ITE Journal on ThE wEb / aprIl 2009 69

Determining the required Minimum Spacing between Signalized Intersections and bus-baysSome of the buS StopS

currently inStalled

in Korea are an

inSufficient diStance

from interSectionS,

which cauSeS traffic

delayS or conflictS. thiS

Study aimS to calculate

the minimum diStance

between a buS Stop near

a Signalized interSection

and an interSection. for

thiS purpoSe, both the

queuing model and the

gap acceptance model

are uSed.

by Je-Jin parK and tae-Jun ha

bacKgroundA distance longer than the length of weav-

ing required should be used when a bus stop is installed near an intersection, according to Korean domestic standards.1 However, it is difficult to apply this rule to intersections because it requires only design speed as the variable to determine the length of weaving. For a signalized intersection with traffic char-acteristics such as cycle and queuing length, the distance required for a lane change is a factor that must be considered.

This study aims to calculate general theo-retical formulas that determine the minimum distance between a bus stop and an intersec-tion, based on buses that make left turns near a signalized intersection. The study also presents a new standard that reflects charac-teristics of bus operation after comparing and examining the existing rules presented by the results of theoretical formula calcula-tion and the length of weaving.

Bus stops are classified according to dis-tance. A near-side bus stop is installed before the passage of an intersection; a far-side bus stop is located after the passage of an inter-section; and a mid-block bus stop is installed in the middle point between one intersection and another.2 This study targeted bus stops that have a bus bay at the near side and far side and established the following assump-tions to calculate theoretical formulas:

•Thetypesofvehiclesappliedarepas-senger cars (except buses).

•Vehicles other than buses do notchange lanes between an intersec-

tion and a bus stop. • Arriving traffic vol-ume at the intersection

does not go beyond its capacity.

Figure 1 shows the classification by lo-cation of bus stops and the process deter-mination of the required minimum spac-ing between signalized intersections and bus bays. The queuing model is shown in

Figure 2, where:q = arrival rater = effective red timeg = effective green timet0 = disappearing time for waiting queuess = discharge rate for waiting queues.3

The larger value produced by the method of using the length of waiting queues and applying the gap acceptance model is used as the value for the distance of a bus stop at the near side.

uSing the length of waiting queueS method

This method applies when a traffic light in time zone A( r + t0 ) is red. The following basic assumptions were applied to this method:

•Thelengthofvehiclesandtheirgapsis constant.

•Dischargerate,s, is constant. •Arrivalratesofvehiclesareconstant.

If a queue waiting at the stop line of the main intersection is located behind a bus stop, buses cannot advance or change a lane until the queue leading up to the bus stop disappears. Therefore, this theory is based on the fact that the distance has to be longer than the longest queue for each lane group at the intersection. If the peak-time traffic volume of each mo-bile flow is qL, qT , qR according to lane grouping and lane groups NL, NT , NR, the distance of the last vehicle in the dis-appearing queue from the intersection can be calculated. The minimum distance of a bus stop at the near side of the intersec-tion is shown in Equation 1:

(1)jLUj

jjjcn fNtrqlL

,,01

11)( ××+××=

=jjj

ij

Nqs

rq 1×−

jt ,0

+−=

×

×= −∞ −

−

∫∫ ττ

τ

τ

qqedtqe

dttqet

qqt

qt

w

110

+−= −

TVlqw V

l

qqet

T

ττ

11)/(

+−=

−T

VlqwB Vl

qqet

T

*

)/(,

11*

τ

τ

+−=

−*

,

11* τ

τ qqet

qwB

ττT

BT

VVV −=*

BwBn tVD ,1 ×=

θtan2

WDn =

70 ITE Journal on ThE wEb / aprIl 2009

where:lQ, max = maximum distance until the last vehicle is at the intersectionlc = length of the vehicle + space of the vehicleqj = arrival traffic volume of j lane groupgj = effective green time of j lane groupsj = discharge rate of j lane grouprj = effective red time of j lane groupfLU,j = coefficient using lane of j lane group

The above calculation determines the longest queue among the lane groups. Lane utilization factor, fLU, reflects the fact that the traffic volume of each lane within the lane groups is not identical. The lane utilization factor is provided in the Highway Capacity Manual.4

application of gap acceptance model method

In the gap acceptance model, traffic flow can be applied to the continuous time zone, B(g–t0 ). The following basic assumptions were applied to this method:

•Abus runs at constant speed fromthe time when it leaves a bus stop until it passes an intersection.

•Lane change angles of the bus areconstant.

•Theintersectionisindependentfromthe influence of neighboring inter-sections and the arrival of vehicles follows Poisson distribution.

For a bus to turn left at the intersection after it leaves the bus stop, it has to wait for a longer interval than the critical ac-ceptance gap (r) among gaps of traffic flow of the near lane. The bus then repeats its straight advance and change of lanes several times until it reaches the lane where it can make a left turn, as shown in Figure 3. The length of straight advance (Dn1) changes depending on traffic flow and speed of the near lane. Lane change distance (Dn2) becomes a function of the discharge angle of the bus and lane width.

The gap acceptance running distance (Dn1) is the average time (tw

__)traversing traf-

fic has to wait, as shown in Equation 2:5

(2)

jLUjjjjcn fN

trqlL,

,01

11)( ××+××=

=jjj

ij

Nqs

rq 1×−

jt ,0

+−=

×

×= −∞ −

−

∫∫ ττ

τ

τ

qqedtqe

dttqet

qqt

qt

w

110

+−= −

TVlqw V

l

qqet

T

ττ

11)/(

+−=

−T

VlqwB Vl

qqet

T

*

)/(,

11*

τ

τ

+−=

−*

,

11* τ

τ qqet

qwB

ττT

BT

VVV −=*

BwBn tVD ,1 ×=

θtan2

WDn =

Equation 3 applies if the running speed

of a vehicle on the near lane to which the bus wants to move is VT and a gap between the two vehicles (the time interval τ) is lT.

(3)

jLUjjjjcn fN

trqlL,

,01

11)( ××+××=

=jjj

ij

Nqs

rq 1×−

jt ,0

+−=

×

×= −∞ −

−

∫∫ ττ

τ

τ

qqedtqe

dttqet

qqt

qt

w

110

+−= −

TVlqw V

l

qqet

T

ττ

11)/(

+−=

−T

VlqwB Vl

qqet

T

*

)/(,

11*

τ

τ

+−=

−*

,

11* τ

τ qqet

qwB

ττT

BT

VVV −=*

BwBn tVD ,1 ×=

θtan2

WDn =

In this example, tw__

is the average wait-ing time for buses trying to change a lane after waiting for a critical acceptance gap to appear. The bus in this situation is presumed to be stopped.

When the buses are waiting for a chance to change a lane and have a slower speed (VB), the relative speed decreases and the time gap of the near vehicle relative to that of the bus increases. Therefore, the distance to make the critical acceptance time (lT) de-creases. If the gap at this time is lτ

* , Equation 4 shows the required average waiting time for a mobile bus to change a lane,

_tB,w

__ .

(4)

jLUjjjjcn fN

trqlL,

,01

11)( ××+××=

=jjj

ij

Nqs

rq 1×−

jt ,0

+−=

×

×= −∞ −

−

∫∫ ττ

τ

τ

qqedtqe

dttqet

qqt

qt

w

110

+−= −

TVlqw V

l

qqet

T

ττ

11)/(

+−=

−T

VlqwB Vl

qqet

T

*

)/(,

11*

τ

τ

+−=

−*

,

11* τ

τ qqet

qwB

ττT

BT

VVV −=*

BwBn tVD ,1 ×=

θtan2

WDn =

where:q = traffic volume

Figure 1. Classification by locations of bus stop and process determination of the required minimum spacing between signalized intersections and bus bays. Figure 2. Queuing model, length of waiting queues and locations of bus stops.

ITE Journal on ThE wEb / aprIl 2009 71

lB = movement distance during τ timelτ

* = 1τ – lB = (VT – VB)τ

The average waiting time of the bus is shown in Equation 5:

(5)

jLUjjjjcn fN

trqlL,

,01

11)( ××+××=

=jjj

ij

Nqs

rq 1×−

jt ,0

+−=

×

×= −∞ −

−

∫∫ ττ

τ

τ

qqedtqe

dttqet

qqt

qt

w

110

+−= −

TVlqw V

l

qqet

T

ττ

11)/(

+−=

−T

VlqwB Vl

qqet

T

*

)/(,

11*

τ

τ

+−=

−*

,

11* τ

τ qqet

qwB

ττT

BT

VVV −=*

BwBn tVD ,1 ×=

θtan2

WDn =

The running distance of the bus is shown in Equation 6:

jLUjjjjcn fN

trqlL,

,01

11)( ××+××=

=jjj

ij

Nqs

rq 1×−

jt ,0

+−=

×

×= −∞ −

−

∫∫ ττ

τ

τ

qqedtqe

dttqet

qqt

qt

w

110

+−= −

TVlqw V

l

qqet

T

ττ

11)/(

+−=

−T

VlqwB Vl

qqet

T

*

)/(,

11*

τ

τ

+−=

−*

,

11* τ

τ qqet

qwB

ττT

BT

VVV −=*

BwBn tVD ,1 ×=

θtan2

WDn =

(6)

Equation 7 shows how to calculate the lane change distance (Dn2), which is the length of the lane required for a bus to start and finish its lane change.

(7)

jLUjjjjcn fN

trqlL,

,01

11)( ××+××=

=jjj

ij

Nqs

rq 1×−

jt ,0

+−=

×

×= −∞ −

−

∫∫ ττ

τ

τ

qqedtqe

dttqet

qqt

qt

w

110

+−= −

TVlqw V

l

qqet

T

ττ

11)/(

+−=

−T

VlqwB Vl

qqet

T

*

)/(,

11*

τ

τ

+−=

−*

,

11* τ

τ qqet

qwB

ττT

BT

VVV −=*

BwBn tVD ,1 ×=

θtan2

WDn =

The formula for the minimum distance from an intersection to a bus stop is ex-plained in Equation 8 when determining the distance of a bus stop at the near side where the gap acceptance model is applied. The number of lane changes the bus has to make to enter the left-turn lane is n.

(8)( )∑=

+=n

iininn DDL

1,2,12

( )21, nnn LLMaxL =

−=aVVnW

DBwB

f 2,

tanmax

0,2

,2

1 θ

+−=

−*

2,

11* τ

τR

qR

B qeqt

R

ττR

BR

VVV −=*

2,,2 BwBf tVD ×=

=

aVW

DwB

f 2,

tanmax

,2

3 θ

321 ffff DDDL ++=

( )( )

2.1

118.12

301128.0

−−

+

+=wD

w

nww

ww SS

SNV

VVV

L

Equation 9 shows the determination of the minimum distance at the near side (Ln). The largest value is chosen after a comparison is done between the values resulting from calculations for a waiting queue case and a non-waiting queue case according to the characteristics of a signal-ized intersection. If traffic volume is large, Ln1 becomes larger and if the number of lanes is large, Ln2 becomes larger.

( )∑=

+=n

iininn DDL

1,2,12

( )21, nnn LLMaxL =

−=aVVnW

DBwB

f 2,

tanmax

0,2

,2

1 θ

+−=

−*

2,

11* τ

τR

qR

B qeqt

R

ττR

BR

VVV −=*

2,,2 BwBf tVD ×=

=

aVW

DwB

f 2,

tanmax

,2

3 θ

321 ffff DDDL ++=

( )( )

2.1

118.12

301128.0

−−

+

+=wD

w

nww

ww SS

SNV

VVV

L

(9)

determination of minimum diStance at the far Side

The following is the description of the process from the time the bus makes a left

Figure 3. running process of a left-turning bus at the near side and critical acceptance gap.

Figure 4. Stop distance of a left-turning bus at the far side.

72 ITE Journal on ThE wEb / aprIl 2009

turn until it reaches the bus stop:•0<t <t1 : The bus reaches its speed

a by running at the angle θ and the deceleration speed VB,w after it makes a left turn and reaches the second lane from the bus stop.

•t1 < t < t2 : The average waiting time required for a bus to find a critical acceptance gap while it runs at the equivalent speed, VB,w.

•t2<t <t3 : The required time from when the bus accepts the gap and begins a lane change until it reaches the bus stop. Its discharge angle, deceleration speed and speed while waiting for gap acceptance are pre-sumed to be constant.

Calculation of Df1Two cases are considered in calculating

Df1:•Case1:Thebuscompletes its lane

change before its speed reaches VB,w.

•Case2:ThebusspeedreachesVB,w before it completes its lane change.

The required length is depicted in Equation10:

(10)

( )∑=

+=n

iininn DDL

1,2,12

( )21, nnn LLMaxL =

−=aVVnW

DBwB

f 2,

tanmax

0,2

,2

1 θ

+−=

−*

2,

11* τ

τR

qR

B qeqt

R

ττR

BR

VVV −=*

2,,2 BwBf tVD ×=

=

aVW

DwB

f 2,

tanmax

,2

3 θ

321 ffff DDDL ++=

( )( )

2.1

118.12

301128.0

−−

+

+=wD

w

nww

ww SS

SNV

VVV

L

where:VB,0 = early speed after bus makes left turnn = number of lane changes by the busW = width of lane

Calculation of Df2 The distance covered by the bus from

the time when it found the critical ac-ceptance gap until it begins a lane change is calculated in the same way as the gap acceptance distance at the near side (see Equation 11).

(11)

( )∑=

+=n

iininn DDL

1,2,12

( )21, nnn LLMaxL =

−=aVVnW

DBwB

f 2,

tanmax

0,2

,2

1 θ

+−=

−*

2,

11* τ

τR

qR

B qeqt

R

ττR

BR

VVV −=*

2,,2 BwBf tVD ×=

=

aVW

DwB

f 2,

tanmax

,2

3 θ

321 ffff DDDL ++=

( )( )

2.1

118.12

301128.0

−−

+

+=wD

w

nww

ww SS

SNV

VVV

L

Calculation of Df3After a comparison is conducted on

the required distance to make the bus reach the speed at VB,w, the deceleration speed and the required distance for the bus to change its lane at the angle of θ, the larger value is used (see Equation 12).

(12)

( )∑=

+=n

iininn DDL

1,2,12

( )21, nnn LLMaxL =

−=aVVnW

DBwB

f 2,

tanmax

0,2

,2

1 θ

+−=

−*

2,

11* τ

τR

qR

B qeqt

R

ττR

BR

VVV −=*

2,,2 BwBf tVD ×=

=

aVW

DwB

f 2,

tanmax

,2

3 θ

321 ffff DDDL ++=

( )( )

2.1

118.12

301128.0

−−

+

+=wD

w

nww

ww SS

SNV

VVV

L

Minimum Distance of Bus Stop at the Far Side

Equation 13 shows that the distance of a bus stop at the far side of the intersec-tion, Lf, is obtained by adding Df1, Df2 and Df3.

( )∑=

+=n

iininn DDL

1,2,12

( )21, nnn LLMaxL =

−=aVVnW

DBwB

f 2,

tanmax

0,2

,2

1 θ

+−=

−*

2,

11* τ

τR

qR

B qeqt

R

ττR

BR

VVV −=*

2,,2 BwBf tVD ×=

=

aVW

DwB

f 2,

tanmax

,2

3 θ

321 ffff DDDL ++=

( )( )

2.1

118.12

301128.0

−−

+

+=wD

w

nww

ww SS

SNV

VVV

L

(13)

compariSon and Verification of the model by mocK eXperimentMock Experiment

For the application of the model formu-

las, several traffic conditions were set at con-stant values and important variables such as traffic volume, bus speed, lane changes and discharge angles were all set as values before the experiment was conducted. The results were calculated through the previously de-scribed distance calculation formulas and are shown in Table 1.

Comparison and Analysis with Existing Models

Comparisonandanalysisbetweenthelength of weaving, the existing regulation and the result of the mock experiment were completed in order to evaluate the result of distance estimation. The model formula applied is an evaluation formula of road capacity with uninterrupted flow. Because this formula was created based on limited data, it is different from traffic conditions of the signalized intersection, which is the main scope of this study. However, mutual comparison was com-pleted because green time without a queue in the signalized intersection can be re-garded as uninterrupted flow.

Table 1. Establishment of the environment for mock experiment and stand-off.

Establishment of environment for mock experiment

Input parameters Set values

Width of lane (W) 3.5 meters (m)

Waiting queue gap (lc) 5 m

Criticalacceptancegap(τ) 3 seconds (sec.)

Decelerationspeedofbus(a) 3 m/sec.2

Signal time (r, g) r=90sec.,g=30sec.

Dischargerate(s) 1,800(vehicles/hour/lane)

Lane utilization factor (fLU) 0.95

Number of access lanes 3 lanes

Average running speed of vehicles (Vt, VR, VB,0) 60kilometers/hour

Bus speed (VB, m / s2) 20 30 40

Traffic volume per lane (q, vehicle / hour / lane) 100 200 300 400

Angle of bus to change lane (θ, °) 5 10 15 20 25 30

Stand-off distance at the near side (queueing model, Ln1)

Traffic volume (vehicles/hour/lane) Ln1(m)

100 13.9

200 29.5

300 47.3

400 67.5

(continued)

ITE Journal on ThE wEb / aprIl 2009 73

The formula to obtain the length of weaving on the road of uninterrupted flow is shown in Equation 14:

(14)

( )∑=

+=n

iininn DDL

1,2,12

( )21, nnn LLMaxL =

−=aVVnW

DBwB

f 2,

tanmax

0,2

,2

1 θ

+−=

−*

2,

11* τ

τR

qR

B qeqt

R

ττR

BR

VVV −=*

2,,2 BwBf tVD ×=

=

aVW

DwB

f 2,

tanmax

,2

3 θ

321 ffff DDDL ++=

( )( )

2.1

118.12

301128.0

−−

+

+=wD

w

nww

ww SS

SNV

VVV

L

where:Lw = minimum length of weavingVw = weaving traffic volumeVnw = non-weaving traffic volumeV = total traffic volume of weavingSw = average speed between weaving vehiclesSD = design speedN = number of lanes

The study model and the existing model were compared after the mock experiment was completed. In the case of a bus stop at the near side, the results showed little change according to traffic volume. The value resulting from the length of weaving was more sensitive to traffic volume. The results showed that it was longer than the distance resulting from the existing length of weaving. The value resulting from the length of weav-ing of distance at the near side is less than20meters(relativelyshort),butthedistance presented by this study showed a larger value overall.

concluSionThis study aimed to develop a model

that reflects the characteristics of bus op-eration on signalized intersections where left-turning buses exist. The contents of the study conducted can be summarized as follows:

•Forthedistanceofabusstopatthenear side, the model formulas were calculated and presented based on the queuing model and the gap ac-ceptance model with consideration of the characteristics of signals and traffic flow of a signalized intersection. Gap acceptance model formulas were cal-culated step-by-step in consideration of lane change angles and speed.

•Forthedistanceofabusstopatthefar side, each model formula was presented by classifying the running characteristics of each progressing lane and the distance was calculated

by adding the values resulting from the three steps.

•Asaresultofcomparingandevalu-ating the developed model formu-las and the existing model formulas (length of weaving), the values of the developed model formulas were longer in most instances.

Several limitations were found while the study was conducted. The study failed to consider drivers’ lane change performance. This study presented the basic assumptions on lane change an-gles and speed changes at the time of a

lane change. Because lane change angles vary depending on drivers and the traf-fic conditions, it was unreasonable to apply constant values. Therefore, it is believed that the development of a model that considers drivers’ performance will be needed.

Furthermore, because the assumptions of the study excluded the conditions of an intersection, such as traffic exceeding its capacity and signals being connected, its application is limited in these cases. Also, because the study was focused on tracing left-turning buses themselves, in-teractive influence among buses created

Table 1 (continued). Establishment of the environment for mock experiment and stand-off.

Stand-off distance at the near side (gap acceptance model, Ln2)

VB(kilometers/

hour)

q(vehicles/hour/

lane)

Stand-off distance by angle of bus to change a lane

5° 10° 15° 20° 25° 30°

20

100 82.3 42.0 28.4 21.5 17.3 14.4

200 84.6 44.3 30.7 23.8 19.6 16.7

300 87.1 46.8 33.2 26.3 22.1 19.2

400 89.6 49.3 35.7 28.8 24.6 21.7

30

100 81.9 41.6 28.0 21.1 16.9 14.0

200 83.9 43.6 30.0 23.1 18.9 16.0

300 85.9 45.6 32.0 25.1 20.9 18.0

400 87.9 47.6 34.1 27.2 22.9 20.1

40

100 81.1 40.8 27.2 20.4 16.1 13.2

200 82.3 42.0 28.4 21.5 17.3 14.4

300 83.4 43.1 29.6 22.7 18.4 15.6

400 84.6 44.3 30.7 23.8 19.6 16.7

Stand-off distance at the far side (Lf)

VB(kilometers/

hour)

q(vehicles/hour/

lane)

Stand-off distance by angle of bus to change a lane

5° 10° 15° 20° 25° 30°

20

100 121.1 62.1 55.3 51.9 49.8 48.3

200 122.3 63.3 56.5 53.1 51.0 49.5

300 123.5 64.5 57.7 54.3 52.2 50.7

400 124.8 65.8 59.0 55.6 53.5 52.0

30

100 121.0 60.5 48.7 47.2 47.2 47.2

200 121.9 61.5 49.7 48.2 48.2 48.2

300 122.9 62.5 50.7 49.2 49.2 49.2

400 124.0 63.5 51.8 50.3 50.3 50.3

40

100 120.6 60.8 47.3 46.9 46.9 46.9

200 121.1 61.4 47.8 47.4 47.4 47.4

300 121.7 62.0 48.4 48.0 48.0 48.0

400 122.3 62.6 49.0 48.6 48.6 48.6

74 ITE Journal on ThE wEb / aprIl 2009

as the traffic volume of left-turning buses increases was not considered. Studies on vehicles arriving at the intersections that consider traffic characteristics of buses must continue in order to calculate more realistic model formulas. n

References1. Rules on Standards for Road Structure Facili-

ties.Seoul,Korea:KoreaMinistryofConstruc-tion&Transportation,2001.

2. Highway Capacity Manual. Washington, DC, USA: Transportation Research Board,2000.

3. Fitzpatrick, K., K. Hall, K. Perkinson, L. Nowlin and Y. Koppa. National CooperativeHighwayResearchProgramReport19:Guide-lines for the Location and Design of Bus Stops. Washington,DC:NationalResearchCouncil,1996.

4. Highway Capacity Manual2000.5. Fitzpatrick, K. et al. 1996.6. Korea Highway Capacity Manual. Seoul:

KoreaMinistryofConstruction&Transporta-tion, 1992.

Je-Jin parK is employed as a senior researcher at Korea Expressway Corpora-tion in South Korea. His research interests include highway design, traffic safety, analysis of

car accidents and traffic planning.

tae-Jun ha is currently an associate professor in the school of transportation engi-neering at Chonnam University. He has also been employed by the Transportation Science

Institute, Road Traffic Safety Authority in South Korea as a research team leader. His research interests include highway design, traffic safety, analysis of car accidents and traffic planning.

Table 2. Establishment of the environment for comparison with the existing models and comparison length of weaving with distance.

Input parameters Set values

Angle of bus to change a lane (θ) 15 degrees

Number of lanes (N) 3 lanes

Designspeed(SD) 60kilometers/hour

Weaving traffic volume (VW) 40pcu

Runningspeedofbus(SW) 40kilometers/hour

Non-weaving traffic volume (vehicles / hour / lane) 100 200 300 400

Comparison length of weaving (LW) with distance (Ln) at the near side

Traffic volume (vehicles/hour/lane )

Indicated distance in this study, Ln (m)

Calculated distance in existing models, LW (m)

100 27.2 12.8

200 29.5 21.8

300 47.3 30.9

400 67.5 39.9

Comparison length of weaving (LW) with distance (Lf) at the far side

Traffic volume (vehicles/hour/lane)

Indicated distance in this study, Lf (m)

Calculated distance in existing models, LW (m)

100 47.3 6.7

200 47.8 9.7

300 48.4 12.8

400 49.0 15.8