城市主干路交通溢流建模及其仿真研究
摘要
随着我国城市化的快速发展,城市人均汽车保有量迅速攀升,给空间极度有限的城市交通网络造成了前所未有的巨大压力。近年来,交通工程学和信息技术取得了突飞猛进的进步,给智能交通系统带来了巨大的发展前景,但目前仍然不能解决交通出行中遇到的所有问题,拥堵仍然是城市交通的顽疾,随之而来的是能源浪费、环境污染和安全事故等衍生问题。
交通拥堵问题是近几年国内外交通领域学者研究的热点问题之一。要解决交通拥堵问题,关键是深入挖掘交通拥堵的产生机理,建立科学合理的数学模型,刻画复杂多变的实际交通问题产生过程,进而选择针对性的控制策略,实现交通供需之间的动态均衡。制约城市主干路通行能力的关键点是路口,由于“瓶颈效应”,路口的通行能力比路段低,交通需求过大就会引发拥堵。交通溢流现象是路口交通拥堵的一种极端现象,会引发以拥堵路口为中心的四个方向的格锁,随着发生格锁路口的增多,整个交通路网会发生大面积的交通瘫痪。因此,探索合理的控制策略避免交通溢流的发生是非常有意义的。
排队长度是造成交通溢流现象的关键因素,也是研究交通溢流发生机理的着手点。基于以上解决交通拥堵的基本思路,本文深入研究了排队增长和消散的动态变化过程,利用冲击波理论对单路口的排队做了分析,并对传统的冲击波理论做了改进,利用视频检测器获取路口的实时排队,提出了基于排队的冲击波理论,提高了冲击波理论在估计排队长度方面的精确度。在路段排队长度的分析中,将冲击波理论和传统的累积的输入输出方法结合起来,利用基于排队的冲击波理论估计路口的排队消散速度,利用累积的原理估计路段某段时间后的排队长度,克服了交通波理论要求稳定交通流的弊端,提高了模型的鲁棒性。最后,在此研究之上挖掘交通溢流形成的机理,提出了基于“物理排队”的交通溢流发生机理模型,并提出了交通溢流持续时间和平均溢出排队长度两个关键参数的估计方法。仿真显示,该方法的基本思路与实际交通溢流发生现状比较吻合。
此外,本文还提出了交通溢流控制模型。在单点信号配时研究和单向绿波协调信号配时方法研究基础上,提出了城市主干路协调控制策略下以避免交通溢流发生为目标的相位差优化的思想,利用基于排队的冲击波理论和累积的输入输出方法,根据从下游路口信号周期开始到最大排队发生时上游到达的车辆数加路段初始排队车辆数等于下游路口消散的车辆数的基本物理原理,推导出了发生交通溢流时的临界相位差模型。算例分析通过改变同一长度路段的流量和同一流量下改变路段的长度,分别计算临界相位差,结果显示,模型能够有效避免交通溢流现象的发生。
本文提出了交通溢流发生机理模型和交通溢流控制模型,揭示了交通格锁拥堵形成的机理原因和控制方法,为解决交通格锁拥堵提供了理论依据,丰富了微观交通流理论,为微观交通流理论与控制策略研究提供了新的理论方法。
With the development of urbanization speeding up, car ownership of per urban person rises rapidly and causes unprecedented pressure to urban traffic network with extremely limited space. In recent years, although transportation engineering science and information technology have made obvious progress and brought great development prospects to the Intelligent Transportation System (ITS), traffic problems still can't be solved and congestion is still the crucial ill of urban traffic and energy waste, environment pollution and safety incidents are consequent derivative problems.
Traffic congestion is one of the hot issues for domestic and foreign traffic region researchers in recent years. To solve the traffic congestion problem, the key is to dig the generation mechanism of traffic congestion deeply, establish a scientific and reasonable mathematical model to portray the complex and changeable traffic generation process, and then select the targeted control strategies, to realizes the dynamic equilibrium between supply and demand of traffic. The key point to restrict traffic capacity of the arterials is intersection. The traffic capacity of an intersection is lower than a link because of the bottleneck effect and excessive traffic demand will lead to congestion. The traffic spillover is an extreme phenomenon of intersection traffic congestion and it will lead to four directions grid lock as the center of the original congestion intersection. With the increase of the grid lock intersection occurs, the entire road network will happen large scale congestion. Therefore, to explore a reasonable control method to avoid traffic spillover is very meaningful.
Queue length is the key factor to traffic spillover and is the start point of the traffic spillover formation mechanism research. Based on the above basic ideas to solve the traffic congestion, this paper studies the dynamic process of queue growth and dissipation deeply, shockwave theory is used to analysis the queue process of a single intersection, and the traditional shockwave theory is improved, real-time queue length is obtained by the video detector, shockwave theory based on queue is proposed and the accuracy of queue length estimation based on shockwave theory is improved. The shockwave theory and cumulative input-output method are combined to analysis the queue process of a link, shockwave theory based on queue is used to estimate queue dissipate velocity, queue length of a link after sometime is estimated according to the cumulative principle, the shortcoming of stable traffic flow acquired of traditional traffic-wave theory is conquered and robustness of the model is increased. At last, traffic spillover formation mechanism is present according to above researches and spillover formation mechanism model based on physical queue is proposed. Estimate method of spillover duration time and average overflow queue length are also proposed. The simulation shows the basic idea of the method is relatively consistent with the actual traffic overflow status.
Moreover, this paper also presents the spillover control model. A signal offset optimal ideology aimed to avoid spillover of arterials under coordinated control strategy is proposed based on previous single-point signal timing and coordinated control strategy researches. According to arrival vehicles from start time of a cycle to the maximal queue time of the downstream intersection add initial queue vehicles equals vehicles dissipates from the downstream, the threshold of offset is derived when a spillover is happening based on shockwaves theory and cumulative input-output method. Example analysis is implemented by change traffic flow and queue, offset thresholds are calculated respectively. Results shows the model can avoid traffic spillovers visibly.
This paper presents a traffic spillover formation mechanism model and a traffic spillover control model, revealed the reasons for the formation and control methods of traffic grid lock congestion, provides a theoretical basis to solve the traffic grid lock congestion, riches the microscopic traffic flow theory and provide a new theoretical method for microscopic traffic flow theory and control strategies.
交通拥堵问题是近几年国内外交通领域学者研究的热点问题之一。要解决交通拥堵问题,关键是深入挖掘交通拥堵的产生机理,建立科学合理的数学模型,刻画复杂多变的实际交通问题产生过程,进而选择针对性的控制策略,实现交通供需之间的动态均衡。制约城市主干路通行能力的关键点是路口,由于“瓶颈效应”,路口的通行能力比路段低,交通需求过大就会引发拥堵。交通溢流现象是路口交通拥堵的一种极端现象,会引发以拥堵路口为中心的四个方向的格锁,随着发生格锁路口的增多,整个交通路网会发生大面积的交通瘫痪。因此,探索合理的控制策略避免交通溢流的发生是非常有意义的。
排队长度是造成交通溢流现象的关键因素,也是研究交通溢流发生机理的着手点。基于以上解决交通拥堵的基本思路,本文深入研究了排队增长和消散的动态变化过程,利用冲击波理论对单路口的排队做了分析,并对传统的冲击波理论做了改进,利用视频检测器获取路口的实时排队,提出了基于排队的冲击波理论,提高了冲击波理论在估计排队长度方面的精确度。在路段排队长度的分析中,将冲击波理论和传统的累积的输入输出方法结合起来,利用基于排队的冲击波理论估计路口的排队消散速度,利用累积的原理估计路段某段时间后的排队长度,克服了交通波理论要求稳定交通流的弊端,提高了模型的鲁棒性。最后,在此研究之上挖掘交通溢流形成的机理,提出了基于“物理排队”的交通溢流发生机理模型,并提出了交通溢流持续时间和平均溢出排队长度两个关键参数的估计方法。仿真显示,该方法的基本思路与实际交通溢流发生现状比较吻合。
此外,本文还提出了交通溢流控制模型。在单点信号配时研究和单向绿波协调信号配时方法研究基础上,提出了城市主干路协调控制策略下以避免交通溢流发生为目标的相位差优化的思想,利用基于排队的冲击波理论和累积的输入输出方法,根据从下游路口信号周期开始到最大排队发生时上游到达的车辆数加路段初始排队车辆数等于下游路口消散的车辆数的基本物理原理,推导出了发生交通溢流时的临界相位差模型。算例分析通过改变同一长度路段的流量和同一流量下改变路段的长度,分别计算临界相位差,结果显示,模型能够有效避免交通溢流现象的发生。
本文提出了交通溢流发生机理模型和交通溢流控制模型,揭示了交通格锁拥堵形成的机理原因和控制方法,为解决交通格锁拥堵提供了理论依据,丰富了微观交通流理论,为微观交通流理论与控制策略研究提供了新的理论方法。
With the development of urbanization speeding up, car ownership of per urban person rises rapidly and causes unprecedented pressure to urban traffic network with extremely limited space. In recent years, although transportation engineering science and information technology have made obvious progress and brought great development prospects to the Intelligent Transportation System (ITS), traffic problems still can't be solved and congestion is still the crucial ill of urban traffic and energy waste, environment pollution and safety incidents are consequent derivative problems.
Traffic congestion is one of the hot issues for domestic and foreign traffic region researchers in recent years. To solve the traffic congestion problem, the key is to dig the generation mechanism of traffic congestion deeply, establish a scientific and reasonable mathematical model to portray the complex and changeable traffic generation process, and then select the targeted control strategies, to realizes the dynamic equilibrium between supply and demand of traffic. The key point to restrict traffic capacity of the arterials is intersection. The traffic capacity of an intersection is lower than a link because of the bottleneck effect and excessive traffic demand will lead to congestion. The traffic spillover is an extreme phenomenon of intersection traffic congestion and it will lead to four directions grid lock as the center of the original congestion intersection. With the increase of the grid lock intersection occurs, the entire road network will happen large scale congestion. Therefore, to explore a reasonable control method to avoid traffic spillover is very meaningful.
Queue length is the key factor to traffic spillover and is the start point of the traffic spillover formation mechanism research. Based on the above basic ideas to solve the traffic congestion, this paper studies the dynamic process of queue growth and dissipation deeply, shockwave theory is used to analysis the queue process of a single intersection, and the traditional shockwave theory is improved, real-time queue length is obtained by the video detector, shockwave theory based on queue is proposed and the accuracy of queue length estimation based on shockwave theory is improved. The shockwave theory and cumulative input-output method are combined to analysis the queue process of a link, shockwave theory based on queue is used to estimate queue dissipate velocity, queue length of a link after sometime is estimated according to the cumulative principle, the shortcoming of stable traffic flow acquired of traditional traffic-wave theory is conquered and robustness of the model is increased. At last, traffic spillover formation mechanism is present according to above researches and spillover formation mechanism model based on physical queue is proposed. Estimate method of spillover duration time and average overflow queue length are also proposed. The simulation shows the basic idea of the method is relatively consistent with the actual traffic overflow status.
Moreover, this paper also presents the spillover control model. A signal offset optimal ideology aimed to avoid spillover of arterials under coordinated control strategy is proposed based on previous single-point signal timing and coordinated control strategy researches. According to arrival vehicles from start time of a cycle to the maximal queue time of the downstream intersection add initial queue vehicles equals vehicles dissipates from the downstream, the threshold of offset is derived when a spillover is happening based on shockwaves theory and cumulative input-output method. Example analysis is implemented by change traffic flow and queue, offset thresholds are calculated respectively. Results shows the model can avoid traffic spillovers visibly.
This paper presents a traffic spillover formation mechanism model and a traffic spillover control model, revealed the reasons for the formation and control methods of traffic grid lock congestion, provides a theoretical basis to solve the traffic grid lock congestion, riches the microscopic traffic flow theory and provide a new theoretical method for microscopic traffic flow theory and control strategies.
引文
[1]京华时报.中国汽车保有量首破亿辆占机动车总量45.88%[EB/OL]北京：中国新闻网,2011, http://www.chinanews.com/auto/2011/09-17/3334312.shtml.
[2]杨佩昆,吴兵.交通管理与控制[M].第二版.北京：人民交通出版社,2005.
[3]张立东.城市交通溢流智能协调控制算法研究[D].济南：山东大学,2012.
[4]全永燊，城市交通控制[M].北京：人民交通出版社,1989.
[5]徐吉谦.交通工程总论[M].北京：人民交通出版社,1991.
[6]Reusch B..Vehicle movements in a platoon[J]. Oesterreichisches Ingenieur-Archir, 1950,4:193-215.
[7]Pipes L.A.. An operational analysis of traffic dynamics[J]. Journal of Applied Physics,1953,24(3):274-281.
[8]李曙光,许宏科.具有物理排队的多模式动态网络加载问题研究[J].交通运输系统工程与信息,2009,1(9)：56-61.
[9]Lighthill M.J.,Whitham G. B.. On kinematic waves I Flood movement in long rivers. Π, A theory of traffic flow on long crowded roads. Proceedings of Royal Society(London),1955, A(229):281-345.
[10]Richards P. J.. Shock waves on the highway[J]. Operations Research 1956,4:42-51.
[11]Stephanopoulos G., Michalopoulos P. G.. Modeling and analysis of traffic queue dynamics at signalized intersections [J]. Transportation Research,1979,13 A.
[12]Michalopoulos P.G., Stephanopoulos G.. An application of shock wave theory to traffic signal control [J]. Transportation Research Part B,1981,15:31-51.
[13]Henry X. Liu, Xinkai Wu, Wenteng Ma, et al. Real-time queue length estimation for congested signalized intersections [J]. Transportation Research Part C,2009, 17:412-427.
[14]Xinkai Wu, Henry X.Liu, Gettman D.. Identification of oversaturated intersections using high-resolution traffic signal data[J], Transportation Research Part C,2010,18:626-638.
[15]陈宁宁,何兆成,余志.考虑动态红灯排队消散时问的改进MAXBAND模型[J].武汉理工大学学报：交通科学与工程版.2005,5(33)：843-847.
[16]Xinkai Wu, Henry X. Liu. A shockwave profile model for traffic flow on congested urban arterials[J]. Transportation Research Part B,2011,45:1768-1786.
[17]Wolfram S.. Statistical mechanics of cellular automata[J]. Reviews of Modern Physics,1983,55(3):601-644.
[18]Geroliminis N.. Queue spillovers in city street networks with signal-controlled intersections[R]. Swiss Transport Research Conference,2009.
[19]Xuegang (Jeff) Ban, Peng Hao, Zhanbo Sun. Real time queue length estimation for signalized intersections using travel times from mobile sensors[J]. Transportation Research Part C,2011,19:1133-1156.
[20]Daganzo C. F.. The nature of freeway gridlock and how to prevent it[A], Proceeding of 13th International Symposium Transportation and Traffic Theory[C], 1996:629-646.
[21]Daganzo C. F.. Improving city mobility through gridlock control 1:an approach and some ideas[R], UC Berkeley Center for Future Urban Transport Working Paper UCB-ITS-VWP-2005-1, Institute of Transportation Studies, University of California, Berkeley July 5,2005.
[22]王殿海.交通流理论[M].北京：人民交通出版社,2002.
[23]Tafti M. F.. The application of artificial neural networks to anticipate the average journey time of traffic in the vicinity of merges[J]. Knowledge-based Systems, 2001,14:203-211.
[24]Khosravi A., Mazloumi E., Nahavandi S., et al. A genetic algorithm-based method for improving quality of travel time prediction intervals[J]. Transportation Research Part C Emerging Technologies,2011,19(6):1364-1376.
[25]刘金,邓卫.基于模糊神经网络的交通事故微观预测方法研究[J].交通运输工程与信息学报,2011,9(4)：69-75.
[26]张玉梅,曲仕茹,温凯歌.基于混沌和RBF神经网络的短时交通流量预测[J].系统工程,2007,25(11)：26-30.
[27]张明,韩松臣,黄林源.基于双重重力模型和人工神经网络的空中交通流量组合预测[J].西南交通大学学报,2009,44(5)：764-770.
[28]王正武,罗大庸,黄中祥,等.线控系统协调优化模型及其改进粒子群算法研究[J].系统工程理论与实践,2007,10：165-171.
[29]王殿海,景春光,曲昭伟.交通波理论在交叉口交通流分析中的应用[J].中国公路学报,2002,1(15)：93-96.
[30]刘智勇,吴今培,李秀平,等.城市交通大系统递阶模糊神经网络控制[J].信息与控制,1997,6(26)：441-448.
[31]Gazis D. C.. Optimum control of a system of oversaturated intersection[J]. Operation research,1964,12(6):815-831.
[32]朱文兴,贾磊.城市交通主干路相位差优化研究[J].计算机工程与应用,2005，20：212-214.
[33]张立东,贾磊,朱文兴.基于模糊理论的交通溢流识别算法[J].计算机应用,2012,32(8)：2378-2380,2384.
[34]高雨,沈国江,叶炜.城市路网交通分散协、调控制[J].信息与控制,2005,34(5)：616-620，626.
[35]王薇,杨兆升,赵丁选.有限阶段马尔科夫决策的可变限速控制模型[J].交通运输工程学报,2011,11(5)：109-114.
[36]张慧哲,王坚,任子晖.一种基于重要度的实值粗糙集及其在交通状态辨识中的应用[J].信息与控制,2009,38(5)：591-595.
[37]齐龙涛.基于粒子滤波的交通流状态预测及其优化研究[D].北京：北京交通大学,2011.
[38]Daganzo C. F.. Queue spillovers in transportation networks with a route choice[J]. Transportation Science,1998,32(1):3-11.
[39]Newell G. F.. Approximation methods for queues with application to the fixed-cycle traffic light[J]. SIAM Review,1965,7(2):223-240.
[40]Webster F.. Traffic signal settings[J]. Road Research Technical Paper 39. Road Research Laboratory, Her Majesty's Stationery Office, London,1958.
[41]Robertson D. I.. TRANSYT:A Traffic Network Study Tool[J]. Road Research Laboratory, Laboratory Report LR 253, Crowthorne,1969.
[42]Catling I.. A time dependent approach to junction delays[J]. Traffic Engineering and Control,1977,18:520-526.
[43]Strong D. W., Nagui M. R., Ken C.. New calculation method for existing and extended HCM delay estimation procedure[R]. Paper #06-0106, Proceedings of the 85th annual meeting of the Transportation Research Board, Washington, D.C., 2006.
[44]Sharma A., Bullock D. M., Bonneson J.. Input-output and hybrid techniques for real-time prediction of delay and maximum queue length at a signalized intersection[R]. Transportation Research Record,#2035, TRB, National Research Council, Washington DC,2007:88-96.
[45]Vigos G, Papageorgiou M., Wang Y.. Real-time estimation of vehicle-count within signalized links[J]. Transportation Research:Part C,2008,16(1):18-35.
[46]Papageorgiou M., Vigos G.. Relating time-occupancy measurements to space-occupancy, density and link vehicle-count[J]. Transportation Research:Part C, 2008,16(1):1-17.
[47]吴兵.交通管理与控制[M].北京：人民交通出版社,2011.
[48]杨晓光.城市道路交通设计指南[M].北京：人民交通出版社,2003.
[49]任福田,刘小明,荣建,等.交通工程学[M].第二版.北京：人民交通出版社,2008.
[50]Khisty C. J., Lall B. K.. Transportation Engineering[M]. Second Edition. New Jersey:Prentice Hall,1998.
[51]李晓红.城市干线交通信号协调优化控制及仿真[D].大连：大连理工大学,2007.
[52]宋现敏.交叉口协调控制相位差优化方法研究[D].长春：吉林大学,2005.
[2]杨佩昆,吴兵.交通管理与控制[M].第二版.北京：人民交通出版社,2005.
[3]张立东.城市交通溢流智能协调控制算法研究[D].济南：山东大学,2012.
[4]全永燊，城市交通控制[M].北京：人民交通出版社,1989.
[5]徐吉谦.交通工程总论[M].北京：人民交通出版社,1991.
[6]Reusch B..Vehicle movements in a platoon[J]. Oesterreichisches Ingenieur-Archir, 1950,4:193-215.
[7]Pipes L.A.. An operational analysis of traffic dynamics[J]. Journal of Applied Physics,1953,24(3):274-281.
[8]李曙光,许宏科.具有物理排队的多模式动态网络加载问题研究[J].交通运输系统工程与信息,2009,1(9)：56-61.
[9]Lighthill M.J.,Whitham G. B.. On kinematic waves I Flood movement in long rivers. Π, A theory of traffic flow on long crowded roads. Proceedings of Royal Society(London),1955, A(229):281-345.
[10]Richards P. J.. Shock waves on the highway[J]. Operations Research 1956,4:42-51.
[11]Stephanopoulos G., Michalopoulos P. G.. Modeling and analysis of traffic queue dynamics at signalized intersections [J]. Transportation Research,1979,13 A.
[12]Michalopoulos P.G., Stephanopoulos G.. An application of shock wave theory to traffic signal control [J]. Transportation Research Part B,1981,15:31-51.
[13]Henry X. Liu, Xinkai Wu, Wenteng Ma, et al. Real-time queue length estimation for congested signalized intersections [J]. Transportation Research Part C,2009, 17:412-427.
[14]Xinkai Wu, Henry X.Liu, Gettman D.. Identification of oversaturated intersections using high-resolution traffic signal data[J], Transportation Research Part C,2010,18:626-638.
[15]陈宁宁,何兆成,余志.考虑动态红灯排队消散时问的改进MAXBAND模型[J].武汉理工大学学报：交通科学与工程版.2005,5(33)：843-847.
[16]Xinkai Wu, Henry X. Liu. A shockwave profile model for traffic flow on congested urban arterials[J]. Transportation Research Part B,2011,45:1768-1786.
[17]Wolfram S.. Statistical mechanics of cellular automata[J]. Reviews of Modern Physics,1983,55(3):601-644.
[18]Geroliminis N.. Queue spillovers in city street networks with signal-controlled intersections[R]. Swiss Transport Research Conference,2009.
[19]Xuegang (Jeff) Ban, Peng Hao, Zhanbo Sun. Real time queue length estimation for signalized intersections using travel times from mobile sensors[J]. Transportation Research Part C,2011,19:1133-1156.
[20]Daganzo C. F.. The nature of freeway gridlock and how to prevent it[A], Proceeding of 13th International Symposium Transportation and Traffic Theory[C], 1996:629-646.
[21]Daganzo C. F.. Improving city mobility through gridlock control 1:an approach and some ideas[R], UC Berkeley Center for Future Urban Transport Working Paper UCB-ITS-VWP-2005-1, Institute of Transportation Studies, University of California, Berkeley July 5,2005.
[22]王殿海.交通流理论[M].北京：人民交通出版社,2002.
[23]Tafti M. F.. The application of artificial neural networks to anticipate the average journey time of traffic in the vicinity of merges[J]. Knowledge-based Systems, 2001,14:203-211.
[24]Khosravi A., Mazloumi E., Nahavandi S., et al. A genetic algorithm-based method for improving quality of travel time prediction intervals[J]. Transportation Research Part C Emerging Technologies,2011,19(6):1364-1376.
[25]刘金,邓卫.基于模糊神经网络的交通事故微观预测方法研究[J].交通运输工程与信息学报,2011,9(4)：69-75.
[26]张玉梅,曲仕茹,温凯歌.基于混沌和RBF神经网络的短时交通流量预测[J].系统工程,2007,25(11)：26-30.
[27]张明,韩松臣,黄林源.基于双重重力模型和人工神经网络的空中交通流量组合预测[J].西南交通大学学报,2009,44(5)：764-770.
[28]王正武,罗大庸,黄中祥,等.线控系统协调优化模型及其改进粒子群算法研究[J].系统工程理论与实践,2007,10：165-171.
[29]王殿海,景春光,曲昭伟.交通波理论在交叉口交通流分析中的应用[J].中国公路学报,2002,1(15)：93-96.
[30]刘智勇,吴今培,李秀平,等.城市交通大系统递阶模糊神经网络控制[J].信息与控制,1997,6(26)：441-448.
[31]Gazis D. C.. Optimum control of a system of oversaturated intersection[J]. Operation research,1964,12(6):815-831.
[32]朱文兴,贾磊.城市交通主干路相位差优化研究[J].计算机工程与应用,2005，20：212-214.
[33]张立东,贾磊,朱文兴.基于模糊理论的交通溢流识别算法[J].计算机应用,2012,32(8)：2378-2380,2384.
[34]高雨,沈国江,叶炜.城市路网交通分散协、调控制[J].信息与控制,2005,34(5)：616-620，626.
[35]王薇,杨兆升,赵丁选.有限阶段马尔科夫决策的可变限速控制模型[J].交通运输工程学报,2011,11(5)：109-114.
[36]张慧哲,王坚,任子晖.一种基于重要度的实值粗糙集及其在交通状态辨识中的应用[J].信息与控制,2009,38(5)：591-595.
[37]齐龙涛.基于粒子滤波的交通流状态预测及其优化研究[D].北京：北京交通大学,2011.
[38]Daganzo C. F.. Queue spillovers in transportation networks with a route choice[J]. Transportation Science,1998,32(1):3-11.
[39]Newell G. F.. Approximation methods for queues with application to the fixed-cycle traffic light[J]. SIAM Review,1965,7(2):223-240.
[40]Webster F.. Traffic signal settings[J]. Road Research Technical Paper 39. Road Research Laboratory, Her Majesty's Stationery Office, London,1958.
[41]Robertson D. I.. TRANSYT:A Traffic Network Study Tool[J]. Road Research Laboratory, Laboratory Report LR 253, Crowthorne,1969.
[42]Catling I.. A time dependent approach to junction delays[J]. Traffic Engineering and Control,1977,18:520-526.
[43]Strong D. W., Nagui M. R., Ken C.. New calculation method for existing and extended HCM delay estimation procedure[R]. Paper #06-0106, Proceedings of the 85th annual meeting of the Transportation Research Board, Washington, D.C., 2006.
[44]Sharma A., Bullock D. M., Bonneson J.. Input-output and hybrid techniques for real-time prediction of delay and maximum queue length at a signalized intersection[R]. Transportation Research Record,#2035, TRB, National Research Council, Washington DC,2007:88-96.
[45]Vigos G, Papageorgiou M., Wang Y.. Real-time estimation of vehicle-count within signalized links[J]. Transportation Research:Part C,2008,16(1):18-35.
[46]Papageorgiou M., Vigos G.. Relating time-occupancy measurements to space-occupancy, density and link vehicle-count[J]. Transportation Research:Part C, 2008,16(1):1-17.
[47]吴兵.交通管理与控制[M].北京：人民交通出版社,2011.
[48]杨晓光.城市道路交通设计指南[M].北京：人民交通出版社,2003.
[49]任福田,刘小明,荣建,等.交通工程学[M].第二版.北京：人民交通出版社,2008.
[50]Khisty C. J., Lall B. K.. Transportation Engineering[M]. Second Edition. New Jersey:Prentice Hall,1998.
[51]李晓红.城市干线交通信号协调优化控制及仿真[D].大连：大连理工大学,2007.
[52]宋现敏.交叉口协调控制相位差优化方法研究[D].长春：吉林大学,2005.