SOUND: A Traffic Simulation Model for
Oversaturated Traffic Flow on Urban Expressways

Toshio Yoshii 1)
andMasao Kuwahara 2)

1: Research Assistant
2: Associate Professor
Institute of Industrial Science,
University of Tokyo

This manuscript is being submitted to the
7th World Conference on Transportation research
to be held at Sydney in July, 1995.

1. INTRODUCTION
This study develops a dynamic traffic simulation model SOUND (a Simulation model On
Urban Networks with Dynamic route choice) which reproduces dynamic traffic condition
such as length of traffic congestion and travel time for an oversaturated expressway
network in urban areas. Importantly, we are frequently required to understand how traffic
condition changes if we, for instance, construct a new route and/or alter traffic regulations.
Especially, for a congested urban expressway network, we must estimate time-dependentchanges of traffic to evaluate impacts of new policies and development of infrastructures.
In developing the SOUND model, special attentions are paid so that it incorporates drivers’route choice behavior and controls traffic density as a function of the flow to simulate theshockwave propagation.

2.

REVIEW OF EXISTING MODELS

Since late 1970’s, several traffic simulations models applicable to areawide networks
incorporating drivers’ route choices have been developed. SATURN (Vliet and Hall,
1991), CONTRAM (Leonard et al., 1989), DYTAN (Kido et al., 1978), the BOX model(Iida et al., 1991), and the Block-Density model (Ueda et al., 1991, Kuwahara et al., 1993)
are all belonging to this category.
A common problem in these model is that traffic
density on a link has not been controlled in relation to the traffic flow.
Morespecifically, let’s consider highway sections with large and small bottleneck capacities asshown in Fig’s 1 and 2. When the capacity is large as in Fig.1, the traffic density in the
upstream section is normally low compared to that with small capacity as in Fig.2.
Therefore, in Fig.1, a vehicle from an on-ramp can enter in the middle of the congestion,
while it must join the queue at the tail in Fig.2. Thus, depending upon the density,
priorities of vehicles change. In general, on- and off- ramps on an urban expressway
locate closely each other, and hence, the control of traffic density is important to maintainpriorities of vehicles especially at merging points.

2

large flow rate
Fig.1 Large Bottleneck Capacity

small flow rate
Fig.2 Small Bottleneck Capacity
The Block-Density model previously developed by us can control traffic density as in themanner described above. However, it consumes huge computer memories, since traffic
density of each block of 50~60 meters must be stored by destinations of vehicles in theblock. Also, these block densities must be updated at every scan interval of a few
seconds, which requires substantial amount of computation time.

3.

OUTLINE OF THE SOUND MODEL

As presented in Fig. 3, SOUND consists of two parts: the vehicle simulation module androute choice module. In the vehicle simulation module, travel time in each link isevaluated by moving vehicles forward along routes determined by the route choice module,
whereas the route choice module evaluates every driver's route at a regular interval basedon travel times estimated by the vehicle simulation module. These two modules arerepeatedly implemented to reproduce dynamic evolution of traffic flow on a network.
This model, therefore, does not reproduce an exact user equilibrium flow but
approximately describes the equilibrium flow dynamically.

Route Choice Module
Determine route choice probability(Dial's LOGIT assignment)
the probability ofchoosing a route

Travel timeat each link

Vehicle Simulation Module(Car-following model)

Fig.3

Model Structure

3.1. Vehicle Simulation Module3(1) Car-following ModelB

flow direction

A
Moving distance L

Time

t

Spacing s

B

A

Time

t+△t

Fig.4 The Car-Following Method
In the vehicle simulation module, at every scanning intervals (△t), each vehicle (or eachgroup of several vehicles) on the network moves discretely in the order from downstreamof each link based on the pre-specified flow-density relationship (q-k curve). Suppose
two vehicles A and B run on a link at time t as shown in the upper picture of Fig.4.
Attime t+△t, the front vehicle A first moves by some distance as in the lower figure. In thissituation, if vehicle B moves by L, the spacing between A and B becomes s at time t+△t.
Since each vehicle memorizes its current position which means that the distance s+L canbe known in the module, moving distance of vehicle B during scanning interval △t is
calculated so that spacing s and velocity L/△t agree with the flow-density relationship prespecified.
First of all, the volume-density relationship,q = f(k),
where

(1)

q = traffic flow [veh/unit time],k = traffic density [veh/unit length],

is converted to the speed-spacing relationship as below:q = k⋅v = k⋅1/s = f(1/s),∴
wherev = s⋅f(1/s)(2)

v = vehicle speed [unit length/unit time],s = vehicle spacing = 1/density [unit length/vehicle].

Also, since the vehicle speed is written as L/△t, Eq.(2) becomes
L/△t = s⋅f(1/s).

(3)

Since s + L is known, the L and s can be determined from Eq.(3).
(2) Motions at Merging and Diverging Nodes

4

At a merging section, we consider following two situations depending on the degree oftraffic congestion:
1. No congestion
When traffics on both upstream links are not in the congested region on the q-k curve,
vehicles on each merging link can be treated independently just as on a simple
section.
2. Congested
When traffic is congested in either or both upstream link(s), vehicles on these links
merge so as to maintain the prespecified merging ratio. The merging ration is here
defined as a ratio of traffic when both upstream links are congested, and the ration
can be determined in advance according to the geometric design at the section.
For example in Fig.5, when both upstream links i and j are congested, capacities ofthese links are calculated as the capacity of link k multiplied by the merging-ratio.
However, note that when traffic demand on either one of the links is less than theabove capacity (which means the link is not congested), the surplus capacity of thelink should be transferred to the another congested upstream link.

i
k
j
Fig.5

Illustration at a Merging Section

At a diverging section, the diverging ratios by vehicle destinations are updated based on
the route choice module mentioned later.
The number of vehicles to each of two
downstream links are monitored by their destinations. And, when one vehicle arrives atthe diverging node, it chooses one of the downstream links which has the least count for a
destination of that vehicle.
(3) Evaluating Travel Time of Each Link
Each vehicle memorizes entrance and exit times of every link on the way to its destination.
When a vehicle exit from a link, travel time of the link is updated as the time difference
between the entrance and exit times. Consequently, travel time of a link carrying large
traffic demand is more frequently updated than a link with less traffic.

3.2. Verification of Shockwave Propagation
In this section, we compare shockwave propagation estimated by the vehicle simulation
module with that by the kinetic theory using a simple highway section with one bottleneck.
A length of the highway section is 3000 [m] and the entire section has the same q-k
relationship shown in Fig.6. However, the most downstream section has traffic capacityof only 1400 [veh/h], which becomes a bottleneck.

5q[veh/h]
2400
2100

b
c

1400
1200

a

0

Fig.6

20 35

90

160
k[veh/km]

Flow-Density Relationship used in Verification

pattern 1 : A heavy traffic comes from upstream, arrives at the bottleneck, and thencongestion occurs from the most downstream bottleneck which propagates
upstream.
Assume that traffic flow is initially 1200 [veh/h] denoted as point a in Fig.7 for the entire
highway section, and then traffic flow of 2100 [veh/h] denoted as point b start entering the
highway at time 0. The forward shockwave 1 propagates and traffic condition of the
upstream section of the shockwave shifts from a to b. After the shockwave reaches thebottleneck, the backward shockwave 2 is generated and propagate to the opposite direction.
Traffic condition of the downstream section of the backward shockwave changes from b toc, since the bottleneck capacity is limited to 1400 [veh/h] less than the traffic demand of
2100 [veh/h]. Speeds of these propagation by the theory are 60 [km/h] for wave 1 and
700/55 [km/h] for wave 2, with which the simulated wave speeds well agree.
△ Simulation

3

Distance [km]― Theory

2wave 1wave 2

1
60[km/h]

700/55[km/h]

0

5

0

10

15

20

Time [min]Fig.7
pattern 2 :Result of Verification (pattern 1)

Traffic congestion disappears due to the reduction of traffic demand.

Starting from condition c for the entire highway section, when traffic demand returns from6

2100 [veh/h] to 1200 [veh/h], the forward shockwave 3 starts as shown in Fig.8. Traffic
condition of the upstream section of the wave changes from c to a. The simulated
shockwave speed exactly fits the theoretical one, 200/70 [km/h] .

Distance [km]3

Simulation― Theory△wave 32

1

200/70[km/h]
0
0

10

20

30

40

50

60

70

Time [min]Fig.8

Result of Verification (pattern 2)

From these results, it is confirmed that the vehicle simulation module can reproduce theshockwave propagation reasonably although the time is dealt with discretely to simplify
the model.

3.3. Route Choice Module
We first divide users into two groups: Route Choice Group sharing the fraction of a (0 < a< 1) who selects routes based upon time varying travel times on a network, and Fixed
Route Group with fraction 1-a who does not change routes regardless of travel times.
For Route Choice Group, the route choice probabilities are updated at every ∆T ( ≥ ∆t)interval by the Dial’s Logit assignment with parameter θ shown blow based on link travel
times obtained from the vehicle simulation module. On the other hand, for Fixed RouteGroup, the fixed probabilities are calculated similarly by the Dial assignment with thesame parameter value of θ based on the free flow link travel times at speed of 60 [km/h].
Pr ob( r ) =exp(−θ ⋅ Tr )(4)

∑ exp(−θ ⋅ Ti )
i

where

Prob(r) = route choice probability of route r,
Tr= travel time of route r,= non-negative parameterθThen, based on the determined route choice probabilities, the diverging ratio at every
diverging node by destinations is reevaluated. Hence, in the vehicle simulation module, avehicle chooses a next link to go based on the above diverging ratio. Consequently, avehicle in Route Choice Group on the network may change the route at every ∆T.
7

As in the above, this model does not strictly follow the dynamic stochastic user
equilibrium principle. To strictly reproduce the equilibrium, the iterative calculation of
the whole study time period may the required as in CONTRAM (Leonard et al., 1989) and
SATURN (Vliet and Hall, 1991), which however needs a substantial amount of computer
memories and computation time. For this reason, we decide to employ the above noniterative procedure to reproduce the equilibrium flow approximately.
3.4

Validation of the SOUND Model

To examine how simulated traffic condition differs from the strict stochastic user
equilibrium flow, several test runs are made using a simple network as shown in Fig.9 withscan interval ∆t of 1 second and ∆T of 10 seconds. The flow-density relationships are
assumed as in Fig.10 in which the free flow travel speed is 60 [km/h] and the slope in thecongested region are 20 [km/h] for every link. However, since the capacity of link 4 is2400 [veh/h] which is smaller than those of other links (= 4200 [veh/h]), the merging node
C becomes bottleneck, where the merging ratio of link 1 to 2 is assumed 1:1.

link 1
1500m
route 1link 3
1500m

O

B

C

traffic demand=3000[veh/h]link 41500m

D

merging ratio 1:1
route 2Fig.9

link 2
3000m
A Simple Network used to ValidationFlow[veh/h]4200

Link 1,2,3
2400

Link 460[km/h] 20[km/h]

0Fig.104070

20[km/h]160

280

Density[veh/km]

Flow-Density Relationship used to Validation

The parameter θ is set as the following two values:81. θ = ∞ (a vehicle in the choice group chooses the shortest route to its destination),
2. θ = 0.011 [1/sec].
Fig.11 shows how the choice probability generally sensitive to the parameter value.
Also, all vehicles are assumed to belong to the choice group; that is, α=1.
For θ=∞, if the demand of 3000 [veh/h] is loaded from the origin, at first every vehicle
chooses route 1 because of the shorter travel distance of 1500 [m] on link 1. However,mean while, traffic congestion starts from C on link 1 and at some time, travel time onroute 1 becomes equal to that on route 2. From that time, the same amount of demands
are assigned on the both routes so as to maintain the equilibrium.
Probability of choosing route 11θ=∞

θ=0.011
θ=0.001

0.5

0
−30

−20

−10

0

10

20

30 [min]Travel time route2 - Travel time route1

Fig.11

Sensitivity of Route Choice Probability to Dial’s Parameter θ

Fig.12 shows the cumulative departure curves from node B on routes in the above situation.
The simulated values is oscillating compared to the strict equilibrium for the following
reason. In the strict equilibrium, a vehicle chooses its route based upon the actuallyexperienced travel time; on the other hand, in the simulation, the assignment is made basedon the current instantaneous travel times when a vehicle passes B. For example, if travel
time on one route is smaller than the another and the demand is assigned on the shorterroute, actually the travel time gets longer due to the newly loaded demand. In the strict
equilibrium, the diverging ratio at B is determined by taking the above effect into
consideration. However, in the simulation, the effect is not fully incorporated and hence
more demand tends to be assigned on the shorter route which causes the oscillation.
[veh]

9

△

Cumulative number

1500

−

Simulation
Equilibrium condition
Route 1

1000

Route 2
500

0
0

10

20

30

40

50

60

Time [min]Fig.12Comparison of the Cumulative Number of Vehicles
entering each Route at Merging Node B (θ = ∞)Fig.13 shows the same cumulative vehicles for θ=0.011. Although the oscillation also
appears in this case, the amplitude is much smaller. This is because the amount ofdemand more than necessary assigned on the shorter route is smaller under the stochastic
equilibrium than under the user equilibrium (θ = ∞). In spite of the local oscillation, thesimulated values seem to globally follow the strict equilibrium in both cases. For a more
precise reproduction, we should improve the evaluation of travel times not just based on
the instantaneous travel times but predicted travel times in near future.

Cumulative number

[veh]
△
1500

−

Simulation
Equilibrium condition

1000

Route 1
Route 2

500

0
0

10

20

30

40

50

60

Time [min]Fig.13Comparison of the Cumulative Number of Vehicles
entering each Route at Merging Node B (θ = 0.011)

4. APPLICATIONS TO METROPOLITAN EXPRESSWAY
4.1 A Case without Route Choices10In the previous section, we learn that the model can reasonably reproduce the shockwave
propagation as well as the route choice behavior. Here, we first apply the model to the
12-km highway section from Yoga to Tanimachi on Route 3 as a case without route
choices.
As the input data, the model requires the time-dependent OD demand between every on
and off ramps along Route 3, the flow-density relationship of every link, and the merging
ratio at every merging node. The OD demand can be obtained in every one-hour period
from the OD survey conducted by the public corporation and the other input data havebeen observed by traffic detectors.
Fig.14 compares the observed travel time from Yoga to Tanimachi with the simulated time.
In this study, the observed travel time is obtained from the average travel time measured
by traffic detectors installed about every 300 meters along Route 3. Without route
choices, the model seems to well reproduce the observed travel time.

Travel Time [min]40

Simulated

30

Observed

20
10
0
4

6

8

10

Time of Day [a.m.]

Fig.144.2.

Comparison of Travel Time (from Youga to Tanimachi)

A Case with Route Choices

Next, the model is applied to a whole network of the Metropolitan expressway where
several alternative routes exist for each of the OD pairs. The Metropolitan expressway isan about 200-km length and approximately carries 1 million vehicle trips per day. To
include the route choice module, Dial’s parameter θ and a fraction of Route ChoiceGroup α must be additionally specified. Figures 15 and 16 show the simulation resultwith various combination of θ and α, in which the vertical axis shows the total absolutedifference in travel times between observed and simulated values. During the simulation
period from 4 am. to 11 am, about 350 thousands vehicle trips are generated and they
spend the total travel time of 10 million minutes. In the simulation, the total travel time isabout 11 million minutes which means 10% larger, but as shown in the figures, the11

Total Absolute Differencein Travel Timedifference in the travel time appears more than 3 million minutes for all the combinationsof θ and α because it counts the absolute values of the travel time difference.

Fig.15[10000min]
500
α=40%

450

α=50%

400

α=60%

350
300
0.0001

0.001

0.01

0.005

0.1

Parameter θ [1/sec]
Changes of Travel Time depending on Dial’s Parameters θ

Total Absolute Differencein Travel Time

From Fig.15 and 16, we see that the optimum value of θ is around 0.001 to 0.005 [1/sec]
and the optimum fraction of Route Choice Group α seems about 60%, which agrees withthe result estimated by a separate research based on an interview survey.

Fig.16[10000min]500450

θ=0.001400
θ=0.002350300

θ=0.005

0

20

40

60

80

100

Fraction of Route Choice Group, α [%]Changes of Travel Time depending upon the Fractionα

Figures 17 and 18 shows the observed and simulated average travel speed by each link
during 8 to 10 am. The correlation coefficient is 0.81 and the average difference in the
speed is 3.5 [km/h].

5.

SUMMARY AND FUTURE RESEARCH NEEDS

This study is briefly summarized as follows:
1) This study develops a dynamic simulation model SOUND which incorporating drivers’route choice behavior.
122) The model consists of the vehicle simulation and route choice modules, which arealternatively implemented in short time intervals to approximately reproduce thedynamic stochastic user equilibrium flow.
3)For the vehicle simulation module, we particularly pay attention to shockwave
propagation. According to several examinations on the wave propagation using asimple network, the simulated wave speed well fits with the theory.
4) As a case without route choices, we apply the model to Route 3 of the Metropolitan
expressway and see that the simulated traffic condition reasonably agree with theobserved one.
5) Finally, the model applications to the entire network of the expressway reveals that thefraction of Route Choice Group is about 60% and the simulated travel time and speed
show similar values of observed ones.
For the future research, the followings are required to improve the SOUND model:
1) For the route choice module, the evaluation of travel time should be further analyzed so
as to describe drivers’ choice behaviors.
2) For the traffic simulation module,- capacity reduction at a diverging node due to complex vehicle motions such as lanechanging should be investigated,

above 40km/h
under 40km/h

Fig.17

Average Speed at each Link (Observed)

13

above 40km/h
under 40km/h

Fig.18Average Speed at each Link (Simulated)- the model should consider vehicle types such as trucks, passenger cars, and so on,- variation in capacity at bottlenecks due to congestion should be included, and- vehicle motions in surface streets with signalized intersections should be modeled.

ACKNOWLEDGMENTS

The authors thank members of Planing Department Survey and Research Division andTraffic Control Engineering Division of the Metropolitan Expressway, Public Corporation
who kindly provide many valuable traffic data.

REFERENCES

Iida, Uchida, Fujii & Washio (1991). A Dynamic Traffic Simulation Model Considering a
Phenomenon of Traffic Jam Lengthening. Proceedings of Infrastructure Planning,vol.14(1), 301-308.
Kido, Ikenoue & Saito (1978). Searching routes in a network, Traffic assignment model(DYTAN-I). Institute of Japanese Police Science Report vol.19, No.1.
Kuwahara, M., Ueda, I., Akahane, H. & Morita, H. (1993). Development of Dynamic
Traffic Simulation Model for Urban Expressways. Traffic Engineering vol.28. 4, 11-20.

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Leonard, D.R., Gower, P. & Taylor, N.B. (1989). CONTRAM : Structure of the Model.
TRRL Research Report RR178.

Ueda, I., Tsubono, S., Kuwahara, M., Akahane, H. & Ozaki, H. (1991). Development of a
Traffic Simulation Model Incorporating Route Choice. Proceedings of Infrastructure
Planning, vol.14(1), 279-286.
Vliet, D.V. and Hall, M.D. (1991). SATURN 8.3-A User’s Manual-Universal Version,
Institute for Transport Studies, University of Leeds.

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