freeway sensor spacing and probe vehicle penetration

of predictions of freeway travel time for traveler information onvariable message signs (VMS). Previous work revealed that the maincauses of travel time prediction–estimation errors were transitionsbetween traffic states, detector failure, and large detector spacing(2). Therefore, the objective of this paper is to study analytically theimpacts of sensor spacing and penetration rate of probe vehicles on theaccuracy of predicting–estimating travel times for traveler informationand traffic management during transitions.

The terms “online travel time prediction” and “offline travel timeestimation” are distinguished in this paper. “Online travel time pre-diction” provides vehicles with forecast travel times based on real-time data across a certain link before they enter that link; “offlinetravel time estimation” uses previously recorded data from a link toestimate vehicle travel times after they have passed through that link.In this paper, the midpoint method is used for online travel time pre-diction and the Coifman method (3) is used for offline travel timeestimation. This paper uses a hypothetical flow–density relationshipto quantify errors in total travel time (TTT) by the midpoint andCoifman methods, so that a mathematical relation between TTTprediction–estimation error and sensor spacing or probe vehicle pen-etration rate can be established. Because inaccurate travel time pre-diction–estimation usually occurs during the onset and dissipation ofqueues, travel times are predicted–estimated during three shockwave types: backward-forming shock waves, backward-recoveryshock waves, and forward-recovery shock waves.

Many factors influence the accuracy of prediction–estimation oftravel times: prediction–estimation method, spatial and temporalfluctuations of traffic conditions, detector spacing, detector noiseand failure rate, probe vehicle penetration rate, and data aggregationlevel. For this analysis, the impacts of the prediction–estimationmethod, detector spacing, probe vehicle penetration rate, and trafficstate transitions are emphasized.

BACKGROUND

Online travel time prediction can be theoretical—usually conceptuallysimple—where a link length is divided by a measured speed (4, 5).Statistical prediction methods may use regression models (6–9),Kalman filtering methods (10, 11), or time series analysis (12, 13)to predict travel time from real-time and historical data.

Offline travel time estimation methods can use estimated trajec-tories from extrapolation of measured point speeds (3, 4, 14–18).Other methods incorporate flow, density, or occupancy data alongwith the speed data (19–24). Some were developed for specificconditions such as free flow (19, 23), queued (20), or both (22).Previous research (3, 25–27 ) has found large travel time estimationerrors during traffic state transitions.

Freeway Sensor Spacing and Probe Vehicle PenetrationImpacts on Travel Time Prediction and Estimation Accuracy

Wei Feng, Alexander Y. Bigazzi, Sirisha Kothuri, and Robert L. Bertini

67

Accurate travel time prediction–estimation is important for advancedtraveler information systems and advanced traffic management systems.Traffic managers and operators are interested in estimating optimalsensor density for new construction and retrofits. In addition, with thedevelopment of vehicle-tracking technologies, they may be interested inestimating optimal probe vehicle percentage. Unlike most studies focus-ing on data-driven models, this paper extends some limited previouswork and describes a concept developed from first principles of trafficflow. The goal is to establish analytical relationships between travel timeprediction–estimation accuracy and sensor spacing by means of twobasic travel time prediction–estimation algorithms, as well as to probevehicle penetration rate. The methods are based on computing themagnitude of under- and overprediction–estimation of total traveltime (TTT) during shock passages in a time–space plane by using themidpoint method for online travel time prediction and the Coifmanmethod for offline travel time estimation. Three shock wave configu-rations are assessed with each method so as to consider representativetraffic dynamics situations. TTT prediction–estimation errors are calcu-lated and expressed as a function of sensor spacing and probe vehiclepercentage. Optimal sensor spacing is calculated with consideration ofthe tradeoff between TTT estimation error and sensor deployment cost.The results from this study can provide simple and effective support fordetector placement and probe vehicle deployment, especially along afreeway corridor with existing detectors. Optimal sensor spacingresults are analyzed and compared for various methods of travel timeestimation during different types of shock waves.

Travel time prediction–estimation accuracy is crucial to a wide rangeof applications, including planning, design, performance measure-ment, operational analysis, traffic management, incident detection,navigation, and traveler information. Data for travel time prediction–estimation can be collected by using fixed sensors, such as loopdetectors, video cameras, microwave detectors, and radar sensors,or from mobile data sources, such as automatic vehicle identification(AVI) toll tags and automatic vehicle location (AVL) probes (1).The motivation for this research was originally provided by an OregonDepartment of Transportation (ODOT) idea to study the impacts ofincreasing the density of the existing sensor network on the accuracy

Department of Civil and Environmental Engineering, Portland State University,P.O. Box 751, Portland, OR 97207-0751. Corresponding author: W. Feng,[email protected].

Transportation Research Record: Journal of the Transportation Research Board,No. 2178, Transportation Research Board of the National Academies, Washington,D.C., 2010, pp. 67–78.DOI: 10.3141/2178-08

Research on optimal sensor spacing for prediction–estimation oftravel time has included empirical methods aimed at minimizingprediction–estimation errors, such as mean absolute error (MAE) orroot mean square error (RMSE), by means of the optimization ofmixed integer programming with genetic solution algorithms (28, 29),dynamic programming optimization with graphical solution algorithms(30, 31), sampling theory (32), and spatial and temporal discretizationmodels (33). With empirical methods the prediction–estimation errorusually has a nonmonotonic relationship with sensor spacing becauseof data variance and fluctuations. Infinitely decreasing sensor spacinghas a limited benefit toward reducing the prediction–estimation error.However, results from these studies only minimize some specificmeasurements such as MAE and RMSE; the tradeoff between prediction–estimation error and sensor density is difficult to balance.

Analytical methods for optimal sensor spacing express prediction–estimation error as a function of spacing. One study empirically andanalytically calculated over- and underestimation error by meansof the midpoint method during queue formation and dissipationby using field data, but estimation errors under various sensorspacings were not provided (26). Previous studies (1, 34) used TTT forquantifying travel time prediction accuracy with the midpoint methodin a time–space plane where a traffic state transition occurs. Under-and overpredicted TTT errors and relative errors were computedwhile changing the number of sensors over a freeway link. Penaltyvalues were also assigned to the under- and overpredicted TTTerrors, and the optimal sensor density was found by minimizing totalpenalty error. The cost of sensors was not considered for determin-ing the optimal sensor spacing. This paper, as an extension of aprevious paper (1), considers spacings and costs of the sensors andfurther extends to offline travel time estimation and probe vehiclepenetration rate.

FRAMEWORK FOR TRAVEL TIME ESTIMATION

An assumed fundamental traffic flow–density relation is shown inFigure 1a (1). The congested state C (flow qc and speed vc) and theuncongested states A, B, D, and E (flows qa, qb, qd, and qe and speed vf)are marked.

Figures 1b and 1c show two time–space planes containing two typesof bottlenecks (bn), either recurrent or nonrecurrent, on a freewaycorridor downstream of link l. In macroscopic traffic dynamic theory,a transition exists between the uncongested state A and the congestedstate C, defined by a shock wave speed vAC. Figures 1b and 1c showthat, for an arbitrary highway link l (separated by two dashed lines),transition AC is bounded by a rectangle as the shock passes. Afterthe recovery begins at time tr, transitions CD and CE propagate throughlink l between the congested state C and the uncongested states Dand E, marked by backward-moving and forward-moving recoverywaves of velocities vCD and vCE, respectively. Shock wave CD happensafter recovery begins because of an increase in supply, and shockwave CE happens after recovery begins because of a decrease indemand. Transitions DA and BE are separated by a forward recoverywave with speed vf.

IMPACTS OF SENSOR SPACING ON TRAVEL TIME PREDICTION AND ESTIMATION

In this section, the midpoint method and the Coifman method areused to calculate the predicted and estimated TTT during shockwave regime AC; for brevity, calculations of regimes CD and CEare not shown.

68 Transportation Research Record 2178

Online Travel Time Prediction with MidpointMethod in Shock Wave Regime AC

Figure 2a illustrates transition AC from an uncongested state withvehicles traveling at vf, to a congested state with vehicles travelingat vc. The backward-forming shock wave AC passes through a linkof length l, with sensor spacing s, at a speed vAC. Vehicles betweenj1 and j3 pass through the link with speeds from vf to vc. The solidlines are actual vehicle trajectories, and the dashed lines are predictedvehicle trajectories. The idea of the midpoint method used for onlinetravel time prediction is that, when a detector located at the midpointin the link measures a speed, it is assumed that vehicles arrivingat the upstream of link l will traverse the entire link at that speed.Therefore, before time tc, vehicles arriving at the upstream end ofthe link would be predicted to traverse the entire segment at constantspeed vf, and vehicles arriving after time tc, with constant speed vc.Travel times of vehicles from j1 to j2 are underpredicted and vehiclesfrom j2 to j3 are overpredicted.

This method of prediction relates to the real-world situation of aVMS (Figure 2) providing travel times for the following section ofroadway. Obviously, this method would be limited to the detectorinformation available at the time a vehicle enters the link. This canbe contrasted with offline estimation of travel time, which can usedetector information from the time that the vehicle has alreadyentered the link. The midpoint method applied for estimation instead

AB C

D

E

qA

qC

qE

vfvc

vAC vCD

vCE

q

k

(a)

x

t

B D

A AC

CD

A

C

vAC

vCD

DA vf

l

tr

bn

vms

(b)

x

t

B

A AC E

vAC

vf

C

CE

vCE

l

tr

bn

vms

(c)

FIGURE 1 Travel time prediction–estimationframework.

of prediction would yield different travel times for vehicles arrivingbetween te and tc in Figure 2a.

The TTTs of under- and overpredicted vehicles are calculated bysumming their predicted or actual travel times. They can be expressedas functions of traffic parameters and detector spacing. When there isonly one sensor in a link l, as calculated in Equation 1, the under- andoverpredicted TTT and actual TTT can be expressed as follows:

where

qA � = number of vehicles underpredicted;

qA � = number of vehicles overpredicted;

= free-flow travel time as vehicle j1;

= congested travel time as vehicle j3;s

vc

s

vf

s

v2

−( )AC

s

v

s

vf

+−( )

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥2

AC

TTTpredAC

oA

c

q

s

v

s

v=

−( )i i2 4( )

TTTactAC

ACoA

c

c f

c f

q

s

v

s

v

s v v v

v v=

−( ) ++ −

i i i2 1

2 2

2

−−( )⎡

⎣⎢

⎤

⎦⎥vAC

( )3

TTTpredAC

uA

f f

qs

v

s

v

s

v= +

−( )

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥i i2 2( )

TTTactAC

uA

f f

cqs

v

s

v

s

v

s v= +

−( )

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥ +i i i2 1

2 2

++ −−( )

⎡

⎣⎢

⎤

⎦⎥

v v

v v vf

c f

21AC

AC

( )

Feng, Bigazzi, Kothuri, and Bertini 69

= actual travel time of vehicle j2, obtained from

with x solved by

TTTu = total travel time for underpredicted vehicles,and

TTTo = total travel time for overpredicted vehicles.

These apply for both actual (act) and predicted (pred) travel times.With changing detectors and traffic conditions, these values willchange not only because of changing predictions but also becausedifferent vehicles fall into each category (under- and overpredicted).

When there are n (= l/s) sensors in a link of length l as shown inFigure 2b, vehicles that enter the upstream of link l before time t1 arepredicted to traverse l with speed vf. Vehicles that enter the upstreamof link l after time t1 (when the most-downstream detector senses theshock wave) are predicted to traverse the distance (n − 1)s withspeed vf and the last s with speed vc; vehicles that enter the upstreamlink l after time t2 are predicted to traverse the distance (n − 2)s withspeed vf and the last 2s with speed vc, and so on. Vehicles that enterthe upstream end of link l after time tn are predicted to traverse thewhole link l with speed vc and therefore are overpredicted. Fromthese constructions, general expressions of under- and overpredictedTTT and actual TTT are shown below:

TTTactAC AC

uA

f

f

c

ql

v

l

v

s

v

l

v

s v

= − +⎛⎝⎜

⎞⎠⎟

+

i

i i

2

1

2 2

++ −−( ) +

−( )⎡

⎣⎢

⎤

⎦⎥

v v

v v v

n s

vf

c f c

2 15AC

AC

( )

x

v

sx

vf

=−⎛

⎝⎜⎞⎠⎟

−( )2

AC

x

v

s x

vf c

+ −⎛⎝⎜

⎞⎠⎟

s v v v

v v vc f

c f2

2i

+ −−( )

AC

AC

(b)

vAC

t1 t2ttf t3 tn

s

L=nsvf

vc

0vms

(a)ttf

tc

vf

vc

vAC under over

t

x j1 j2 j3

te

L=s

xvms

FIGURE 2 Online travel time prediction in regime AC.

where

TTTuact = underpredicted actual TTT,

TTTupred = underpredicted TTT,

TTToact = overpredicted actual TTT, and

TTTopred = overpredicted TTT.

A hypothetical link of unit length l = 1 mi will be considered forcomparison purposes. For the sake of numerical examples, a rangeof sensor spacing (s = 0.1 to 1 mi) will be used, and assumed trafficflow parameters will be qa = 2,000 vph, qc = 1,800 vph, vf = 60 mph,vc = 30 mph, vAC = −7.5 mph, vCD = −17.1 mph, and vCE = 6 mph. Thismethod could be replicated by using other assumptions.

Table 1 and Figure 3 show the impacts of sensor spacing on pre-dicted TTT and TTT errors. As sensor spacing decreases towardzero, TTTo

pred converges to zero but TTTupred approaches 7.22, still a

4% underprediction. This prediction error with zero spacing is causedby (a) the time lag between arrival of the first vehicle at the upstreamend of link l and arrival of the shock wave AC at the downstreamend of link l and (b) the difference in the shock location between whena vehicle enters the link and when it reaches the shock (a function

TTTpredAC

oA

c

q

s

v

l

v=

−( )i i2 8( )

TTTactAC

ACoA

c

c f

c f

q

s

v

l

v

s v v v

v v=

−( ) ++ −

i i i2 1

2 2

2

−−( ) +−( )⎡

⎣⎢

⎤

⎦⎥v

n s

vcAC

17( )

TTTpredAC AC

uA

f fA

fAq

l

v

l

vq

s

v

l

vq

l

v

n

= − −

−

i i i i i

i

2

11

2

1 16

( )+

⎛⎝⎜

⎞⎠⎟

s

v vf c

i ( )


of the shock wave speed and the free-flow speed). Even with exactknowledge of traffic conditions in the link when the vehicle enters,changing conditions lead to a prediction error.

The change in actual over- and underpredicted TTT with differentdetector spacing is due to changes in which vehicles’ travel timesare over- or underpredicted. With shorter spacing, more vehicles areunderpredicted, though the TTT for all vehicles remains constant.The absolute TTT error reflects the total size of errors without dis-tinguishing over- and underprediction and decreases nonlinearly to4% as sensor spacing decreases toward 0 (due to the time lag andtraffic dynamics discussed earlier). Additive TTT error reflects thebias in the estimates by weighing overprediction against under-prediction error and remains constant with shorter spacing due to theoffsetting errors of under- and overpredicted TTT. Similar analysesof regimes CD and CE yielded different results but are not shownhere for brevity.

Offline Travel Time Estimation with CoifmanMethods in Shock Wave Regime AC

The Coifman method can be used offline to estimate travel timeswith data from either or both upstream and downstream sensors(3). This is a distinct process from the prediction performed in theprevious section, and it uses data after the vehicles have passedthrough the link. Three variations were used to estimate TTT duringtransitions:

Coifman 1. Sensor at upstream end of link; estimate travel timeover downstream link.

Coifman 2. Sensor at downstream end of link; estimate traveltime over upstream link.

Coifman 3. Sensor at middle of link; estimate travel time overhalf of downstream and half of upstream link.

TABLE 1 TTT Prediction Performance Measures in Regime AC with Midpoint Method

Underprediction (before tc) Overprediction (after tc) Total Prediction

TTT/mi TTT/mi Under % TTT/mi TTT/mi Over % TTT/mi TTT/mi Additive Absolutes (mi) Pred (h) Act (h) Errora Pred (h) Act (h) Errorb Pred (h) Act (h) % Errorc % Errord

1.00 2.78 3.55 22 4.44 3.95 −13 7.22 7.50 4 17

0.50 5.00 5.40 7 2.22 2.10 −6 7.22 7.50 4 7

0.33 5.74 6.07 5 1.48 1.43 −4 7.22 7.50 4 5

0.25 6.11 6.42 5 1.11 1.08 −3 7.22 7.50 4 5

0.10 6.78 7.06 4 0.44 0.44 −1 7.22 7.50 4 4

0 7.22 7.50 4 0.00 0.00 0 7.22 7.50 4 4

aUnder % error =

bOver % error =

cAdditive % error =

dAbsolute % error =TTT TTT TTT TTT

TTT TT

act act pred

act

upredu o o

u

− + −

+ TTacto

TTT TTT TTT TTT

TTT

act pred act pred

act

u u o o−( ) + −( )uu o+ TTTact

TTT TTT

TTT

act pred

act

o o

o

−

TTT TTT

TTT

act pred

act

u u

u

−

Coifman 1 Estimate Downstream

The Coifman 1 method estimates each vehicle’s downstream trajectory by adding subsequent vehicles’ trajectories. As shownin Figure 4a, during the uncongested state, for vehicles that departbefore time T1, each vehicle’s trajectory is estimated by adding allthe subsequent trajectories (calculated by Σi vi ti, vi is the detectedsubsequent speed, ti is the headway between two detected speeds) inthe t–x plane until the estimated trajectory (e.g., j1) reaches the endof the link. While in the congested state, for vehicles that enter theupstream of link after time T2, each vehicle’s trajectory is estimatedby adding the subsequent chords that are truncated as soon as theyreach the next observed shock wave signal (calculated by Σi vi′ti′)until the estimated chord reaches the end of the link. Travel timesfor vehicles that cross the upstream sensor before time T1 are under-estimated with the same travel time l/vf; travel times for vehiclesthat cross the upstream sensor after time T2 are estimated withouterror. For those vehicles that cross the upstream sensor betweentimes T1 and T2, travel times are first estimated in the uncongestedstate with speed vf and then in the congested state with speed vc; thus,each estimated vehicle travel time in this time period is different,and the estimated TTT for these vehicles can be calculated by anintegral. Because there is no overestimated error in this case, theunderestimated TTT and actual TTT for all vehicles can be calculatedas follows:

TTTestAC

uA

fA

v

f

qv

l

vq

l

vtf= −⎛

⎝⎜⎞⎠⎟

+ −⎛⎝⎜∫i i i

10

1 ⎞⎞⎠⎟

+− −

⎛⎝⎜

⎞⎠⎟

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

= −

11

vv

t

vdt

ql

v

ff

c

Ai

AC

⎛⎛⎝⎜

⎞⎠⎟

+ +⎛⎝⎜

⎞⎠⎟

iil

v

q l

v

v

vf

A

f

f

c

2

221 10( )

TTTactAC

uA

f f c

ql

v

l

v

l

v

l

v= −

⎛⎝⎜

⎞⎠⎟

+⎛⎝⎜

⎞⎠⎟

i i1

29( ))


Similarly, as shown in Figure 4b, when there are n sensors in the unitlength link l, the n boxes along shock wave AC have shapes similar tothose in Figure 4a. The general expressions of Equations (9) and (10)can be shown as Equations (11) and (12) (for brevity, equations forregimes CD and CE are not shown):

Coifman 2 Estimate Upstream

The Coifman 2 method estimates upstream link travel times byadding previous vehicles’ trajectories. As shown in Figure 4c,each vehicle chord is estimated by adding the truncated free-flowspeed vf and the truncated congested speed vc. Here, the estimatedTTT is equal to the actual TTT, which means that there is no TTTestimation error in regime AC when the Coifman 2 method is used.While estimated TTT in regimes CD and CE are not equal to theactual TTT, calculations of regimes CD and CE are omitted forbrevity.

Coifman 3 Estimate Half Upstream and Half Downstream

Because the Coifman 3 method estimates half upstream and halfdownstream when the sensor is located in the middle of the link,this method is equal to estimating the downstream half by the

TTTestAC

u A

f

f

c

A

f

nq s

v

v

v

q s

v v= +

⎛⎝⎜

⎞⎠⎟

−⎡

⎣i

i 2

2

2

21⎢⎢

⎤

⎦⎥

+−( )

−⎛⎝⎜

⎞⎠⎟

+⎛⎝⎜

⎞⎠

n nq

s

v

s

v

s

v

s

vAf f c

1

2i i

AC⎟⎟

⎡

⎣⎢

⎤

⎦⎥ ( )12

TTTactAC

uA

f f c

q ns

v

s

v

s

v

s

v= −

⎛⎝⎜

⎞⎠⎟

+⎛⎝⎜

⎞⎠

i i i2 1

2 ⎟⎟ ( )11

Add error %

Actual TTT

Abs error %

Under predicted TTT

Predicted TTT

Over predicted TTT

4% 4% 5% 5% 7%

0.00

00%

10%

20%

30%

TT

T E

rro

r/m

ile40%

50%

60%

-6

17%

-3

2.78

0 TT

T/m

ile

4.44

3

67.227.50

9

0.2 0.4

Sensor Spacing (miles)0.6 0.8 1

0.441.11

1.482.22

7.22 6.786.11

5.745.00

FIGURE 3 TTT prediction errors with midpoint method in regime AC.


(a)

(b)

(c)

(d)

T1T2

l = s

vf

vc

vAC

j1

t1t2 t3t1

't2't3

'

j3

vms

vAC

s = l/n

n

n(n-1)/2

n(n-1)/2

vms

vAC

vf

l = s

vms

vc

l = s

vf

vc

vAC

s/2Coifman 1

Coifman 2

vms

FIGURE 4 Coifman methods: (a) Coifman Method 1 estimates downstream traffic inregime AC with one detector, (b) Coifman Method 1 estimates downstream traffic inregime AC with multiple detectors, (c) Coifman Method 2 estimates upstream traffic in regime AC with one detector, and (d ) Coifman Method 3 estimates half upstream and half downstream traffic in regime AC with one detector.

Coifman 1 method and the upstream half by the Coifman 2 method(see Figure 4d). Formulas for estimated and actual TTT for regimesCD and CE are omitted here for brevity.

Tables 2 through 4 and Figure 5 show the effects of sensor spacingon the estimated TTT and TTT errors with the Coifman 3 method.

Because there is no overestimated TTT error, the estimated TTTis the underestimated TTT. As sensor spacing decreases, the esti-mated TTT in regime AC + CD (here AC + CD means either the sumof the under- or overestimated TTT or the actual TTT in regime ACand CD, respectively) increases from 10.85 to actual TTT 11.63.The additive error and the absolute error are the same due to nooverestimated error; they linearly decrease from 7% to 0 as the sensorspacing decreases to 0.


OPTIMAL SENSOR SPACING

As described earlier, higher sensor density on a certain link leads tomore-accurate TTT prediction–estimation but involves higher sen-sor construction cost. Therefore, it is desirable to find an optimalsensor spacing that balances the TTT prediction–estimation errorsand sensor construction cost.

To solve this problem, an objective function can be proposed tominimize the total cost of sensor construction and TTT predictionor estimation error. TTT errors that reflect the social mistrust aboutVMS information systems or performance measurement unreliabilityfor traffic management are converted into dollars to retain consistentunits with the sensor cost. Two cost coefficients are created here toconvert the TTT errors into dollars. The function is proposed as

under the following constraints:

0 < ≤

−( )<

−

s l

u u

u

o

TTT TTT

TTT

TTT TT

acu pred

acu

pred

�

TT

TTTacu

acu

o

o

( )<

⎧

⎨

⎪⎪⎪

⎩

⎪⎪⎪ �

min TTT TTT TTT TTTacu pred pred aC C Cuu u

oo= −( ) + −i i

ccuo

dCl

s( ) + ⎛

⎝⎜⎞⎠⎟

i

( )13

TABLE 3 TTT Performance Measures in Regimes CD with Coifman Method 3

Overestimation Before tr Underestimation After tr Total Estimation

TTT/mi TTT/mi Over % TTT/mi TTT/mi Under % TTT/mi TTT/mi Additive Absolutes (mi) Pred (h) Act (h) Error Pred (h) Act (h) Error Pred (h) Act (h) % Error % Error

1.00 0 0 — 3.91 4.13 5 3.91 4.13 5 5

0.50 0 0 — 4.02 4.13 3 4.02 4.13 3 3

0.33 0 0 — 4.06 4.13 2 4.06 4.13 2 2

0.25 0 0 — 4.08 4.13 1 4.08 4.13 1 1

0.10 0 0 — 4.11 4.13 1 4.11 4.13 1 1

0 0 0 — 4.13 4.13 0 4.13 4.13 0 0

NOTE: — = data not applicable.

TABLE 2 TTT Performance Measures in Regimes AC with Coifman Method 3

Underestimation Before tc Overestimation After tc Total Estimation

TTT/mi TTT/mi Under % TTT/mi TTT/mi Over % TTT/mi TTT/mi Additive Absolutes (mi) Pred (h) Act (h) Error Pred (h) Act (h) Error Pred (h) Act (h) % Error % Error

1.00 6.94 7.50 7 0 0 — 6.94 7.50 7 7

0.50 7.22 7.50 4 0 0 — 7.22 7.50 4 4

0.33 7.31 7.50 2 0 0 — 7.31 7.50 2 2

0.25 7.36 7.50 2 0 0 — 7.36 7.50 2 2

0.10 7.44 7.50 1 0 0 — 7.44 7.50 1 1

0 7.50 7.50 0 0 0 — 7.50 7.50 0 0

NOTE: — = data not applicable.

TABLE 4 TTT Performance Measures in Regimes ACand CD with Coifman Method 3

Under % Over % Additive Absolutes (mi) Error Error % Error % Error

1.00 7 0 7 7

0.50 3 0 3 3

0.33 2 0 2 2

0.25 2 0 2 2

0.10 1 0 1 1

0 0 0 0 0

where

Cu = cost coefficient of underpredicted–underestimated TTT error[$/vehicle hour (veh h)],

Co = cost coefficient of overpredicted–overestimated TTT error($/veh h),

Cd = sensor cost per shock wave [construction and maintenancecost of each sensor in its life cycle divided by number of shockwaves over this time period ($/sensor/shock wave)], and

� = tolerance of relative TTT error.

To solve this function, one can first cut the feasible region bysolving the constraints and then substituting the predicted–esti-mated and actual TTT in the objective function with the formulasdescribed earlier. Next, all parameters Cu, Co, Cd, l, qA, qC, qE, vf ,vc, vAC, vCD, and vCE are treated as a combined constant, isolatingthe decision variable s. Then, by taking first and second deriva-tives, the optimal sensor spacing s* can be calculated. Equation14 is the optimal sensor spacing solution in combined regime AC+ CD (sum of the under- and overestimated TTT errors of bothregimes in the same objective function, with an assumption madehere that all forming shock waves are AC and all recovering shockwaves are CD, not CE, in the life cycle) by using the Coifman 1method:

where s* is the optimal sensor spacing without constraint and

where s** is the optimal sensor spacing combined with constraints.Similar results for regimes AC + CD and AC + CE are calculated

with other Coifman methods for travel time estimation.Equation 14 shows that the optimal sensor spacing is independent

of the link length and only depends on the ratios Co /Cd and Cu /Co,and on traffic parameters. The value of sensor cost coefficient Cd canbe adjusted by decision makers according to different situations;it varies according to the type of sensor, the life of the sensor, themaintenance frequency, and the number and type combinations ofshock waves crossing over a certain period. In addition, the values

s l s** min , *,= marginal value determined by connstraints[ ] ( )15

sC v v v

c q v vd f c

u A c f

* ( )=−( )

214AC


of the cost coefficients Cu and Co vary according to the diversityamong trips at different times of day, trip duration, and trip purpose.Thus, to evaluate reasonably the cost coefficient values is beyond thescope of this research. However, a possible range of each coefficientcould be estimated so that a sensitivity analysis can be performed bychanging the ratio Co/Cd.

Figure 6 shows the TTT estimation error and sensor cost pershock wave in the combined regime AC + CD by using the Coifman1 method, under the assumption that the ratio Co /Cd is equal to 1(which assumes a $4/veh h overestimation error, $10,000 sensorconstruction and maintenance cost for a 10-year life cycle, and250 shock waves per year at a recurrent bottleneck location) andCu = 3Co (1). All parameters are consistent with the above for a unitlength l = 1 mi, and the marginal spacing value determined by theconstraints is 0.25 mi with a tolerance � = 10%. Figure 6 shows that,when sensor spacing is around 0.5 mi, the total cost per shock wavehas a minimum value of about $12.44. But because 0.5 mi is outsidethe constraints (estimation error is 14.81%), the optimal sensorspacing in this case remains 0.25 mi (where � = 10%). This spacingseems small because of the high underestimated TTT error in regimeAC + CD by the Coifman 1 method. This result also shows that

FIGURE 5 TTT estimation errors with Coifman Method 3 in regimes AC � CD.

Sensor Spacing (miles)

To

tal C

ost

($

per

sh

ock

wav

e)

FIGURE 6 Total cost per shock wave by Coifman Method 1 inregime AC � CD.

decreasing sensor spacing in regime AC by using the Coifman 1method has limited benefits for improving TTT estimation accuracy.

Figure 7 shows the optimal sensor spacing with a varying Co/Cd

ratio for regimes AC + CD and AC + CE by using all travel timeestimation methods (under the assumption that Cu = 3Co and toler-ance � = 10%) and by considering the constraints. All parameters areconsistent with the above.

Figure 7 also shows that, for all values of the Co/Cd ratio, the opti-mal sensor spacing for travel time estimation calculated by the Coif-man 1 method is relatively short because of the high underestimatedTTT error.

Figure 7a shows that, for most parts of the range of the Co/Cd ratios,the optimal sensor spacing calculated from the objective functionis within the constraints. The optimal sensor spacings calculated bythe Coifman 2 and 3 methods are larger than those calculated by theCoifman 1 method, which means that the lower TTT estimation errorby the Coifman 2 and 3 methods are obtained in regime AC + CD.A different set of optimal sensor spacings is found in regime AC + CE,as shown in Figure 7b. The Coifman 2 method generates the largestoptimal sensor spacing, followed by the Coifman 3 and 1 methods.In both Figures 7a and 7b, the Coifman 1 method generates theshortest optimal sensor spacing, which indicates that, in these two combined regimes, the travel time estimated by the Coifman 1method has the largest error. This finding is consistent with theresults tested on the basis of the field data in Coifman’s originalpaper (3).


IMPACTS OF PENETRATION RATE OF PROBEVEHICLE ON TRAVEL TIME PREDICTION

Probe vehicles are another data source for travel time prediction–estimation, often recorded by a Global Positioning System (GPS)device in the vehicles. Today, with the development of enhancedcommunications technologies, GPS installed in smart phones can pro-vide useful AVL data on transportation networks, as long as a systemis established to gather, organize, and process these data efficiently.By using information from probe vehicles running between detectorsalong with detector data, more accurate travel times can be predictedor estimated.

From the theoretical framework described earlier, the relation-ship between accuracy of travel time prediction and probe vehiclepercentage will be developed analytically.

The calculation of predicted TTT in regime AC can be describedas follows. First, the information from probe vehicles only is usedto predict TTT. Second, when any probe vehicle encounters the shockwave signal in its trajectory reported by a change of speed, time andlocation can be recorded; simultaneously, this information is trans-ferred to the vehicles that are entering the upstream edge of the link,and therefore these entering vehicles are predicted to travel (a) withspeed vf until the position where the shock wave signal was detectedby the probe vehicle and (b) with speed vc for the remainder of the link.

Figure 8 shows the assumed vehicle trajectories in shock waveregime AC with uniformly distributed probe vehicle headways

(a) (b)

FIGURE 7 Optimal sensor spacing (a) regimes AC � CD and (b) regimes AC � CE.

vf vc

vAC

l

t

xj1 j2 j3

h h hhh h h

t t t t t t t t<=t

j4 j5 j6 j7 j8

l

ttf

m

x h<=h

t1

vms

FIGURE 8 TTT prediction with probe vehicles in regime AC.

(formulas for exponentially distributed probe vehicle headways areomitted for brevity).

The variables shown in Figure 8 are explained as follows:

h = assumed evenly distributed headway of probe vehicles;μ = percentage of probe vehicles among all vehicles, 0∼100%;t = time gap between probe vehicles reporting shock wave

signals;x = time that the first probe vehicle enters the upstream edge of

the link; because of the uniformly distributed probe vehicleheadway, x could be any value in the range 0 ≤ x ≤ h with auniform probability distribution;

m = distance traveled by the first probe when it reports detectionof the shock wave signal;

Δh = difference between the time that the last probe enters theupstream edge of the link and the time that the shock wavesignal passes the upstream edge of the link, 0 ≤ Δh ≤ h;

Δ t = difference between the time that the last probe reports ashock signal detection and the time that the shock wavesignal passes the upstream edge of the link, 0 ≤ Δ t ≤ t;

Δl = distance between the positions where two adjacent probevehicles report detection of shock wave signals, 0 ≤ Δ l ≤ l;

t1 = time difference between the time that the shock wave sig-nal passes the downstream edge of the link and the time thatthe first probe vehicle reports detection of the shock wavesignal;

N = number of probe vehicles within the study time period plusone (green boxes); and

ttf = free-flow travel time for vehicles to pass the entire link l withspeed vf.

Other variables are consistent with the earlier discussion.To calculate predicted TTT, all these variables have to be either

transferable to the single variable μ with traffic parameters (flow andspeed) or canceled in the final TTT equation; thus the connectionsbetween these variables, the variable μ, and traffic parameters canbe expressed as follows:

If x = 0, then t1 = 0; m = l, then

Then even when x ≠ 0, Equation 18 still holds. Then

m

l

lv

t

tv

tm

vx

m lv

ff

=

− −

−

= + −

⎧

⎨

⎪⎪⎪

⎩

⎪⎪⎪

⇒ = +

AC

AC

tt

1

1

ff

f

v

v vxAC

AC−i ( )20

Δ tm

vn t= − − −( )

AC

1 19( )

∵i

itt andAC

ff

f

ff

l

vt

h v

v vl h t v= ∴ =

−= −( )Δ ( )18

t h v vl

v

l

vt lf f

ff+ −( ) = − − − +( )⎡

⎣⎢

⎤

⎦⎥ = −tt ttAC

AC

i i ΔΔt ( )17

hqA

= 116

i μ( )


With the definitions and connections between variables and trafficparameters, the predicted TTT can be calculated (because x could beany value between 0 and h with uniform probability, the predictedTTT is a mathematical expectation):

Because m is a function of x, m can be substituted with x, the integrandtied up, and then the integral solved:

Figure 9 shows the impacts of the penetration rate of the probevehicle on TTT prediction accuracy.

Figure 9 also shows that the predicted TTT approaches the actualTTT with an increasing percentage of probe vehicles but with limitedimprovement at larger penetration rates. In regime AC, the reasonsare the shock wave time lag (between the time that the first vehiclearrives at the upstream end of link l and the time that shock waveAC arrives at the downstream end of link l) and the movement of theshock wave upstream between its detection and its being encounteredby the entering vehicle.

The optimal percentage of probe vehicles is not resolved in thispaper, but by attaching cost coefficients to the TTT prediction errorand probe vehicle cost, an objective function for minimizing the totalcost of TTT prediction error and probe vehicles can be proposed,and an optimal solution can be computed with methods similar tothose described earlier. In addition, TTT prediction using infor-mation from both sensors and probe vehicles can be formulated.The relationship between TTT prediction error, sensor spacing, andthe penetration rate of the probe vehicles can be established, and anoptimal combination of penetration rate for probe vehicles andsensor spacing can be solved by aiming at minimizing the total cost

TTTpredAC

AC

uA

f

f f c

c f

ql

vh

v v v v

v v v= +

−( )−( )

i i2 3

2ii

iih

v v l

v v vh

v v v

f c

c f

f c

2

2

1

2

1 1

⎡

⎣⎢⎢

+−( )

−( )

+ −

AC

ACC

AC

AC AC

⎛⎝⎜

⎞⎠⎟ −

⎛⎝⎜

⎞⎠⎟

−−( )

i i i

i

lv v

v vh

N l

vf

f

1

2

Δ

11 1

v v

v v

v vh

v v

c f

f

f

f c−⎛⎝⎜

⎞⎠⎟ −

⎛⎝⎜

⎞⎠⎟

+−( )

i ii

AC

AC

ll

v v v v v v

lN l

v

c f f c

2

2

2

1 1

1

AC AC

AC

+ −⎛⎝⎜

⎞⎠⎟

−−( )

i iΔ 11 1 1 2

2

1 1

v vl

N N l t

v v

c f

c f

−⎛⎝⎜

⎞⎠⎟

+−( ) −( )

−⎛⎝⎜

ii

i

Δ

⎞⎞⎠⎟

− −( ) −⎛⎝⎜

⎞⎠⎟

⎤

⎦⎥N l t

v vc f

11 1

232

i i iΔ ( )

TTTpredu

A

h

f f f f

qdx

hx

l

v

m

v

l

vt

m

v

l m

v= + + + −∫0

i i i i i

cc

f c

m l

v

l m l

v

m l

v

⎛⎝⎜

⎞⎠⎟

⎡

⎣⎢

⎧⎨⎩

+ − + − +⎛⎝⎜

⎞⎠⎟

+ −Δ Δ Δ2

ff c

f

l m l

v

m N l

v

l m N

+ − +⎛⎝⎜

⎞⎠⎟

+ +− −( )

+− + −( )

Δ

Δ. . . 2 2 ΔΔΔ

Δ Δ

l

vt

m N l

v

l m N l

v

c

f c

⎛⎝⎜

⎞⎠⎟

⎤

⎦⎥ +

− −( )+

− + −( )i

1 1⎡⎡

⎣⎢

⎤

⎦⎥⎫⎬⎭

( )22

N

lv

lv

hf=

−⎛⎝⎜

⎞⎠⎟

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥ ( )AC rounding up (21))

of TTT error, sensors, and probe vehicles through an approach thatoptimizes nonlinear programming.

CONCLUSION

This paper established analytical relationships among travel timeprediction–estimation measurements (by using TTT as a primarymetric), influencing variables (sensor spacing and percentage of probevehicles), and assumed traffic parameters (flow and speed in eachtraffic state) over several representative types of shock wave regimeson the basis of a simplified triangular flow–density relationship. Forease of establishing these functions, the midpoint method was usedfor online prediction of travel time on the basis of data from sensorsfor the purpose of traveler information; the Coifman method wasused for offline estimation of travel time on the basis of data fromsensors for the purpose of managing traffic performance; and a simplemethod based on probe vehicle trajectories was used for online predic-tion of travel time that used data from probe vehicles. Predicted andestimated TTT values were expressed as a function of sensor spacing,probe vehicle percentage, and traffic parameters such as flow andspeed in each traffic state. Numerical examples were tested to revealthe impacts of sensor spacing and probe vehicle percentage on TTTprediction–estimation accuracy.

Optimal sensor spacing was calculated for each shock wave regimeon the basis of minimizing both the total cost of TTT estimationerror and sensor cost within certain constraints. Comparisons amongthe three Coifman methods for estimating travel time were performed,and the result showed that (a) the Coifman 3 method generatedthe largest optimal sensor spacing in regime AC + CD and (b) theCoifman 2 method generated the largest optimal sensor spacing inregime AC + CE.

The results shown here are based on a triangular flow–densityframework (1) and other assumptions, such as no failure or systemerror for sensors or probes and no time delay for processing sensoror probe data. In reality, temporal and spatial speed variance, sensorand probe data noise, transformation delay, and inconsistent shockwave speeds would lead to deviations from the results shown earlier.Incorporating all the uncertainties is difficult, but some assumptionscould be relaxed with further efforts. Despite the assumptions, theseresults can provide simple and effective support for placement of


detectors and deployment of probe vehicles. For example, becausea set of detectors along northbound I-5 in Portland, Oregon, exists(with mean spacing greater than 1 mi), the number of additionalsensors to be added into each current link along the corridor can becalculated according to different traffic conditions; considerationsfor probe vehicles can also be included. These results will providedifferent sensor spacings for links between existing detectors alonga corridor, which means that optimal sensor positions can be foundby considering existing sensors instead of using uniform spacing fora whole corridor.

ACKNOWLEDGMENTS

The authors gratefully acknowledge Galen McGill, Dennis Mitchell,and Jack Marchant of ODOT for support of this freeway travel timeestimation research. Kristin Tufte, Benjamin Zielke, Rafael Fernández,and Miguel Figliozzi of Portland State University assisted in devel-opment of this work. The authors are also grateful to David Lovell,Arne Kesting, and previous anonymous reviewers for their helpfulcomments and criticisms.

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Error (Add)

Error (Abs)

Predicted TTT

Actual TTT

25%

20%

15%

TT

T E

rro

r/m

ile10%

5%

0%0% 2% 4%

Probe Vehicle Penetration Rate6% 8% 10%

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T/m

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The Freeway Operations Committee peer-reviewed this paper.

freeway sensor spacing and probe vehicle penetration

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