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: fcastrillon3@gatech.edu (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