Bicycle capacity of borrowed－priority merge at unsignalized intersections in China

YANG Xiao－bao（楊小寶）， HUAN Mei（環(huán)梅）， SI Bing－feng（四兵鋒），GAO Liang（高亮）

（MOE Key Laboratory for Urban Transportation Complex Systems Theory and Technology，Beijing Jiaotong University，Beijing 100044，China）

As a developing country，China has its own traffic characteristics.A mix of non－motor vehicles and motor vehicles is an important traffic type in China.Cycling is still an important travel mode in China and this tendency may last for a long time［1］.On the mixed traffic of bicycles and motor vehicles，many studies have been completed.Wang and Nihan［2］studied the conflict characteristics of motor vehicle and bicycle at signal intersection.Hunter et al.［3］analyzed the effect of wide curb lanes on bicycle and motor vehicle at interactions.Wardman et al.［4］developed a mode choice model for a journey，with special emphasis on the propensity to cycle.Xie et al.［5］proposed a new two－di－mensional car－following model to depict mixed traffic flow characteristics at unsignalized intersections.

This paper proposes a capacity model which assumes borrowed priority for bicycle stream at unsignalized intersections based on the additionconflict－flow （ACF）procedure that was proposed by Brilon and Wu for application to unsignalized intersections［6－8］.This procedure can easily handle the intersection between different streams［9］.In this paper，the major－road bicyclists are assumed to borrow the priority of major－road cars to enter the intersections when consecutive headway for major－road cars is lower than the critical gap for minor－road cars.We developed the equations for bicycle capacity of the borrowed priority system.The results indicate that bicycle capacity at the intersection is the function of two major－road car stream and minor－road car stream.

1 Borrowed priority phenomenon

In developed countries，there are few non－motor vehicles at the intersections.All minor－road vehicles give way to major－road vehicles.However，traffic behaviors in developing countries are largely different from those in developed countries.Current mixed traffic behaviors in China are very difficult to analyze with the common rules used in developed countries.In many Chinese cities，majorroad bicycles usually give way to minor－road cars because bicyclists may suffer more casualties in the traffic accident.Based on the field observations，some traffic phenomenon can be found.First，bicyclists often decelerate（even stop）and allow minor－road cars to enter the intersection.In addition，minor－road cars decelerate and allow major－road cars to enter the intersection if consecutive headway for major－road cars is lower than the critical gap for minor－road cars.In this case，bicyclists on the major road can borrow the priority of majorroad cars to pass the intersection.

Fig.1shows the mixed traffic streams at a simple and typical unsignalized intersection.Firstly，there are two lanes on the major road—the bicycle lane and the motor lane.And there is only a motor lane on the minor road.Secondly，there are three types of traffic streams：major－road cars，minor－road cars，and major－road bicycles.Thirdly，major－road cars have the priority over minor－road cars，and major－road bicycles usually give way to minor－road cars.But，when minor－road cars give way to major－road cars，bicycles can borrow the priority of major－road cars to enter the intersec－tion.Similar phenomena may be found in other developing countries in Asian，such as India，Malaysia，Vietnam and Cambodia.

Fig.1 Illustration of a typical unsignalized intersection with mixed traffic flow

2 Bicycle capacity model for borrowed priority merge

The Highway Capacity Manual 2000（HCM）［10］defines that the capacity of a facility is the maximum hourly rate at which persons or vehicles can reasonably be expected to traverse a point or a uniform section of a lane or roadway during a given time period under prevailing roadway，traffic，and control conditions.Accordingly，the capacity for major－road bicycles at the intersection with mixed traffic flow can be defined as the maximum hourly rate at which bicycles can pass an unsignalized intersection during agiven time period with other flow rates（including the arrival rates of both major－road and minor－road cars）remaining unchanged.This section describes the basic theoretical formulas used to estimate the capacity for major－road bicycles at the intersection.Only a simple and typical unsignalized intersection is discussed（Fig.1）.

Consider an unsignalized intersection as shown in Fig.1.Major－road cars are assumed to arrive at random，i.e.negative exponentially distributed arrival headways.This distribution is based on the assumption that vehicles arrive at random without any dependence on the time when the previous vehicle arrived.It can be derived from the Poisson distribution which gives the probability ofnvehicles arriving at timetas follows：

whereq1is the volume of major－road cars in veh／h.Forn＝0，Eq.（1）gives the probability that no cars at timet.According to Eq.（1），the probability that the headway，T，is greater thant，is

We classify traffic conditions into two types on the basis of whether consecutive headway for major－road cars is lower than the critical gap for minor－road cars.According to Eq.（2），the probability that consecutive headway for major－road cars is larger than the critical gap for minor－road cars（tc2）is

Whereas，the probability that consecutive headway for major steam is not larger than the critical gap for minor－road cars is

If consecutive headway for major－road cars is lower than the critical gap for minor－road cars，bicycle stream can borrow the priority of major－road cars to pass the intersections.In this case，bicycle capacity is not affected by minor－road cars and it reaches its maximum capacity，Cmax.Otherwise，the car－bicycle conflict takes place.Bicyclists give way to minor－road drivers，and bicycle capacity is affected by minor－road cars.Thus，the total capacity of major－road bicycles at the intersection，can be expressed by

whereCnis the total capacity of major－road bicycles at the intersection，（veh／h）；Cn1is the capacity of major－road bicycles when consecutive headway for major－road cars is lower than the critical gap for minor－road cars（Cn1＝Cmax），andCn2is the capacity of major－road bicycles when consecutive headway for major－road cars is not lower than the critical gap for minor－road cars.As the critical gap for minor－road can be estimated by the surveyed data，we only need analyze the capacity during the car－bicycle conflict，Cn2.

Based on field observations，most of majorroad bicycles give way to the minor－road cars because bicyclists may suffer more casualties in the traffic accident.Thus，we assume that minor－road cars have the priority over major－road bicycles.In this paper，the capacityCn2is derived by the addi－tion－conflict－flow procedure.

When the car－bicycle conflict exists，bicycle capacity is the maximum discharge flow rate of bicycle stream （Cmax）multiplied by the proportion（p0）of time available for bicycle queue discharge：

wherep0is also the probability that the car－bicycle conflict area is not occupied by minor－road cars，which can be calculated as

wherepdis the probability of no queuing or discharging minor－road cars andpais the probability of no approaching minor－road cars.

The probabilitypdthat the conflict area is not blocked by queuing or discharging minor－road cars is

where ρ2istrafficdensityforminor－roadcars；q2istrafficflowrateforminor－roadcars；andts2istheservicetimeforminor－roadcarpassing theintersection，thatis，thelengthoftimethat theconflictareaisoccupiedbythedischarging minor－roadcar.Anditcanbeobtainedbythesurveyeddata.

Theprobabilitypathattheconflictareaisnot blockedbyanapproachingminor－roadcarisexpressedastheprobabilitythatagapavailablefor bicyclesislargerthanatimeperiodta：

where the time periodtais the time period that the conflict area is blocked by an approaching minorroad car.It is comparable to the shortest acceptable headwayto＝tc，3－tf，3／2，wheretc，3andtf，3are the critical gap and follow－up time for bicycles，respectively.

Combining Eqs.（5）－（9），we can obtain the total capacity of major－road bicycles at the unsignalized intersection as follows

3 Empirical observations

Here a bicycle capacity model based on borrowed priority at an unsignalized intersection is proposed.To validate our model，traffic data were collected using a videotape recorder.Some photos for data investigation were taken in Beijing for 1h during the morning peak period.These photos coincide with the assumptions in our model.Two typical situations from these photos are shown in Fig.2.Fig.2acorresponds to the situation that bicyclists stop and allow minor－road cars to enter the intersection.Fig.2bcorresponds to another situation that bicyclists on the major road borrow the priority of major－road cars to pass the intersection.However，it is noted that the proposed capacity model need to be further calibrated by surveyed data.Empirical investigation and related calibration are in progress.

Fig.2 Conflict between bicycles and cars at an unsignalized intersection

4 Numerical analysis

The total capacity formula for major－road bicycles indicates that both major－road cars and minor－road cars have significant effects on bicycle capacity.Fig.3shows the effects of different arrival rate of subject－approach cars on bicycle capacity.Here，major－road cars headways followed the negative exponential distribution.The critical gap and the service time for minor－road cars were 4.0sand 1.8s，respectively.For major－road bicycles，the critical gap was 3.0s，and the follow－up time was 1.1s.The maximum capacity for bicycle stream is assumed to 3 200veh／h.As shown in Fig.3，at the same major－road car flow rate，bicycle capacity decreases with the increasing arrival rates of minorroad cars.Particularly，when minor－road cars flow is zero，the car－bicycle conflict disappears.Thus，bicycle stream can reach its maximum capacity.Whereas，when minor－road is at its maximum capacity（here 2 000veh／h），there are no acceptable gap for bicycles in the minor road.In this case，bicycle capacity becomes zero.

Fig.3 Bicycle capacity with different arrival rates for minor－road cars，t c ，2 ＝4s，and t c ，3 ＝3s

Fig.4shows that bicycle capacity increases with the increasing major－road car flow rate.This is because that the increasing major－road cars can provide more borrowed priority for bicycle stream at the intersections.The effects of major－road cars on bicycle capacity can also be obtained from Fig.3 at the same flow rate of minor－road cars.

Fig.4 Bicycle capacity with different arrival rates for major－road cars，t c ，2 ＝4s，and t c ，3 ＝3s

5 Conclusion

Bicycle capacity at a typical unsignalized intersection under mixed traffic flow is investigated on the basis of borrowed priority.Borrowed priority is defined that the major－road bicyclists borrow the priority of major－road cars to enter the intersections when consecutive headway for major－road cars is lower than the critical gap for minor－road cars.Total capacity for bicycle stream at the intersection is the function of major－road car stream and minor－road car stream.Bicycle capacity increases with increasing major－road cars but decreases with increasing minor－road cars.To improve the model for the mixed traffic system and experimentally validate it，empirical investigation and related calibration are in progress.Further research can include① more complex intersections，②delay models for major－road bicycles and minor－road cars at the intersections，and ③ the plan and design problems of unsignalized intersections with mixed traffic flow.

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（Edited byCai Jianying）