#### Abstract

In order to accurately evaluate the level of service of pedestrians and provide the basis for the optimized design, the pedestrian delay of the continuous flow intersection was analyzed. According to the characteristics of streams of pedestrians’ arriving and leaving, the pedestrian delay models of different directions (namely, straight and diagonal) were established for three pedestrian passing patterns of the continuous flow intersection. The accuracy of the models was verified by VISSIM. The deviation was less than 3%. The effects of three key factors, namely, the vehicle demands, pedestrian demands and percentage of diagonal crossing, on the delay of the pedestrian under three modes were discussed by sensitivity analysis. The results show that the traditional pedestrian passing pattern mainly applies on the conditions that vehicle and pedestrian demands are low. The pattern of interspersed pedestrian passing is mainly applicable to the conditions of high vehicle and pedestrian demands. Although the pattern of exclusive pedestrian passing phase was least selected as the optimal design, it can apply to traffic demand fluctuating condition for its insensitive to volume and pedestrian demand pattern.

#### 1. Introduction

The intersection is the bottleneck node of the urban road network, in which the traffic problems are mainly caused by the complicated traffic flow. To better deal with the conflict between left-turn and through vehicles, a series of unconventional intersection designs were proposed, including median U-turn intersections [1, 2], superstreet intersections [1, 3, 4], uninterrupted flow intersections [5], special width approach lanes [6], dynamic reversible lane control [7], displaced left-turn intersections [8, 9], exit-lanes for left-turn intersections (EFL) [10–12], and tandem intersections [13–15].

The continuous flow intersection (CFI) is one of these unconventional intersections, which was first proposed by Mier [16]. It can eliminate the conflict between left-turn and through vehicles at the main signal by transferring the conflict point between the left-turn vehicles and opposite through vehicles to the upstream presignal, so that the capacity of intersections will be improved. A series of theoretical research and practical tests about CFI have been carried out, including geometric design, signal control, and operational evaluation. The CFI has been taken into application in real life in many countries [17], such as United States, Australia, and China.

From the perspective of the geometric design, Inman [18] studied the traffic sign and lane marking system of the continuous flow intersection to improve its visual recognition based on driving simulator experiments. Hughes [17] gave a series of detailed suggestions on layout design, including left-turn lane length, presignal intersection width, turning radius, and other detail structure at continuous flow intersections. The distances between the main signal and presignal under different traffic demands were further recommended by Tanwanichkul [19] based on the operational efficiency comparison under different distances between the main signal and the presignal using the VISSIM simulation.

From the perspective of signal control, Tarko [20] developed the basic strategy of signal timing according to the traffic characteristics of continuous flow intersections. On this basis, considering the conditions of the balance and unbalance of traffic flow in each approach and exit lane, Esawey [21] proposed a signal control method composed of six phases and realized it by Synchro software. Zhao [22, 23] established an integrated optimization model for the geometric layout and signal timing in which many key parameters, such as intersection form, lane function, left-turn lane length, and signal timing, were optimized in a unified framework.

From the perspective of operational evaluation, Chang et al. developed a well-calibrated CFI traffic simulators using VISSIM to evaluate the operational properties under various constraints and traffic conditions, which can assist engineers in identifying potential bottlenecks and estimating delays for CFI designs at the planning stage [24]. Moreover, in practice, the average waiting time of vehicles decreased by 50% and the traffic capacity increased by 31% after the continuous flow intersections of Utah's 3500 South Highway and Bangerter Highway opened to the public in 2007, which further verified the practical significance of continuous flow intersections [25].

The above research on continuous flow intersections mainly focuses on vehicles. However, in developing countries, walking is an important travel mode [26–28], so it is a necessary condition for its application in developing countries to deal with pedestrian crossing street at continuous flow intersections. For this, Jagannathan [29] proposed a new design for pedestrians, namely, interlaced crossing. The performance of vehicles and pedestrians was discussed based on Vissim simulation, in which the signal timing was optimized for vehicular traffic performance. Coates [30] further proposed a flexible signal control program under the interlaced pedestrian crossing design to reduce vehicle delay while prioritizing pedestrian crossing. The signal control procedure dynamically chooses the appropriate phase and green time combination to minimize delay by considering pedestrian wait time and existing queue length. However, these studies did not provide in-depth discussions about pedestrian delay at continuous flow intersections.

A series of calculation models for pedestrian delay at conventional intersections have been established in the past. One commonly used pedestrian delay model is the one presented in the highway capacity manual (HCM) [31]. It is a function of the ratio of the length of the pedestrian effective green time and the length of the signal cycle for one-stage crossings based on uniform arrival rates. Based on this model, many influencing factors were deeply analyzed by researchers to enhance the accuracy of the calculation model. These factors include the pedestrian-vehicle conflict considering the variety of the motorist yielding behavior [32], the pedestrians’ arrival rate considering the irregularity of the pedestrians’ arrival rate [33–35], the signal noncompliance considering pedestrians who entered crosswalks during clearance phases or red phases [36–39], the information of remaining green time [40], the two-stage crossing design considering the effect of the signal timing at the first-stage crossing on the pedestrian arrivals at the second-stage crossing [41, 42], and the bidirectional pedestrian flow considering the opposing pedestrian flow on the walking speed [43, 44].

Pedestrian service level is an important factor for the application of continuous flow intersections on urban roads. At present, the research on continuous flow intersections mainly focuses on vehicles and the evaluation of pedestrian delay is mainly based on simulation, which limits the application, especially in developing countries. Therefore, this paper aims to establish a calculation model of pedestrian delay at CFI, in which three design patterns of pedestrian crossing are considered. Considering the proposed delay models can be used to provide the guidance for engineering practice in the future, all possible pedestrian crossing patterns are discussed, even though some of them have not been taken into practice in the real world.

The rest of the paper is organized as follows. In Section 2, the three design patterns of pedestrian crossing at CFI are described. The detailed pedestrian delay calculation model is established in Section 3. The proposed model is validated in Section 4. The sensitivity of the key parameters in the proposed model is discussed in Section 5. Conclusions and recommendations are given at the end.

#### 2. Design Patterns of Pedestrian Crossing at CFI

The basic geometric design concept of continuous flow intersection is shown in Figure 1(a). The presignal is set at the upstream of the intersection so that left-turn vehicles can be guided to the left of opposite traffic flow, which can eliminate the conflicts between left-turn and opposite through vehicles at the main signals. Therefore, the two-phase signal plan can be used in the main-signal and presignals. Combining the main-signal and presignals, the CFI can run under a six-phase plan, as illustrated in Figure 1(b).

**(a)**

**(b)**

For the pedestrians, there are three design patterns, namely, the conventional pattern, the exclusive phase pattern, and the interlaced crossing pattern.

Under the conventional pattern (pattern 1), the pedestrians cross the CFI as the same way as the conventional intersections, as shown in Figure 2. There are conflicts between pedestrians and left/right turn vehicles when pedestrians are crossing the intersection at the green light through vehicles, because the continuous flow intersection runs in two phases at the main signal.

Under the exclusive phase pattern (pattern 2), the pedestrians can only cross the street during the exclusive phase, as shown in Figure 3. The exclusive phase for pedestrians is added on the basis of two phases for vehicles at the main signal. Pedestrians can cross the intersection in each direction (including diagonally) without any conflicts.

Under the interlaced crossing pattern (pattern 3), the pedestrians can cross the intersection between through vehicles in the same direction and left-turn vehicles in the opposite direction on the designed crosswalk, as shown in Figure 4. By this way, the conflicts between pedestrian and left-turn vehicles can be eliminated while pedestrians crossing the street will be retarded by multiple signals.

#### 3. Pedestrian Delay Model Establishment

##### 3.1. Model Parameters

To facilitate the model presentation, the notations used hereafter are summarized in Table 1.

##### 3.2. Pedestrian Delay Model under Conventional Crossing Pattern

There are two aspects of pedestrian delay in this pattern, including delay caused by signal control: pedestrians have to wait at the roadside due to the pedestrian phase, which leads to signal delay; delay caused by conflicts: pedestrians must pass through conflict zones of left-right turning vehicles, which causes conflict delays.

*(1) Delay Caused by Signal Control*. Under the conventional crossing pattern, only the through movement of pedestrian is allowed. The diagonal movement can be accomplished by crossing the street twice. Therefore, the through and diagonal movement have to meet the signal control once and twice, respectively. The calculation diagrams of the two conditions are shown in Figures 5 and 6, respectively.

For through movement, as shown in Figure 5, the segment “AC” represents the arrival of pedestrians to the intersection and its slope is the arriving rate. The segment “BC” represents the departure of pedestrians at the beginning of the green phase and its slope is the saturation flow rate. The segment “BD” is the pedestrian dissipation time. The area of the shadow triangle “ABC” is the total pedestrian delay, which can be calculated by (1). The length of segments “AB” and “CD” can be calculated by (2) and (3), respectively. Therefore, the average pedestrian delay of signal control can be calculated by (4).For diagonal movement, as shown in Figure 5, the segment “AD” represents the arrival of pedestrians to the intersection. The segment “BC” represents the departure of pedestrians at the beginning of the green phase. The segment “EFG” represents the arrival of pedestrians to the second stage of crossing. The segment “HI” represents the departure of pedestrians at the beginning of the green phase along the crossing street. The length of horizontal segment in triangle “ABC” denotes the delay when pedestrians waiting for green light at the first stage of crossing; the length of horizontal segment in polygon “BCDEFG” denotes the pedestrian walking time on the crosswalk; the length of horizontal segment in polygon “EFGHI” denotes the delay when pedestrians waiting for green light at the second stage of crossing. Therefore, the area of the shadow part is the total pedestrian delay, which can be calculated by (5). The length of segments “AB”, “GH”, “EI”, and “IK” can be calculated by (6), (7), (8), and (9), respectively. Therefore, the average pedestrian delay of signal control can be calculated by (10).

*(2) Delay Caused by Conflicts. *The delay caused by conflicts is directly related to the left-turn and right-turn traffic flow and the distance distribution between vehicles, which can be calculated according to the equation in the literature [45]. Therefore, the delay caused by conflicts for through movement and diagonal movement can be calculated by (11) and (12), respectively, in which the acceptable gap for pedestrians to cross, , can be calculated by (13) [45].Combining the delay caused by signal control and the delay caused by conflicts, the pedestrian delay under conventional crossing pattern can be calculated by (14).

##### 3.3. Pedestrian Delay Model under Exclusive Phase Pattern

Pedestrians have exclusive phases so that the through and diagonal pedestrians can cross the street simultaneously, and its delay is only caused by signal control which can be illustrated by Figure 5. The shadow triangle abc is the total pedestrian delay and the calculation equation of average pedestrian delay is shown in (15).

##### 3.4. Pedestrian Delay Model under Interlaced Crossing Pattern

Pedestrian delay in this pattern is mainly caused by signal control and conflict delay of right-turn vehicles.

*(1) Delay Caused by Signal Control. *The setting of interlaced crosswalk is illustrated in Figure 7. Under the interlaced crossing pattern, the through and diagonal movement have to meet the signal control twice and thrice, respectively. The calculation diagrams of the two conditions are shown in Figures 6 and 8, respectively.

For through movement, pedestrians should meet the signal control twice. For easy discussion, using the route from point 1 to point 4 in Figure 7 as an example, pedestrians have to wait for green signal of the east-west direction at point 1 and wait for green signal of the south-north direction at point 3. As shown in Figure 6, the segment “AD” represents the arrival of pedestrians to point 1. The segment “BC” represents the departure of pedestrians at the beginning of the green phase. The segment “EFG” represents the arrival of pedestrians to point 3. The segment “HI” represents the departure of pedestrians at the beginning of the green phase along the crossing street. Therefore, the average pedestrian delay of signal control can be calculated by For diagonal movement, pedestrians should meet the signal control thrice. For easy discussion, using the route from point 1 to point 6 in Figure 7 as an example, pedestrians have to wait for green signal of the east-west direction at point 1, wait for green signal of the south-north direction at point 3, and wait for green signal of the east-west direction at point 5. As shown in Figure 8, the area of the shadow part is the total pedestrian delay, which can be calculated by (17). The length of segments “AB”, “GH”, “KM”, “EI”, “JN”, and “NK” can be calculated by (18), (19), (20), (21), (22), and (23), respectively. Therefore, the average pedestrian delay of signal control can be calculated by (24).

*(2) Delay Caused by Signal Control. *The calculation principle of the delay caused by signal control under this design pattern is the same as the conventional crossing pattern. Therefore, the delay caused by right-turn vehicles can be calculated by Combining the delay caused by signal control and the delay caused by conflicts, the pedestrian delay under conventional crossing pattern can be calculated by

#### 4. Model Validation

The accuracy of models was tested by VISSIM simulation. The geometric design of CFI to be tested is shown in Figure 9. The distance of the main signal and presignal is 100 m. The pedestrian speed is 1.2 m/s. The pedestrian volume is 720 ped/h (the arriving rate is 0.2 ped/s). The saturation flow rate of the crosswalk is 8 ped/s. The traffic volume of is shown in Table 2. The average pedestrian crossing delay can be measured by calculating the weighted average delay of all pedestrian crossing routes.

The experiment was simulated by VISSIM and the simulation results were then compared with the calculation results of the model proposed, as is shown in Figure 10. The average error for the conventional pattern (pattern 1), the exclusive phase pattern (pattern 2), and the interlaced crossing pattern (pattern 3) are 1.98 s, 1.24 s, and 2.56 s, respectively. Moreover, the results of paired t-test, as shown in Table 3, further show no significant difference between the results of the proposed model and that from simulation (p value = 0.363 > 0.05), which indicates the accuracy of the proposed pedestrian delay model is acceptable.

#### 5. Sensitivity Analyses

The impact of three pedestrian crossing patterns on the delay of pedestrian is related to vehicular volume, pedestrian volume, and the proportion of diagonal crossing. The sensitivity analyses of the three design patterns were conducted below to further reveal the applicability of the three patterns.

The impact of vehicular volume on pedestrian delay is shown in Figure 11. The vehicular volume is changed from 300 veh/h/ln to 540 veh/h/ln. The pedestrian delays under the three patterns increase with the increase of vehicular volume. The conventional pattern is most affected and the exclusive phase pattern is least affected. In the condition of low vehicular volume (less than 380 veh/h/ln), the conventional pattern has the least pedestrian delay. It is due to the fact that there are lots of gap for pedestrians to cross in the vehicular flow when the volume is not high, so it is the most efficient for pedestrians to cross the street by using the gap between vehicles. However, in the condition of medium vehicular volume (between 380 and 480 veh/h/ln) the interlaced crossing pattern has the shortest average pedestrian delay. In the condition of high vehicular volume (more than 480 veh/h/ln), the exclusive phase pattern has the shortest average pedestrian delay, while the pedestrian delay of conventional pattern turns to be the longest. It is due to the fact that the conflict delay of conventional pattern increases dramatically with the increase of traffic volume and it increases a little in the other two patterns.

The impact of pedestrian volume on pedestrian delay is illustrated in Figure 12. The pedestrian volume is changed from 500 ped/h to 1500 ped/h. It shows a linear growth trend with slight increase intensity. It is due to the fact that the saturation flow rate of the crosswalk is quite large. All the waiting pedestrians can depart in the first several seconds of the green phase.

The impact of the proportion of diagonal crossing on pedestrian delay is shown in Figure 13. The proportion of diagonal crossing is changed from 0.2 to 0.8. It can be found that it has the most effect on conventional pattern. The average pedestrian delay of conventional pattern is the shortest when the proportion of diagonal crossing is less than 0.6 and the average delay of interlaced crossing pattern is the shortest when the proportion of diagonal crossing is from 0.6 to 0.75, while exclusive phase pattern is the shortest when the proportion is more than 0.75.

#### 6. Conclusions

As for three design patterns of pedestrian crossing at the continuous flow intersection, the delay model of different pedestrian flow directions (including through and diagonal crossing) was established, respectively, in this paper according to the characteristics of pedestrian arriving and leaving the intersection. The model was validated by VISSIM simulation. The impact of vehicular volume, pedestrian volume, and the proportion of diagonal crossing on the pedestrian delay in the three patterns were discussed by sensitivity analyses. This provided guidance for decision-makers to better select pedestrian crossing patterns at signalized intersections.

The accuracy of the proposed model for the three design patterns is acceptable. The deviation is less than 3%. There is no significant difference between the results of the proposed model and that from simulation.

Among these three design patterns, the conventional crossing pattern is mainly applicable to the case of low vehicular and pedestrian volume while the pattern of interlaced pedestrian crossing is mainly applicable to the case of heavy traffic and high pedestrian demand.

Although the exclusive phase pattern is seldom selected as the optimal one (with the shortest delay) in the tested examples, it has the least sensitivity to the change of traffic flow rate and direction ratio, so it is more suitable for the case of high traffic demand volatility.

Please note the proposed delay models are established based on the assumptions that the traffic operates in order and the walking speed and saturation flow rate of pedestrians do not fluctuate, which should be considered in the real-life application. Moreover, the effect of the traffic fluctuation should be considered. Moreover, for the person with vision impairment (blind), it is very important to give them the requisite guidance for crossing, such as the typhlosole, which can be the direction of future work to improve the applicability of the CFIs.

#### Data Availability

The data used to support the findings of this study are included within the article.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Acknowledgments

This research is supported by the National Natural Science Foundation of China under Grant no. 51608324.