the optimization of lane assignment and signal timing at the tandem intersection with pre-signal

The optimization of lane assignment and signal timing at the tandemintersection with pre-signal

Yaping Zhou and Hongbin Zhuang*

School of Management, University of Science and Technology of China, Hefei, Anhui Province 230026, China

SUMMARY

This paper presents an integrated model for optimizing lane assignment and signal timing at tandemintersection, which is introduced recently. The pre-signal is utilized in the tandem intersection toreorganize the traffic flow; hence, the vehicles, regardless of whether left-turns or through vehicles, canbe discharged in all the lanes. However, the previous work does not consider the extra traffic disruptionand the associated delay caused by the additional pre-signal. In the paper, the extra delay aroused by thecoordination is incorporated in a lane assignment and signal timing optimization model, and the problemis converted into a mixed-integer non-linear programming. A feasible directions method is hence introducedto solve the mixed-integer non-linear programming. The result of the optimization shows that the performanceof the tandem intersection is improved and the average delay is minimized. The comparison between thetandem and the conventional configuration is presented, and the results verify that the former shows betterperformance than the latter. In addition, the optimal sequence corresponding to the turning proportion andthe optimal lane assignment at the upstream approach of the pre-signal are presented. Furthermore, if thenumber of lanes is equal in all arms, the paper proves that the average delay will be reduced if lane assignmentis proportional to the turning proportion and the vehicles with low proportion are discharged in advance.Copyright © 2013 John Wiley & Sons, Ltd.

KEY WORDS: tandem intersection; lane assignment; signal timing; delay estimation

1. INTRODUCTION

Reorganizing traffic movements is a frequently used method to increase the capacity of urbanintersections. By regulating the traffic flows, the vehicles maneuver in the expected manner tolower the average delay or reduce the stops. Unconventional intersections such as median U-turns,jughandles, superstreets, continuous flow intersections, and bowties are most mentioned in theregulation. The method is shown effective in the previous researches (Rodegerdts et al., 2004;Autey et al., 2012) and proved to increase the capacity of intersections in different extent. How-ever, the unconventional design requires extra infrastructure, which may not be available in urbanintersections. Besides, exclusive lanes do not operate efficiently because the lanes devoted to left-turn or through lane discharge and the discharging capability is not fully utilized. Recently, Xuanet al. (2011) proposed a new manner to explore the potential capacity of intersections. They put for-ward a tandem intersection that all (or partial) lanes can fully discharge during either left-turn phase orthrough phase. Without loss of generality, as is shown in Figure 1, the left-turn phase is assumed tolead the through phase. The left-turn and through vehicles separately enter the sorting area during

*Correspondence to: Hongbin Zhuang, School of Management, University of Science and Technology of China, Hefei,Anhui Province 230026, China.E-mail: [email protected]

Copyright © 2013 John Wiley & Sons, Ltd.

JOURNAL OF ADVANCED TRANSPORTATIONJ. Adv. Transp. 2014; 48:362–376Published online 15 February 2013 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/atr.1222

the sub-phase of pre-signal. The signal cycle of pre-signal is the same as the main signal, and the greentime is allocated alternatively for the through or left-turn vehicles. As a result, the through vehiclesfollow the left-turns in the sorting area, and the lanes in the sorting area are fully utilized. The sortingarea combined with the pre-signal produces approximately the same capacity as if no left turns existedand requires small construction; hence, the improvement is significant.However, compared with the unconventional intersections mentioned earlier, the number of phases

in the main signal of the tandem intersection is not reduced. Besides, the coordination between mainsignal and pre-signal will arouse extra delay. Hence, the average delay experienced by vehicles willnot be decreased. Furthermore, as the turning proportion is varied upstream, the green spilt of pre-sig-nal for left-turns and through vehicles will play an important role in exploring the capacity of themethod. The research gap, therefore, is expected to be filled by integrating the signal timing and laneassignment optimization, which is the main focus of the paper.The general signal timing methods were first presented in the stage-based method (Webster, 1958;

Allsop, 1972) and later developed into the group-based method (Improta and Cantarella, 1984;Gallivan and Heydecker, 1988; Silcock, 1997). Meanwhile, the lane assignment was previouslyproposed in the highway systems and traffic assignment context (Fisk, 1990; Hall and Lotspeich, 1996;Hall and Caliskan, 1999; Ramaswamy, et al., 1997). The signal timing together with lane assignmentwas later investigated by other researches (Lam et al., 1997; Wong and Wong, 2003; Wong andHeydecker, 2011). However, few studies focus on the optimization on the basis of geometric improve-ment proposed by Xuan et al. (2011).In view of this, the paper presents a model that jointly optimizes the main signal and pre-

signal, together with the lane assignment of the approach. To minimize the average delay, afeasible directions method is introduced to solve the mixed-integer non-linear programming.The comparison between the optimized results of the tandem intersection and the conventionalintersection verifies the efficiency of the method and prevents the decrease of capacity arousedby inappropriate signal timing and lane assignment. Therefore, the contribution of this paperlies in the minimization of the average delay in the tandem intersection by considering theeffect of an additional traffic signal, together with the optimization of the lane assignment, signaltiming, and signal sequence.

2. MODEL DEVELOPMENT

Without loss of generality, three assumptions are presented to simplify the analysis. Firstly, thesymmetric signal timing is considered in the paper. As there are only two phases in the main signalof a tandem intersection, the asymmetric signal timing will not be considered in the paper. Secondly,the lost time between each phase at the pre-signal and the main signal are assumed the same. Finally,the right-turns are assumed to maneuver in the exclusive rightmost lanes or behave in the same manneras the through movements; hence, the analysis of right-turns is neglected in the paper.

Figure 1. An example of an approach at the tandem intersections

363TANDEM INTERSECTION OPTIMIZATION

Copyright © 2013 John Wiley & Sons, Ltd. J. Adv. Transp. 2014; 48:362–376DOI: 10.1002/atr

2.1. Arm label and movement direction

The intersection with four arms is investigated, and the arms are marked by using clockwise rotation. Todepict the movement direction, two functions that reflect the label transition of arms are introduced below.

m1 ið Þ ¼ mod iþ 1; 4ð Þ þ 4 1� sgn mod iþ 1; 4ð Þ½ �f g; i ¼ 1; 2; 3; 4 (1)

m2 ið Þ ¼ mod iþ 2; 4ð Þ þ 4 1� sgn mod iþ 2; 4ð Þ½ �f g; i ¼ 1; 2; 3; 4 (2)

where sgn(�) is the sign function. sgn(x) = 1 if x> 0, sgn(x) = 0 if x= 0, and sgn(x) =� 1 if x< 0. mod(�) isa modulo operation to find the remainder of division of one number by another. Arm m1(i) is the armlocated on the left side of arm i, and arm m2(i) is the one located at the opposite direction of arm i. Thefirst part of the equations reflects the label transition, and the second part prevents zeros in some modulooperation. For example, m2(3) = 1 indicates the flow from arm 3 moving to arm 1. Besides, the approach,which consists of the entry lanes on a road, shares the same label with the corresponding arm.

2.2. Signal timing constraints for the main signal

The purpose of the tandem intersection is to utilize more lanes for vehicles, hence discharging morevehicles without producing large delays. To ensure that the configuration operates efficiently, theconstraints for the lane assignment and signal timing will be presented. In the tandem intersection,the asymmetric signal timing is inappropriate because the vehicles are reorganized. Hence, symmetricsignal timing is considered in the paper, and we have

gi;m1 ið Þ ¼ gm2 ið Þ;m1 m2 ið Þð Þ (3)

gi;m2 ið Þ ¼ gm2 ið Þ;i (4)

where gij is the length of green phase and i= 1, 2, 3, 4, j= 1, 2, 3, 4, j 6¼ i. Equation (3) implies that thegreen length for left-turns from arm i is equal to that from arm m2(i). Equation (4) states the symmetricgreen length for through vehicles.Besides, it is obvious that the signal cycle is equal to the sum of all the phases; hence, we have

C ¼ g12 þ g13 þ g23 þ g24 þ 4tlost (5)

where tlost is the lost time between consecutive phases and is the sum of the clearance lost time plusstart-up lost time.

2.3. Signal timing constraint for the pre-signal

Two phases are alternatively provided for the left-turns and through vehicles. To ensure synchronizationbetween the pre-signal and main signal, the cycle should be equal. The constraint is given by

gfi þ gli þ 2tlost þ ri ¼ C (6)

wheregfi andgli are the green length for through and left-turn vehicles, respectively. ri is the red time and is

a slack variable in the optimization. Therefore, ri=0 if the red time is not needed in the pre-signal. Theminimum green length, denoted by gmin

i , is presented to prevent the green time from being too short.

Hence, it is obvious gfi≥gmini and gli≥gmin

i .

364 Y. ZHOU AND H. ZHUANG


2.4. Lane matching constraint

In the tandem intersection, the lane matching is critical to the exploration of the capacity. Thenumber of lanes in the approach i, denoted by ai, shall not be less than the total lanes utilizedby vehicles in the sorting area. Hence, the number of effective lanes for through vehicles in thesorting area is given by

afi;m2 ið Þ ¼ min ai; em2 ið Þ� �

(7)

where em2 ið Þ is the number of exit lanes on the destination arm. Similarly, the number of effectivelanes for the left-turn vehicles is given by

ali;m1 ið Þ ¼ min ai; ; em1 ið Þ� �

(8)

whereem1 ið Þ is the number of exit lanes on the destination arm. Besides, the sum of lanes in one approach isa constant, which is presented as

afi þ ali ¼ ai (9)

where afi and ali are the number of lanes allocated upstream of the pre-signal for through and left-turnvehicles, respectively. ai is assumed to be given in the local intersection.When considering the left-turn bay, the situation is slightly different. If the stop line of pre-signal is

marked inside the left-turn bay, constraint (9) is still valid, because the left-turn bay can be deemed as anormal lane in this condition. If the stop line of pre-signal is located at the upstream of the left-turnbay, another constraint is needed. The sum of lanes utilized by vehicles in the upstream of arm i should

satisfy afi þ ali þ nbayxbay=xi≤ai , where nbay is the number of lanes in the left-turn bay, xbay is thelength of left-turn bay, and xi is the length of the sorting area.

2.5. Capacity matching constraint

To prevent the accumulation of vehicles in the intersection, the vehicles in the sorting area are requiredto be cleared in each cycle. Therefore, the number of vehicles discharged during the green interval ofpre-signal shall be limited to avoid the overflow. Hence, we have

afiifg ≤ afi; m2 ið Þgi;m2 ið Þ (10)

aligli ≤ ali;m1 ið Þgi;m1 ið Þ (11)

The constraint (10) and (11) ensure that even the number of lanes in the destination approaches isless than that in the sorting area; the vehicles can merge into the destination approaches and bedischarged during the green phase.

3. AVERAGE DELAY ESTIMATION

The average delay that vehicles experienced in the tandem intersection consists of two parts, which canbe separately estimated. The delay at the pre-signal is the same as the conventional intersection and isexpected to be determined by the Webster (1958) formula, which is given by



dijpre ¼rij2

2C 1� lij=mij� �� þ Rij

2

2lij 1� Rij

� �� 0:65C

lij2

� �1=3

Rij2þ5gij=Cð Þ (12)

where gi is green length that allows movement from arm i into arm j. rij is effective red time from arm ito arm j, and rij=C� gij, i= 1, 2, 3, 4, j= 1, 2, 3, 4. li is average arrival rate of traffic flow in arm i. lij isaverage arrival rate of movement from arm i to arm j, where lij= liaij and aij is the proportion ofvehicles in arm imoving into arm j. mij is saturation flow rate per lane. Rij is degree of saturation, whereRij= lij/(mijgij/C). In what follows, the average delay of left-turns and through vehicles in pre-signal aredepicted as di;lpre and di;fpre, respectively.Because of the signal coordination, the average delay experienced in the main signal shall be recon-

sidered because the vehicles in the sorting area are reorganized. The delay, however, is dependent onthe signal sequence and can be estimated as follows. Figure 2 presents the average delay estimation atthe main signal of one tandem approach where left-turns lead through vehicles. The offset between thepre-signal and main signal is denoted by ’i, which is set to be the free flow travel time through the

sorting area. The red time in the main signal for the approach is denoted by rm. di;lmain and di;fmain are

the average delay for the left-turns and through vehicles, respectively. The additional delay is inducedby the signal coordination. Hence, we have

di;fmain ¼12jgfi � gi;m2 ið Þj (13)

di;lmain ¼12jgli � gi;m1 ið Þj þ jgfi � gi;m2 ið Þj (14)

The absolute value ensures the green length for main signal can be longer than that for pre-signal. Atime–space diagram that illustrates coordination relationships between the pre-signal and main signalis presented in Figure 3. The leading left-turn phase is operated in the figure. The black bold line repre-sents the left-turn vehicles, whereas the gray bold line depicts the through vehicles.In Equations (13) and (14), the leading left-turn phase is assumed in the pre-signal. If the lagging

left-turn phase is operated in the pre-signal, the delay can be similarly deduced. Considering the signalsequences, the expression can be rewritten as

di;fmain ¼12gfi � gi;m2 ið Þ�� þ 1� dið Þ gli � gi;m1 ið Þ

�� (15)

di;lmain ¼12gli � gi;m1 ið Þ�� þ di g

fi � gi;m2 ið Þ

�� (16)

Figure 2. Extra delay between the pre-signal and the main signal.



where di is binary variables. di = 1 if left-turns lead through vehicles, and di= 0 if left-turns followthrough vehicles. The average delay that vehicles experience in approach i is hence summarized as

di ¼ ai;m1 ið Þ di;lpre þ di;lmain

þ ai;m2 ið Þ di;fpre þ di;fmain

(17)

where aij is the proportion of vehicles in approach i moving into arm j. If the right-turns are neglected,ai;m1 ið Þ þ ai;m2 ið Þ ¼ 1.Therefore, the average delay of an intersection is given by

D ¼

X4i¼1

lidi

X4i¼1

li

(18)

where li is the arrival rate of approach i.The optimal lane assignment and signal timing at the tandem intersection are expected to be

obtained by minimizing Equation (18). Constraints (3) to (11) ensure that vehicles can orderly andsafely discharge from intersections. The problem is hence converted to a mixed-integer non-linearprogramming, and the algorithm is presented below.

4. ALGORITHM FOR THE MIXED-INTEGER NON-LINEAR PROGRAMMING

The average delay above is dependent on the decision variables, which include four green length

variables gij at the main signal, eight green length variables gfi and gli at the pre-signal, two binary

variables of signal sequence di, and the discrete lane assignment variables afi (or ali). The non-linear

programming with continuous and discrete variables is difficult to be solved because the computational

complexity increases exponentially with the combination of lane assignment variable afi . However, thenumber of lanes in one approach is limited and can be enumerated on the basis of the local geometric

configuration. For example, if there are five lanes in an approach, the combination of variables afi of the

Figure 3. A time–space diagram.



intersection is less than 44 because there are only four feasible assignments available in one approach.Therefore, the problem is converted to a non-linear programming with given value of lane assignmentvariables. The given input includes the total number of lanes in each approach ai, the destinationexit ei, the lost time tlost, the arrival rate li, the turning proportion aij, and the saturation flow rate mij.A feasible directions method is introduced in this paper to solve the non-linear programming. The

method of feasible directions was first proposed by Zoutendijk (1960) and later improved by Topkisand Veinott (1967) and Zangwill (1969). The steps are presented as follows:

Step 1 Let Km be the total number of lane assignment plans and Dmin be the minimized average delaywith the initial value Dmin = +1.

Step 2 Given a lane assignment, assign K= 1, where K is for the different lane assignment plans andK≤Km.

Step 3 Define G = (g13,g12,g24,g23) as a continuous variable set of main signal and the average delayfunction f(G) =D(G).

Step 3.1 Predetermine two sufficient small e1> 0 and e2> 0, and choose an initial value G(0)2R andlet k = 0.

Step 3.2 Judge the state of the active constraints indicator set J(G(k)) = {j|gj(G(k)) = 0, 1≤ j≤ l} to

check if all the inequalities are binding, where l is the number of inequality constraintsand gj(G

(k)) is the jth inequality. If J(G(k)) 6¼∅ (∅ is the empty set), go to step 3.3. If J(G(k)) =∅ and ||r f(G(k))||2≤ e1, the iteration is halted, and the final value G(k) is obtained.If J(G(k)) =∅ and ||r f(G(k))||2> e1, choose the direction vector z(k) =�r f(G(k)) and jumpto step 3.5.

Step 3.3 Solve the following linear programming to attain the optimal z(k) and �k.

min �k

s:t:

rf G kð Þ� �Τz≤�k

�rgj G kð Þ� �Τz≤�k;

j 2 J G kð Þ� �� 1≤zi≤1; i ¼ 1; 2; . . . ; n

8>>><>>>:

where z= (z1,z2, . . .,zn) is the feasible direction.Step 3.4 If |�k|≤ e2 is satisfied, the iteration is halted, and we have G(k); else go to next step.Step 3.5 Solve the following one-dimensional optimization problem lk : min0≤l≤�l f G kð Þ þ lz kð Þ� �

,

where �l ¼ max ljgj G kð Þ þ lz kð Þ

≥0; j ¼ 1; 2; . . . ; ln o

.

Step 3.6 Let G(k+ 1) =G(k) + lkz(k) and replace k by k+ 1 and jump to step 3.2.

Step 4 Compare D with Dmin. If D<Dmin, then update Dmin with D and save the optimal lane assign-ment K* and signal timing G*. If else, go to the next step.

Step 5 If K<Km, update K :=K + 1 and jump to step 2.Step 6 Save minimal average delay Dmin, optimal lane assignment K*, and signal timing G*.

Figure 4 presents a flowchart of the algorithm above. All steps except for step 3 are depicted inFigure 4(a), whereas step 3 is shown in Figure 4(b). The flowchart of Figure 4(b) should be insertedinto the process of “use the direction method” in Figure 4(a).

5. NUMERICAL EXAMPLES

An at-grade intersection with four arms is considered to demonstrate the advantage of tandem intersec-tion and the delay analysis. The optimal shared-use lane assignment and signal timing are presented,and the sensitivities of the optimized average delays are investigated, which include the turningproportion, the arrival rate, and the signal sequence. The geometric layout and the value of relevantparameters are as follows.



As is shown in Figure 5, three lanes are available in arm 1 and arm 3, whereas four lanes in arm 2and arm 4. The lane marks are not fully identified because the remaining lanes are to be determinedthrough the joint optimization with signal timing. The right-turns, which is easy to be incorporatedin practice, are neglected to simplify the analysis. The sorting area is not divided because the signalsequence will be considered as well. The purpose, therefore, is to obtain the optimal green length,signal sequence, and lane assignment, given the arrival rate and turning proportion in each arm.Two cases are hence presented with arrival rate and turning proportion varied, respectively.In the first case, the proportion of through vehicles is assumed to be 60% in all directions, and the

arrival rate in arm 1 (or arm 3) is fixed to 1080 veh/h, whereas the arrival rate in arm 2 (or arm 4)varies. Figure 6 depicts the average delay comparison between the conventional intersection and thetandem intersection. Both the results are optimized. The dash line represents the average delay withoptimal signal timing at the conventional intersection, whereas the solid line is that of the tandemintersection. The delay of conventional intersection increases with the arrival rate and sharply leapsinto high level because of the high degree of saturation. The tandem intersection, however, is ableto handle the high arrival rate with 2880 veh/h from both arm 2 and arm 4. The advantage lies inthe high capacity of the configuration, which ensures the relative low degree of saturation beingmaintained. Table I summarizes the optimal results of tandem intersection. As symmetric signaltiming is operated, the green length of the main signal in arm 1 and arm 2 is shown. Although theoptimal green length in main signal and the signal cycle are increased with the arrival rate, the greenlength of pre-signal in the perpendicular arm, however, are less affected.The turning proportion’s effect on the average delay is shown in the second case. The arrival rate in arm

1 and arm 2 are set to 1080 and 2700 veh/h, respectively. The proportion of through vehicles in arm 2 isvaried from 20% to 90%, whereas the proportion of arm 2 is fixed to 60%. As shown in Table II, the greenlength for through vehicles at main signal and pre-signal is increased with the ratio of through vehicles.The signal cycle first decreased and later increased with the through proportion in arm 2 and arm 4. If

Figure 4. The flowchart of the algorithm.



the proportion of through vehicles is 90%, it seems absurd that two lanes are still provided for left-turns,considering that only four lanes are available. This is due to the value of minimum green length, which isset to 8 seconds for the pre-signal signal in the example. If the minimum length is enhanced, 10 secondsfor instance, the optimal lane assignment and signal sequence is the same as the scenario with 80%through proportion. The optimal sequence and lane assignment obtained imply that the signal timingand lane assignment are complementary in the tandem intersection. However, if all the lanes are effectivelanes and the turning ratio are proportional to the lane assignment, the optimal sequence can be pre-determined, which is addressed in the following discussion.

6. DISCUSSION ON SIGNAL SEQUENCE

The model results present the optimization of tandem intersection and the sensitivities of relevant para-meters on the average delay. In what follows, we will explicitly discuss the optimal signal sequence

Figure 5. An example of a tandem intersection with a blank lane mark.

Figure 6. The average delay comparison between the conventional intersection and the tandem intersection.



Table

I.The

optim

alresults

ofthetandem

intersectio

nwith

thearrivalrate

ofarm

2(and

arm

4)varies.

Arrival

rate

(veh/h)

Green

length

ofthemainsignal

(s)

Green

length

ofthe

pre-signal

atarm

1andarm

2(s)

Signalcycle(s)

Lanes

allocated

forleft-turns

atthe

upstream

ofpre-signal

Signalsequence

inarm

1andarm

2

g 2,4

g 2,3

g 1,3

g 1,2

gf 1gl 1

gf 2gl 2

al 1al 2

Arm

1Arm

2

720

1010

1110

1630

1330

571

1FLLT

FLLT

1080

1010

1010

1530

2015

561

2FLLT

FTLL

1440

1010

1010

1530

2015

561

2FLLT

FTLL

1800

1211

1010

1530

1633

591

1FLLT

FLLT

2160

1614

1110

1630

3221

661

2FLLT

FTLL

2520

2018

1210

1830

4027

761

2FLLT

FTLL

2880

2728

1611

2433

5436

981

2FLLT

FTLL

Note:

Sym

metricarrivalrate

isassumed

inthescenario.

FLLT,F

irst-Left-Last-Through;FTLL,F

irst-Through-Last-Left.



Table

II.The

optim

alresults

ofthetandem

intersectio

nwith

theturningproportio

nof

arm

2(and

arm

4)varies.

The

ratio

ofthrough

vehicles

inarm

2(orarm

4)

Green

length

ofthe

mainsignal

(s)

Green

length

ofthe

pre-signal

atarm

1andarm

2(s)

Signalcycle(s)

Lanes

allocatedfor

left-turns

attheupstream

ofpre-signal

Signalsequence

inarm

1andarm

2

g 2,4

g 2,3

g 1,3

g 1,2

gf 1gl 1

gf 2gl 2

al 1al 2

Arm

1Arm

2

0.20

1066

1813

2838

4066

123

13

FLLT

FTLL

0.30

1047

1510

2230

4047

981

3FLLT

FTLL

0.40

1340

1510

2231

2760

941

2FLLT

FTLL

0.50

1631

1410

2130

3247

871

2FLLT

FTLL

0.60

2119

1310

1930

4228

791

2FLLT

FTLL

0.70

2314

1210

1830

4721

751

2FLLT

FTLL

0.80

3110

1310

2030

4131

801

1FLLT

FLLT

0.90

3310

1310

2030

668

821

2FLLT

FTLL

Note:

Sym

metricarrivalrate

isassumed

inthescenario.

FLLT,F

irst-Left-Last-Through;FTLL,F

irst-Through-Last-Left.



and lane assignment. Two assumptions are presented to simplify the discussion: one is that the numberof lanes in all arms is assumed to be equal and the other one is that the number of lanes utilizedupstream at the approach is proportional to the turning proportion.If the left-turns lead through vehicles, the sequence is denoted as First-Left-Last-Through

(FLLT, i.e., leading left turn phase). Conversely, First-Through-Last-Left (FTLL, i.e., lagging leftturn phase) represents the left-turns follow through vehicles. Because symmetric signal timing isrequired in the tandem intersection, only four signal sequences are available. For example, FLLTin approach i and FTLL in approach m1(i) are a feasible signal sequence. Despite the optimizedresults proposed earlier, a simple guideline for choosing the signal sequence in a special case willbe introduced. We consider a full tandem intersection with all lanes being utilized by through andleft-turn vehicles. The number of lanes in all the arms is assumed to be equal; hence, we have

ali;m1 ið Þ ¼ afi;m2 ið Þ ¼ ai ¼ ei. To ensure that the space in the sorting is being fully used, the following

equation should be satisfied:

gli ¼aialigi;m1 ið Þ (19)

g fi ¼ ai

a fi

gi;m2 ið Þ (20)

where a fi and ali are the number of lanes allocated upstream of the pre-signal for through and left-

turn vehicles, respectively. gi;m1 ið Þ and gi;m2 ið Þ are the green length at the main signal, whereas gfi and

gli are green length at the pre-signal. Equations (19) and (20) depict that the number of vehicles dis-charged at the pre-signal should be equal to the number of vehicles discharged at the main signal inone cycle.For FLLT, by substituting Equations (19) and (20) into (15) and (16), the average delay aroused by

the signal coordination can be obtained. The delay of through vehicles is di; fFLLT ¼ 12alia fi

gi;m2 ið Þ, and the

left-turns is di;lFLLT ¼ 12afialigi;m1 ið Þ þ ali

a fi

gi;m2 ið Þ . Therefore, the average delay arouse by coordination in

sequence FLLT is

diFLLT ¼ 12alia fi

gi;m2 ið Þai;m2 ið Þ þ 12a fi

aligi;m1 ið Þai;m1 ið Þ þ ali

a fi

gi;m2 ið Þai;m1 ið Þ (21)

For FTLL, the delay are di;fFTLL ¼ 12afialigi;m1 ið Þ and di;lFTLL ¼ 1

2aliafigi;m2 ið Þ þ afi

aligi;m1 ið Þ , respectively. The

average delay arouse by coordination in sequence FTLL is given by

diFTLL ¼ 12afi

aligi;m1 ið Þai;m1 ið Þ þ 1

2aliafi

gi;m2 ið Þai;m2 ið Þ þ a fi

aligi;m1 ið Þai;m2 ið Þ (22)

The delay comparison between (21) and (22) is expected to determine the signal sequence. It is

obvious that the FLLT is superior to FTLL ifafialigi;m1 ið Þai;m2 ið Þ >

aliafigi;m2 ið Þai;m1 ið Þ is satisfied. For the

conventional main signal, if we assume equal degrees of saturation, which means the ratio of arrivalflow rate to saturation flow rate is equal, the following equation is satisfied:



gi;m1 ið Þgi;m2 ið Þ

¼ ai;m1 ið Þai;m2 ið Þ

(23)

Hence, we have gi;m1 ið Þai;m2 ið Þ ¼ gi;m2 ið Þai;m1 ið Þ . Therefore, the FLLT is dominant if only ali < afi is

satisfied. The result implies that vehicles allocated with fewer lanes at the upstream approach of thepre-signal should be discharged in advance. In the following analysis, we will verify that fewer lanesare consistent with low proportion of vehicles. Hence, if the left-turns account for large share of thearrival, the through vehicles should lead the left-turns, and conversely, if the proportion of throughvehicles is high, the through vehicles should follow the left-turn vehicles. An approach with three orfour lanes, which are the most common scenarios in urban intersections, will be analyzed.Without loss of generality, we assume

gi;m2 ið Þgi;m1 ið Þ

¼ ai;m2 ið Þai;m1 ið Þ

¼ x and gi;m1 ið Þai;m1 ið Þ ¼ p. Then gi;m2 ið Þai;m2 ið Þ ¼x2p, and gi;m2 ið Þai;m1 ið Þ ¼ gi;m1 ið Þai;m2 ið Þ ¼ xp. Equations (21) and (22) can be rewritten as

diFLLT ¼ 12

alia fi

x2 þ afi

aliþ 2

alia fi

x

!p (24)

diFTLL ¼ 12

aliafi

x2 þ afiali

þ 2afialix

!p (25)

For the intersection with three lanes in all the arms, that is, ai = 3, it is obvious that if x> 1 (or

ai;m2 ið Þ > ai;m1 ið Þ), the optimal lane assignment is afi ¼ 2 and ali ¼ 1. Meanwhile, FLLT is superior toFTLL, which verifies that the vehicles with relative low proportion should be discharged inadvance to reduce the delay. Conversely, if x< 1 (or ai;m2 ið Þ < ai;m1 ið Þ ), the lane assignment with

afi ¼ 1 and ali ¼ 2 is preferred. The signal sequence makes no difference if x=1 (or ai;m2 ið Þ ¼ ai;m1 ið Þ) .For the intersection with four lanes in all the arms, that is, ai= 4, three lane assignments are avail-

able. Therefore, it is easy to prove that the optimal lane assignment is afi ¼ 3 for the scenario x> 1,and the corresponding sequence is the FLLT, because the other options are with higher average delay.The results for the other two scenarios are presented accordingly in Table III. The results verify thatthere is more chance to fill the sorting area and fully utilize the green time by letting the vehicles withlow proportion discharge first.Because the green spilt and the lane assignment are complementary at the upstream of the pre-

signal, the signal sequence may change according to the local configuration. For instance, shortgreen time with more lanes may serve the same function as long green time with fewer lanes insome conditions. Therefore, the judge is relatively rough and only recommended in the earlier stageof assessment.

Table III. Optimal sequence for all arms with four lanes.

afi ¼ 3; ali ¼ 1 afi ¼ 2; ali ¼ 2 afi ¼ 1; ali ¼ 3

x> 1 FLLTa FLLT FLLTx= 1 FLLT or FTLL FLLTa or FTLLb FLLT or FTLLx< 1 FTLL FTLL FTLLb

Note: FLLT, First-Left-Last-Through; FTLL, First-Through-Last-Left.aRepresents the optimal sequence for all the lane assignment, given the turning proportion.bThe sequence for the scenario with lowest average delay.



7. CONCLUSION

In this paper, an integrated model for lane assignment and signal timing optimization at tandemintersections is proposed. The model aims to minimize the average delay that vehicles experiencedin the pre-signal and main signal. The constraints required to maintain the orderly movements in theintersection are proposed to ensure that the capacity is being fully explored. The optimal results provethat the tandem intersections outperform the conventional intersections, and the average delay isreduced compared with that of the previous research. Besides, the sensitivities analysis, which includesthe arrival rate and turning proportion, is investigated. Furthermore, the optimal sequence correspondingto the turning proportion and the optimal lane assignment is discussed. If the number of lanes is equal inall directions, the results conclude that vehicles with low proportion and fewer lanes should be dischargedin advance to reduce the average delay.Some limitations of the experimented tandem intersection are addressed below. Firstly, the paper

only investigates the configuration without left-turn bay. Although the left-turn bay is mentioned inthe lane matching constraint, the paper mainly focuses on the configuration that the number of lanesin sorting area and upstream approach is the same. The general case including the left-turn bay willbe investigated in the future work. Secondly, there is lack of the empirical evidence in the study.Because the tandem intersection is recently proposed in Xuan et al. (2011), the site that utilizes tandemintersection is few. The comparison between the empirical data and the results in the model may beconducted in the future. Thirdly, if the lanes in an approach are more than that in the destinationarm, the vehicles have to merge into the limited lanes. Although the green length is adequate to releaseall the vehicles in the sorting area, a merging process in the paper is ideally considered. Finally, the modelin the paper requires that the tandem configuration is operating at capacity. When applied in practice, thecapacity may have reservation in case of unexpected situations.

ACKNOWLEDGEMENTS

This work is supported by the Natural Science Foundation of Anhui Province (grant no.11040606M19) and Humanities and Social Science Foundation of Ministry of Education of China(grant no. 12YJA630201). The authors would like to thank the three anonymous reviewers for theirvaluable comments.

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the optimization of lane assignment and signal timing at the tandem intersection with pre-signal

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