Multi-Path Signal Synchronization Model with Phase Ratio and Dispersion Constraints

Proper designs of offsets forsignalized intersections in an urban arterial system can help improve the level of service (LoS) significantly. The MAXBAND model has been the most widely used approach for this purpose. In this paper, a MAXBAND model in which the lengths of phases in a cycle are variable is proposed. In other words, the lengths of phases in a cycle, as well as the cycle itself, are no longer fixed.Besides, theincorporation oftraffic dispersion module in the proposed model allows that the travel times from upstream intersection to the downstream one are taken into account .Real-world application on an arterial system in Taiwanfor evaluating the performance of the proposed model is conducted to validate the methodology. Actual traffic flow data are collected through on-site experiments. Results suggest that the improvements are around 44%, on average, in terms of total delay of the entire network.


Introduction
An effective signal control strategyis crucial in imp roving the LoS fo r coordinated signalized intersections.By adjusting the offsets of the coordinated intersections in an arterial system, the control delays and the stopping times of vehicles can be well controlled, thereby providing better LoS.The MAXBA ND model, proposed by Morgan and Little 1, is one effect ive approach for this purpose.The concept of the MAXBAND model is to maximizethe total progression bandwidth of a two-way coordinated signalized system by adjusting the offsets at each intersection over atime interval.
The very first MAXBAND model was proposed by Little et al.2 in 1966.The model encompasses time constraintsdescribing timerelationship between the upand down-stream intersections.Chang et al.3proposed the MAXBAND-86 that extendsLittle's model to solve mu lti-arterial closed network problems.
Yet, neithermodel has considered the lead-lag left-turn ing phases and the queuing clearance times.Gartner et al.4solved the problem by developing a mult iband model thatincorporates the left-turn phase sequence optimizat ion and initial queue clearance t imes .Themajor contribution of multiband model resides in the consideration of the flow volume in the model.
Based on the multiband model, several extensions have been proposed thereafter.Stamatiadis et al. [1]67proposed the Multiband-96 model, extending the mu ltiband model to arterial networks.Lu et al. 8extended the mult iband model by incorporating the travel time constraints with traffic dispersion module9.Zhang et al.10improved Gartner's model by relaxing the limitation of symmetrical progression.However, all these models above do not consider the turning flo w fro m the minor legs crossing the arterial.In this case, the efficiency of the arterial systemmay increase, whereas that of the entire network may not necessarily.Most recently, Yang et al.11proposed a multi-path progression model featuring the optimizat ion of the phase sequences and the determination of crit ical paths in the associated network.This model can handle the intersections with high turn ing volumes on the minor legs by adding a new path through the arterial.
A major restriction in Yang's model resides in that the length of the co mmon cycle, as well as that of the phases, for the coordinated intersections are fixed.Such a restriction may possibly lead to unsatisfactorylocal solutions.Therefore, based upon this observation, an extension of their model by relaxing such restrictionsis proposed so that the lengthsof phases in a cycle, as well as the cycle itself, become variable.Such flexib ilit ies can certainly lead to better solutions, compared to the fixed case.Besides, theincorporation ofRobertson's traffic dispersion module9allows oneto take into accountthe travel times fro m the upstream intersection to the downstream one, thereby providing more realistic results.Consequently, the resultant model beco mes a mixed-integer quadratically constrained programming (MIQCP) problem wh ich is NP-hard 1213in essence.We then exploit Gurob i solver14to search for an approximate feasible solution.Plus,emp irical e xperiments are conducted for evaluating our methodology, and the results suggest that the performance of our model co mes with an improvement being around 44% on average, in terms of total control delay per minute.

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This is how the rest of the paper is organized.In Section 2, we first review of base model proposed by Yang et al.11, and then introduce the revised model thereafter.In Sect ion 3, we elaborate on how the experiments as well as the numerical simulat ions are performed.Conclusions will be given in Section 4.

Model Specifications
In this section, the review of the mu lti-path progression model proposed by Yang et al.11is given.Thereafter, tworevised models, one by relaxing the phase ratios in the model, and the other by adding the dispersion module to the model, are introduced.

Multi-path progression model -A review
Variables to be used in the mult i-path progression modelare given inTable 1. .M is a penalty parameter.The objective function ( 1) is to maximize the sum of bandwidths of outbound ( ) and inbound paths .The weights can be adjusted in accordance with the volumes of the associated paths.Here we have .Constraints (2) and (3) are interference constraints.They guarantee that the progression band of path will only use the available green t ime at intersection i. Constraints (4) and ( 5) are loop constraints, representing the relationship of the intersection i and i + 1 while they are synchronized.
Constraints ( 6)-( 17) are phase sequenceconstraints at each intersection so that the green band competition amongdifferent paths are controlled.Specifically, Constraints ( 6)-( 11) are the boundary conditions for the left/right side red t imes of p rogression band.Constraints ( 12)-( 17) ensure that the resulting phase sequence is feasible.

Revised model 1: Making a variable
In the progression model introduced in Section 2, the phase ratio is a constant.Here this restriction is relaxed and is now made a variab le.By doing so, constraints associated with become nonlinear in essence.The problem then beco mes a nonlinearly constrained mixed integer optimization problem, o r more specifically, a mixed integer quadratically constrained programming (MIQCP) problem.Besides,constraints ICTTE 2017 (15)-(17) will not be used, similar to the man ipulation in Yang et al.11 Along with the relaxation onthe phase ratio, the restriction that the cycle length is also fixed is also relaxed.To do this, we addconstraint (18) to ensure that the cycle length is now varying butbounded. (18 where Z is the reciprocal o f the cycle length, and e (f) is the upper (lower) bound of cycle length.Based on (18), the travel time fro m intersection i to intersection i+1 can then be formulated as where ( ) is the distance from outbound (inbound) intersection i to i+1; and ( )is the free flo w speed of the link.
The sum of the phase lengthsatany intersection must equal the cycle length; in other words, the total ratio of phases at an intersection must equal 1.Thus, (21) To ensure that the resulting MIQCP is solvable,according to13, the model must come withlo wer boundson the phase ratio variable , to begiven in (22).To correctly set the bounds for the ratio variab le, we resort to the capacity equation discussed in Roess et al. 15, given by where is the saturation flow rate for lane or lane group q, is the effect ive green time for lane or lane group q, and C is signal cycle length.This equation can be rewritten as .Since the critical volu me of lane will never be bigger than the capacity ,we then have .Here,the ratio can be considered as aproportion of the phase length to the cycle length, which isalso the ratio of phase( ).Consequently, we have Therefore, the lower bounds for the phase ratio variable are given below The threshold 0.9 is set to evade over-saturation, so that the remain ing one-tenth of the entire cycle can be used as further adjustments, if needed.

Revised model 2: adding traffic dispersion
In RMPPM 1, the parameter is mainly defined by the ratio of distance between an intersection to thedownstream to the free flo w speed of the link .In the development of MAXBAND model, there aretwo methods to process .One is to make it a constant, and the other is to set a boundary for it, see LP2 in Little et al.2.Both methods have their disadvantages.For the first method, the real free flo w speed might never be reached even when traffic is moderate to dense; and the second method requires new variables, thereby introducing more complexit ies to the problem.Based upon this, Lu where , . is thetravel time of fastest vehicle between intersectionsi and i+1 in an outbound(inbound) path, while is thetravel time of slowest vehicle between intersection iand i+1 in an outbound(inbound) path.

Empirical study
Areal-world applicationis conducted on the proposed models.The actual flow data of an arterial system,containing a highway ramp and surface signalized intersections,in Chubei City in Taiwan are analysed.Figure 1shows the topology for the underlying arterial system.The traffic data were collected during 17:00 p.m. -21:00 p.m. in April 13th, 2013.Fivecritical paths are chose for evaluation in accordance with the traffic data.Path 1 contains the traffic flow fro m the northbound offramp through the major arterial; paths 2 and 3 pass through the major arterial in opposite directions; and paths 4 and 5 are fro m minor legs of the surface signalized intersections to the major arterial.We mark the three intersections, fro m right to left , as 1, 2 and 3,

Numerical implementations
For nu merical imp lementations, both problem solving and traffic simu lations for performance evaluations are carried out.For problem solving, a C++ program is coded for connecting theGurobisolver for obtaining a feasible timing plan for the proposed models introduced in Section 2. Obtained signal timing p lansthen serve as the input to the traffic software integrated system (TSIS) for random simu lations.A total of 500 independent replicat ions are performed and output the associated performance measures.
On-going timing plans as well as those suggested in Yang et al.11are used for co mparisons.Designs of phasesat each intersection of the two plansaregiven inTable 2and Table 3. Fro m Tab le 4, we see that the cycle length suggested by the proposed model is now longer than the orig inal timing plan.Fro m the t iming plan in Plan b, we observe that the lengths of the phase 1'sat intersections 1 and 2 have increased.In part icular, this allows mo re traffic on the major arterial in outbound directions (path 1 and path 2).On the other hand, fro m the result of Plan c, the cycle length decreases in Plan a,while it increases in Plan b.Also, comparing Plan b and Plan c, the ratio of the phase 1 decreases while the ratio of phase 2 increases.Thissuggests that sacrificing the benefit of outbound pathscan improve that of the inbound paths.
Fro m Table 5, as forthe plan suggested by Yang et al., we can see that comparing the original p lan (Plan x), the cycle length decreases both in the results of RMPPM 1 (Plan y) and RMPPM 2 (Plan z).For intersection 3 with high right-turn traffic, the model result suggests a longer length of that phase so that the right-turn traffic can dissipate more therein.This can be observed fro m the results that the ratio of phase 4 at intersection 3 increases.The cycle length of Plan z is longer than Plan y, but all the phase ratios are closely the same.

Performance evaluations
To evaluatethe system performance under different signal timing plans, we adopt the total control delay per minute in the network as the performance measure.Figure 2 andFigure 3 show the results for the timing plan containing in Table 4 and Table 5, respectively.
Colored lines in Figure 2, fro m the top to the bottom,correspond tothe total control delay per minute of plans a, b, and c, respectively.It is apparent fro m Figure 2 that Plan c outperforms the rest planswith respect to the on-going timing plan.Co mpared to the original signal (Plan a), the imp rovement with Plan c is around 44.39% in terms of total control delay per minute of the network; and around 15.59% co mpared to plan b.It can be noticed that Plan c, as a matter of fact, co mes with the shortest cycle length.We infer that while the cycle length becomes longer, the phase lengthsbecome longer.In this case, at the intersection 1, the left-turning flow fro m the westbound will form a queue if the red time is too long, preventing the eastbound left-turning traffic flow at intersection 2 from dissipating.
Lines on Figure 3 are the total control delay per minute of p lans x, y, and z, respectively.Results show that Plan y is the best amongst the rest.Theimprovement is around 59.57% of delay co mpared to Plan x; and 13.12% of delay compared to Plan z.
We further compare the performance of each path under the Plan a to Plan c and plan y.Th is is because Plans c and y are respectively the best alternatives derived fro m solving the RMPPM 1and RM PPM 2. When we co mpare Plan c to Plan a, the improvements of the paths are around 20-52% on average, resulting basically fro m the imp rovements on the arterial paths (Path 2 and 3).In terms of delay, the improvements are around 37%-49%.Yet, when co mpared to Plan y, even though the performance of Path 2 and 3 has improved around 17%-DOI: 10.1051/matecconf/201712401002 ICTTE 2017 65%, the performance of the minor paths is less satisfactory, being around -7%-33%.

Conclusions
In this paper, extensions of the signal synchronization model proposed in Yang et al.11are g iven by letting the ratios of phases become variab le, and by incorporating the traffic flow dispersion module for adjusting the travel times.Nu merical evidences suggest a total improvement of around 44.39% and 59.57% with on-going t iming plan and those suggested by Yang et al.11, respectively, in terms of the total control delay per minute of the entire network.We observe that in this experiment, the results of RM PPM 2 are not always better than that of RMPPM 1. Th is may be due to interference of the paths that we have not considered in the experiment -such as the path flow fro m the northbound of intersection 2 and the leftturning flow at intersection 2 -or the red time is too longer to remove the queue in each cycle and keep the progression.
Co mparisons on the performance of each path under different plans are carried out as well.Results show that compared to the on-going timing plan (Plan a), with Plan c, the total control delay of each path can be improved around 20% to 52%; with Plan y, the total control delay of paths main ly in the major, such as path 2 and path 3, will be imp roved around 17% to 65%.But when it co mes to paths through the branches, the results of our model have a smaller contribution for the total control delay of them, due to the difference of the signal design.
As of this moment, d iscussions of the signal synchronization problem still are limited by assuming that the design of the phases comprising a cycle is predefined.A potential future research direction resides in the optimal design of phases for signalized intersections.A proper design of phases can certain help improve the LoS of both the underlying signalized intersections and the entire network associated.

Figure 2 Figure 3 .
Figure 2 Delay of Control of Chubei Network with On-going Signal Timing Plan

Table 1 .
Notations et al.8proposed a model that co mbined the mu ltiband model proposed by Gartner et al.4and the traffic flow d ispersion model proposed by Robertson9, so that the calculation of the link travel timecan be more realistic.

Table 4
M odel Results For On-going Timing Plan

Table 5 M
odel Results Timing Plan Suggested inYang et al.11