TOP: Vehicle Trajectory based Driving Speed Optimization Strategy forTravel Time Minimization and Road Congestion Avoidance

Li Yan and Haiying ShenDepartment of Electrical and Computer Engineering

Clemson University, Clemson, SC 29631Email: {lyan, shenh}@clemson.edu

Abstract—Traffic congestion control is pivotal for in-telligent transportation systems. Previous works optimizevehicle speed for different objectives such as minimizingfuel consumption and minimizing travel time. However,they overlook the possible congestion generation in thefuture (e.g., in 5mins), which may degrade the performanceof achieving the objectives. In this paper, we propose avehicle Trajectory based driving speed OPtimization strat-egy (TOP) to minimize vehicle travel time and meanwhileavoid generating congestion. Its basic idea is to adjustvehicles’ mobility to alleviate road congestion globally. TOPhas a framework for collecting vehicles’ information to acentral server, which calculates the parameters depictingthe future road condition (e.g., driving time, vehicle density,and probability of accident). The server then formulatesa non-cooperative Stackelberg game considering these pa-rameters, in which when each vehicle aims to minimizeits travel time, the road congestion is also proactivelyavoided. After the Stackelberg equilibrium is reached, theoptimal driving speed for each vehicle and the expectedvehicle density that maximizes the utilization of the roadnetwork are determined. Our real trace analysis confirmssome characteristics of vehicle mobility to support thedesign of TOP. Extensive trace-driven experiments showthe effectiveness and superior performance of TOP incomparison with other driving speed optimization methods.

I. INTRODUCTION

In recent decades, Intelligent Transportation Systems(ITSs) have received much attention. The ITSs summa-rize advanced applications aiming at providing innova-tive services related to different modes of transportationand traffic management. To support the operation ofvarious ITS applications, traffic congestion control isvery important for urban road networks [1]–[3] whentrying to maximize their utilization. For example, theroad management authorities hope that the density ofvehicles simultaneously passing through each road islower than a threshold so that the overall road networkkeeps operable. Also, the public transit service vehiclesrequire their covered routes to be non-congested sothat they can follow their schedule on time. However,due to the high mobility of vehicles and difficulty incontrolling vehicle speeds, congestion control in urbanroad networks is a non-trivial task.

In recent years, many methods have been proposedto reduce vehicles’ travel time by adaptively controllingtraffic signal [4], [5] or suggesting optimal speeds tovehicles for different objectives such as minimizing fuel

consumption and travel time [6]–[10]. In the formergroup of methods [4], [5], the controller at a roadintersection properly schedules the passing of vehiclesto minimize the vehicles’ total travel time caused by redlights or long queues. In the latter group of methods [6]–[10], the optimal driving speed of a vehicle is determinedbased on the vehicle’s real-time driving information (e.g.,fuel consumption, traffic state). However, these methodsoverlook the possible road congestion generation in thefuture (e.g., in 5mins), which may degrade the perfor-mance of achieving the objectives. In other words, thesemethods cannot avoid the generation of road congestionglobally in the road network in the future. By “in thefuture”, we mean in a future time during a vehicle’sdriving time period. For example, before “rush hours”,arterial roads may be non-congested. However, if legionsof vehicles drive by the currently “optimal speeds” intheir individual routes, they may crowd into the arterialroads simultaneously, which results in congestion.

However, solving this neglected problem is non-trivial.The road congestion is measured by vehicle density; ahigher vehicle density increases the utilization of theroad network but generates congestion and decreasesvehicle speed, and vice versa. Therefore, it is a challengeto maximize the utilization of the road network whileproactively avoiding congestion and maximizing the ve-hicle speed. In this paper, we aim to tackle this challengeby proposing a vehicle Trajectory based driving speedOPtimization strategy (TOP) that uses game theory to letvehicle drive as fast and safely as possible, and mean-while proactively avoid generating road congestion in thefuture. Its basic idea is to periodically adjust vehicles’mobility to alleviate road congestion globally. The vehi-cles report their information to a central server throughroad-side-units (RSUs) located alongside the roads. Thecentral server calculates each vehicle’s trajectory in thenext time slot (denoted by Tc+1) and determines theparameters depicting the future utilization of the roadnetwork (e.g., vehicle density, driving time and probabil-ity of accident). This is based on the previous observationthat vehicles’ trajectories can soundly illustrate the futuremobility of the vehicles [11]–[15]. To maximize the uti-lization of the road network while minimizing the proba-bility of road congestion, the central server formulates anon-cooperative Stackelberg game, in which each vehicleaims at minimizing its travel time and maximizing safety

while avoiding generating congestion in the future. Afterthe Stackelberg equilibrium is reached, when vehiclesfollow their optimal speeds, it also proactively avoidsgenerating congestion in the future. Moreover, the roadnetwork can be fully utilized without imbalanced or highvehicle density (i.e., congestion) in road segments. Insummary, our contributions include:(1) Our analysis on two real vehicle traces [16], [17]

confirms the characteristics of vehicle mobility andlays the foundation for the design of TOP.

(2) We propose a non-cooperative Stackelberg gamebased vehicle speed optimization strategy to findthe optimal speed for each vehicle that enables it todrive as fast and safely as possible, while avoidingthe generation of congestion in the future.

(3) We have conducted extensive trace-driven experi-ments to show the effectiveness of TOP in max-imizing the utilization of road network, avoidingcongestions, and satisfying drivers’ need of drivingas fast and safely as possible.

Within our knowledge, this work is the first to providevehicle speed optimization that aims at letting vehiclesdrive as fast and safely as possible, while proactivelyavoiding the generation of road congestion in the future.The remainder of the paper is organized as follows.Section II presents related works. Section III presents thetrace analysis and findings that support TOP. Section IVpresents the detailed design of TOP. Section V presentstrace-driven experimental results. Section VI concludesthe paper and marks future research direction.

II. RELATED WORK

Real-time traffic based vehicle speed optimization.Several methods for vehicle speed optimization withdifferent objectives have been proposed. Kouvelas etal. [4] proposed a hybrid approach for traffic signalcontrol considering the saturation status of the road.Pandit et al. [5] proposed a vehicular network basedmethod that collects and aggregates real-time trafficinformation to optimize signal control. Tseng et al. [10]proposed a vehicle density estimation scheme usingneighbor tables communicated between vehicles. Chen etal. [9] proposed to use VANETs to send queries betweensource and destination back and forth, and selects thepath with the shortest time. Ozatay et al. [7] used cloudcomputing [18], [19] in optimizing vehicle speed profileby solving a dynamic programming problem. Asadi andVahidi [8] proposed a control algorithm to enable vehi-cles to approach traffic light at green as much as possible,thereby saving fuel and reducing travel time. Groot etal. [6] proposed to model vehicle-congestion relationshipas reverse Stackelberg games to optimally distributetraffic over road network, and meanwhile ensure thateach vehicle can finish travel within its expected traveltime. For the vehicles with the same origin-destination,the central server uses different pricing of freeways (e.g.,longer route has lower price) to induce these vehicles to

choose different routes to distribute the traffic. However,this method overlook that the vehicles with differentorigin-destination pairs may compete for the same roadsegment. Also, it does not aim to minimize the traveltime of the vehicles. As indicated previously, the abovemethods do not consider whether the currently suggestedspeed will cause congestion to certain road segments inthe future. To solve this problem, TOP firstly utilizesvehicles’ trajectories to extract the parameters of futureroad traffic, and then uses the parameters in formulatinga non-cooperative Stackelberg game that aims to let ve-hicles drive as fast and safely as possible, and meanwhileavoid the generation of congestion in future.

Vehicle future mobility based routing. Manyworks [20], [21] focus on using vehicles’ current orhistorical mobility statistics to predict the vehicles’ fu-ture mobility. Some other works [11]–[15] found thatutilizing vehicles’ GPS trajectory to deduce the vehi-cles’ future mobility is reliable for data delivery invehicular networks. Wu et al. [11] found the spatio-temporal correlation in vehicle mobility and noted thatthe future trajectory of a vehicle is correlated withits past trajectory. In Trajectory-based Data ForwardingScheme (TBD) [12], Trajectory-based Statistical For-warding Scheme (TSF) [13], [14] and Shared-Trajectory-based Data Forwarding Scheme (STDFS) [15], trajectoryinformation of vehicles is collected through access pointsand used to predict the vehicle mobility for data for-warding. Our work is based on the observations in theseworks that vehicles’ trajectories can soundly illustratethe future mobility of vehicles, which is used to estimateroad vehicle density in the future.

III. TRACE ANALYSIS

In this section, we present our trace analysis on theRome trace [16] and the San Francisco trace [17], whichdemonstrates the characteristic of vehicle mobility inurban area and provides the rationale of the designof TOP. Both of the traces are 30-day taxi traces.Since taxis move persistently and cover almost theentire road networks, their movement can illustrate thetraffic state [2], [22]–[24]. In these traces, each taxireports location record (timestamp, ID, GPS position)to a central server every 7 seconds. We filtered out thepositions with precision larger than 20 meters and taxiswith few appearances (

0 1 2 3 40

0.5

1Rome

0 10 20 300

0.5

1

Vehicle densities of road segments

San Francisco

CD

F o

f veh

icle

s

(a) Vehicle densities.

0 50 1000

0.5

1Rome

0 100 200 3000

0.5

1

Vehicle flow rates of road segments

San Francisco

CD

F o

f veh

icle

s

(b) Vehicle flow rates.Fig. 1: Concurrent competition for road usage.

0 5 10 15 200

2

4

6Rome

0 5 10 15 200

20

40San Francisco

Time

Ave

rage

veh

icle

den

sity

(a) Vehicle densities over time.

0 5 10 15 200

5

10

15Rome

0 5 10 15 200

10

20

30San Francisco

Time

Ave

rage

veh

icle

spe

ed

(b) Vehicle speeds over time.Fig. 2: Vehicles’ temporal preference on roads.

A. Concepts and Problem IntroductionWe define a road segment (denoted by si) as the

road link between two neighboring intersections (i.e.,landmarks). Vehicle density of road segment si (denotedby di) is defined as the average number of vehicles permile in the road segment (veh/mile), and the flow rate ofroad segment si (denoted by fi) is defined as the averagenumber of vehicles driving through the segment per unittime [6], [25] (veh/h). The vehicle flow rate of segmentsi equals to the product of vehicle density and averagevehicle speed on si (denoted by vi) [26];

fi = di · vi. (1)The road congestion is measured by vehicle density, andthe utilization of the road network is measure by flowrate. Therefore, in order to increase the utilization of theroad segment si (i.e., fi), we need to increase vehicledensity (di) and/or vehicle speed (vi). However, highervehicle density may lead to congestion and hence lowervehicle speed. Therefore, it is a challenge to maximizethe utilization of road network and meanwhile maximizevehicle speed, which is the objective of this paper.

B. Concurrent Competition for Road UsagePrevious methods locally control traffic or compute

suggested speed based on current traffic state on eachvehicle’s scheduled route. If a vehicle follows the speedindividually optimized for it, due to the ignorance ofother vehicles’ mobility, some arterial roads may be-come crowded with many vehicles, that is, the vehi-cles concurrently compete for these roads. To confirmthis conjecture, we measured the cumulative distributionfunction (CDF) of the vehicle density and the CDF of theflow rate on all road segments as shown in Figure 1(a)and Figure 1(b). We calculated the vehicle density andvehicle flow rate for 30 days and draw their averagevalues. The vehicle density and vehicle flow rate aresampled every 30mins on every road segment per day.We see that for the Rome trace, the vehicle densityof 90% of road segments is less than 0.5veh/mile, andthe vehicle flow rate of 90% of road segments is lessthan 10veh/h. But the other 10% of road segments havevehicle density and vehicle flow rate as high as 3veh/mileand 60veh/h, respectively. For the San Francisco trace,the vehicle density of 95% of road segments is lessthan 3veh/mile, and the vehicle flow rate of 95% ofroad segments is less than 25veh/h. But the other 5%

of road segments have vehicle density and vehicle flowrate as high as 24veh/mile and 50veh/h, respectively.These results demonstrate that in the urban road network,vehicles usually concurrently compete for usage on fewpopular roads, resulting in their excessive utilization.Therefore, we can try to distribute traffic evenly in theroad network, i.e., achieve similar vehicle density inall road segment, in order to avoid the congestion andincrease the utilization of road network. The cause torepeated congestion on arterial roads is excessive con-current utilization of vehicles [6]. Therefore, we furtheranalyze vehicles’ temporal preference on driving roads.

C. Vehicles’ Temporal Preference on Roads

If the vehicle density on a road exceeds a threshold,the driving speed of vehicles on the road is likely tobe affected due to congestion. This is especially truefor arterial roads since they are quite likely to be over-utilized during rush hours. To verify such intuition,we measured the average vehicle density and averagevehicle speed on the most highly utilized road segmentof the two traces (road segment 4433 in the Rometrace, road segment 0 in the San Francisco trace) hourlyduring each day in the 30 days, which are shown inFigure 2(a) and Figure 2(b), respectively. We see that forthe Rome trace, the vehicle densities during 6:00∼13:00and 17:00∼20:00 are higher than the other hours. Incontrast, the average vehicle speeds during these twoperiods are lower than the other hours. For the SanFrancisco trace, the vehicle densities during 1:00∼4:00and 12:00∼15:00 are higher than the other hours. Incontrast, the average vehicle speeds during these twoperiods are generally lower than the other hours. Theseresults demonstrate that excessively high vehicle den-sity deteriorates road driving condition, which causesreduced driving speed. The results confirm that avoidingcongestion is important to increasing driving speed andreducing travel time, especially in rush hours.

IV. SYSTEM DESIGN

A. Overview

ITSs support the installation of RSUs alongside roadsegments to provide communication between vehiclesand the central server [14], [27], [28]. As shown inFigure 3, we establish a three-layer information col-lection and dissemination framework, which consists of

vehicles as the service layer, RSUs as the communicationbackbone and a central server as the computation layer.Each vehicle contacts the central server through RSUs.

Vehicl

e

densit

y

Tx DataRx Data(trajectory,speed)

RSU

Central Server Cloud

Vehicle 1

Prefer

red

speed

Vehicle density

Preferred speed

(vehicle density)

Vehicle 2

...

Fig. 3: System structure.

As in the traffic managementpapers in [29], we considerroad segments have equal ve-hicle density limits. In this pa-per, we focus on optimizing thevehicles’ speed on their origi-nal route. We leave the optimalroute selection as future incre-mental work. To let vehiclesdrive as fast and safely as pos-sible while avoiding generat-ing congestion on the road net-work, we use the Stackelberggame [30] between the vehicles and the central server todetermine the expected vehicle density that maximizesthe utilization of the road network and optimal drivingspeed for each vehicle. The gaming process is executedperiodically with period T (e.g., 5mins) as follows:1) Through a nearby RSU, the vehicle reports its currentposition and intended destination to the central server.2) Based on the information collected from vehicles, thecentral server calculates the trajectory of each vehiclein Tc+1 and predicts the vehicle density in each roadsegment at the next time slot. Then, a gaming process isconducted between the central server and each vehicle.3) Based on the predicted average vehicle density perroad segment in the road network in Tc+1, the centralserver determines a set of expected average densities thatare achievable by vehicle speed adjustment.4) Based on each expected average density, each vehicledetermines its speed that maximizes its utility (speed andsafety) and reports the speed to the central server.5) The central server determines the final expectedaverage density that maximizes its utility (maximizingflow rate of the road network) and notifies all vehicles.6) Each vehicle chooses its speed corresponding to thefinal expected average density.

With the optimal speeds, the vehicle density of eachroad segment will be approximately the determinedvehicle density. Thus, the total traffic in the road networkis well balanced with no congestion and its utilizationis maximized. We will first explain how the centralserver predicts the vehicle density of road segments(Section IV-B) and then present the non-cooperativeStackelberg gaming (Section IV-C).

B. Future Road Vehicle Density Prediction

The gaming process runs after each time slot T (e.g.,5mins). For example, when the central server starts thegame at 00:00, it needs to estimate the vehicle density ofeach road segment in [00:00,00:05] in order to determinean achievable vehicle density in the entire road networkfor vehicles to choose their optimal speeds.

In this section, we present how to estimate the vehicledensity of each road segment in Tc+1 with current ve-hicle speeds. First, the central server needs to determineeach vehicle’s trajectory in Tc+1. It consists of the roadsegments it will pass in Tc+1 and their correspondingtravel times {(si, T̃i)|i = 1, 2, . . . ,M}, where si denotesthe ith road segment, T̃i denotes the estimated travel timefrom current position to si and M denotes the numberof road segments that the vehicle will pass in Tc+1.

Then, by modeling the arrival times as normal randomvariables, TOP sums up the vehicles’ probabilities ofappearance on each road segment as its vehicle densityat the next time slot. The average vehicle density per roadsegment will be used in the driving speed optimizationgaming presented in Section IV-C. After each vehicledetermines its speed in gaming, the vehicle density willbe updated and used for the next gaming process.

1) Trajectory Calculation: A vehicle periodically re-ports its current position and its destination to the centralserver. To generate the vehicle’s trajectory in Tc+1, thecentral server first determines the sequence of road seg-ments connecting the current position and the destinationbased on road topology [31]. It then calculates the traveltime of each road segment that will be passed in Tc+1 bythe vehicle. After a gaming process, a vehicle’s optimalspeed on si is determined, denoted by vi. Then, for eachroad segment si, the estimated travel time on si (denotedby t̃i) can be calculated by

t̃i = li/vi, (2)

where li is the length of si. A problem is how toestimate the travel time of si initially when no game hasbeen played. To handle this problem, we use the currentvehicle density of the road segment to estimate the speedfor the vehicle as in traditional vehicle density basedspeed estimation methods [26]. It has been indicated thatfor a road segment si, its reachable speed is related toa vehicle density limit dmi . When the vehicle density isbelow dmi , vehicles on the road segment can drive withthe free flow speed (i.e., speed limit, denoted by vmaxi ).If the vehicle density exceeds dmi , the road segment willbe congested and the vehicles have to drive with thecongested speed (denoted by vmini ). d

jami is the vehicle

density that will cause si to be completely jammed. dmican be obtained from field observation, and djami can beobtained from the road network’s designed capacity [26].Currently, the vehicle density of each road segment canbe well monitored [2], [22]–[24], [32]. Then, we canroughly estimate the allowed vehicle speed under currentvehicle density for each road segment as below:

t̃i =

⎧⎨⎩li/v

maxi , 0 � di < dmi

li/vmini , d

mi � di < djami

∞, di � djami(3)

The trajectories generated by GPS do not consider theroad congestion condition and hence may not be suffi-ciently accurate. In contrast, TOP calculates the trajec-tories of vehicles considering future road congestion.

The travel times of a road segment can be describedby normally distributed and statistically independentrandom variables with acceptable precision [33], [34].Therefore, the estimated travel time from the currentposition to road segment si is the sum of the travel timesof the composing road segments from current positionto si, T̃i =

∑Mim=1 t̃m, where Mi is the number of road

segments from current position to si. When T̃i ≥ T ,the trajectory for Tc+1 has been generated. Based on thehistorical data of the estimated travel time of sm and realtravel time on sm of all vehicles, the central server cancalculate variance σ2m. Then, the standard deviation of T̃iis calculated by summing the variances of the composingroad segments, Δ2i =

∑Mim=1 σ

2m.

2) Road Vehicle Density Calculation: The estimatedtravel time in {(si, T̃i)|i = 1, 2, . . . ,M} only has a cer-tain probability to be accurate. Then, we have two stepsto calculate the vehicle density of each road segmentin Tc+1. First, we use a vehicle’s trajectory in Tc+1to estimate the probability that the vehicle will appearat each road segment in its trajectory in Tc+1. Then,we calculate the sum of all the vehicles’ appearanceprobabilities at a road segment in Tc+1 as the vehicledensity of the road segment in Tc+1.

Suppose the next time slot is the jth time slot in aday, represented by Tc+1 = [tsj , tej ] (e.g., [00:00,00:05]),where tsj and t

ej are the starting time and ending time of

the time slot, respectively. For each vehicle, TOP usesits estimated travel time to si to measure its appearanceprobability at si during [tsj , t

ej ]. Therefore, the vehicle’s

appearance probability at si during [tsj , tej ] is

P (Ti� tej − tsj)=Φ(tej − tsj − T̃i

Δi)−Φ(−T̃i

Δi) (4)

where Ti denotes the vehicle’s actual travel time fromcurrent position to si, and Φ(·) is the CDF of thestandard normal distribution with mean T̃i and standarddeviation Δi. The CDF for each vehicle on each roadsegment si is calculated based on the historical recordsof all vehicles’ travel times on the road segment. Then,for each road segment si, the central server estimates itsvehicle density in Tc+1 by summing up the appearanceprobabilities of vehicles (Pk) on si during Tc+1:

dsii+1 =

N∑k=1

Pk(Ti � tej − tsj) (5)

where N is the number of vehicles that will pass siduring [tsj , t

ej ]. For example, given current time 00:00,

the estimated vehicle density of College Ave for Tc+1,namely 00:00∼00:05, is 26.16 vehicles/mile.

3) Safety Estimation: Each road segment has a prob-ability of accident occurrence. The probability dependson the structure feature of the road segment (e.g., thedegree of straightness, sharp turn, road surface bump)and the traffic condition. It has been verified that trafficconditions (e.g., heavy traffic volume, speeding) affectthe likelihood of accident occurrence [35]. The trafficcondition of a road segment has a long-term pattern,

that is, the vehicle flow rate at each time slot remainssimilar irrespective of days. For example, people arelikely to encounter congestion on their way to workduring morning rush hours every workday.

TOP relies on historical records of accidents to depictthe likelihood of accident for each road segment in eachtime interval in a day [36], [37]. Considering that theprobability of accident is time-varying (e.g., some roadsegments are more likely to have accidents in Winterthan in Spring), TOP uses a time window to control thenumber of days for consideration. The larger the windowsize is, the more accident events that can be captured.Specifically, the accident probability of road segment siduring jth time interval [tsj , t

ej ] is calculated by:

pji =

∑Ww=1 T

wj

W · (tej − tsj)(6)

where pji is the accident probability of si during the jthtime interval, W is the window size, and Twj is the lengthof time that si is affected by accident during the jth timeinterval in the wth day. Finally, for each road segment,the central server generates a table summarizing acci-dent probability during each time interval, as shown inTable I. Since higher vehicle density leads to shorterdistance between vehicles, which renders higher risks ofaccident, we relate pji with vehicle density in determiningthe utility of drivers in gaming in Section IV-C.TABLE I: Table of accident probability of road segment College Ave.

Time Accident probability00:00∼00:05 0.0500:05∼00:10 0.02

... ...

C. Driving Speed Optimization Gaming1) Overview: On one hand, traveling quickly (i.e.,

short driving time and no congestion) and safely (i.e.,no accident) is desired by drivers. On the other hand,the transportation authority hopes to maximize the uti-lization of the road network (i.e., maximum vehicle flowrate). Based on Equation (1), to increase road networkutilization, we need to increase the vehicle density, whichhowever may lead to road congestion. Then, the vehiclespeed drops down and results in low flow rate and hencelow utilization of the road network.

It is found that drivers may drive slower given a higherspecified vehicle density in order to keep a safe inter-vehicle distance to keep safety, especially for the driverswith high probability of accidents [35]. Therefore, thedrivers will adjust speeds in response to a given vehicledensity. Thus, we can formulate the speed optimizationas a non-cooperative Stackelberg game [6], [38] betweenthe central server and the drivers, where the centralserver is the leader and the drivers are followers.

In the Stackelberg game, the leader considers thepredicted average vehicle density of a road segment (in-troduced in Section IV-B2), and then chooses a set of ex-pected vehicle densities (denoted by D={d1, d2, ..., dn})that are achievable by vehicle speed adjustment. The

central server hopes to evenly distribute the vehiclesover all road segments by properly assigning a d value.The followers receive D from the leader and picks aspeed in response to each di to maximize its own utility(driving as fast and safely as possible while minimizingthe risk of congestion). Next, the central server selectsthe vehicle density that maximizes its utility (i.e., vehicleflow rate of the road network), denoted by dl and thenthe vehicles choose their speeds corresponding to theselected dl. Finally, we solve the Stackelberg equilibriumof the game, i.e., the game reaches a state that theroad network utilization is maximized while the driversare satisfied with the driving status (judged by drivingspeed and associated risk of congestion). The gaming isexecuted periodically. In the following, we first introducethe utility of a driver and the utility of the central server,and then introduce the gaming between them.

2) Utility Function of Drivers: For drivers, we definea utility function to quantify the level of benefit that adriver obtains from driving by a speed on road segmentsi. It is calculated by subtracting the potential risk ofcongestion (Ur(·)) from a vehicle’s satisfaction degree(Us(·)), as shown in Equation (8).

F (vi, αi, pji ) = Us(vi, αi, p

ji )− Ur(d, vi, pji ) (7)

s.t. vi � vmaxi

where vi is the vehicle’s speed for optimization; αi isa scale factor to make Us(·) and Ur(·) comparable; pjiis calculated by Equation (6).Us(·) ought to be non-decreasing as each driver de-

sires high speed (i.e., short driving time). Meanwhile,the marginal satisfaction degree of the driver is non-increasing because the driver’s satisfaction degree grad-ually gets saturated when the vehicle speed increases tosome level [26]. Moreover, Us(·) is inversely related withthe probability of accident because a lower possibility ofaccident corresponds to higher level of satisfaction [39].Considering these properties, we design Us(·) as a con-cave function. Since the Natural Logarithmic Functionsare representative concave functions [40], we define:

Us(vi, αi, pji ) = αi · ln(vi + pji

−1) (8)

A driver’s potential risk of congestion is determinedby the accident probability of the road segment (pji ) andvehicle flow rate [35] (Equation (1)).

Ur(d, vi, pji ) = p

jidvi (9)

The utility of a driver decreases with a higher vehicledensity and vice versa. Combining Equation (8) andEquation (9) into Equation (8), we have:

F (vi, αi, pji ) = αi · ln(vi + pji

−1)− pjidvi (10)

s.t. vi � vmaxiNote the gaming is executed periodically, so it is

possible that a vehicle may enter other road segmentsduring the current time slot. We use γi to denote thepercentage of T that the vehicle will spend on segmentsi. Then, the utility of the vehicle is calculated by:

∑i

γiF (vi, αi, pji ) (11)

s.t. vi � vmaxi3) Utility Function of Central Server: The central

server always aims at maximizing the vehicle flow rateon overall road network:

L(d) =

Ns∑i=1

di · vi (12)

where Ns is the total number of road segments.4) Optimal Driving Speed Selection: Recall that

based on Equation (5), the central server predicts thevehicle densities of all road segments. It then calculatesthe average estimated vehicle density of the road networkduring next period of gaming: dc+1 =

∑Nsk=1 d

ski+1/Ns.

Based on dc+1, the central server determines a range ofexpected vehicle densities that are achievable by vehiclespeed adjustment, and offers these densities to eachvehicle for selection, which is defined as:

du = ln(u+ 1) · dc+1, u ∈ [1, ..., n] (13)We use D={d1, d2, ..., dn} to denote the n levels ofexpected vehicle densities for Tc+1. In practice, n shouldbe at least larger than the exponent so that the vehi-cle has multiple selections around dc+1. The centralserver notifies drivers of the D. If dc+1 leads to anincreased expected vehicle density (du), the drivers willbe encouraged to decrease driving speed in order todrive safely. Otherwise, the drivers will be encouragedto increase driving speed in order to increase benefitwhile maintaining driving safety. Note the incrementrate of Us(·) (Natural Logarithmic Function) is slowerthan Ur(·) (Linear Function) when speed vi increases.Therefore, according to Equation (8), increasing drivingspeed on current road segment (vi) will reduce a driver’sutility because Ur(·) will increase faster than Us(·).Thus, driving at a slower speed can prevent the vehicledensity of the road network from further increasing, i.e.,prevent traffic congestion.

For each du ∈ D, if a driver will drive in its currentroad segment si during the next time slot, it chooses anew speed that maximizes its utility F (·), denoted byviu, as shown in Equation (14).

viu = argmaxvi�vmaxi

F (vi, αi, pji ) (14)

If a driver will drive through more than one roadsegment si, sj ,..., it chooses a set of speeds in eachof the segments to maximize its utility F (·), denoted by{viu, vju, ...} as shown in Equation (15).

{viu, vju, ...} = argmaxvk�vmaxk

∑k

γkF (vk, αk, pjk) (15)

Finally, the driver reports the n candidate speeds to thecentral server. The central server finalizes the expectedvehicle density (dl) that maximizes its utility L(·) basedon the candidate speeds from all drivers.

dl = argmaxdu∈D

L(du) = argmaxdu∈D

du∑Ns

viu (16)

The central server then notifies all drivers of the new ex-pected vehicle density dl. Then, among the n candidatespeeds, each driver picks the speed corresponding to dl.

V. PERFORMANCE EVALUATION

A. Experimental Settings

We conducted trace-driven experiments based on theRome [16] and the San Francisco [17] traces introducedin Section III. Unless otherwise specified, the experimentsettings are the same as those in Section III. The numberof accidents occurred in Rome and San Francisco ineach month are obtained from [41] and [42], respectively.The window size W was set to 7 days and Twj = 1h.The scale factor αi was set to 2.85 for Rome and 5for San Francisco. We measure a driver’s satisfactiondegree when (s)he drives on road segment si withspeed vi by ln(vi + p

ji

−1)/ ln(vmaxi + p

ji

−1) (deduced

from Equation (8)). The gaming procedures are launchedevery 15mins in the two traces. To simulate that vehiclesdrive by their optimal speeds, we dynamically updatethe timestamps of arrivals at landmarks according to thevehicles’ optimal speeds. Therefore, in the experiment,the vehicles follow the movement paths recorded in thetraces but with modified timestamps.

We compared TOP with the traffic signal controlmethod [5] (Signal in short) and the vehicle speedoptimization method [7] (RealSpeed in short). Signaluses vehicular ad hoc networks (VANETs) to formulatevehicles into platoons. The controller at each intersectionuses the oldest-arrival-first scheduling algorithm toarrange the passing of platoons so that the vehicles’total travel time is minimized. In RealSpeed, by aimingat reducing fuel consumption and satisfying driver withreduced travel time, the vehicle speed is optimized bydynamic programming constrained by speed limit, real-time traffic and driver’s destination. To make RealSpeedcomparable to the other methods, we excluded its fuelconsumption constraint in our experiments. Signal andRealSpeed cannot proactively avoid the generation ofroad congestion in the future. Each experiment is for 30days. In each hour throughout each day, we measuredthe following metrics and report the average value ineach hour for the 30 days.• Average vehicle speed: The average speed of all the

speeds determined by the games during an hour.• Average flow rate: After each game, we calculate

the flow rate per road segment by∑Ns

i=1 di · vi/Ns.Then, we calculate the average flow rate per roadsegment in all the games during an hour.

• Average driving time: The average driving time oneach road segment for all the travels on segmentsduring an hour.

• Average driver satisfaction: The average satisfac-tion degree of the drivers after travels per hour.

B. Experimental Results

1) Average Vehicle Speed:Figure 4(a) and Figure 5(a) show the average speed

of vehicles during different time intervals with theRome and San Francisco traces, respectively. We seethat for Rome, the average vehicle speeds follow:TOP>RealSpeed>Signal. While for San Francisco, theaverage vehicle speeds follow: TOP>Signal>RealSpeed.

TOP always has much higher average vehicle speedthan others in both traces. Before optimization, the futuretraffic on the scheduled route has been deduced by thecentral server from the vehicle trajectories. Althoughthe results might have deviation from the true results,they are still effective in predicting the future movementof vehicles. With gaming using the predicted vehicledensity, TOP enables the central server to maximallyavoid road congestion caused by confluent vehicle flows.Meanwhile, each vehicle can drive by a speed as fastand safely as possible. As a result, TOP generates thehighest average speed of vehicles. Both RealSpeed andSignal do not proactively avoid generating congestionsin the future and congestions decrease vehicle speeds,thus producing lower average speed of vehicles.

RealSpeed has the secondary performance in Rome,but the lowest performance in San Francisco. In Re-alSpeed, for a vehicle, the server generates a routebased on the collected information of the intendedtravel. Then, the server collects the associated trafficand geographical information, and calculates optimalvehicle speed aiming at reducing travel time throughdynamic programming. However, since San Franciscohas many uniformly distributed road segments with shortlengths [17], the vehicle flow on the vehicle’s scheduledroute can be easily congested by vehicle flows from otherintersected road segments. In contrast, the road segmentsin Rome have fewer intersections [16]. Therefore, thevehicle traffic in the road network is less likely to becongested than that in San Francisco.

Signal has the lowest average vehicle speed in Rome,but the second lowest performance in San Francisco.This is because Signal aims at reducing vehicles’ traveltime near intersections rather than in the global roadnetwork. In Signal, the vehicle flows on the road networkare partitioned into several platoons of vehicles. Byviewing each platoon as a job, the traffic managementproblem is formulated as a job scheduling problem atintersections. To minimize the time of vehicles passingthe intersection, Signal utilizes the oldest-arrival-firstscheduling algorithm. However, Rome’s road segmentsare quite crowded at popular sites and have shortdistance [16], which make streets near popular sitesheavily utilized. Locally minimizing the time at certainintersections inevitably exacerbates congestion at theother intersections. Therefore, Signal cannot achieve anoptimal solution in the whole road network in Rome. Incontrast, the road segment distribution of San Franciscois more uniform than that in Rome [17]. Therefore,

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Signal can better schedule vehicles passing through theintersections in San Francisco than in Rome, resultingin the better performance of Signal in San Francisco.

2) Average Flow Rate:Figure 4(b) and Figure 5(b) show the average flow

rate during different time intervals with the Romeand San Francisco traces, respectively. We see thatfor Rome, the average vehicle flow rates follow:TOP>RealSpeed>Signal. While for San Francisco, theresults follow: TOP>Signal>RealSpeed.

The average flow vehicle rates follow the same trendas the average vehicle speed result. Higher speed meansthat the vehicle flow can move faster on road segment aslong as the vehicle density does not result in congestion.Although some road segments may be too crowdedto let vehicles maintain high speeds, their flow rateis still large as long as their vehicle density does notexceed the jam level. Through comparing Figure 4(a)and Figure 5(a) with Figure 4(b) and Figure 5(b), we cansee that although the average speed keeps above 15km/h,the vehicle flow rate is as low as 1veh/h. This shows thatwhen the road network is non-congested, the vehicles inSignal and RealSpeed can drive as fast as possible (i.e.,speed limit), which results in acceptable average drivingspeed. While without proactively avoiding congestion,the vehicle flows may generate congestion.

3) Average Driving Time:Figure 4(c) and Figure 5(c) show the average

driving time during different time intervals with theRome and San Francisco traces, respectively. We seethat for Rome, the average vehicle driving time fol-lows: Signal>RealSpeed>TOP. While for San Fran-cisco, the average vehicle driving time follows: Real-Speed>Signal>TOP.

TOP always has the lowest driving time because eachvehicle can drive by a fast speed with low probability ofsuffering from congestion. Signal has the highest driving

time in Rome, and the second highest driving time in SanFrancisco. Correspondingly, RealSpeed has the secondhighest driving time in Rome, and the highest drivingtime in San Francisco. These results are consistent withthose of the average vehicle speed due to the samereasons. It is noticeable that in San Francisco, thereis a heap between 0h and 1h. This is because thereis a drop of speed during this time interval. Whenmultiple vehicles simultaneously enter an intersection,but traffic signals cannot schedule their passing in time,the vehicles then wait in queues at the intersection.

4) Average Driver Satisfaction:Figure 4(d) and Figure 5(d) show the average

driver satisfaction during different time intervals withthe Rome and San Francisco traces, respectively. Wesee that for Rome, the average satisfaction follows:TOP>RealSpeed>Signal. While for San Francisco, theaverage satisfaction follows: TOP>Signal>RealSpeed.

Driver satisfaction is jointly determined by vehiclespeed and accident probability. Since the accident prob-ability is calculated offline and does not change duringvehicles’ movement, drivers’ satisfaction is solely deter-mined by the vehicle speed. Since TOP generates thehighest vehicle speed, it always ranks the highest andachieves over 80% satisfaction in both traces. The sat-isfaction results are consistent with the average vehiclespeed results due to the same reasons.

VI. CONCLUSIONPrevious works for speed optimization does not

proactively avoid the generation of congestion inthe future. We proposed TOP, a vehicle trajectorybased driving speed optimization strategy aiming atminimizing each vehicle’s travel time while avoidinggeneration of congestion. Our analysis on the vehiclemobility and congestion based on two real-world tracessupport the motivation for the design of TOP. TOP uses

vehicle trajectories to estimate the vehicle density ofeach road segment in the near future. Then, by using anon-cooperative Stackelberg game between each vehicleand the central server, the vehicle’s driving speed isoptimized so that it can drive as fast and safely aspossible while proactively avoiding generating conges-tion. We have conducted extensive experiments basedon the two traces. The experiment results validate thehigh effectiveness of TOP and its superior performancecompared to previous methods in terms of the utilizationof road network, congestions, and driver satisfaction. Inour future work, we plan to consider vehicles’ social re-lationship in avoiding road congestion and develop morereasonable schemes to motivate vehicle cooperation.

ACKNOWLEDGEMENTSThis research was supported in part by U.S. NSF

grants NSF-1404981, IIS-1354123, CNS-1254006, andMicrosoft Research Faculty Fellowship 8300751.

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