COMPUTING TURN-DEPENDENT DELAY TIMES
AT SIGNALIZED INTERSECTIONS BASED ON FLOATING CAR DATA
Thorsten Neumann*, Elmar Brockfeld, Alexander Sohr
German Aerospace Center (DLR), Institute of Transportation Systems
Rutherfordstr. 2, 12489 Berlin, Germany
1. INTRODUCTION
Current link-based travel times for (urban) road networks play an important
role in dynamic navigation and routing. They are basic ingredients when
searching for fastest paths between given locations on a digital map. In this
context, typical routing applications make use of static travel times for each
link which are based on road types and speed limits. However, also static
daily variation curves with traffic-state-dependent travel times are being considered more and more. Sometimes, even current (instead of historical) traffic
states are taken into account if there are suitable traffic data available (cf.
Cohn and Rutton, 2009).
Nevertheless, in most instances no distinction is drawn between different turning directions at (signalized) intersections or there are only some very simple
blanket assumptions realized (cf. Brockfeld et al., 2010). The true travel time,
however, significantly depends on whether a vehicle is driving straight on or
turns right or left at an intersection as the length of the turning queue may
strongly vary among these three cases. Furthermore, the specific layout of the
studied road junction influences the turn-dependent travel times, too. That is,
when turning left (in case of right-hand driving) vehicles normally have to give
way to oncoming traffic first. And, particularly in urban areas, they sometimes
have to wait for pedestrians crossing their way when turning left or right.
These assumptions indicate that considering turn-dependent delay times
might result in a much more accurate navigation and routing performance.
This paper describes a theoretical approach how to compute the required
turn-dependent delay times based on common floating car data. Here, a number of probe vehicles provide information about their current positions every
few (typically about 30-60) seconds so that travel times can be obtained for
each pair of data points received from the same car (cf. Schäfer et al., 2002).
Filtering for different turning directions at a given (signalized) intersection finally allows computing travel times for each traffic stream separately (cf.
Brockfeld et al., 2010).
The challenge is that these travel times even in case of the same turning direction are related to very different travel distances as well as they are originally not link-based (cf. Fig. 1). This is a crucial point since the decomposition
into so-called link travel times or delay times is not trivial as has been shown
in the literature (cf. Hellinga et al., 2008; Zheng et al., 2010).
The new idea is to solve this task by introducing a simple linear model for the
measured travel times which arises from the superposition of two types of
© Association for European Transport and contributors 2010
1
turn-dependent delay times and a free flow travel time depending on travel
distance and average flow resistance on the relevant road sections. The unknown parameters are then computed by linear regression.
Figure 1: Measured vs. link based travel times.
The theoretical approach is checked against simulation as well as real data. In
this context, two examples from Berlin are considered where floating car data
of more than 4000 taxis are continuously available. As a result, the new approach yields mostly plausible estimates for turn-dependent delay times as
well as free flow speeds as is demonstrated. By that, the conjecture is confirmed that turn-dependent delay times are an interesting and important topic
in dynamic navigation and routing – worthy to be studied in more detail.
2. THE THEORETICAL APPROACH
Assume there is an intersection as depicted in Figure 2 with vehicles coming
from the left and passing the traffic signal into different directions. As is quite
common, vehicles driving straight on are supposed to be free to cross the intersection during the green phase while turning vehicles have to give way to
oncoming traffic and/or pedestrians first (cf. dark blue conflict areas in Fig. 2).
Consequently, there are two different types of delay which are mostly related
to the yellow and orange areas in Figure 2. On the one hand, that is the waiting time emerging in front of the traffic light because of a red signal which may
vary for each turning direction. It is denoted by d S(1,)L, R in the following depending on whether the corresponding vehicle is driving straight on (S), turning left
(L) or turning right (R). On the other hand, d S( 2, L) , R describes the additional delay for turning vehicles when waiting on the intersection for oncoming traffic
and/or pedestrians (cf. yellow waiting areas in Figure 2).
Figure 2: Theoretical intersection scenario.
That is, the travel time t of a vehicle passing the intersection is given by the
superposition of free flow travel time and these two types of delay. With regard to given GPS data from a common floating car system, for example, the
measured travel time t between two data points of the same vehicle – one of
them upstream, the other one downstream the intersection (cf. Fig. 2) with
known distance x – can be written as
t 
x
 d S(1,)L, R  d S( 2, L) , R
v
(1)
where v is the average free flow speed of the considered vehicle.
In other words, one obtains a linear model for the travel times provided by the
floating car system. Hence, assuming a constant network resistance (i.e. a
constant free flow speed v) for all vehicles, it is easy to compute 1/v and the
average total delay d S , L, R : d S(1,)L, R  d S( 2, L) , R for each turning direction separately
by linear regression. For this purpose, just consider all suitable pairs of GPS
points in the trajectories of the available floating cars (t and x are known in
this case) where the first GPS point is located upstream and the other one
downstream the studied intersection.
3. A SIMPLE SIMULATION STUDY
In many fields of transportation research, it is helpful to evaluate new algorithms and methods by simulation first before applying them to real-world
data. For, due to simplified and reproducible conditions, simulations facilitate
the detection of drawbacks of the considered approach regarding theory and
implementation. Moreover, they provide important reference values for the
results of the proposed methods which are not available in real life quite often.
Thus, simulations regularly enable the systematic evaluation of new ideas
which is hardly possible without such tools.
For these reasons, also the intersection scenario from Figure 2 has been
transferred into a simulation model. Using a simple cellular traffic flow model
(see Nagel and Schreckenberg, 1992) in its deterministic version but extended by traffic signals and interactions between crossing traffic streams, the
complete setup has been realized (cf. Figure 3). In this context, motorized traffic has been modelled at microscopic level completely while interactions with
pedestrians are implemented in a simplified manner. That is, every time step
both pedestrian crossings are assumed to be blocked independently of traffic
states and traffic signal phases with a certain probability pPedestrian each. Moreover, left turners have to wait in front of the red marked cell in this case (cf.
Figure 3) in order not to hinder oncoming traffic. Finally, lane changing at the
inflow section is omitted. Instead, vehicles are inserted into the model directly
according to their turning direction following independent Poisson distributions
for each lane.
Figure 3: Simulation framework (Cell length = 7.5 m).
As there is a nearly infinite number of constellations regarding traffic demand
for each turning direction and traffic signal parameters, the following results
focus on an exemplary but more or less typical setup for simplicity. All relevant
effects are expected to be covered by this specific scenario. The considered
parameter values are summarized in Table 1.
Table 1: Parameter setup for the simulation.
Parameter
Traffic volumes
Inflow (straight on)
Inflow (left)
Inflow (right)
Oncoming traffic
Speed parameter
Maximum speed
Traffic signal setting
Green time
Red time
Amber time
Pedestrian traffic
Blocking probability
FCD system parameters
Penetration rate
Sampling interval
Symbol
Value
qS
qL
qR
qO
360 veh/h
360 veh/h
360 veh/h
540 veh/h
vmax
54 km/h
g
r
(neglected)
25 sec
35 sec
(neglected)
pPedestrian
0.15
ρ
t0
1%
5 sec
Then, 30 independent simulation runs were performed each representing one
hour of traffic at days or times of day with similar traffic conditions. After that,
the computed virtual floating car data were filtered according to turning directions and were aggregated along all runs which – translated to real life – corresponds to the common approach of integrating current with suitable historical data. By that, sufficiently large data sets can be obtained even at very low
penetration rates ρ.
Finally, all virtual GPS points were recombined vehicle-wisely into pairs where
the first point is always located upstream and the second one downstream the
intersection. Floating car messages from the inner part of the intersection (i.e.
the yellow parts of Figures 2 and 3) were ignored. Considering all valid combinations then yields a number of travel distances x (i ) between the GPS
points together with known travel time t (i ) each (cf. Figure 4).
Figure 4: Combination of GPS points (Example).
Clearly, this is the information to be used for estimating the parameters of the
linear model from Equation (1). Figure 5 shows some typical examples including the lines of best fit computed by least-square linear regression. Moreover,
Table 2 compares the resulting average delay times d S ,L,R  d S(1,)L,R  d S(2,L) ,R and
free flow speeds v with the true ones measured during simulation.
(a) Straight-on traffic
(b) Left turning traffic
Fitted model: t = 0.5038 ∙ x + 11.775
Fitted model: t = 0.521 ∙ x + 35.712
Figure 5: Typical estimation results (Examples).
Table 2: Comparison between estimated and simulated delays and speeds.
Turning direction
Average total delay
Estimated
Simulated (true delay)
Error
Difference
Relative Error
Free flow speed
Estimated
Simulated (true speed)
Error
Difference
Relative Error
S
L
R
Average
11.775 sec
12.261 sec
35.712 sec
37.684 sec
11.84 sec
13.017 sec
19.776 sec
20.987 sec
– 0.486 sec
– 4.0%
– 1.972 sec
– 5.2%
– 1.177 sec
– 9.0%
– 1.212 sec
– 5.8%
53.6 km/h
54 km/h
51.8 km/h
54 km/h
52.0 km/h
54 km/h
52.5 km/h
54 km/h
– 0.4 km/h
– 0.7%
– 2.2 km/h
– 4.1%
– 2.0 km/h
– 3.7%
– 1.5 km/h
– 2.8%
As can be seen, the estimates fit in with the true values very well most of the
time yielding only small absolute and relative errors. Nonetheless, in particular, the estimated delay times seem to be systematically lower than the measured correct ones. The reason for this bias is the fact that some of the data
points from, for instance, Figure 5 refer to pairs of GPS positions of which the
first position is indeed located upstream the intersection but not outside the
current traffic light queue (cf. the orange queuing area in Figure 2). That is,
when computing the travel time between these two GPS points, a certain fraction of the total intersection delay (i.e. the waiting time prior to the time stamp
of the first GPS position) is not covered. Consequently, it is not surprising that
the linear regression method slightly underestimates the true delay.
Of course, the bias can be reduced or even eliminated by taking only those
GPS points into account which are located outside any (possibly traffic-statedependent) queuing or waiting area instead of ignoring just GPS messages
from the effective crossing area. With regard to Figure 2, that means that only
the gray, but not the yellow and orange parts of the road are relevant data
locations for the described approach of delay time estimation.
Unfortunately, in particular the extent of the (orange) traffic-state-dependent
queuing area in front of the traffic signal is usually not known or needs further
intense analyses to be done (cf. Neumann, 2009; Neumann, 2010). For simplicity, however, it is mostly sufficient to define a fixed but suitable large virtual
queue length L instead of a detailed investigation of the traffic signal queues.
Table 3 shows the corresponding simulation results in case of L = 262.5 m to
be compared with the biased values from Table 2.
Table 3: Comparison between estimated and simulated delays
and speeds with reduced bias.
Turning direction
Average total delay
Estimated
Simulated (true delay)
Error
Difference
Relative Error
Free flow speed
Estimated
Simulated (true speed)
Error
Difference
Relative Error
S
L
R
Average
12.725 sec
12.261 sec
36.33 sec
37.684 sec
13.55 sec
13.017 sec
20.868 sec
20.987 sec
+ 0.464 sec
+ 3.8%
– 1.354 sec
– 3.6%
+ 0.533 sec
+ 4.1%
– 0.119 sec
– 0.6%
54.5 km/h
54 km/h
53.3 km/h
54 km/h
53.7 km/h
54 km/h
53.8 km/h
54 km/h
+ 0.5 km/h
+ 0.9%
– 0.7 km/h
– 1.3%
– 0.3 km/h
– 0.6%
– 0.2 km/h
– 0.4%
Obviously, the systematic error is significantly lower than before and can be
found to be close to zero now. Needless to say, this is a strong evidence for
the theoretical accuracy of the proposed linear regression approach even if
just a single scenario (see Table 1) has been analyzed so far.
Despite that, there is another interesting aspect besides studying the error
behaviour when looking at the plots from Figure 5. As can be seen, the depicted data points display a very clear structure. Particularly in case of
straight-on traffic (see Figure 5a), there is a regular stripe showing a constant
gradient and a fixed vertical width which is approximately equal to the red time
of the considered traffic signal. Of course, this effect is not surprising as vehicles are modelled to move with maximum speed constantly if not queued.
Moreover, the red time (nearly) exactly indicates the possible range of delays
vehicles may enter when they do not have to wait for oncoming traffic and/or
pedestrians and when traffic signal queues completely dissolve during each
green phase. Obviously, these conditions are met in case of vehicles driving
straight on in the considered simulation scenario (cf. Figure 3 and Table 1).
However, Figure 5b implies a different behaviour for left turning traffic
streams. Apparently, there is more than one of the above mentioned stripes
which overlap in a certain way. Moreover, having a closer look, each of these
seems to have a larger width than in Figure 5a. In particular, the lower stripe
suggests a vertical range of about 60 seconds in this specific case (see Figure
5b) which corresponds to the cycle time of the considered traffic signal.
Again, that is not surprising as left turning vehicles have to wait for oncoming
traffic in addition to the delay in front of the intersection. Hence, assuming fully
saturated traffic at the oncoming lane during a given cycle, this maximum extra delay adds up to a complete green time. That is finally, the maximum total
delay in case of a single stop at the traffic light is indeed equal to the cycle
time of the signal. Moreover, the overlapping of several stripes in Figure 5b is
explained by the fact that due to the reduced capacity for left turning traffic
(i.e. long signal queues) vehicles sometimes have to wait more than one traffic light cycle before being able to pass the intersection.
4. TWO REAL-LIFE EXAMPLES
Obviously, the results from Section 3 are quite reasonable. Nonetheless, it is
important to check whether the simplifying assumptions of the model hold
when applying the proposed linear regression method in real-life conditions.
For this purpose, two exemplary intersections from Berlin were chosen where
floating car data of more than 4000 taxis are available. The sample period
were defined more or less randomly as from July 21, 2010 to August 6, 2010.
The selected junctions are displayed in Figure 6.
(a) Intersection No. 12
Flughafen Tegel / Saatwinkler Damm
(b) Intersection No. 69
Landsberger Allee / Petersburger Str.
Figure 6: Exemplary intersections (from Google Earth).
In this context, as previous studies (see Brockfeld et al., 2010) indicated that
the Eastern inflow branches (Branch No. 12-2 and 69-2) would be interesting
in both cases, the following analyses focus on these specific parts of the depicted intersections. Figure 7 shows the fitted linear models in sense of Equation (1) for each turning direction. Moreover, Table 4 summarizes the estimated delay times as well as the computed average free flow speeds.
(a) Branch No. 12-2 (straight on)
(b) Branch No. 12-2 (left turning)
Fitted model: t = 0.0844 ∙ x + 36.381
Fitted model: t = 0.12 ∙ x + 92.256
(c) Branch No. 12-2 (right turning)
(d) Branch No. 69-2 (straight on)
Fitted model: t = 0.072 ∙ x + 29.333
Fitted model: t = 0.0914 ∙ x + 19.19
(e) Branch No. 69-2 (left turning)
(f) Branch No. 69-2 (right turning)
Fitted model: t = 0.0793 ∙ x + 37.042
Fitted model: t = 0.0851 ∙ x + 25.828
Figure 7: Fitted linear models for both exemplary intersections.
Table 4: Estimated delays and speeds for both exemplary intersections.
Branch / Turning direction
Intersection No. 12
Branch No. 12-2 (straight on)
Branch No. 12-2 (left turning)
Branch No. 12-2 (right turning)
Branch No. 12-2 (Average)
Intersection No. 69
Branch No. 69-2 (straight on)
Branch No. 69-2 (left turning)
Branch No. 69-2 (right turning)
Branch No. 69-2 (Average)
Average total delay
Average free flow speed
36.381 sec
92.256 sec
29.333 sec
52.657 sec
42.7 km/h
30.0 km/h
50.0 km/h
40.9 km/h
19.19 sec
37.042 sec
25.828 sec
27.353 sec
39.4 km/h
45.4 km/h
42.3 km/h
42.4 km/h
As can be seen, the results are quite plausible in most cases. Estimated delays as well as computed speeds are totally within the typical range to be expected for urban road traffic. Unfortunately, it is very difficult to verify the values above exactly as suitable reference measurements are missing.
Despite that, it is likely to guess that at least the displayed free flow speeds
are slightly too low in some cases. The reason might be that the proposed
approach assumes undisturbed traffic apart from the studied intersection as in
the simulation example from Section 3. A fact that has been ignored in case of
the exemplary intersections considered here.
Instead, as to guarantee the availability of sufficiently much floating car data
for each turning direction, more or less long intersection branches with lengths
between 200 m and 1500 m has been defined avoiding the influence of too
large further crossroads only. Small junctions, however, might indeed affect
the above estimation results as they of course amplify the resistance of the
road network, too, and thus slightly reduce the observed speeds.
Because of that, future research should try to find the optimal trade-off between lots of input data on the one hand and the requirement of having undisturbed traffic on the other hand. That is, while high data availability typically
calls for the consideration of long intersection branches which provide much
space for acquiring large floating car data sets with at least two obligatory
data points of each probe vehicle in short time, undisturbed traffic (in the
sense above) is often limited to quite short parts of the road network upstream
of urban intersections. Obviously, attaining a suitable compromise is not a
trivial task.
5. CONCLUSION
Turn-dependent delay times are an interesting and important topic in dynamic
navigation and routing. As is indicated by the results from Section 3 and 4,
there are significant deviations in travel time according to whether a vehicle is
driving straight on at an intersection or turns left or right.
The linear regression method proposed in this paper provides a simple but
effective approach how to automatically compute the corresponding delay
times for each traffic stream separately based on common floating car data. It
avoids the critical problem of decomposing travel times to individual road
segments as it is addressed in the literature (cf. Hellinga et al., 2008; Zheng et
al., 2010), but directly yields a fitted linear model for the observed travel times
in case of undisturbed traffic apart from the intersection under investigation.
However, since long undisturbed road sections are a quite rare phenomenon
in urban traffic, future research should try to find a way to overcome this more
or less strong restriction. At least, there should be a reliable localization of
these parts of the road network as to assess the quality of estimation depending on the spatial context of the studied intersection. Finally, the obtained results as in Section 4 have to be verified by suitable reference measurements
in addition to the plausibility arguments given above.
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