CAPACITY OF FREEWAY MERGE AREAS WITH DIFFERENT ON-RAMP TRAFFIC FLOW

This paper is aimed at investigating the influence of different types of traffic flows on the capacity of freeway merge areas. Based on the classical gap-acceptance model, two calculating models were established specifically considering randomly arriving vehicles and individual difference in driving behaviours. Monte-Carlo simulation was implemented to reproduce the maximum traffic volume on the designed freeway merge area under different situations. The results demonstrated that the proposed calculating models have better performance than the conventional gap-acceptance theory on accurately predicting the capacity of freeway merge areas. The findings of research could be helpful to improve the microscopic traffic flow simulation model from a more practical perspective and support the designing of freeway merge areas as well.


INTRODUCTION
On freeways, all entering maneuvers take place on ramps that are designed to facilitate smooth merging of on-ramp vehicles into the freeway traffic stream [1].Interactions are dynamic in freeway merge areas, and so the operating conditions on the freeway can affect the operating conditions on the ramp, and vice versa [2].Thus, understanding the behaviour of on-ramp flows and estimating the capacity of freeway merge areas are of critical importance in developing effective operational strategies for freeway management [3][4][5].
In the recent twenty years, many scholars have focused their studies on the capacity of freeway merge areas.The most widely used method of calculating the capacity of freeway merge areas has been known as the gap-acceptance model [6][7][8][9].Previous researchers have suggested that gap-acceptance models cannot only provide a scientific approach to studying traffic control maneuvers [10,11], but they can also calculate vehicle queue length on the ramp [12,13] and the service quality of the freeway merge areas [14,15].In those studies, gap-acceptance is defined as the process in which a driver accepts an available gap between two vehicles.The model requires that drivers of on-ramp vehicles inevitably choose to merge onto the freeway as soon as they come across the best headway.In the existing studies, on-ramp vehicles are supposed to be homogeneous and the best headway is usually considered to be the minimum headway into which the on-ramp vehicles could merge onto the freeway.According to this hypothesis, drivers of on-ramp vehicles often make similar choices when encountering the same headway.They either choose to merge onto the freeway or wait for a more reasonable headway.
However, Winnie Daamen et al. concluded that the existing gap-acceptance theories have not accurately taken driver behaviour into consideration during the entire research process [16].Vehicles which merge onto the freeway from a ramp follow a certain probability and show up randomly.Sometimes, there is no vehicle on the ramp when the best headway appears.Be-sides, Alexandra Kondyli and Lily Elefteriadou suggest that driver behaviour can have a significant impact on the capacity of freeway merge areas [17].Moreover, various studies have been conducted to analyze the influence of the behaviour of drivers.Cheng-Chen Kou and Randy B. Machemehl certified that the ramp vehicle merging position in relation to the freeway has a statistically insignificant effect upon individual traffic parameters [18].J. Wu, M. McDonald and K. Chatterjee carried out a detailed investigation to study the potential impacts of ramp metering on driving behaviour [19].They believed that ramp metering indeed results in changes of driver behaviour when considered in the overall context of traffic on the motorway, the carriageway and the on-ramp.Alexandra Kondyli and Lily Elefteriadou developed a ramp-merging model to consider the merging process as perceived by drivers and to investigate the contribution of individual drivers' merging behaviour to the breakdown event [20].The study indicated that when applying the gap-acceptance model to estimate the capacity of a freeway merge area, if different driving behaviours are seriously considered, more valid results will be achieved.
Accordingly, as a branch of those studies, this paper takes random arrivals of vehicles and different types of driving behaviour into account when evaluating the capacity of the freeway merge areas.In the rest of this paper, at first, based on the gap-acceptance model, the study compares two calculation models in order to assess the capacity of freeway merge areas.After that, the characteristics of different models are further studied by applying the Monte-Carlo simulation technique.The main research achievements and potential directions for future research are presented in the section of conclusions.

Gap-acceptance model
Gap-acceptance is defined as a process by which a driver accepts an available gap between two vehicles [21].A gap is the time between two successive vehicles from the rear bumper of the front vehicle i to the front bumper of the second vehicle i 1 -.It can be described as tmgi in Figure 1.
In the gap-acceptance model, drivers will never merge onto the freeway if the gap of freeway stream is smaller than the critical gap tmilag .Conversely, drivers will normally merge onto the freeway as long as the freeway stream gap is larger than tmilag .
In practical use, it is difficult to detect the vehicle gap data.In this paper, headway, as the most widely used variable is introduced into the model.Headway is defined as the time gap between two successive vehicles from the front bumper of the front vehicle i to the front bumper of the following vehicle i 1 and is represented as tmi in Figure 1.So, in a strict sense, tmgi is not same as tmi and the relationship between the two variables can be described as

= -
For a time interval of duration T, the total number of freeway vehicles is T m m .Meanwhile, the amount of time gap where a k vehicle platoon could merge onto freeway from the ramp is TP m k m .In this situation, the total volume of vehicles that can merge onto the freeway from the on-ramp could be calculated by Equation (4).
where n is the flow rate of vehicles that can merge onto freeway from the ramp in headway tmi .Substituting Equation (3) into Equation (4) yields There is a geometric progression that can be found in the right side of Equation (5).According to the geometric sequence sum calculation, Equation ( 5) can be represented as The maximum flow volume of vehicles on ramp can be calculated as Therefore, the formula for calculating the capacity of the merge area can be inferred as where, qR12 is the capacity of freeway merge area; q1 is the capacity of freeway lane 1; q2 is the capacity of freeway lane 2. However, there are two problems that have not been sufficiently considered in Equation ( 8).On the one hand, although vehicles which merge onto freeway from the ramp, turn up following a particular distribution; in general their emergence takes place at random.This kind of randomness cannot ensure the vehicle merging onto freeway when the best headway comes across.This means that the actual traffic which can merge onto freeway from the ramp is smaller than the theoretical value.On the other hand, different types of driving behaviours will make great impact on the capacity of freeway merge areas.Therefore, in order to improve the accuracy of the equation, random arrivals of vehicles and different types of driving behaviour will be further studied in this paper.

Case A: on-ramp vehicles arrive at random
In this case, on-ramp vehicles arrive at random.The headway of ramp vehicles is exponentially distributed with mean / 1 r r n m = . So, at time interval tmi , the probability of the occurrence of a k vehicle platoon on the ramp is given by So, the probability for k vehicles which can merge onto freeway in the platoon is For time interval of duration T, the maximum actual ramp flow rate that can merge onto the freeway can be calculated as Therefore, the capacity of freeway merge areas can be calculated as In this section, all variables are the same as previously defined.

Case B: on-ramp vehicles with different driving behaviours
Apparently, different types of drivers choose different headways to merge onto the freeway [20].As Figure 2 shows, aggressive drivers are more likely to force their cars onto the freeway even when the headway is not long enough.In this type of situation, the drivers of freeway vehicles have to decelerate remarkably or change lanes suddenly in order to avoid traffic conflict.Average drivers are likely to choose relatively modest headway to merge onto the freeway, while conservative drivers are more likely to choose a relatively longer headway before they choose to initiate merging.
From Figure 2, aggressive, average and conservative drivers make totally different decisions in the practical traffic stream.For example, the conservative drivers often refuse the headway that aggressive drivers often recognize.As a result, the calculation model could be promoted when the behaviour of drivers is taken into consideration.Based on the previous research [20], drivers' behaviour can be divided into three types, which was observed in in-vehicle experiments.The influence of different types of driving behaviour is expressed by the specific type of headway that is chosen.Suppose tjr m represents the minimum headway that can allow on-ramp vehicle j to merge onto the freeway.It is tjr that represents the minimum headway of ve- In this case, specific headway hmi which allows a k vehicle platoon to merge onto the freeway can be expressed as Equation ( 15): The probability of k vehicles in the platoon that can merge into the traffic flow on the freeway from the ramp can be shown as 6 @ / (16) At the time interval tmi , the probability of k vehicle running on the ramp is given, So, the probability for k vehicles which can merge onto freeway from the ramp in the platoon is At a time interval of duration T, the maximum actual ramp flow rate that succeed in merging onto freeway can be calculated as Therefore, the capacity of freeway merge areas can be calculated as 20) In this section, all variables are the same as previously defined.

DATA COLLECTION
In reality, it is difficult to detect all the parameters of dynamic behaviour in the proposed models.Previous researchers have conducted studies in which the qualities of gap-acceptance model can be checked by simulation model [22].Microscopic simulation could be used to study on-ramp metering [23] and enhance the effectiveness of traffic control technique [24].In this study, to predict the dynamic behaviour of those parameters, Monte-Carlo simulation was applied to analyze the susceptibility and validity of the proposed models.
From Chapter 13 in Highway Capacity Manual (HCM) 2010, it is known that the total maximum volume of lane 1 and lane 2 (see Figure 1) should be maintained below 3,600 pcu/h to ensure no worse than a LOS of F on the freeway.That is to say, based on HCM2010, if the maximum traffic volume of twolane freeway exceeds 3,600 pcu/h, the level of service (LOS) of freeway will fall below F. In addition, based on Zong Z. Tian's research [7], the maximum traffic volume of lane 1 should not exceed 1,200 pcu/h.So, in the process of simulation analysis, the maximum vol- ume of lane 1 is expected to reach 1,200 pcu/h to ensure that LOS of freeway is not lower than F. Based on the above assumptions, a simulation study has been conducted to test different gap-acceptance models.The two traffic streams (see Figure 3) have been generated by Monte-Carlo simulation.In this chart, the headway of ramp vehicles is exponentially distributed with mean 6 r n = s and the headway of vehicles on lane 1 is exponentially distributed with mean 3 m n = s.For each simulation run, a combination of constant traffic volumes on freeway has been given.Traffic volumes on the ramp varied based on Case A and Case B model.In addition, repeated simulations have been conducted to reduce uncertainty.T-test has also been applied to analyze the validity of simulated data.The headway that measures less than 0.5 s is excluded in order to raise the validity of the simulated headway.

NUMERICAL ANALYSIS 4.1 Case A model
To study the influence of random arrivals of ramp vehicles on the capacity of freeway merge areas, the demand volume of ramp vehicles has been assumed as 1,138 pcu/h, 483 pcu/h, 177 pcu/h and 112 pcu/h, respectively.The minimum headway that an on-ramp vehicle requires in order to merge onto the freeway from the ramp is 6.5 s.The minimum headway of on-ramp vehicles that can merge onto freeway in the platoon is 3 s.In the simulation analysis, the first vehicle on the ramp would miss the headway of vehicles on the freeway as long as the headway was smaller than 6.5 s.The second vehicle would repeat the same process if the headway of vehicles on the freeway were less than 8.5 s.Otherwise, the second vehicle would merge onto the freeway following the first vehicle.This continued until the sum of total headways of the freeway was equivalent to one hour.
In Table 1, V1 is the simulated traffic volume of lane 1. Cr-g is the maximum traffic volume that can merge onto the freeway calculated by Equation ( 8), in which the values are all maintained at 499 pcu/h.Qrp (i) is the traffic volume of the on-ramp vehicles.C-rp (i) is the actual on-ramp volume calculated by Case A model with different mean headways on the ramp.Avg.Delay(i) is the average delay of ramp vehicles that succeed in merging onto the freeway.Avg.row is the average value of simulation data.The statistical results of T-test are displayed in the last two lines in Table 1.
In our study, T-test is selected to analyse the significance of difference in simulation data.The criterion for rejecting the hypothesis is 0.05.It means the simulation data are valid by a significance test with p > 0.95.The results of the analysis show that all the simulation data were proven to be valid according to the T-test.The results indicate that the changes of the demand volume on the ramp greatly affect the actual volume that can merge into the freeway.Especially, this effect will expand significantly when the demand volume on the ramp drops below 148 pcu/h.The results also show that the average delay of vehicles that can merge onto the freeway from ramp is enhanced remarkably with the increase of the demand volume on the ramp.The optimal volume that can merge onto the freeway from the ramp is 148 pcu/h.The increase of the demand volume on the ramp will not raise the actual ramp flow rate.Otherwise, this change could result in much more delays of the vehicles on the ramp.Therefore, when the demand volume of the ramp exceeds 148 pcu/h, ramp metering should be implemented to reduce the travel delay of vehicles.So, the actual traffic volume that can merge into the freeway from the ramp can be as described in Figure 4.
That is to say, their relationship can be presented as follows , / , / C q when q p cu h At this moment, the capacity of the freeway merge area is , ,

Case B model
In the analysis, the traffic volume of lane 1 is kept at 1,200 pcu/h.Based on the research of Case A model, the capacity of traffic flow is kept at 500 pcu/h in order to study the maximum traffic volume on freeway merge areas with different headways.
In addition, each type of drivers on the ramp was assigned a specific headway.With reference to previous studies [21], the related parameters for the analysis are assumed as, .
s. Vehicle i on the ramp will miss the headway of freeway vehicles until the headway is larger than the driver's minimum headway t r m 1 -.This continued until the sum of the total headway on the freeway was equivalent to one hour.The maximum actual volume that can merge onto freeway from the ramp is illustrated in Figure 5.
In Figure 5, it is known that the actual volume that can merge onto freeway from the ramp varies with the proportions of driving behaviours, especially sensitive to the proportions of aggressive and conservative drivers.The relationship between actual ramp volume and these different types of vehicles can be illustrated in As shown in Figure 6, the correlation of the two regression models is acceptable, in which all the coefficients or R values exceed 0. + + = .Thus, the capacity of the freeway merge area can be expressed as By Equation ( 24), the maximum traffic volume of the two lanes freeway merge area can be calculated as 3,839 pcu/h when p 1 i = , and 4,320 pcu/h by the traditional gap-acceptance model.The maximum traffic volume of the two lanes freeway merge area can be calculated as 3,717 pcu/h when p 1 c = , the result is enlarged to 3,948 pcu/h by the traditional gap-acceptance model.

CONCLUSION
On the basis of gap-acceptance models, this paper establishes a more practical model for calculating the capacity of freeway merge areas.The most relevant findings of the study presented in this paper are as follows: From the analysis results of the model in Case A, the uncontrolled random arrivals of ramp vehicles have a significant influence on the maximum actual volume that can merge into freeway from the ramp.The capacity of two lanes freeway merge areas can be calculated at 4,099 pcu/h by the traditional gapacceptance model when the traffic flow of freeway lane 1 is maintained at 1,200 pcu/h.Under the impact of the arrival randomness of on-ramp vehicles, the maximum actual volume of the two lanes freeway merge areas was estimated at 3,748 pcu/h.In this situation, the increase of the demand volume on the ramp will not raise the actual ramp volume.On the contrary, this change could trigger much more delays to the vehicles on the ramp.Therefore, when the demand volume of the ramp exceeds 148 pcu/h, effective traffic management methods should be applied in controlling the vehicles which are planning to merge onto the freeway from the ramp.Otherwise, the traffic congestion will be inevitable.
According to the research results of Case A model, the study of Case B in this paper shows that different driving behaviours exert a significant influence on the capacity of freeway merge areas.There is a linear relationship between the maximum actual volume of the ramp and the proportions of different types of drivers.The maximal capacity of two lanes freeway merge area obtained from Equation ( 24) with a completely aggressive driver population ( p 1 i = ) is 3,839 pcu/h while the corresponding value from the gap-acceptance model is 4,320 pcu/h.Similarly, for maximal capacity from Equation (24) with only conservative drivers along, is 3,717 pcu/h while the value from gap-acceptance model is 3,948 pcu/hr.From the research above, it is found that the influence of arrival randomness of on-ramp vehicles and characteristics of different drivers should be highly considered in the practical engineering application.For example, the driving behaviour of local drivers should be an important attribute and greatly considered when assessing the capacity of freeway merge areas, which could help relieve the traffic congestion brought by excessive estimation of the capacity.However, the achievements inferred from the proposed models depended on the given parameters.Although those parameters were carefully selected from previous results, it may be questionable to use those values in specific traffic conditions.Therefore, there should be a greater level of importance attached to the calibration of these parameters on different occasions while using these formulas.

Figure 3 -
Figure 3 -Flow rates of ramp and freeway per minute

Table 1 -
Simulation results of Case A model Project for Postgraduates in Jiangsu Province (Grant No. CXZZ13_0121).