8 different color-coded MFD curves, each with a different mean dimensionless headway, γ. Recall thatto obtain a headway with dimensions, eqn. (3) must be used, i.e. h = γg. Simulationparameters include: g = 36 s, v = 40 km/h, p = 1/3, d = 20 s, b = 75 blocks, n = 1 lanes.
8
Experiment 2- The effect of corridor length189
This experiment examines how the MFD changes as the corridor length190
increases. The results on Figure 4(a) suggest the following:191
F(d) As the number of blocks increases, the shape of the MFD in the left192
vicinity of capacity converges towards a straight line, similar to the193
moving bottleneck line on unsignalized corridors.194
F(e) This convergence is almost independent of bus headway.195
The time-space diagram on Figure 4(b) provides a simplified explanation for196
this behavior, by ignoring the effects of traffic lights. When buses enter or197
exit the corridor, traffic states A and C are induced at the beginning and end198
of the corridor, whose area is independent of the corridor length. It follows199
that for long corridors the effect of these areas vanishes, and traffic states U200
and D of the moving bottleneck model dominate. Notice that a similar effect201
was found in the case of trucks on two-lane rural roads Laval (2006a).202
Experiment 3- The effect of bus operational parameters203
This experiment examines the effects of operational parameters v, p and204
d while keeping bus average free-flow speeds approximately constant. Three205
different roadway cases are shown on Figure 5. Each case is depicted by206
two diagrams: the MFD and the average bus speed vs. density. Average bus207
speeds on the y-axis are normalized to the maximum car speed u = 80 km/h.208
For each case, four different simulation runs are chosen with similar bus209
average free-flow speeds, but with varying bus operational parameters. For210
instance, on the first case (top two diagrams), the normalized average free-211
flow speed is close to 0.12. Notice that all curves converge to this value when212
densities are close to zero. The results on Figure 5 indicate the following:213
F(f) As bus input parameters differ, bus average speeds also differ dur-214
ing free-flow and capacity conditions; during congested conditions they215
converge.216
F(g) The bus free-flow average speed appears to be essential when determin-217
ing the MFD. The left column of the figure suggests that the average218
bus free-flow speed might be the most important bus input operational219
parameter to consider even as other bus operational parameters vary.220
The objective now is to develop an analytical method able to capture the221
insights revealed in this section. This is shown next.222
9
(a)
A
U D
A
v
u
s
bustrajectory
headway
space
time
corridor length
(b)
blocks = 10
flow
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
blocks = 50
flow
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
blocks = 75
flow
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
blocks = 150
flow
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
C u
U D
C
1 2 4 6 8 10 20 US
Legend:
normalized headway ( = headway/green)?
k
q
kUkD
AU
vv
u -w
s-w
C
D
Figure 4: (a)Experiment 2: Simulation runs with varying number of blocks. The inputparameters are: λ = 1.0, ρ = 0.5, τ = 0, g = 30 s, v = 30 km/h, p = 1/3, d = 20 s, n = 2lanes; recall that dimensional variables can be obtained with eqn. (3). (b) Kinematic wavetime-space diagram example showing the boundary effects of buses.
10
Figure 5: Experiment 3: Three different roadway configurations, with four different sim-ulation runs each. For a given configuration, all runs have similar bus average free-flowspeeds but varying bus operational paramaters. For instance, for the top case, the inputparameters (v,p,d) are (20 km/h, 0, 0 s), (40 km/h, 0.5, 18 s), (60 km/h, 0.5, 24 s), (80km/h, 1, 12 s).
11
(a) (b)
capacity cut
moving bottleneck cut
Figure 6: (a) MFD approximation (green line) is given as the lower boundary of VT ”cuts”(gray). (b) MFD approximation (red) is given by the lower bound of the MFD cuts (gray)and the two new cuts (black)
5. Modified method of cuts223
In this section we develop the mMOC using the insight learned from the224
simulation experiments to include buses as moving bottlenecks. From finding225
F(a) we observe that the MFD for the ”NO BUS” case is an upper bound to226
the MFD when buses are introduced. However, this upper bound is not tight227
for regions of moderate to high flow. The new method consists in adding228
two additional cuts to provide a tighter upper bound to the MFD; see Figure229
6(b). The moving bottleneck cut is based on the endogenous determination of230
the (i) average bus free-flow speed, which captures the effects of operational231
parameters, and (ii) its maximum passing rate, which depends on signal232
settings. A second cut is needed because buses trigger an additional capacity233
reduction similar to the ”short block” effect identified in Laval and Castrillon234
(2015). These additional cuts are presented below.235
The proposed method extends Xie et al. (2013) by including the capacity236
reduction effect, the endogenous computation of average bus speed and the237
signal timing dependency of bus maximum passing rates.238
5.1. Moving bottleneck cut239
Finding F(d) gives the strong indication that the moving bottleneck effect240
is indeed present in signalized corridors, and that it can be predicted by241
12
moving bottleneck theory. In order to test this hypothesis, we find the moving242
bottleneck ”line” using the VT framework. In the following two subsections,243
we propose methods to estimate both the bus average speed and its maximum244
passing rate.245
5.1.1. Bus average free-flow speed246
Recall that the bus travels along a homogeneous corridor at free-flow247
speed v and stops at red signals or at bus stops with probability p. The time248
to board/alight passengers is d, deterministic. The total distance traveled on249
a particular block is equal to the length of the block l, while the time elapsed250
on the block is equal to the free-flow travel time l/v plus the delay from bus251
stops, ∆s, and the delay if this signal is red, ∆r. These delays are random252
variables and will be approximated by their expected value, E[·]. According253
to renewal theory, the long-term average bus free-flow speed is given by:254
v =l
l/v + E[∆s] + E[∆r](4)
where E[∆s] = pd since the number of blocks that the bus travels before255
stopping can be modeled as a geometric process with mean 1/p. To find256
E[∆r] we neglect the effect of bus stops on whether or not the bus will stop257
at the following red light. In such case, we can use the result in Daganzo258
and Geroliminis (2008), which states that the number of blocks an observer259
travels before hitting a red phase is given by:260
Nmax = 1 + max{N : [N(l/v − o+ c)] mod c ≤ g/c} (5)
where o is the offset and c is the cycle length. The time it waits at a traffic261
signal is R = c − [Nmax(l/v − o+ c)] mod c. The approximated average262
wait time per block is therefore E[∆r] = R/Nmax.1
263
In the sequel we use the following dimensionless version of (4) is:264
α = v/u =l/u
l/v + pd+R/Nmax
(6)
We have tested the quality of this approximation method. We generated265
1,000 values for each of the parameters ρ, λ, τ , p, and d. For each set of266
1Notice that obtaining a general expression for Nmax that accounts for dwelling is dif-ficult because traffic signal settings are deterministic and therefore renewal theory cannotbe applied.
13
Figure 7: y-y plot of simulated (”real”) vs. modeled (”estimated”) α speeds
parameter values, the average bus speed was estimated using (6) and also267
using a simulated trajectory with the specified parameters over a very long268
corridor. Remarkably, the proposed approximation is unbiased, as shown in269
Fig. 7, which means that we have correct estimates of the expected values270
in (4), as needed.271
5.1.2. Bus maximum passing rate272
Maximum passing rates in a VT time-space network are challenging to273
calculate analytically because of the intricate interactions between signals274
and buses. In order to overcome this issue we separate the corridor into two275
sets of lanes: one mixed car-bus lane containing all buses, and the remaining276
car-only lanes. Here, we are assuming that at any given location there is no277
more than one bus on all lanes, which is reasonable because buses usually278
operate and make stops on the outside lane.279
The advantage of separating the lanes is that for the mixed lane, the280
14
flo
w
v
v
vR (v)car-only
R(v)
R (v)mixed
~
~
~~
~
~
density
1
Figure 8: The moving bottleneck cut can be constructed using a lane weighted average ofthe maximum passing rate (R) from mixed lanes and car-only lanes.
maximum passing rate for the moving bottleneck is zero, since no cars can281
overtake buses on that lane. Also, the maximum passing rate on the car-only282
lane, Rcar−only(v), can be readily obtained using the moving observer formula283
(Newell, 1998). This can be accomplished graphically by placing a line with284
slope v tangent to the MFD curve and calculating its y-intercept; see Figure285
8. The final maximum passing rate for all lanes is therefore:286
R(v) =n− 1
nRcar−only(v) (7)
where n is the total number of lanes. Notice that for piecewise linear MFDs287
chances are that this line will pass by a discontinuity point such as point ”1”288
on Figure 8.289
5.2. The bus short-block cut290
Estimating the capacity of a homogeneous corridor with buses becomes291
challenging because of combined effects of running buses, bus stops and sig-292
nalized intersections. Even in the absence of buses, Laval and Castrillon293
(2015) describe how short blocks make it increasingly difficult to calculate294
minimum paths on a VT network. The main problem is with the stationary295
cut (with zero average speed) since its related minimum path is not straight296
but takes “detours” to neighboring intersections; see Fig. 9a. As mentioned297
in that reference, this happens whenever a red phase falls in the influence298
triangle depicted in the figure. To date, analytical approximations to this299
phenomenon have not been found. Recall, however, that this phenomenon300
15
Figure 9: (a) Minimum paths are altered by short-blocks in the form of detours, (b) busesalter minimum paths similar to short-blocks.
16
is automatically captured within numerical solution methods such as the301
simulation model presented here (in appendix), or VT networks.302
Buses exacerbate the short-block effect by providing more opportunities303
for detours due to both the slower travel speed and the stops; see Fig. 9b.304
To overcome the problem of analytical intractability of this “bus short-block305
cut”, a regression model is proposed.306
The regression is based on the simulation results of a single mixed car-bus307
lane. The model includes the topological dimensionless parameters, λ, ρ and308
τ and the dimensionless bus parameters, γ, α. Notice that bus parameters309
such as the bus free-flow speed v, the stop probability p and the dwell time310
d are not necessary as long as α is known; see finding F(g).311
In order to develop the model, 1000 different training runs and 500 testing312
runs are saved and the maximum capacity is obtained for each one. For each313
simulation run, random variables are generated for each of the parameters.314
Four of the parameters are uniformly distributed: λε{0.25, 2.25}, ρε{0.2, 2},315
τ ε {0, 1}, and γ ε {0, 24}. Notice that the average speed of buses cannot be316
manipulated in the simulation as it is a resulting function of the network317
parameters, as well as the bus free-flow speed v, the stop probability p and318
the dwell time d. Therefore, bus input parameters are uniformly distributed319
with ranges p ε {0, 1} and d ε {0, 60} sec, while v can only take four discrete320
values {20, 40, 60, 80} km/h. The final regression equation is obtained by321
adding relevant variables and keeping the significant ones at the 0.001 level:322
Qmixed = a1α− a2α2 + a3α3 + a4λ− a5ρα + log(γa6) (8)
where a1 = 2.781282, a2 = 4.328240, a3 = 2.589717, a4 = 0.032333, a5 =323
0.544471, and a6 = 0.034573.324
The R2 value for the training data set is 0.98. For the test data, the mean325
absolute error is 0.039 and the root mean square error is 0.0464 (recall that326
capacity ranges from 0 to 1). Notice that the offset τ turned out to be not327
significant, and the model only requires the remaining four parameters.328
The model suggests that α has a cubic polynomial effect on capacity as329
well as an interactive effect with ρ. Also, λ has a positive linear effect and γ330
has a positive logarithmic effect on capacity. The marginal effect of the input331
parameters is given on the first row of Table 2. The variable ρ has a negative332
marginal effect, which is expected since the red phase increases compared to333
the green phase. Also, λ has a positive marginal effect which is expected334
since the queues from bus stops have a larger block length before they spill335
17
x = ρ x = λ x = γ x = α
dQmixed
dx−a5α a4 a6/γ a1 − 2a2α + 3a3α
2 − a5ρ
mean of x 1.053 1.229 11.924 0.224
x-elasticity of Qmixed −0.298 0.092 0.080 0.341
Table 2: Marginal effect and elasticity of inputs
back to upstream intersections. Furthermore, γ has a positive marginal effect336
on capacity which is also expected since an increase in headway translates to337
a decrease in the number of buses on the corridor.338
The last row of the table gives the elasticity of input parameters evaluated339
at their mean values. The results suggest that ρ and α are the most important340
parameters when determining capacity at the mean input values.341
To estimate the capacity of all lanes, we obtain the capacity of the mixed342
lane from Equation 8 and the capacity of the car-only lanes is obtained from343
the method of cuts; see Figure 8. The final equation is:344
Qtotal =1
nQmixed +
n− 1
nQcar−only (9)
which defines the bus short-block cut.345
Finally, notice that the regression model in this section should be appli-346
cable to any facility regardless of the fundamental diagram because (i) the347
variables are dimensionless and (ii) the symmetry property of the kinematic348
wave model outlined in Laval and Chilukuri (2016).349
5.3. Results350
In section we investigate the accuracy of the proposed method relative351
to (i) the original MOC, and (ii) the traffic simulation described in the Ap-352
pendix.353
The results on Fig. 10 indicate that the original MOC (in green) provides354
an upper bound approximation to the simulation results (in black), which355
in many cases is not tight. However, when the proposed bus cuts are in-356
troduced (the bus moving bottleneck cut and the bus short-block cut shown357
in purple), the combined lower envelope improves the agreement with the358
18
simulation results. Notice that including the moving bottleneck cut alone is359
not sufficient in some cases, but the short-block cut is necessary to provide360
a good approximation to the capacity. Conversely, the two write-most cases361
on the top row of the figure indicate that on some network configurations the362
proposed cuts are not necessary. This happens when v is greater than the363
speed of the free-flow cut near capacity; this can be explained by the duality364
of traffic lights and moving bottlenecks revealed in Laval (2006b).365
In closing, our results suggest that the proposed mMOC method provides366
an accurate approximation to the MFD, and that the main parameter is the367
bus average free-flow speed, v, which dictates the magnitude of both the368
moving bottleneck and bus short-block cuts.369
6. Discussion370
Our findings suggest that the interaction between buses, block lengths and371
signal parameters is essential when determining the MFD of an urban arterial372
corridor where buses operate. Notably, our results suggest that these complex373
interactions can be boiled down to a single parameter, the bus average free-374
flow speed. This paper provides both a method to estimate it based on375
observable parameters, and a method to derive its associated impacts on the376
MFD. We found that the two major effects that buses have on a corridor are377
the moving bottleneck effect and the reduction in capacity due to the ”short378
block” effect.379
This research constitutes an initial step in understanding the effects of380
buses on the MFD. Although homogenous corridors are a simplification of381
real life corridors, they help to understand the interactive effects between382
traffic signal bands (red and green waves) and bus operations. The model383
proposed here will probably not be directly applicable to heterogeneous cor-384
ridors but (i) the methodology to calculate, endogenously, the bus average385
travel speed in free flow, and (ii) the insights obtained from the moving bot-386
tleneck effect and the capacity reduction, should be. A follow-up paper is387
being developed to apply what was learned here to heterogeneous corridors.388
Future research can focus on extending homogeneous corridors to hetero-389
geneous corridors with varying block lengths and signal parameters as well390
as traffic networks with origins and destinations and routing for each vehi-391
cle. Networks introduce local heterogeneities whose traffic properties can be392
obtained from averaging local corridor properties with varying topologies,393
signal timings and traffic demand. For instance, if one considers a given ρeast394
19
Figure 10: Comparisons between simulation results, method of cuts (no buses) and buscuts. The same ρ, λ are used from Experiment 1, but only one γ, τ and n value are usedfor each roadway case. The remaining parameters are: b = 75 blocks, g = 36 s, v = 40km/h, p = 1/3, d = 20 s.
20
(red/green ratio) on the eastbound direction, this automatically assumes a395
ρwest=1-ρeast on the westbound direction. Therefore, we now have two cor-396
ridors with varying properties. Other topics are to examine passenger flow397
capacity, dedicated bus lanes, or how demand-driven transit operations such398
as bus-bunching interact with the MFD of a corridor. The authors are cur-399
rently investigating these topics.400
Acknowledgements401
This research was supported by NSF research project 1301057. The au-402
thors are grateful to three anonymous referees who provided comments that403
greatly improved the quality of this paper.404
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Appendix: Simulation model500
This appendix describes the numerical model to simulate the experiments501
in this paper, which are limited to arterial corridors (not networks).502
Cellular Automata traffic simulation503
We use a simulation model that gives the exact solution of the Kinematic504
Wave model Lighthill and Whitham (1955); Richards (1956) with a triangular505
fundamental diagram, using cellular automaton (CA) Daganzo (2006). Time506
and space are discretized by ∆t and ∆x, where each vehicle occupies one507
cell. A variable with a dimensionless quantity is described with a ”hat”, e.g.508
x = x/∆x. For a single link, the dimensionless position is given by:509
x(i, t+ 1) = min{x(i, t) + u/w, x(i− 1, t)− 1} (10)
Where x is the dimensionless position measured in units of the jam spac-510
ing κ, i is the vehicle number, t is dimensionless time measured in units of511
1/(κw), and u/w is an integer.512
24
Cars travel at free-flow speed 4 cells/time unit which corresponds to 80513
km/hr when ∆x = 6.67 and ∆t = 1.2sec.. These parameters are similar to514
the range of values used in the field. Flow and density averages for a time515
aggregation period τ and a lane-distance aggregation area X are calculated516
by using Edie’s generalized definitions Edie (1963):517
q =D
τX(11a)
k =T
τX, (11b)
where q is the flow, k is the density, D is the sum of the distance covered by518
all vehicles, and τ is the sum of the time spent by all vehicles on the system.519
Lane-changing model520
We implemented the lane-changing rules in Nagel et al. (1998), which521
consist on determining whether there is an incentive to change lanes, and522
then looking for sufficient gap on the target lane before making the change.523
The incentive to change lanes has one important parameter called the “look524
ahead” value, which is the number of cells in front to look for another vehicle.525
If there happens to be a vehicle within the look ahead value, then the velocity526
of the vehicle ahead is compared to its own velocity. If the velocity is slower527
than its own, then the incentive to change lanes becomes active and it looks528
for a large enough empty gap on the target lane to make the change. The529
”look ahead” value used is 6 cells and the lane changing gap is 5 cells where a530
vehicle looks at 4 cells behind and 1 lane in front of the current cell location.531
These values were calibrated to ensure that the model replicates the single-532
pipe moving bottleneck model described in figure 1.533
This simulation model has been implemented as a network simulation to534
include traffic intersections, turns, routing and lane-changing. A beta version535
of the software can be accessed online 2. The simulation software has user-536
friendly controls, a real-time visualization component to see how vehicles537
interact in the system, and an MFD output component538
2MFD multimodal simulation: http://felipecastrillon.github.io/mfd simulation bus/
25