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Journal of the Eastern Asia Society for Transportation Studies, Vol.7, 2009

The Delay Estimation under Heterogeneous Traffic Conditions

Chu Cong MINHD.Eng., LecturerDepartment of Bridge and Highway

EngineeringSchool of Civil EngineeringHochiminh city University ofTechnology268 Ly Thuong Kiet street, District 10Hochiminh city, VietnamFax: +84-58-524241Email: [email protected]

Tran Hoai BINHGraduate StudentDepartment of Bridge and Highway

EngineeringSchool of Civil EngineeringHochiminh city University of Technology268 Ly Thuong Kiet street, Dist. 10Hochiminh city, VietnamFax: +84-58-524241Email: [email protected]

Tran Thanh MAIUniversity graduateHanoi University of TechnologyNo.1, Dai Co Viet Street,

Hanoi, VietnamFax: +84-58-3524241Email: [email protected]

Kazushi SANOAssociate ProfessorSchool of Civil and EnvironmentalEngineering

Nagaoka University of TechnologyKamitomioka1603-1, Nagaoka Niigata940-2188, JapanFax: +81-258-47-9650E-mail: [email protected]

Abstract: Delay is an important factor in the optimization of traffic signals and the

determination of the level of service of a signalized intersection. The paper proposes the

modified Websters formula to estimate the delay of vehicles at pre-timed signalized

intersections under mixed traffic conditions. Traffic volumes approaching signalized

intersections are classified into four groups: Motorized two-wheeler, Car, Minibus, and Bus.The passenger car unit is estimated by using the multiple regression analysis. Distributions of

the saturation flow are developed and proved to follow the normal distributions at three

studied intersections. The expectation function method and Taylor series expansion are then

utilized to estimate the mean and variance of delay. A comparison between the proposed

model and (i) the conventional Websters model and (ii) observed delays are performed to

present the improvements of this model.

Key Words:Delay, traffic operation, mixed traffic

1. INTRODUCTION

A system of signalized intersections is a critical element in the smooth operation and control

of traffic streams in a transportation network. Vehicles approaching a signalized intersection

and stopping at the stop line during red lights produce traffic delays. Traffic delays at

signalized intersections increase travel time, travel costs, as well as reduce speeds of vehicles.

The increase of delay may bring about air and noise pollution. The ability to quantify vehicle

delays at signalized intersections accurately is, therefore, a critical and important component

in planning, designing and analyzing traffic signal controls.

Several studies have been conducted to estimate the traffic delay at signalized intersections inthe existing literature. Most of them were based on homogeneous traffic characteristics. In

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Proceedings of the Eastern Asia Society for Transportation Studies, Vol.7, 2009

developing countries, many types of vehicles including motorized and non-motorized, two-

wheeled and four-wheeled vehicles have the same right of traveling on roadways.

Intersections with heavy traffic are usually controlled by the pre-timed signalized systems.

Various vehicles with different shapes and sizes and different static and dynamic

characteristics cause a variety of lateral and longitudinal distances among them on the

carriageway. The conventional methodologies to estimate delay for homogeneous traffic ifapplied directly may lead to a biased estimation. The proposed methodology deals with the

stochastic characteristics to estimate the delay of vehicles at signalized intersections under

mixed traffic conditions. The saturation flow, which is defined as the maximum number of

vehicles passing the stop line of an intersection approach, is a key factor when computing

delay. Since heterogeneous traffic conditions with various static and dynamic characteristics,

the saturation flow distributions are developed. Then, an expectation function method and

Taylor series expansion are utilized to estimate the mean and variance of delay.

2.LITERATURE REVIEW

Many researchers have conducted delay studies at signalized intersections. Webster (1958)

developed the delay formula, which can be considered as the foundation for most delay

models developed afterwards, as follows:

d = +

)X1(2

)1(C 2]X[

q

c65.0

)X1(q2

X 523/1

2+

(1)

where d : average overall delay per vehicle (sec); C : cycle time (sec); g : effective green time

(sec); q : traffic volume (vehicle/sec); = g/C : proportion of the cycle that is effectively green

for the approach (g/C); X = q/c= q.C/(S.g) : degree of saturation; c = S.: capacity of the lanegroup (vehicle/sec); S : Saturation flow (vehicle/sec).

The first term of the above formula expresses the average delay assumed that all vehicles are

uniform arrivals. The second term represents the additional delay due to random arrivals and

signal time failures. The third term is an adjustment factor to compare theoretical with

practical data for specific field conditions. The model has been commonly used in delay

estimation at signalized intersections for under-saturated traffic conditions. The improvements

of this model were developed by several studies sequentially such as Miller (1963), Hurdle

(1984).

Several studies has focused on developing models to estimate the mean delay but very few

works have been done to quantify the variability of delay or to propose methods of estimating

the variance delay at a signalized approach. Park et al (2003) developed an analytical

methodology that estimated the HCM delay confidence interval (2000). The authors applied

an expectation function method to estimate the variance of the HCM delay. Since the

expectation function method was applicable only to power functions, the HCM delay equation

was approximated by using Taylor Series expansion. Applicable input distributions included

normal, triangle, uniform, lognormal, and gamma distributions. The output was mean,

variance and higher moments of HCM delay, which were used to estimate the HCM delay

confidence interval. Their model was based on the assumed volume distribution and therefore

it might not match actual demand variations. Arasan et al (1995) developed a probabilisticapproach based on first-order second-moment method to estimate the saturation flow and

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delay at signalized intersections under mixed traffic conditions. The different types of vehicles

in traffic streams were divided into five groups, pedal cycles, motorized two-wheelers,

autorickshaws, cars, and buses. To take the random composition of vehicles in heterogeneous

traffic into account, the authors applied a truncated Taylor series to estimate stream parameters

at an intersection approach. The results obtained by the model represented that traffic delays

were better estimated by the probabilistic method than the conventional Websters method atstudied intersections. Akgungor (2008) proposed a methodology and a new formulation to

identify the delay parameters in the delay model. In this approach, random delays were drawn

from total delay which was obtained from simulation or observed in the field. The delay

parameters were calculated by using random delays. The delay model was developed by

means of estimated results. A comparison between the proposed time-dependent model and

four existing models was performed to present the improvements in this model. Olszewski

(1993) developed a numerical method to calculate the average delay and time-dependant

distribution of the average cyclic delay. He stated that for the uniform delay part, the delay

ratio was a function of red time signal and acceleration-deceleration delay was fluctuant. This

model required substantial and computational resources for calculating and storing data and

therefore is not well suited practical situations. Fu and Hellinga (2000) developed an

analytical model to estimate the variance of overall delay, including the variance of uniform

delays and the variance of random delays. The variance of delays was directly calibrated from

the simulation data. A discrete cycle-by-cycle simulation model was used to generate data for

calibrating and validating the model. Because their model was developed on the basis of

simulation data, it might not take actual demand variations into consideration.

3. METHODOLOGY

3.1 Expansion of Taylor Series

If Y = f(X) is a nonlinear function, in which X is the random variable and X is the mean, f(X)can be expanded by Taylor series (Ang et al, 1979) as follows:

Y = f(x) + (X - x)dX

df+ ...

dX

fd)X(

2

12

2

2

X+ (2)

The expectation value of Y = f(X) approximating to the second order can be represented as:

E(Y) f(x) + 2

2

dX

fd)X(Var

2

1(3)

where Var(X) : the variance of X; Var(X) = E(X2) - x

2

The variance of Y = f(X) approximating to the second order can be expressed as:

Var(Y)

2

dX

df)X(Var

- +

2

2

22 )(

4

1

dX

fdXVar +

2

2

3

XdX

fd

dX

df)X(E

2

2

2

4

XdX

fd)X(E

4

1

+ (4)

The expectation function method is an analytic procedure that overcomes shortcomings of

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sampling procedures. It is used to estimate functions, which include random variables. In this

study, the uncertain variable is considered as a random variable following a given distribution.

Tyagi and Haan (2001) developed expectation functions, which are functions of the mean and

coefficient of variance (COV) of an input random variable. The input random variable can be

a uniform, triangular, lognormal, gamma, exponential, or normal distribution. With a priorknowledge of the mean, variance and distribution of the input random variable, the

expectation value of the higher power of a variable can be computed. In this study, because

the degree of saturation of all locations follows normal distributions, the expectation value

under various power of input variable X that follows a normal distribution with the mean

value ofxand the coefficient of variance of CVx is obtained as follows:

]z[ECVn2

r]X[E n2n2

X

2/r

0n

r

X

r =

= where r is an even number

(5)

]z[ECVn2

r]X[E n2n2

X

2/)1r(

0n

r

X

r

=

= where r is an odd number (6)

where, r : Power value of input variable X and a positive integer; z = (X - x)/x; CVx = x/

x; n = r + 1.

3.2 Estimation of Saturation Flow

Saturation flow is defined as the maximum number of vehicles passing through the stop line

of an intersection approach. In heterogeneous traffic conditions, traffic volumes approaching a

signalized intersection are classified into four groups: Motorized two-wheeler (Motorcycles,

Mopeds and Scooters); Car (Passenger cars, jeeps); Minibus (less than 25 seats); and Bus(more than 25 seats). According to Turner and Harahap (1993), the passenger car unit (PCU)

is estimated by a multiple regression analysis. The regression function expresses the

relationship between the saturated green time and the total number of vehicles for all groups

passing the approach. Assumed that the relationship between dependent and independent

variables is linear, the regression function can be represented as follows:

ts = a1n1 + a2n2 + a3n3 + a4n4 (7)

where ts : the saturated green time (sec); a1, a2, a3, a4: Parameters of groups of motorized

two-wheeler, car, minibus and bus, respectively; n1, n2, n3, n4 : Number of vehicles passing thestop line at time t at the subject approach for each group of motorized two-wheeler, car,

minibus and bus, respectively.

In order to estimate the saturated green time, all vehicles passing an approach are counted for

every five-second interval. They are converted into passenger car unit briefly in Table 1. It is

noted that the values in this table are not the result of this study. Those values are used only to

estimate the saturated green time. From the survey, in every five seconds in green time if more

than 3 PCUs passing through the stop line for an approach with 6 (m) width, more than 4

PCUs passing through the stop line for an approach with 7 (m) 7.5 (m) width, more than 5.8

PCUs passing through the stop line for approach with 9 (m) width, and more than 6 PCUs

passing through the stop line for an approach with 12 (m) width, then that time is considered

as the saturated green time. The saturated green time in this study varies from 5(sec) 25 (sec).

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Table 1 Passenger Car Unit (PCU) for Vehicle Groups

Vehicle group PCU

Motorized two-wheeler 0.15

Car 1.0Minibus 2.0

Bus 2.5

Adopted from Vinh (2006)

From the regression analysis, PCU values of each group are obtained by the ratio of the

coefficient of each group to the coefficient of the car group as below:

PCU(Motorized two-wheelers) = a1/a2 (8)

PCU(Minibuses) = a3/a2 (9)

PCU(Buses) = a4/a2 (10)

In mixed traffic, vehicles of different types may cause various headways. Especially,

motorcycles and bicycles move abreast in a single lane, creating a significant number of zero

headways. Therefore, the average headway in the paper is also known as the imaginary

headway. It is defined as the ratio of a saturated green time to the total number of vehicles of

different types which are converted into passenger car unit. The formula for estimating the

average headway is shown as below:

H =44332211

s

pnpnpnpn

t

+++(sec) (11)

where, ts : Saturated green time (s); p1, p2, p3, p4 : PCUs values of motorized two-wheeler,

car, minibus and bus group, respectively; n1, n2, n3, n4 : Number of vehicles passing the stop

line at time t at a intersection approach of motorized two-wheeler, car, minibus and bus group,

respectively.

The saturation flow is then obtained: S =H

3600(PCU/hr) (12)

3.3 Estimation of delayIn Websters formula (1963), the degree of saturation (X) is estimated by the traffic volume

which is a random value. The degree of saturation is the ratio of the traffic volume (q) to

capacity (c). Assuming that at an approach, the green time (g), the cycle time (C) and the

saturation flow (S) are constant; the degree of saturation (X) depends only on the random

variable, the traffic volume (q). Therefore, the degree of saturation (X) is also the random

variable and may follow some distribution, such as a normal distribution, lognormal

distribution, etc. In order to estimate the distribution of the degree of saturation, the degree of

saturation is classified into groups with interval (max min)/(1+3.322log10n), where n :

number of samples; max, min : maximum and minimum values, respectively (Arasan and

Koshy, 2003).

In Websters formula, the average overall delay is computed from three terms. The first term

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expresses the average delay assuming that all vehicles are uniform arrivals. The second term

represents the additional delay due to random arrivals and signal time failures. The third term

is adjustment factor to compare theoretical with practical data for specific field conditions.

Among three, the first term mainly contributes to the average overall delay. In the present

study, all vehicles were assumed uniform arrivals. In order to apply the concept of Taylor-

series expansion, the conventional Websters formula was simplified by removing the secondand third terms. The first term of Websters formula, the average delay to vehicles, is

expressed as:

d=)X1(2

)1(C 2

(13)

where, C : Cycle time (s); X = q/(S): Degree of saturation; = g/C : Green ratio; S :saturation flow (PCU/h); q : Vehicles approaching the subject approach (PCU/h). In other

words, it can be rewritten as: d = f(X). Since the degree of saturation (X) is the stochastic

variable, Taylor series expansion can be applied to the delay estimation. From (2) and (3), theexpectation value of the delay can be written as:

E(d) =2

22

dX

)d(d)X(Var

2

1

)X1(2

)1(C+

(14)

where,

3

22

2

2

)X1(

*)1(C

dX

)d(d

= (15)

and Var (X) = E(X2) - x2 (16)From equation 4, the variance of the delay is given by:

Var(d)

2

dX

)d(d)X(Var

- +

2

2

2

2

dX

)d(d)X(Var

4

1+

2

2

3

XdX

)d(d

dX

)d(d)X(E

2

2

2

4

XdX

)d(d)X(E

4

1

+ (17)

When the degree of saturation follows a normal distribution, using (5) and (6) one can obtain

the followings:

E(X2) = x

2(1 + CV

2) (18)

E(X3) = x

3(1 + 3*CV

2) (19)

E(X4) = x

4(1 + 6*CV

2+CV

4) (20)

According to the characteristics of expectation value:

E(X - x)3

= E(X3) - 3xE(X

2) + 2x

2(21)

and E(X - x)4 = E(X4) - 4xE(X3) + 6x2E(X2) - 3x4 (22)

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where, X : Degree of saturation; x : Mean of degree of saturation X; CV = /.

4. DATA COLLECTION

In order to evaluate the proposed methodology, data with regard to delay, volume, etc. of three

signalized intersections which are advantageous to observing in Hochiminh city, Vietnamwere applied. The locations are not near any bus stop or gasoline station, etc. to eliminate

modification maneuvers from road users. The entire data were collected under a condition of

clear weather, dry pavement, and low magnitude of wind on February 2008.

All study approaches have two lanes in each direction with raised medians. The lane widths

for all locations are 3.50 (m), 3.50 (m) for the inner and outer lanes, respectively. The first

location has the fixed-time signalized control installed with 62 (sec) of a cycle time, including

24 (sec), 3 (sec), and 35 (sec) for green, yellow, and red times, respectively. There is no all-red

time for this signalized intersection. The second location is the pre-timed signalized control

with 67 (sec) of a cycle time, 28 (sec), 4 (sec) and 35 (sec) for green, yellow and red times,respectively. It also has no all-red time. The third location is the pre-timed controlled

intersection with the cycle time of 65 (sec), including 36 (sec), 3 (sec), and 26 (sec) for green,

yellow, and red times respectively.

A digital video recorder was set up at the top of the high buildings nearby the studied sites and

captured over 30 (m) long of roadways. The camera recorded traffic streams during green,

yellow and red times. From the video images, vehicle types and passing time are captured

later by interpreting in the traffic laboratory. These observations with varying saturated green

times (from 5 seconds to 45 seconds) are recorded to estimate PCUs, average headway, and

saturation flow. More than thirty of such observations were taken at each approach.

The measurement of delay was made by observing the traveling of vehicles to intersections.

According to Canadian Guideline for Intersection (2005), it needs at least two people for

measuring average delay. The first one stood at the head of the queue and the second one

stood at the stop line. The first one counted the number of vehicles approaching the

intersection, while the second one counted the number of vehicles passing the stop line. Both

of them started counting simultaneously and in the same time interval, usually 10 (sec). Figure

1 illustrates the method for measurement of delay.

From Canadian Guideline for Intersection (2005), average delay at a signalized intersection is

computed by the formula given as:

d = [tm(Xi, in - Xi, out )/ Xi, in ] tt (23)

where, d : Average delay at an intersection, (s/vehicle); tm : Observing time (sec), usually 10

30 (sec); Xi, in : Number of vehicles approaching an intersection; Xi, out:Number of vehiclespassing the stop line; Xi, in : Total accumulating vehicles approaching the intersection at

observing time; Xi, out : Total accumulating vehicles passing the stop line at observing time; tt: Travel time from vehicles location to the stop line (sec).

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Second observer

(Counting at the stop line)

First observer

(Counting the coming vehicles)

Figure 1 Measurement of Average Delay

5. ESTIMATION RESULTS

The degree of saturation for heterogeneous traffic is evaluated using the multiple linearregression analysis, in which the saturated green time is a function of vehicles passing the stop

line during that green time. Assuming that the relationship between dependent and

independent variables is linear, the regression formula, therefore obtains by using SPSS

software for all study intersections with 30 observations for each location.

The first intersection: t = 0.21n1 + 1.06n2 + 2.12 n3 + 2.76n4, R2

= 0.99 (24)

The second intersection: t = 0.17n1 + 1.00n2 + 2.18 n3 + 2.88n4, R2

= 0.98 (25)

The third intersection: t = 0.16n1

+ 1.11n2

+ 2.25 n3

+ 2.95n4, R

2= 0.99 (26)

Where, t : Saturated green time (s);

n1, n2, n3, n4 : Number of vehicles passing the stop line at time t at the subject approach for

each group of motorized two-wheeler, car, minibus and bus, respectively.

Estimated results with t value of the models are presented in Table 2.

Table 2 Parameter Estimates for Regression Functions

Intersection Coefficient Std. Error tstatistic%95

criticalt

1

2-wheeler 0.21 0.0034 62.36

1.697Car 1.06 0.1197 8.85

Minibus 2.12 0.2028 10.44

Bus 2.76 0.1496 18.48

2

2-wheeler 0.17 0.0071 24.48

1.697Car 1.00 0.2109 4.74

Minibus 2.18 0.3202 6.79

Bus 2.88 0.1538 18.75

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3

2-wheeler 0.16 0.0040 41.18

1.697Car 1.11 0.0559 19.82

Minibus 2.25 0.1484 15.16

Bus 2.95 0.1073 27.52

From regression analysis, the PCU value of each group is obtained when the coefficient of

that group is divided by the coefficient of the car group. Table 3 represents PCU values and

the traffic composition at studied sites.

Table 3 Traffic Composition and Passenger Car Unit (PCU) of Vehicle Groups

LocationTwo-wheeler Car Minibus Bus

% PCU % PCU % PCU % PCU

1

23

97.87

97.1591.84

0.20

0.170.15

1.40

1.816.92

1.00

1.001.00

0.27

0.190.25

2.00

2.182.03

0.47

0.860.99

2.61

2.882.66

Average

PCU0.17 1.00 2.07 2.72

The average headways and the average saturation flows are obtained as shown in Table 4.

Table 4 Headway and Saturation Flow

IntersectionHeadway (sec) Saturation flow (PCU/hr)

Have H Save S Save 1.28S Save + 1.28S

1 1.07 0.056 3384.87 175.38 3160.40 3609.35

2 1.01 0.067 3565.12 226.89 3274.70 3855.53

3 1.12 0.043 3226.83 124.39 3067.60 3386.05

where, Have and Save: average headway and average saturation flow, respectively; H and S:standard deviation of headway and saturation flow, respectively.

By using SPSS software, according to KolmogorovSmirnov (KS) test for assumed

saturation flow distribution, the maximum discrepancy of locations 1, 2, and 3 are 0.69, 0.64,

and 0.9, respectively. The critical value at the 5% significance level is 0.24, 0.25, and 0.30.Since the maximum discrepancies are less than the critical value, all frequency distributions of

saturation flows at intersections 1, 2, and 3 correspond to normal function at the 5%

significance level as shown in Table 5.

Table 5 K-S test for Assumed Headway Distribution as a Normal Distribution with 5% Level

of Significance

Intersection Observations (K- S)statistic (K-S)critical

1 32 0.69 0.24

2 30 0.64 0.253 20 0.90 0.30

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The values of average delay with various degrees of saturation at different intersections by

using the proposed model, conventional Websters method and surveying are presented in

Table 6, 7 and 8.

Table 6 Estimated and Observed Data for Delay at Intersection 1

X 0.76 0.84 0.88 0.93 0.98 1.01

q(PCU/hr) 1042 1104 1150 1217 1284 1318

d 0.78 0.93 1.87 1.11 0.65 1.28

Proposed 16.56 17.30 17.83 18.24 18.78 19.16

Websters 20.57 23.92 26.87 35.30 83.79 4.23

Observed 15.44 16.83 17.62 19.67 20.45 21.23

where, X: Degree of saturation; q (PCU/hr) : Average traffic volume at subject approach in anhour; d : Standard deviation of delay in the proposed method; Proposed: Delay of proposedmethod (sec); Websters: Delay of Websters method (sec); Observed: Delay from observation

(sec).


X 0.58 0.61 0.63 0.69 0.99 1.07

q(PCU/hr) 874 914 947 1035 1481 1592

d 1.10 0.93 0.59 0.38 1.51 1.76

Proposed 15.12 15.32 15.48 16.00 19.53 20.74

Websters 16.61 17.01 17.38 18.49 199.16 1.92

Observed 16.06 16.77 17.01 17.69 21.03 21.87


X 0.68 0.72 0.74 0.76 0.812 0.974

q(PCU/hr) 843 884 913 939 1000 1200

d 0.842 0.72 0.50 1.03 0.870 1.275

Proposed 19.48 19.83 20.09 20.36 20.939 23.173

Websters 22.35 23.19 23.91 24.60 26.732 76.860

Observed 20.00 20.90 21.89 22.05 22.76 24.04

The comparisons of estimated data from proposed model, conventional Websters model and

observed data at intersection 1, 2 and 3 are shown respectively in Figure 2, 3 and 4 as below.

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0

10

20

30

40

50

60

70

80

90

0.7 0.8 0.9 1 1.1

Degree of Saturation

OverallAvera

geDelay

Estimated

Observed

Webster's

Figure 2 Comparison of Estimated, Observed and Websters Delay at Intersection 1

0

10

20

30

40

50

60

70

80

90

0.5 0.6 0.7 0.8 0.9 1 1.1


OverallAverageDelay

Estimated

Observed

Webster's


0

10

20

30

40

50

60

70

80

90

0.6 0.7 0.8 0.9 1


OverallAverage

Delay

Estimated

Observed

Webster's


The estimation resulting from the proposed model is comparatively close to the observed data

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at all studied intersections. When the degree of saturation (X) is less than 1, average delay can

be estimated by either the conventional Websters model or the proposed model. The results

obtained from two methods at that time are reasonably similar. When the degree of saturation

approaches 1 or is greater than 1, the conventional Websters model has a problem. When the

degree of saturation is approximately equal to 1, average delay becomes considerably high but

when the degree of saturation is greater than 1, average delay falls to a small value. Thismakes the denominator of Websters formula become unidentified. It is contradictory to the

practice that, average delay is directly proportional to the traffic volume. This problem has not

occurred in the proposed model. When the traffic volume approaching the intersection is low,

average delays of the proposed model, the conventional Websters model and observation are

inconsiderably different. However, when the traffic volume approaching the intersection is

moderately high, average delay of the proposed model is better than the conventional

Websters model since it is closer to the observed data.

6. CONCLUSION

The paper proposes the modified Websters model to estimate delay for heterogeneous traffic

conditions at pre-timed signalized intersections. Traffic volumes approaching the signalized

intersection are classified into four groups: Motorized two-wheeler, Car, Minibus, and Bus.

The passenger car unit (PCU) is estimated by using the linear multiple regression analysis.

The regression function expresses the relationship between the saturated green time and the

total number of vehicles of all groups passing the approach. Saturation flows are estimated

with the consideration of different types of vehicles traveling together. Distributions of

saturation flow are computed to follow the normal distribution at all observed intersections.

Then, an expectation function method and Taylor series expansion are utilized to estimate the

mean and the variance of delay. The model is evaluated at three pre-timed signalized

intersections, then compared with the conventional Websters and observed delays. The resultsidentify that the output of the proposed methodology is close to the observed data and better

than that of the conventional Websters, especially when the degree of saturation is close to 1.

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