On the fundamental diagram of signal controlled urban roads
Y. Lia , N. Aroraa and C. Osorioa,b,∗
a Google Research, Mountain View CA, USAb HEC Montreal, Mountreal QC, Canada∗ Corresponding author

Extended abstract submitted for presentation at the 11th Triennial Symposium onTransportation Analysis conference (TRISTAN XI)
June 19-25, 2022, Mauritius Island

April 3, 2022
Keywords: fundamental diagram; signal controlled urban roads; traffic flow theory

1

INTRODUCTION

Fundamental diagrams, as their name indicates, are fundamental analysis tools in the field oftransportation. Transportation researchers have attempted to elucidate these functions for urban
areas for the last five decades. This paper addresses this challenge and builds our understandingof how urban congestion forms and propagates.
Fundamental diagrams (FDs) are link (i.e., segment, roadway) models that describe at an
aggregate level the relationship, for a given road, between two out of the three fundamental traffic
variables: (space-mean) speed, flow and density. The FD has been, and continues to be, the mostimportant traffic modeling tool since the emergence of traffic data collection and flow modeling.
The seminal work of Greenshields (1935) is, arguably, considered as the work that founded andpioneered the field of traffic flow theory (Kühne, 2008). The work proposed the first FD that
stipulated a linear relationship between speed and density. The majority of the FD analysis has
focused on uninterrupted traffic, including highways. The assumption of uninterrupted traffic isviolated in most urban roadways that are controlled by traffic lights or signs (e.g., stop, yieldsigns). Research on FD formulations for interrupted traffic of urban signalized roads is limited(Elmar Brockfeld & Wagner, 2008, Wagner et al., 2009, Wu et al., 2011, Dakic & Stevanovic,2018).
Past work on signalized urban roadways has attempted to map density to speed (Wu et al.,2011). Given the difficulty and cost of collecting density data, commonly used proxies include
occupancy (Wu et al., 2011) and degree of saturation (Pascale et al., 2015), estimated from stopbar (or stop line) detectors. For these detectors, the queue-over-detector (QOD) phenomenon canmake the proxies unreliable estimates of density. Hence, the work of Wu et al. (2011) proposed amechanism to remove the impact of QOD. The method involves the use of both advance detectors(placed well upstream, e.g., 400 feet, of the stop-line), stop-bar detectors, and detailed signal
event data. Advance detectors can yield more reliable occupancy estimates (Hall et al., 1992).
The work of Dakic & Stevanovic (2018) recognized the broad availability of stop-bar detectors
and proposed the use of density saturation as a more reliable density proxy. The estimationof density saturation requires both stop-bar detector data along with detailed signal plan eventdata.
Our proposed work overcomes these challenges by proposing a mapping between the more
broadly available flow data to speed. The flow data is robust to the detector placement, as it
is not impacted by the QOD phenomenon. The commonly available stop-bar detectors can be
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Y. Li, N. Arora and C. Osorio2

used to reliably estimate flow. Moreover, it is a direct estimation of the traffic state variable, asopposed to resorting to the use of a proxy.
Additionally, unlike past work, this work shows that an FD can be derived without knowledgeof demand on competing movements nor of detailed signal event data (e.g., phase actuation times,
phase durations). This is unexpected, since the flow of an actuated movement relies on demandfor other lower and higher priority movements at that intersection, as well as on the specificactuation logic of an intersection (Wu et al., 2011).
To the best of our knowledge, past FD insights for interrupted urban traffic have considered
a two-to-one mapping between flows and densities, or equivalently between flows and speeds. In
other words, a given flow measurement is consistent with two traffic states (i.e., two different
speed values or two different density values). We postulate that these mappings are one-to-one.
Moreover, the functional form we use is closely related to one commonly used for highways.
Together these findings contradict past understandings and findings in the field. For instance,
Elmar Brockfeld & Wagner (2008) state: "We hypothesize that in fact there is a fundamental
diagram in urban roads, however, there is no one-to-one correspondence to the fundamental
diagram on freeways.” Our current results indicate plenty of similarities with freeway or highway
FDs.

2METHODOLOGY

2.1

Proposed fundamental diagram (FD)To formulate the FD, we consider the major street of an actuated signalized intersection. We
introduce the following variables: (space-mean) speed v, flow q; and the following parameters:
maximum speed (i.e., speed limit) vmax , flow capacity qcap , FD power coefficients α and β. The
proposed FD has the following functional form:v = vmax 1 −!α !βq.

qcap

(1)

To facilitate comparison across roadways with different speed limits, we rewrite Eq. (1) as:v
vmax=1−q!α !βqcap.

(2)Eq. (1) is inspired from the commonly used functional form for highway FDs that relate speed
and density (rather than flow) (May & Keller, 1967):v = vmin + (vmax − vmin ) 1 −kkjam

!α !β

,

(3)

where k represents the density, vmin the minimum speed and kjam jam density.
For highways, the FD mapping between speeds and density is a one-to-one mapping (e.g.,Eq. (3)). This leads to a one-to-two mapping between speeds and flow. Our findings on urban
signal controlled roads indicate that the speeds to flow mapping is one-to-one (Eq. (1)). Thisimplies that the measurement of flow, which is the most commonly available type of measurement,is sufficient to infer the traffic state. This is an important finding since flow data is the mostcommonly available traffic data in urban areas.

2.2

Data sources

We consider all movements of the major street of an urban traffic actuated intersection. The
definition of major street is obtained from the Utah Department of Transportation (UDOT)
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3

Automated Traffic Signal Performance Metrics (ATSPM) (Utah Department of Transportation(UDOT), 2020). No signal plan data, such as phase durations or actuation information, isrequired to derive the FD. Through the ATSPM portal, we have access to lane-level vehicularcount (i.e., flow) data obtained at stop-bar detectors, aggregated at the roadway-level (i.e.,aggregated across lanes) and at the hourly level. We use aggregated and anonymized spacemean speeds of a given roadway by looking at Google Maps driving trends (Lau, 2020). The
speed estimates are obtained for the entire roadway, they are not lane-specific. Space-mean (i.e.,harmonic mean) speeds are obtained for each volume interval. The volume values are segmented
into intervals of 30 veh/hr. To enable comparison across segments and intersections, we workwith a normalized speed that considers the ratio of speed to speed-limit, as defined by Eq.(2).
The speed limit of each roadway is obtained from Google Maps.

3RESULTS

Figure 1 considers an actuated intersection. Each of the row of plots considers one of the twotravel directions of the major street. For each plot, the x-axis displays the flow (in veh/hour-lane)
and the y-axis displays the (unitless) normalized speed as defined in Eq.(2). In the first columnof plots, the average measurements curve is in blue with error bars that have a total width of
two standard deviations. The fitted FD, based on Eq.(2), is in yellow. The blue curve indicatesa flow-to-speed mapping that is one-to-one. Fitting of the yellow curve does not depend on or
require any signal data.

Figure 1 – Fundamental diagrams of the two travel directions of the major street of an actuatedurban intersection.
The second column of plots consider FDs that exploit signal plan information. The green(resp. red) curves consider only speed measurements observed while the thru-movement phasehas a green (resp. red) signal. These second column plots illustrate how the FD of Figure 1
that is derived without any signal data (i.e., first column plots) can be decomposed into two
signal-specific FDs. These green and red FDs can also be approximated by the functional formof Eq.(2). The corresponding fitted curves are not displayed for the sake of clarity. For a given
row of plots, the error bars of the green and red FDs tend to be smaller than those of the blue
FD. This illustrates how the large error-bars of the blue FD are mainly due to the large speed
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differences between the red signal phase (where vehicles are mostly stopped) and the green signal
phase (where signals are mostly moving).
In the third column of plots, the blue curve is the same curve as that of the first columnof plots. The yellow curve is obtained from a weighted combination of the data that is used todefine the green and the red FDs of the second column. The third column of plots illustrates
how for a given travel direction (i.e., a given row of plots), the blue FDs of the first column plots
can be seen as a mixture of the green and the red FDs of the second column. The two weightparameters of this mixture are estimated as the proportion of observations that fall in the green
signal phase, and the proportion that fall in the red signal phase.
More detailed analysis of our findings, as well as the relationship between highway FDs andsignal-controlled urban FDs will be presented and discussed at the conference.

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