- Email: [email protected]
A new concept of occupancy: Dynamic time occupancy
Transportation Research Part C 31 (2013) 51–61
Contents lists available at SciVerse ScienceDirect
Transportation Research Part C journal homepage: www.elsevier.com/locate/trc
A new concept of occupancy: Dynamic time occupancy Chong Zi Xiao a, Jin Xin Cao b,⇑, Zheng Yu Wang b a b
Key Laboratory for Urban Transportation Complex Systems Theory and Technology of Ministry of Education, Beijing Jiaotong University, China Institute of Transportation, Inner Mongolia University, Hohhot, China
a r t i c l e
i n f o
Article history: Received 20 November 2011 Received in revised form 17 March 2013 Accepted 21 March 2013
Keywords: Occupancy Dynamic time occupancy Traffic flow Average time gaps Minimum safety distance
a b s t r a c t Occupancy is widely used in traffic flow theory. As a static parameter, however, the traditional occupancy cannot provide a comprehensive description of traffic performance. Based on previous studies, given the minimum safety distance between two adjacent vehicles in the same lane, the dynamic space occupancy (DSO) is redefined and a new parameter, namely the dynamic time occupancy (DTO), is proposed in this paper. It is found that the reaction time and the average time gaps are equivalent under congested condition, and this results in an equivalence relationship between the DTO model and the average time gaps model. Also, a new congested traffic fundamental diagram is derived, which is similar to the triangular traffic flow model. The analysis of a large set of the field data shows that DTO is always equal or close to 1 under congested condition. It is found that the reaction time has a great influence on highway capacity when DTO equals 1, and this may be useful for the capacity analysis or improvement. Different from the traditional static occupancy, DTO is a dynamic parameter that can provide a more fitting description of the occupying relationship between vehicles and highways. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction Density, as one of the basic parameters in traffic flow theory, is frequently used to indicate highway congestion. In 1930s, Greenshields proposed a method to determine the density (Greenshields et al., 1934; Greenshields, 1935). After that, a number of ways of detecting density were developed (Gazis and Knapp, 1971; Gazis and Szeto, 1974; Coifman, 2003; AlvarezIcaza et al., 2004; Artimy, 2007; Jerbi et al., 2007). All these methods are very expensive, complicated and less applicable, which leads to a limited application of density. Occupancy, as a new parameter, is widely used instead, and the research indicates that the replacement is feasible (Lin et al., 1996). Compared with density, occupancy has achieved a widespread use. The earliest studies on occupancy are conducted by Greenshields (1960) and Athol (1965). Afterwards, the research on occupancy appears to be rare (Arasan and Dhivya, 2010; Hsu and Banks, 1993; Papageorgiou and Vigos, 2008). Most of the researchers focused on the relationship of density and occupancy, and some linear and non-linear relationships were established (Hall, 1986; Hall and Persaud, 1989; Banks, 1995; Cassidy and Coifman, 1997; Kim and Hall, 2004). In recent years, occupancy has been widely used in many areas (Payne and Tignor, 1978; Chew and Ritchie, 1995; Dudek et al., 1974; Zhang and Ye, 2006). Despite these applications, occupancy cannot give a comprehensive description of the traffic performance, especially under congested conditions. The space occupancy and the time occupancy refer to the portion of lane length covered by vehicles or the portion of time a local sensor covered by vehicles. They ignore the fact that there is a
⇑ Corresponding author. Tel./fax: +86 471 499 6800. E-mail address: [email protected] (J.X. Cao). 0968-090X/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.trc.2013.03.006
52
C.Z. Xiao et al. / Transportation Research Part C 31 (2013) 51–61
space in front of the moving vehicle which cannot be occupied by other objects. From this point of view, this paper takes the space into consideration and proposes a new concept of occupancy, namely the dynamic time occupancy (DTO). The paper is composed of seven sections. Previous studies on occupancy models are reviewed in the section that follows. Then the earliest concept of the dynamic space occupancy (DSO) is introduced and discussed in Section 3, and a new concept of the dynamic time occupancy is proposed in Section 4, which is followed by the mathematic inferences in the same section. The proof of the relationships between reaction time and average time gaps is conducted in Section 5. In Section 6, a large set of field data is used to analyze the application of DTO and its realistic characteristics. Finally, a summary of this paper is provided in Section 7. 2. The traditional occupancy model The traditional occupancy measurement techniques can be divided into the manual method, the loop detector method and the photo-sensor method. The mechanisms of occupancy measurement can be categorized as time-based measurement and space-based measurement. In the space-based measurement, occupancy is defined as the ratio of the total area occupied by vehicles. In reality, area is often replaced by length for simplicity. The model of the space occupancy can be expressed by Eq. (1) (Bham and Benekohal, 2004).
os ¼
n 1X li L i¼1
ð1Þ
where os is the space occupancy; L is the length of the detected road section (m); li is the length of the ith vehicle (m); and n is the number of vehicles detected. When vehicle length distribution is known, the number of vehicles can be derived from the density investigation, and the space occupancy can be determined by using a large sample of data. However, this method is not only costly but also very complicated, and is not suitable for the automatic measurement. Relatively speaking, the time-based method is more suitable for the automatic measurement, and its basic idea is: the ratio of the time occupied by vehicles at a specific point in detecting time interval can also indicate the occupied situation of the road section. Based on this principle, the automatic measurement of occupancy can be easily realized. The time occupancy model can be written as (Greenshields, 1960; Athol, 1965):
ot ¼
n 1X ti T i¼1
ð2Þ
where ot is the time occupancy; T is the detecting time interval (s); ti is the time occupied by the ith vehicle at the detecting point (s); and n is the number of vehicle detected. In the field survey process, the length of the detector also impacts the detecting data, but for occupancy, the parameter of detector length can be considered by applying Eq. (3) (Wang, 2002).
ot ¼
n 1X ti ¼ T i¼1
P
i ðli
þ ld Þ=v i 1 X li ld X 1 þ ¼ T i vi T i vi T
ð3Þ
where ld is the length of the detector (m); and vi is the speed of the ith vehicle (m/s). Compared with space occupancy, time occupancy is far more popular today, and this should attribute to the development of automatic detection techniques for time occupancy. 3. The initial DSO model Although the time-based and space-based occupancy have achieved wide applications, a fact is neglected that a part of the space between two consecutive vehicles also occupies the road. Under this precondition, the concept of DSO was firstly proposed by Yang et al. (2005). DSO consists of two parts: the space occupied by vehicles (real occupancy) and the space in front of moving vehicles (fictitious occupancy), as is shown in Fig. 1. The DSO model is expressed as:
P oDS ¼
0
þ li Þ
i ðli
ð4Þ
L 0
where oDS is DSO; li is the length of fictitious part of the ith vehicle (m). The length of fictitious part of the ith vehicle is: 0
li ¼ v i t ri þ
v 2i
ð5Þ
max
2di
max
where tri is the reaction time of the driver in the ith vehicle (s), and di (m/s2).
is the maximum deceleration rate of the ith vehicle
C.Z. Xiao et al. / Transportation Research Part C 31 (2013) 51–61
53
Fig. 1. Occupancy relationships.
In Yang’s research, the fictitious space in front of moving vehicles cannot be occupied by other objects. It actually indicates the total stop distance from speed vi to 0, which includes the distance vehicle has traveled during the reaction time period, namely vitri. If the preceding vehicle can achieve a complete stop without a braking distance, the fictitious distance and minimum safety distance are equal, but the situation is impossible in reality. That is to say, the fictitious space is always larger than the minimum safety distance and this often makes the fictitious space be occupied by the preceding vehicles under high flow rate condition. The value of DSO calculated from the model is always on the high side, even over 2. However, the value of DSO should be no greater than 1 in theory, so it is very hard to find the boundary of the DSO value with this model. Besides, Yang proposed a video-based method for the data collection to determine DSO. The full view of the road is firstly recorded and the data of vehicle length and speed are extracted respectively. However, the video-based method is costly and complicated, and highly affected by the weather, so it is infeasible for large area detection. Different from Yang’s hypothesis of the fictitious occupancy, the minimum safety distance is used for instead, and DTO is defined in the following part. 4. The definition of DTO and its mathematical analysis In recent years, traffic safety is getting more and more important. Drivers, vehicles, and traffic environment have a great impact on traffic safety, while drivers are the main factor causing accidents (Kuge et al., 1995). The speed of a vehicle is controlled by the driver, and there should be a proper space between two adjacent vehicles in the same lane. If the space between vehicles are too small, a rear-end collision can easily happen. According to the traffic accident statistical data, rearend collision accounted for more than half the total accidents (Wilson et al., 1997), and the countless traffic safety problems have attracted wide attention. Scholars agree that there should be a space, namely the minimum safety distance, between two following vehicles for safety concern. Under normal circumstances, the distance between the two following vehicles should be larger than the minimum safety distance, so it is reasonable to believe that the space of minimum safety distance should also occupy a road section. That is to say, the occupancy of the road includes two parts, namely the vehicle length (real occupancy) and the minimum safety distance (fictitious occupancy), which is different from Yang’s DSO model. Some of the early studies on the minimum safety distance have been conducted by Fukuhara (1994) and Hamed and Jaber (1997). After that, some minimum safety distance models have been proposed by Carbaugh et al. (1998) and Zhou (2003). All the philosophies of the minimum safety distance models above are similar, so a strong representative minimum safety distance model is chosen in this paper (Kenue, 1995), which is expressed as:
d ¼ v fo t r þ
v 2fo 2af
v 2l
ð6Þ
2al
where d is the minimum safety distance (m); vfo is the speed of the following vehicle (m/s); vl is the speed of the leading vehicle (m/s); tr is driver’s reaction time (s); af is the deceleration rate of the following vehicle (m/s2); and al is the deceleration rate of the preceding vehicle (m/s2). Suppose the preceding vehicle encompasses a sudden full brake, and there should be a reaction time for the following vehicle to make a stop decision. In this short period, the speed of the following vehicle is invariable, so the moved distance is vfotr. After that, the following vehicle begins to brake, and the braking distance is v 2fo =ð2af Þ, which is the same to the preceding vehicle. Thus, the minimum safety distance is the difference of v fo tr þ v 2fo =ð2af Þ and v 2lo =ð2al Þ. As is illustrated in Fig. 2, suppose that there are n vehicles in a same lane, and the moving speed and maximum deceleration rate for the ith vehicle are vi (m/s) and ai (m/s2), respectively, so the occupied space of the ith vehicle at the specific time is:
si ¼ li þ di ¼ li þ v i tr þ
v 2i 2ai
v 2i1 2ai1
ð7Þ
54
C.Z. Xiao et al. / Transportation Research Part C 31 (2013) 51–61
Fig. 2. The status of vehicle flow.
So the total occupied space of the n vehicles is:
S¼
n n n X X X si ¼ ðli þ di Þ ¼ li þ v i t r þ i¼1
i¼1
i¼1
v 2i 2ai
v 2i1
n X
¼
2ai1
ðli þ v i tr Þ þ
i¼1
v 2n 2an
v 21 2a1
ð8Þ
Pn In general, the total number of vehicles (n) is a large value, so the value of i¼1 ðli þ v i t r Þ is far greater than v 2n =ð2an Þ v 21 =ð2a1 Þ. However, the speed and the maximum deceleration rate of the vehicles in the counting period are similar, so v 2n =ð2an Þ v 21 =ð2a1 Þ 0. Eq. (8) can be simplified as:
S
n X ðli þ v i tr Þ
ð9Þ
i¼1
Making use of Eqs. (1) and (9), the formula of DSO can be written as:
oDS ¼
S ¼ L
Pn
þ v i tr Þ L
i¼1 ðli
ð10Þ
As is expressed in Eq. (10), making use of Eqs. (3) and (9), the formula of DTO can be written as:
Pn
i¼1 ðli
oDTm ¼
þ ld þ v i t r Þ=v i T
ð11Þ
where oDTm is the measured DTO, including the effects of detector length. From the equations above, it can be concluded that for a certain number of vehicles, the measured DTO depends on the length and the speed of vehicles, and the driver’s response time, but is independent of the deceleration characteristics. By simplifying Eq. (11), a new formula can be obtained as:
Pn
i¼1 ðli
oDTm ¼
þ ld þ v i t r Þ=v i ¼ T
n n 1X li ld X 1 þ T i¼1 v i T i¼1 v i
! þ
n 1X tr T i¼1
ð12Þ
Use Eq. (3) to simplify Eq. (12), then a new relationship occurs.
oDTm ¼ ot þ qtr
ð13Þ
According to the equations above, it shows that the vehicle lengths directly impact the value of time occupancy, and have certain indirect impacts on DTO. Nevertheless, time occupancy contains all the information of vehicles lengths. Consequently, if DTO is to be detected, it is not necessary to determine the length of each vehicle, but to get the value of time occupancy. Although Eq. (13) shows a way to calculate DTO, it is still necessary to reveal the relationship between DTO and speed/density/flow rate. See the following derivation. If there are n vehicles in the counting time interval T, the flow rate q (veh/s) can be written as:
q¼
n n 1 1 ¼ Pn ¼ 1 Pn ¼ T h h h i¼1 i i¼1 i n
ð14Þ
is the mean headway in the countering interval (s). where hi is the headway of the ith vehicle and the preceding vehicle (s), and h The space mean speed v (m/s) is:
1
v ¼ 1 Pn n
i¼1
ð15Þ
1
vi
Combine Eqs. (3), (14), and (15), and a result is as below:
ot ¼
1 X li q þ ld T i vi v
ð16Þ
where T can be treated as the sum of headways of all vehicles. Suppose that the vehicle length is a constant l (m), so:
ot ¼
Pn l 1 i¼1 v i n P n 1 i¼1 hi n
þ ld
q
v
¼
1 n
Pn
i¼1
h
l
vi
þ ld
q
v
¼
n l 1X 1 q þ ld v h n i¼1 v i
ð17Þ
C.Z. Xiao et al. / Transportation Research Part C 31 (2013) 51–61
55
Substitute Eqs. (15)–(17), then:
ot ¼
l 1 h
v
þ ld
q
ð18Þ
v
On the basis of Eqs. (14) and (18) can be derived as:
ot ¼ l
q
v
þ ld
q
¼ ðl þ ld Þ
v
q
v
¼ ðl þ ld Þk
ð19Þ
According to Eqs. (14) and (19), another expression of DTO is:
oDTm ¼ ðl þ ld Þ
q
v
þ qtr
ð20Þ
Note that in Eq. (20), the model actually indicates the measured DTO value, which includes the effect of the detector length. If ld is removed, as can be seen below, the physical DTO can be determined:
oDTp ¼ l
q
v
þ qtr ¼ oDTm ld
q
ð21Þ
v
where oDTp is the physical DTO, namely the portion of time a local detector are covered by total vehicles lengths and minimum safety distances which do not include the detector effect. Under totally congested traffic condition, drivers tend not to stop bumper-to-bumper. There must be some space buffer between vehicles for both stopping and running vehicles. Let DTO equal to 1, when traffic totally jammed, and the buffer space is actually the jam gap. According to the abovementioned, the behavior DTO, which includes the effect of the buffer length, is:
oDTb ¼ ðl þ bÞ
q
v
þ qtr ¼ o þ qtr þ ðb ld Þ
q
v
ð22Þ
where oDTb is the behavior DTO, namely the portion of time a local detector are covered by total vehicles lengths, minimum safety distances, and total buffer lengths which do not include the detector effect; b is the buffer space or jam gap (m). 5. Reaction time determination For the DTO model, detector length, average vehicle length, and buffer length/jam gap are easy to be determined, except the reaction time. For a moving vehicle, the time gap represents the driver’s window of opportunity for reacting to the motion of the preceding vehicle, and the time gap needs to be at least as large as the driver’s reaction time (Banks, 2003). This point of view puts forward an idea to calculate the reaction time. The proof of the relationship between the reaction time and the average time gaps is shown below. Most vehicles follow the preceding vehicles closely under congestion and there is not enough space for them to accelerate or to overtake. For the safety consideration, the average time gaps must contain the drivers’ reaction time, tg P tr. However, the speed of vehicles is much less than the desired speed under congestion condition, thus drivers tend to minimize the time gaps. Nevertheless, the reaction time is the threshold value of the average time gaps.
Fig. 3. Vehicles trajectories diagram.
56
C.Z. Xiao et al. / Transportation Research Part C 31 (2013) 51–61
In case that there are two consecutive vehicles that move under congested condition with speed v1, the preceding vehicle suddenly decelerates its speed to v2 and maintains this speed, so the following vehicle should also change its speed to v2 at a certain point. The time–space diagram is shown in Fig. 3, where the points A(tA, xA), B(tB, xB), C(tC, xC), D(tD, xD), B1(tB1, xB1), and B2(tB2, xB2) are all on the trajectories lines, and the vehicle length equals to xD xA or xB xC. The reaction time is tB tA, where tg1 and tg2 are the time gaps before and after the speed change, respectively. Let xA = xB2, and xA = xB1, where point B1 is on the line of the head trajectory of the preceding vehicle after the speed change. It is clear that tg1 = tB2 tA, and tg2 = tB1 tA. From Banks’ analysis (Banks, 2003), it is known that tg1 = tg2, so tB1 = tB2. This indicates that points B1 and B2 are the same point, namely point B. For moving vehicles under congestion, the signal of the sudden deceleration of the preceding vehicle can be captured by the driver in the following vehicle immediately, so the reaction time tr = tB tA = tB2-tA = tg1 = tg2. The relationship between reaction time and average time gaps is (see Fig. 4):
tg ¼ tr
ð23Þ
In Banks’ work, the average time gaps is divided into three categories, namely the physical time gaps, the measured time gaps and the behavioral time gaps (Banks, 2003).
gm ¼
1o q
g ¼ gm þ
ð24Þ
ld
ð25Þ
v
gb ¼ gm þ
ld b
ð26Þ
v
where gm is the measured time gaps (s); g is the physical time gaps (s); and gb is the behavioral time gaps (s). In Eq. (23), both tr and g are physical values. Comparing Eq. (21) with Eq. (25), the following relationship is obtained:
(
oDTp ¼ o þ qt r ld vq
ð27Þ
1 ¼ o þ qg ld vq
Because tr = g, it is obviously that the physical value of DTO equals 1 under the congested traffic condition. That is to say, the physical DTO model and the physical average time gaps model are identical under the congested condition while the average time gaps is a constant. In the DTO model, ot and q can be easily captured by devices, and v can be calculated from the measured space mean speed, while ld is a known parameter. Although the reaction time is difficult to detect directly, from the analysis above, the average time gaps can be used instead. In general, the average time gaps are different for different lanes, so choosing the minimum one for the implication of the reaction time is reasonable.
tr ¼ minfg i g
ð28Þ
where gi is the average time gaps of the ith lane (s).
Fig. 4. Congested traffic flow fundamental diagram.
C.Z. Xiao et al. / Transportation Research Part C 31 (2013) 51–61
57
6. The application of DTO and its characteristics According to the above analysis, the relationships between traffic flow rate, speed and density can be derived when DTO equals 1. From Eq. (27), we can obtain the following formulation:
1 ¼ o þ qt r ld
q
v
¼l
q
v
þ qtr
ð29Þ
So the relationships of the three parameters are:
8 lq v ¼ 1qt > > r < v ¼ 1lk ktr > > : q ¼ 1lk tr
ð30Þ
Note that the models above only apply to the congested condition, and the flow–density model is similar to the triangular flow–density model (Banks, 1989). Supposing speed is a constant for non-congested condition, the fundamental diagram can be obtained as below. For further analysis, the traffic flow data from the I80 expressway at California is used. The data represents all 30 s intervals for a day, which includes the information of all five lanes at a fixed location. In Fig. 5, it is evident that the average time gaps form a horizontal band from 15 to 60 km/h which represents the congested traffic condition. Compares the average time gaps of five lanes (see Fig. 6), and the reaction time can be easily derived according to Eq. (28), so tr = min{g5} 1.5 s. Because DTO is a constant under congested condition, there should be a horizontal liner relationship between the three traffic parameters and DTO. From Figs. 7 and 8, it is clear that there is a linear or near-linear relationship while traffic is congested. When the occupancy is over 20%, traffic congests, and the physical DTO for each lane remains the same value or decreases slightly. Only the values of Lanes 2 and 5 exceed 1 or fluctuate around 1. Moreover, there is a sudden fall for the physical DTO of Lane 2 and Lane 4 while the occupancy is over 70%, and all of them go to the congested point in theory, namely the congestion occupancy. For the behavior DTO, the fall no longer exists. Compared with the physical DTO, the mean values of the behavior DTO are more closer to 1 and become more stable under congestion (excepting for Lane 2 under very low speed). All data cases show that DTO equals a constant under congestion. Consider the fact that traffic has a finite movement even under the congested condition (Drake and May, 1967), and a stop-and-go speed is used as an additional parameter, which is referred to the research by Wang et al. (2011). Two sets of the congested speed–density data are used to analyze the derived traffic flow models, expressed by Eq. (30). The free-flow speed of I-80 is smaller than that of US-101, and this leads to the different length of speed–density regime. Despite this, they both have the same or similar shape. The field data provides a proof that traffic has a similar character under the congested condition. Taking the congested speed–density data as an example, the numerical relationship between the two parameters is more like the fractional function. DTO, as a new traffic flow parameter, can provides a reasonable explanation for this. No matter what conditions are, the congested traffic maintains a constant DTO value, which is always equals 1 or close to it. Consequently, the congested traffic flow fundamental diagram can be determined by Eq. (20). That is to say, the shape of congested traffic fundamental diagram is up to the safety consideration of drivers.
Fig. 5. Scatter points of average time gaps and speed.
58
C.Z. Xiao et al. / Transportation Research Part C 31 (2013) 51–61
Fig. 6. Average congested time gaps and speed.
Fig. 7. The relationship between the occupancy and the physical DTO.
Fig. 8. The relationship between speed and the behavior DTO.
C.Z. Xiao et al. / Transportation Research Part C 31 (2013) 51–61
59
Fig. 9. Speed-density raw data fitting by the proposed model.
Fig. 10. Congested speed–density relationship with variable vehicle length and reaction time.
Suppose that DTO equals 1 under congested condition, so the relationship between speed and flow rate can be described as Eq. (30). Through the above analysis, it is known that all congested traffic has a common character, namely that DTO equals 1 or very close to 1. In spite of non-congested traffic, the following two figures are used to indicate the speed-flow relationship. Obviously, flow rate always increases as speed increases, but the increasing becomes more and more invisible, and finally goes to a fixed value, namely q = 1/tr (veh/s/ln). Supposing the speed is a constant for non-congested traffic, it is easy to find out the fact that vehicle length changes the capacity of the road, but the influence is not obvious. However, the change becomes very obvious with fixed vehicle length and variable reaction time, as is shown in Fig. 10. All these state clearly that reaction time, vehicle length and free flow speed are the three factors which determine the highway capacity. Reducing vehicle length or reaction time, increasing free flow speed, or a combination of them, can lead to capacity improving. However, the vehicle length is hard to change, and tiny length change cannot improve the capacity obviously. For free-flow speed, there is not enough space for the improvement of capacity while free-flow speed is over 100 km/h, as can be seen in Fig. 10. Nevertheless, a small reaction time is achievable through some intelligent methods such as automated driving.
7. Conclusion Density and occupancy have been broadly applied in transportation as parameters of traffic performance, but the application of density is not as good as occupancy for the difficulties of data collection. Density has been replaced by occupancy in many fields. Occupancy takes the length of vehicles into consideration and can give a more accurate description of the traffic situation than density sometimes. However, occupancy is still a static parameter in essence. In reality, road is occupied by vehicles, but there is still some space cannot be occupied by other objects, and this kind of space also occupies the road. So the ‘‘occupancy’’ here consists of two parts: the real occupancy and the fictitious occupancy. In order to give a more reasonable description of this phenomenon, a new concept of occupancy has been proposed in this paper, and many of the characteristics have been discussed. There are several main findings in the paper:
60
C.Z. Xiao et al. / Transportation Research Part C 31 (2013) 51–61
(1) Combining with the minimum safety distance, DSO is redefined and a more reasonable parameter, namely DTO, is proposed. With the support of a large set of field data, it is observed that the value of DTO is equal or close to 1 under congested condition, and this may be a nature of the moving traffic. DTO equals to 1 is reasonable and acceptable in theory, which is indicted by field data results in Fig. 9. (2) It is proved that reaction time and average time gaps are identically equal under congested condition, and this results in an equivalence relationship between the physical DTO model and the physical average time gaps model. (3) A congested traffic fundamental diagram is proposed based on DTO, and the proposed model has a simple expression which is very similar to triangular speed–flow–density model. Besides, theoretical analysis shows that the reaction time plays an important role in highway capacity improvement.
Different from the traditional occupancy, DTO is a dynamic parameter and can give a more fitting description for traffic performance. However, the application of DTO still need further studies, and more application results should be put forward to back it up. Acknowledgements This work is supported by the National Natural Science Foundation of China (Grant Nos. 71061010, 71262008), the research start-up project for the high-level talent introduction of Inner Mongolia University (Grant No. 210221) and the Scientific Research Foundation for the Returned Overseas Chinese Scholars. References Alvarez-Icaza, L., Munoz, L., Sun, X.T., Horowitz, R., 2004. Adaptive observer for traffic density estimation. In: Proceedings of the 2004 American Control Conference, Boston, MA, pp. 2705–2710. Arasan, V.T., Dhivya, G., 2010. Methodology for determination of concentration of hetrogeneous traffic. Journal of Transportation Systems Engineering and Information 10 (4), 50–61. Artimy, M., 2007. Local density estimation and dynamic transmission-range assignment in vehicular ad hoc networks. IEEE Transactions on Intelligent Transportation Systems 8 (3), 400–412. Athol, P., 1965. Interdependence of certain operational characteristics within a moving traffic stream. Highway Research Record 72, 58–87. Banks, J.H., 1989. Freeway speed-flow-concentration relationships: more evidence and interpretations (with discussion and closure). Transportation Research Record 1225, 53–60. Banks, J.H., 1995. Another look at a priori relationships among traffic flow characteristics (with discussion and closure). Transportation Research Record 1510, 1–10. Banks, J.H., 2003. Average time gaps in congested freeway flow. Transportation Research Part A 37, 539–554. Bham, G.H., Benekohal, R.F., 2004. A high fidelity traffic simulation model based on cellular automata and car-following concepts. Transportation Research Part C 12 (1), 1–32. Carbaugh, J., Godbole, D.N., Sengupta, R., 1998. Safety and capacity analysis of automated and manual highway systems. Transportation Research Part C 6C (1–2), 69–99. Cassidy, M.J., Coifman, B., 1997. Relation among average speed, flow, and density and analogous relation between density and occupancy. Transportation Research Record 1591, 1–6. Chew, R.L., Ritchie, S.G., 1995. Automatic detection of lane-blocking freeway incidents using artificial neural networks. Transportation Research Part C 6 (3), 371–388. Coifman, B., 2003. Estimating density and lane inflow on a freeway segment. Transportation Research Part A 37 (8), 689–701. Drake, J.L.S.J.S., May, A.D., 1967. A statistical analysis of speed–density hypotheses. Highway Research Record 156, 53–87. Dudek, C.L., Messer, C.J., Nuckles, N.B., 1974. Incident detection on urban freeway. Transportation Research Record 495, 12–24. Fukuhara, H., 1994. Vehicle Collision Alert System. US Patent No: 5355118. Gazis, D., Knapp, C., 1971. On-line estimation of traffic densities from time-series of flow and speed data. Transportation Science 5 (3), 283–301. Gazis, D., Szeto, M., 1974. Design of density-measuring systems for roadways. Transportation Research Record 495, 44–52. Greenshields, B.D., 1935. A study of traffic capacity. In: Proceedings of the 14th Annual Meeting of the Highway Research Record, Washington, DC, pp. 448– 477. Greenshields, B.D., 1960. The density factor in traffic flow. Traffic Engineering 30 (6), 26–28, 30. Greenshields, B.D., Thompson, J.T., Dickinson, H.C., Swinton, R.S., 1934. The photographic method of studying traffic behavior. In: Proceedings of the 13th Annual Meeting of the Highway Research Board, Washington, DC, pp. 382–399. Hall, F.L., 1986. The relationship between occupancy and density. Transportation Forum, 46–52. Hall, F.L., Persaud, B.N., 1989. Evaluation of speed estimated made with single-detector data from freeway traffic management systems. Transportation Research Record 1232, 9–16. Hamed, M., Jaber, S., 1997. Modeling vehicle-time head ways in urban multilane highways. Road and Transport Research 6 (4), 32–44. Hsu, P., Banks, J.H., 1993. Effects of location on congested-regime flow–concentration relationships for freeways. Transportation Research Record 1398, 17– 23. Jerbi, M., Senouci, S., Rasheed, T., Ghamri-Doudane, Y., 2007. An infrastructure-free traffic information system for vehicular networks. In: Proceedings of the 66th IEEE Vehicular Technology Conference, Baltimore, MD, pp. 2086–2090. Kenue, S.K., 1995. Selection of range and azimuth angle parameters for a forward looking collision warning radar sensor. In: Proceedings of the Intelligent Vehicles ‘95 Symposium, Boston, pp. 494–499. Kim, Y., Hall, F.L., 2004. Relationships between occupancy and density reflecting average vehicle lengths. Transportation Research Record 1883, 85–93. Kuge, N., Ueno, H., Ichikawa, H., Ochiai, K., 1995. Study on the causes of rear-end collision based on an analysis of driver behavior. JSAE Review 16 (1), 55–60. Lin, F.B., Su, C.W., Huang, H.H., 1996. Uniform criteria for level of service analysis of freeways. Journal of Transportation Engineering 122 (2), 123–130. Papageorgiou, M., Vigos, G., 2008. Relating time-occupancy measurements to space-occupancy and link vehicle-count. Transportation Research Part C 16, 1–17. Payne, H.J., Tignor, S., 1978. Freeway incident detection algorithms based on decision trees with states. Transportation Research Record 682, 30–37. Wang, D.H., 2002. Traffic Flow Theory. China Communications Press, Beijing (in Chinese). Wang, H.Z., Li, J., Chen, Q.Y., Ni, D.H., 2011. Logistic modeling of the equilibrium speed–density relationship. Transportation Research Part A 45, 554–566.
C.Z. Xiao et al. / Transportation Research Part C 31 (2013) 51–61
61
Wilson, T., Butler, W., McGehee, D.V., Dingus, T.A., 1997. Forward-Looking Collision Warning System Performance Guidelines. SAE Technical Paper 970456. Yang, J.G., Zhang, J.H., Yi, X.Q., Wang, Z.A., 2005. Dynamic space occupancy and measuring method. Journal of Tongji University (Natural Science) 33 (11), 1474–1478 (in Chinese). Zhang, Y.L., Ye, Z.R., 2006. A derivative-free nonlinear algorithm for speed estimation using data from single loop detectors. In: Proceedings of the 2006 Intelligent Transportation Systems Conference, pp. 1035–1040. Zhou, W., 2003. Analysis of distance headway. Journal of Southeast University (English Edition) 19 (4), 378–381.
Contents lists available at SciVerse ScienceDirect
Transportation Research Part C journal homepage: www.elsevier.com/locate/trc
A new concept of occupancy: Dynamic time occupancy Chong Zi Xiao a, Jin Xin Cao b,⇑, Zheng Yu Wang b a b
Key Laboratory for Urban Transportation Complex Systems Theory and Technology of Ministry of Education, Beijing Jiaotong University, China Institute of Transportation, Inner Mongolia University, Hohhot, China
a r t i c l e
i n f o
Article history: Received 20 November 2011 Received in revised form 17 March 2013 Accepted 21 March 2013
Keywords: Occupancy Dynamic time occupancy Traffic flow Average time gaps Minimum safety distance
a b s t r a c t Occupancy is widely used in traffic flow theory. As a static parameter, however, the traditional occupancy cannot provide a comprehensive description of traffic performance. Based on previous studies, given the minimum safety distance between two adjacent vehicles in the same lane, the dynamic space occupancy (DSO) is redefined and a new parameter, namely the dynamic time occupancy (DTO), is proposed in this paper. It is found that the reaction time and the average time gaps are equivalent under congested condition, and this results in an equivalence relationship between the DTO model and the average time gaps model. Also, a new congested traffic fundamental diagram is derived, which is similar to the triangular traffic flow model. The analysis of a large set of the field data shows that DTO is always equal or close to 1 under congested condition. It is found that the reaction time has a great influence on highway capacity when DTO equals 1, and this may be useful for the capacity analysis or improvement. Different from the traditional static occupancy, DTO is a dynamic parameter that can provide a more fitting description of the occupying relationship between vehicles and highways. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction Density, as one of the basic parameters in traffic flow theory, is frequently used to indicate highway congestion. In 1930s, Greenshields proposed a method to determine the density (Greenshields et al., 1934; Greenshields, 1935). After that, a number of ways of detecting density were developed (Gazis and Knapp, 1971; Gazis and Szeto, 1974; Coifman, 2003; AlvarezIcaza et al., 2004; Artimy, 2007; Jerbi et al., 2007). All these methods are very expensive, complicated and less applicable, which leads to a limited application of density. Occupancy, as a new parameter, is widely used instead, and the research indicates that the replacement is feasible (Lin et al., 1996). Compared with density, occupancy has achieved a widespread use. The earliest studies on occupancy are conducted by Greenshields (1960) and Athol (1965). Afterwards, the research on occupancy appears to be rare (Arasan and Dhivya, 2010; Hsu and Banks, 1993; Papageorgiou and Vigos, 2008). Most of the researchers focused on the relationship of density and occupancy, and some linear and non-linear relationships were established (Hall, 1986; Hall and Persaud, 1989; Banks, 1995; Cassidy and Coifman, 1997; Kim and Hall, 2004). In recent years, occupancy has been widely used in many areas (Payne and Tignor, 1978; Chew and Ritchie, 1995; Dudek et al., 1974; Zhang and Ye, 2006). Despite these applications, occupancy cannot give a comprehensive description of the traffic performance, especially under congested conditions. The space occupancy and the time occupancy refer to the portion of lane length covered by vehicles or the portion of time a local sensor covered by vehicles. They ignore the fact that there is a
⇑ Corresponding author. Tel./fax: +86 471 499 6800. E-mail address: [email protected] (J.X. Cao). 0968-090X/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.trc.2013.03.006
52
C.Z. Xiao et al. / Transportation Research Part C 31 (2013) 51–61
space in front of the moving vehicle which cannot be occupied by other objects. From this point of view, this paper takes the space into consideration and proposes a new concept of occupancy, namely the dynamic time occupancy (DTO). The paper is composed of seven sections. Previous studies on occupancy models are reviewed in the section that follows. Then the earliest concept of the dynamic space occupancy (DSO) is introduced and discussed in Section 3, and a new concept of the dynamic time occupancy is proposed in Section 4, which is followed by the mathematic inferences in the same section. The proof of the relationships between reaction time and average time gaps is conducted in Section 5. In Section 6, a large set of field data is used to analyze the application of DTO and its realistic characteristics. Finally, a summary of this paper is provided in Section 7. 2. The traditional occupancy model The traditional occupancy measurement techniques can be divided into the manual method, the loop detector method and the photo-sensor method. The mechanisms of occupancy measurement can be categorized as time-based measurement and space-based measurement. In the space-based measurement, occupancy is defined as the ratio of the total area occupied by vehicles. In reality, area is often replaced by length for simplicity. The model of the space occupancy can be expressed by Eq. (1) (Bham and Benekohal, 2004).
os ¼
n 1X li L i¼1
ð1Þ
where os is the space occupancy; L is the length of the detected road section (m); li is the length of the ith vehicle (m); and n is the number of vehicles detected. When vehicle length distribution is known, the number of vehicles can be derived from the density investigation, and the space occupancy can be determined by using a large sample of data. However, this method is not only costly but also very complicated, and is not suitable for the automatic measurement. Relatively speaking, the time-based method is more suitable for the automatic measurement, and its basic idea is: the ratio of the time occupied by vehicles at a specific point in detecting time interval can also indicate the occupied situation of the road section. Based on this principle, the automatic measurement of occupancy can be easily realized. The time occupancy model can be written as (Greenshields, 1960; Athol, 1965):
ot ¼
n 1X ti T i¼1
ð2Þ
where ot is the time occupancy; T is the detecting time interval (s); ti is the time occupied by the ith vehicle at the detecting point (s); and n is the number of vehicle detected. In the field survey process, the length of the detector also impacts the detecting data, but for occupancy, the parameter of detector length can be considered by applying Eq. (3) (Wang, 2002).
ot ¼
n 1X ti ¼ T i¼1
P
i ðli
þ ld Þ=v i 1 X li ld X 1 þ ¼ T i vi T i vi T
ð3Þ
where ld is the length of the detector (m); and vi is the speed of the ith vehicle (m/s). Compared with space occupancy, time occupancy is far more popular today, and this should attribute to the development of automatic detection techniques for time occupancy. 3. The initial DSO model Although the time-based and space-based occupancy have achieved wide applications, a fact is neglected that a part of the space between two consecutive vehicles also occupies the road. Under this precondition, the concept of DSO was firstly proposed by Yang et al. (2005). DSO consists of two parts: the space occupied by vehicles (real occupancy) and the space in front of moving vehicles (fictitious occupancy), as is shown in Fig. 1. The DSO model is expressed as:
P oDS ¼
0
þ li Þ
i ðli
ð4Þ
L 0
where oDS is DSO; li is the length of fictitious part of the ith vehicle (m). The length of fictitious part of the ith vehicle is: 0
li ¼ v i t ri þ
v 2i
ð5Þ
max
2di
max
where tri is the reaction time of the driver in the ith vehicle (s), and di (m/s2).
is the maximum deceleration rate of the ith vehicle
C.Z. Xiao et al. / Transportation Research Part C 31 (2013) 51–61
53
Fig. 1. Occupancy relationships.
In Yang’s research, the fictitious space in front of moving vehicles cannot be occupied by other objects. It actually indicates the total stop distance from speed vi to 0, which includes the distance vehicle has traveled during the reaction time period, namely vitri. If the preceding vehicle can achieve a complete stop without a braking distance, the fictitious distance and minimum safety distance are equal, but the situation is impossible in reality. That is to say, the fictitious space is always larger than the minimum safety distance and this often makes the fictitious space be occupied by the preceding vehicles under high flow rate condition. The value of DSO calculated from the model is always on the high side, even over 2. However, the value of DSO should be no greater than 1 in theory, so it is very hard to find the boundary of the DSO value with this model. Besides, Yang proposed a video-based method for the data collection to determine DSO. The full view of the road is firstly recorded and the data of vehicle length and speed are extracted respectively. However, the video-based method is costly and complicated, and highly affected by the weather, so it is infeasible for large area detection. Different from Yang’s hypothesis of the fictitious occupancy, the minimum safety distance is used for instead, and DTO is defined in the following part. 4. The definition of DTO and its mathematical analysis In recent years, traffic safety is getting more and more important. Drivers, vehicles, and traffic environment have a great impact on traffic safety, while drivers are the main factor causing accidents (Kuge et al., 1995). The speed of a vehicle is controlled by the driver, and there should be a proper space between two adjacent vehicles in the same lane. If the space between vehicles are too small, a rear-end collision can easily happen. According to the traffic accident statistical data, rearend collision accounted for more than half the total accidents (Wilson et al., 1997), and the countless traffic safety problems have attracted wide attention. Scholars agree that there should be a space, namely the minimum safety distance, between two following vehicles for safety concern. Under normal circumstances, the distance between the two following vehicles should be larger than the minimum safety distance, so it is reasonable to believe that the space of minimum safety distance should also occupy a road section. That is to say, the occupancy of the road includes two parts, namely the vehicle length (real occupancy) and the minimum safety distance (fictitious occupancy), which is different from Yang’s DSO model. Some of the early studies on the minimum safety distance have been conducted by Fukuhara (1994) and Hamed and Jaber (1997). After that, some minimum safety distance models have been proposed by Carbaugh et al. (1998) and Zhou (2003). All the philosophies of the minimum safety distance models above are similar, so a strong representative minimum safety distance model is chosen in this paper (Kenue, 1995), which is expressed as:
d ¼ v fo t r þ
v 2fo 2af
v 2l
ð6Þ
2al
where d is the minimum safety distance (m); vfo is the speed of the following vehicle (m/s); vl is the speed of the leading vehicle (m/s); tr is driver’s reaction time (s); af is the deceleration rate of the following vehicle (m/s2); and al is the deceleration rate of the preceding vehicle (m/s2). Suppose the preceding vehicle encompasses a sudden full brake, and there should be a reaction time for the following vehicle to make a stop decision. In this short period, the speed of the following vehicle is invariable, so the moved distance is vfotr. After that, the following vehicle begins to brake, and the braking distance is v 2fo =ð2af Þ, which is the same to the preceding vehicle. Thus, the minimum safety distance is the difference of v fo tr þ v 2fo =ð2af Þ and v 2lo =ð2al Þ. As is illustrated in Fig. 2, suppose that there are n vehicles in a same lane, and the moving speed and maximum deceleration rate for the ith vehicle are vi (m/s) and ai (m/s2), respectively, so the occupied space of the ith vehicle at the specific time is:
si ¼ li þ di ¼ li þ v i tr þ
v 2i 2ai
v 2i1 2ai1
ð7Þ
54
C.Z. Xiao et al. / Transportation Research Part C 31 (2013) 51–61
Fig. 2. The status of vehicle flow.
So the total occupied space of the n vehicles is:
S¼
n n n X X X si ¼ ðli þ di Þ ¼ li þ v i t r þ i¼1
i¼1
i¼1
v 2i 2ai
v 2i1
n X
¼
2ai1
ðli þ v i tr Þ þ
i¼1
v 2n 2an
v 21 2a1
ð8Þ
Pn In general, the total number of vehicles (n) is a large value, so the value of i¼1 ðli þ v i t r Þ is far greater than v 2n =ð2an Þ v 21 =ð2a1 Þ. However, the speed and the maximum deceleration rate of the vehicles in the counting period are similar, so v 2n =ð2an Þ v 21 =ð2a1 Þ 0. Eq. (8) can be simplified as:
S
n X ðli þ v i tr Þ
ð9Þ
i¼1
Making use of Eqs. (1) and (9), the formula of DSO can be written as:
oDS ¼
S ¼ L
Pn
þ v i tr Þ L
i¼1 ðli
ð10Þ
As is expressed in Eq. (10), making use of Eqs. (3) and (9), the formula of DTO can be written as:
Pn
i¼1 ðli
oDTm ¼
þ ld þ v i t r Þ=v i T
ð11Þ
where oDTm is the measured DTO, including the effects of detector length. From the equations above, it can be concluded that for a certain number of vehicles, the measured DTO depends on the length and the speed of vehicles, and the driver’s response time, but is independent of the deceleration characteristics. By simplifying Eq. (11), a new formula can be obtained as:
Pn
i¼1 ðli
oDTm ¼
þ ld þ v i t r Þ=v i ¼ T
n n 1X li ld X 1 þ T i¼1 v i T i¼1 v i
! þ
n 1X tr T i¼1
ð12Þ
Use Eq. (3) to simplify Eq. (12), then a new relationship occurs.
oDTm ¼ ot þ qtr
ð13Þ
According to the equations above, it shows that the vehicle lengths directly impact the value of time occupancy, and have certain indirect impacts on DTO. Nevertheless, time occupancy contains all the information of vehicles lengths. Consequently, if DTO is to be detected, it is not necessary to determine the length of each vehicle, but to get the value of time occupancy. Although Eq. (13) shows a way to calculate DTO, it is still necessary to reveal the relationship between DTO and speed/density/flow rate. See the following derivation. If there are n vehicles in the counting time interval T, the flow rate q (veh/s) can be written as:
q¼
n n 1 1 ¼ Pn ¼ 1 Pn ¼ T h h h i¼1 i i¼1 i n
ð14Þ
is the mean headway in the countering interval (s). where hi is the headway of the ith vehicle and the preceding vehicle (s), and h The space mean speed v (m/s) is:
1
v ¼ 1 Pn n
i¼1
ð15Þ
1
vi
Combine Eqs. (3), (14), and (15), and a result is as below:
ot ¼
1 X li q þ ld T i vi v
ð16Þ
where T can be treated as the sum of headways of all vehicles. Suppose that the vehicle length is a constant l (m), so:
ot ¼
Pn l 1 i¼1 v i n P n 1 i¼1 hi n
þ ld
q
v
¼
1 n
Pn
i¼1
h
l
vi
þ ld
q
v
¼
n l 1X 1 q þ ld v h n i¼1 v i
ð17Þ
C.Z. Xiao et al. / Transportation Research Part C 31 (2013) 51–61
55
Substitute Eqs. (15)–(17), then:
ot ¼
l 1 h
v
þ ld
q
ð18Þ
v
On the basis of Eqs. (14) and (18) can be derived as:
ot ¼ l
q
v
þ ld
q
¼ ðl þ ld Þ
v
q
v
¼ ðl þ ld Þk
ð19Þ
According to Eqs. (14) and (19), another expression of DTO is:
oDTm ¼ ðl þ ld Þ
q
v
þ qtr
ð20Þ
Note that in Eq. (20), the model actually indicates the measured DTO value, which includes the effect of the detector length. If ld is removed, as can be seen below, the physical DTO can be determined:
oDTp ¼ l
q
v
þ qtr ¼ oDTm ld
q
ð21Þ
v
where oDTp is the physical DTO, namely the portion of time a local detector are covered by total vehicles lengths and minimum safety distances which do not include the detector effect. Under totally congested traffic condition, drivers tend not to stop bumper-to-bumper. There must be some space buffer between vehicles for both stopping and running vehicles. Let DTO equal to 1, when traffic totally jammed, and the buffer space is actually the jam gap. According to the abovementioned, the behavior DTO, which includes the effect of the buffer length, is:
oDTb ¼ ðl þ bÞ
q
v
þ qtr ¼ o þ qtr þ ðb ld Þ
q
v
ð22Þ
where oDTb is the behavior DTO, namely the portion of time a local detector are covered by total vehicles lengths, minimum safety distances, and total buffer lengths which do not include the detector effect; b is the buffer space or jam gap (m). 5. Reaction time determination For the DTO model, detector length, average vehicle length, and buffer length/jam gap are easy to be determined, except the reaction time. For a moving vehicle, the time gap represents the driver’s window of opportunity for reacting to the motion of the preceding vehicle, and the time gap needs to be at least as large as the driver’s reaction time (Banks, 2003). This point of view puts forward an idea to calculate the reaction time. The proof of the relationship between the reaction time and the average time gaps is shown below. Most vehicles follow the preceding vehicles closely under congestion and there is not enough space for them to accelerate or to overtake. For the safety consideration, the average time gaps must contain the drivers’ reaction time, tg P tr. However, the speed of vehicles is much less than the desired speed under congestion condition, thus drivers tend to minimize the time gaps. Nevertheless, the reaction time is the threshold value of the average time gaps.
Fig. 3. Vehicles trajectories diagram.
56
C.Z. Xiao et al. / Transportation Research Part C 31 (2013) 51–61
In case that there are two consecutive vehicles that move under congested condition with speed v1, the preceding vehicle suddenly decelerates its speed to v2 and maintains this speed, so the following vehicle should also change its speed to v2 at a certain point. The time–space diagram is shown in Fig. 3, where the points A(tA, xA), B(tB, xB), C(tC, xC), D(tD, xD), B1(tB1, xB1), and B2(tB2, xB2) are all on the trajectories lines, and the vehicle length equals to xD xA or xB xC. The reaction time is tB tA, where tg1 and tg2 are the time gaps before and after the speed change, respectively. Let xA = xB2, and xA = xB1, where point B1 is on the line of the head trajectory of the preceding vehicle after the speed change. It is clear that tg1 = tB2 tA, and tg2 = tB1 tA. From Banks’ analysis (Banks, 2003), it is known that tg1 = tg2, so tB1 = tB2. This indicates that points B1 and B2 are the same point, namely point B. For moving vehicles under congestion, the signal of the sudden deceleration of the preceding vehicle can be captured by the driver in the following vehicle immediately, so the reaction time tr = tB tA = tB2-tA = tg1 = tg2. The relationship between reaction time and average time gaps is (see Fig. 4):
tg ¼ tr
ð23Þ
In Banks’ work, the average time gaps is divided into three categories, namely the physical time gaps, the measured time gaps and the behavioral time gaps (Banks, 2003).
gm ¼
1o q
g ¼ gm þ
ð24Þ
ld
ð25Þ
v
gb ¼ gm þ
ld b
ð26Þ
v
where gm is the measured time gaps (s); g is the physical time gaps (s); and gb is the behavioral time gaps (s). In Eq. (23), both tr and g are physical values. Comparing Eq. (21) with Eq. (25), the following relationship is obtained:
(
oDTp ¼ o þ qt r ld vq
ð27Þ
1 ¼ o þ qg ld vq
Because tr = g, it is obviously that the physical value of DTO equals 1 under the congested traffic condition. That is to say, the physical DTO model and the physical average time gaps model are identical under the congested condition while the average time gaps is a constant. In the DTO model, ot and q can be easily captured by devices, and v can be calculated from the measured space mean speed, while ld is a known parameter. Although the reaction time is difficult to detect directly, from the analysis above, the average time gaps can be used instead. In general, the average time gaps are different for different lanes, so choosing the minimum one for the implication of the reaction time is reasonable.
tr ¼ minfg i g
ð28Þ
where gi is the average time gaps of the ith lane (s).
Fig. 4. Congested traffic flow fundamental diagram.
C.Z. Xiao et al. / Transportation Research Part C 31 (2013) 51–61
57
6. The application of DTO and its characteristics According to the above analysis, the relationships between traffic flow rate, speed and density can be derived when DTO equals 1. From Eq. (27), we can obtain the following formulation:
1 ¼ o þ qt r ld
q
v
¼l
q
v
þ qtr
ð29Þ
So the relationships of the three parameters are:
8 lq v ¼ 1qt > > r < v ¼ 1lk ktr > > : q ¼ 1lk tr
ð30Þ
Note that the models above only apply to the congested condition, and the flow–density model is similar to the triangular flow–density model (Banks, 1989). Supposing speed is a constant for non-congested condition, the fundamental diagram can be obtained as below. For further analysis, the traffic flow data from the I80 expressway at California is used. The data represents all 30 s intervals for a day, which includes the information of all five lanes at a fixed location. In Fig. 5, it is evident that the average time gaps form a horizontal band from 15 to 60 km/h which represents the congested traffic condition. Compares the average time gaps of five lanes (see Fig. 6), and the reaction time can be easily derived according to Eq. (28), so tr = min{g5} 1.5 s. Because DTO is a constant under congested condition, there should be a horizontal liner relationship between the three traffic parameters and DTO. From Figs. 7 and 8, it is clear that there is a linear or near-linear relationship while traffic is congested. When the occupancy is over 20%, traffic congests, and the physical DTO for each lane remains the same value or decreases slightly. Only the values of Lanes 2 and 5 exceed 1 or fluctuate around 1. Moreover, there is a sudden fall for the physical DTO of Lane 2 and Lane 4 while the occupancy is over 70%, and all of them go to the congested point in theory, namely the congestion occupancy. For the behavior DTO, the fall no longer exists. Compared with the physical DTO, the mean values of the behavior DTO are more closer to 1 and become more stable under congestion (excepting for Lane 2 under very low speed). All data cases show that DTO equals a constant under congestion. Consider the fact that traffic has a finite movement even under the congested condition (Drake and May, 1967), and a stop-and-go speed is used as an additional parameter, which is referred to the research by Wang et al. (2011). Two sets of the congested speed–density data are used to analyze the derived traffic flow models, expressed by Eq. (30). The free-flow speed of I-80 is smaller than that of US-101, and this leads to the different length of speed–density regime. Despite this, they both have the same or similar shape. The field data provides a proof that traffic has a similar character under the congested condition. Taking the congested speed–density data as an example, the numerical relationship between the two parameters is more like the fractional function. DTO, as a new traffic flow parameter, can provides a reasonable explanation for this. No matter what conditions are, the congested traffic maintains a constant DTO value, which is always equals 1 or close to it. Consequently, the congested traffic flow fundamental diagram can be determined by Eq. (20). That is to say, the shape of congested traffic fundamental diagram is up to the safety consideration of drivers.
Fig. 5. Scatter points of average time gaps and speed.
58
C.Z. Xiao et al. / Transportation Research Part C 31 (2013) 51–61
Fig. 6. Average congested time gaps and speed.
Fig. 7. The relationship between the occupancy and the physical DTO.
Fig. 8. The relationship between speed and the behavior DTO.
C.Z. Xiao et al. / Transportation Research Part C 31 (2013) 51–61
59
Fig. 9. Speed-density raw data fitting by the proposed model.
Fig. 10. Congested speed–density relationship with variable vehicle length and reaction time.
Suppose that DTO equals 1 under congested condition, so the relationship between speed and flow rate can be described as Eq. (30). Through the above analysis, it is known that all congested traffic has a common character, namely that DTO equals 1 or very close to 1. In spite of non-congested traffic, the following two figures are used to indicate the speed-flow relationship. Obviously, flow rate always increases as speed increases, but the increasing becomes more and more invisible, and finally goes to a fixed value, namely q = 1/tr (veh/s/ln). Supposing the speed is a constant for non-congested traffic, it is easy to find out the fact that vehicle length changes the capacity of the road, but the influence is not obvious. However, the change becomes very obvious with fixed vehicle length and variable reaction time, as is shown in Fig. 10. All these state clearly that reaction time, vehicle length and free flow speed are the three factors which determine the highway capacity. Reducing vehicle length or reaction time, increasing free flow speed, or a combination of them, can lead to capacity improving. However, the vehicle length is hard to change, and tiny length change cannot improve the capacity obviously. For free-flow speed, there is not enough space for the improvement of capacity while free-flow speed is over 100 km/h, as can be seen in Fig. 10. Nevertheless, a small reaction time is achievable through some intelligent methods such as automated driving.
7. Conclusion Density and occupancy have been broadly applied in transportation as parameters of traffic performance, but the application of density is not as good as occupancy for the difficulties of data collection. Density has been replaced by occupancy in many fields. Occupancy takes the length of vehicles into consideration and can give a more accurate description of the traffic situation than density sometimes. However, occupancy is still a static parameter in essence. In reality, road is occupied by vehicles, but there is still some space cannot be occupied by other objects, and this kind of space also occupies the road. So the ‘‘occupancy’’ here consists of two parts: the real occupancy and the fictitious occupancy. In order to give a more reasonable description of this phenomenon, a new concept of occupancy has been proposed in this paper, and many of the characteristics have been discussed. There are several main findings in the paper:
60
C.Z. Xiao et al. / Transportation Research Part C 31 (2013) 51–61
(1) Combining with the minimum safety distance, DSO is redefined and a more reasonable parameter, namely DTO, is proposed. With the support of a large set of field data, it is observed that the value of DTO is equal or close to 1 under congested condition, and this may be a nature of the moving traffic. DTO equals to 1 is reasonable and acceptable in theory, which is indicted by field data results in Fig. 9. (2) It is proved that reaction time and average time gaps are identically equal under congested condition, and this results in an equivalence relationship between the physical DTO model and the physical average time gaps model. (3) A congested traffic fundamental diagram is proposed based on DTO, and the proposed model has a simple expression which is very similar to triangular speed–flow–density model. Besides, theoretical analysis shows that the reaction time plays an important role in highway capacity improvement.
Different from the traditional occupancy, DTO is a dynamic parameter and can give a more fitting description for traffic performance. However, the application of DTO still need further studies, and more application results should be put forward to back it up. Acknowledgements This work is supported by the National Natural Science Foundation of China (Grant Nos. 71061010, 71262008), the research start-up project for the high-level talent introduction of Inner Mongolia University (Grant No. 210221) and the Scientific Research Foundation for the Returned Overseas Chinese Scholars. References Alvarez-Icaza, L., Munoz, L., Sun, X.T., Horowitz, R., 2004. Adaptive observer for traffic density estimation. In: Proceedings of the 2004 American Control Conference, Boston, MA, pp. 2705–2710. Arasan, V.T., Dhivya, G., 2010. Methodology for determination of concentration of hetrogeneous traffic. Journal of Transportation Systems Engineering and Information 10 (4), 50–61. Artimy, M., 2007. Local density estimation and dynamic transmission-range assignment in vehicular ad hoc networks. IEEE Transactions on Intelligent Transportation Systems 8 (3), 400–412. Athol, P., 1965. Interdependence of certain operational characteristics within a moving traffic stream. Highway Research Record 72, 58–87. Banks, J.H., 1989. Freeway speed-flow-concentration relationships: more evidence and interpretations (with discussion and closure). Transportation Research Record 1225, 53–60. Banks, J.H., 1995. Another look at a priori relationships among traffic flow characteristics (with discussion and closure). Transportation Research Record 1510, 1–10. Banks, J.H., 2003. Average time gaps in congested freeway flow. Transportation Research Part A 37, 539–554. Bham, G.H., Benekohal, R.F., 2004. A high fidelity traffic simulation model based on cellular automata and car-following concepts. Transportation Research Part C 12 (1), 1–32. Carbaugh, J., Godbole, D.N., Sengupta, R., 1998. Safety and capacity analysis of automated and manual highway systems. Transportation Research Part C 6C (1–2), 69–99. Cassidy, M.J., Coifman, B., 1997. Relation among average speed, flow, and density and analogous relation between density and occupancy. Transportation Research Record 1591, 1–6. Chew, R.L., Ritchie, S.G., 1995. Automatic detection of lane-blocking freeway incidents using artificial neural networks. Transportation Research Part C 6 (3), 371–388. Coifman, B., 2003. Estimating density and lane inflow on a freeway segment. Transportation Research Part A 37 (8), 689–701. Drake, J.L.S.J.S., May, A.D., 1967. A statistical analysis of speed–density hypotheses. Highway Research Record 156, 53–87. Dudek, C.L., Messer, C.J., Nuckles, N.B., 1974. Incident detection on urban freeway. Transportation Research Record 495, 12–24. Fukuhara, H., 1994. Vehicle Collision Alert System. US Patent No: 5355118. Gazis, D., Knapp, C., 1971. On-line estimation of traffic densities from time-series of flow and speed data. Transportation Science 5 (3), 283–301. Gazis, D., Szeto, M., 1974. Design of density-measuring systems for roadways. Transportation Research Record 495, 44–52. Greenshields, B.D., 1935. A study of traffic capacity. In: Proceedings of the 14th Annual Meeting of the Highway Research Record, Washington, DC, pp. 448– 477. Greenshields, B.D., 1960. The density factor in traffic flow. Traffic Engineering 30 (6), 26–28, 30. Greenshields, B.D., Thompson, J.T., Dickinson, H.C., Swinton, R.S., 1934. The photographic method of studying traffic behavior. In: Proceedings of the 13th Annual Meeting of the Highway Research Board, Washington, DC, pp. 382–399. Hall, F.L., 1986. The relationship between occupancy and density. Transportation Forum, 46–52. Hall, F.L., Persaud, B.N., 1989. Evaluation of speed estimated made with single-detector data from freeway traffic management systems. Transportation Research Record 1232, 9–16. Hamed, M., Jaber, S., 1997. Modeling vehicle-time head ways in urban multilane highways. Road and Transport Research 6 (4), 32–44. Hsu, P., Banks, J.H., 1993. Effects of location on congested-regime flow–concentration relationships for freeways. Transportation Research Record 1398, 17– 23. Jerbi, M., Senouci, S., Rasheed, T., Ghamri-Doudane, Y., 2007. An infrastructure-free traffic information system for vehicular networks. In: Proceedings of the 66th IEEE Vehicular Technology Conference, Baltimore, MD, pp. 2086–2090. Kenue, S.K., 1995. Selection of range and azimuth angle parameters for a forward looking collision warning radar sensor. In: Proceedings of the Intelligent Vehicles ‘95 Symposium, Boston, pp. 494–499. Kim, Y., Hall, F.L., 2004. Relationships between occupancy and density reflecting average vehicle lengths. Transportation Research Record 1883, 85–93. Kuge, N., Ueno, H., Ichikawa, H., Ochiai, K., 1995. Study on the causes of rear-end collision based on an analysis of driver behavior. JSAE Review 16 (1), 55–60. Lin, F.B., Su, C.W., Huang, H.H., 1996. Uniform criteria for level of service analysis of freeways. Journal of Transportation Engineering 122 (2), 123–130. Papageorgiou, M., Vigos, G., 2008. Relating time-occupancy measurements to space-occupancy and link vehicle-count. Transportation Research Part C 16, 1–17. Payne, H.J., Tignor, S., 1978. Freeway incident detection algorithms based on decision trees with states. Transportation Research Record 682, 30–37. Wang, D.H., 2002. Traffic Flow Theory. China Communications Press, Beijing (in Chinese). Wang, H.Z., Li, J., Chen, Q.Y., Ni, D.H., 2011. Logistic modeling of the equilibrium speed–density relationship. Transportation Research Part A 45, 554–566.
C.Z. Xiao et al. / Transportation Research Part C 31 (2013) 51–61
61
Wilson, T., Butler, W., McGehee, D.V., Dingus, T.A., 1997. Forward-Looking Collision Warning System Performance Guidelines. SAE Technical Paper 970456. Yang, J.G., Zhang, J.H., Yi, X.Q., Wang, Z.A., 2005. Dynamic space occupancy and measuring method. Journal of Tongji University (Natural Science) 33 (11), 1474–1478 (in Chinese). Zhang, Y.L., Ye, Z.R., 2006. A derivative-free nonlinear algorithm for speed estimation using data from single loop detectors. In: Proceedings of the 2006 Intelligent Transportation Systems Conference, pp. 1035–1040. Zhou, W., 2003. Analysis of distance headway. Journal of Southeast University (English Edition) 19 (4), 378–381.