Transport Capacity Limit of Urban Street Networks

Transcription

1 bs_bs_banner Research Article Transactions in GIS, 2017, 21(3): Transport Capacity Limit of Urban Street Networks Gang Liu, Peichao Gao and Yongshu Li Key Laboratory of Geoscience Spatial Information Technology, Ministry of Land and Resources of the P. R. China, Chengdu University of Technology Department of Land Surveying and Geo-Informatics, Hong Kong Polytechnic University Faculty of Geosciences and Environmental Engineering, Southwest Jiaotong University Abstract Network transport is an important aspect of geographical information science, transportation, complex networks, etc. Previous studies have shown that the transport capacity of street networks can be enhanced by improving routing algorithms. However, the upper throughput limit of street networks is unknown in detail. This article studies the transport process of networks and finds that any connected network has a maximum throughput depending on the topological and structural properties of the network. Based on this, the maximum throughput of street networks is obtained. Experiments show that when the street network remains unchanged, the maximum throughput of the street network is limited and is dependent on road capacity and average path length, regardless of adopted routing algorithms. Our findings suggest that the throughput of networks can be improved by increasing node capacity or decreasing average path length, but the maximum transport capacity of the network cannot be greater than the ratio of the sum of all the nodes capacities to the average path length of the network, no matter what routing strategies are adopted. This study is expected to be a starting point for more sophisticated research in network transport, such as evaluating the inherent throughput of an urban street network. 1 Introduction Network transport is a dynamic problem encountered in a large number of natural and humanmade transport and communication systems (Goh et al. 2001; Newman 2003, 2010; Guimera et al. 2005; Barbosa et al. 2010; Barthelemy 2011). Path optimization and congestion control are at the core of several communication and logistics applications. In recent years, scientists have explored many efficient routing strategies that can enhance the transport capacity of networks. Traffic congestion is an extensively studied phase transition phenomenon that develops from a free-flow state into a congestion state (Toroczkai and Bassler 2004); it occurs in many complex systems (Arenas et al. 2001; Sole and Valverde 2001; Boccaletti et al. 2006). Traffic congestion begins when the number of packets on the network exceeds network capacity in unit time, and if no effective routing protocol is adopted, the congestion will deteriorate continuously and gradually spread to all the other nodes in the network. In order to diminish traffic jams and improve the transport capacity of networks, the research mainly focuses on two aspects. One is to improve the topological structure of the Address for correspondence: Gang Liu, Key Laboratory of Geoscience Spatial Information Technology, Ministry of Land and Resources of the P. R. China, Chengdu University of Technology, China. liuganggis@sina.com Acknowledgements: This work was supported by the National Natural Science Foundation of China (NSFC) (Grant No ), the Key Laboratory of Geosciences Spatial Information Technology, Ministry of Land and Resources of China (Grant No. KLGSIT ), the Cultivation Plan of Outstanding Innovative Teams of Chengdu University of Technology, the Scientific Research Foundation of the Education Department of Sichuan Province of China (Grant No. 16ZB0107) and Self-Topic Fund of State Key Laboratory of Geohazard Prevention and Geoenviroment Protection, China. VC 2016 John Wiley & Sons Ltd doi: /tgis.12218

2 576 G Liu, P Gao and Y Li network, and some studies have proved that the network structure may have a large influence on traffic jams and their spread (Li et al. 2011). The other is to explore efficient routing strategies (Danila et al. 2007; Kawamoto and Igarashi 2012; Liu et al. 2013, 2014). For many complex systems such as transportation networks and power networks, it is too costly to update the structure of the network and it will greatly damage the surroundings. Echenique et al. (2005) found in numerical simulations that the nature of the congestion transition depends on the type of routing rules. Therefore, in recent years, efficient routing strategies have become a main topic in research on complex networks and traffic congestion (Boccaletti et al. 2006; Newman 2010). In GIS, scholars proposed many path selection methods that consider the road class, traffic distribution, taxi drivers experience, etc. (Krisp et al. 2015; Tang et al. 2010). Although the transportation network is a human-made complex system, it is still difficult to improve the topological structure of the street network because it is too costly and difficult to change the structure of the street network. Therefore, the question becomes whether we can enhance the throughput of the street network continuously through developing more and more efficient routing algorithms. This is an important subject worth studying. Studies on network transport have primarily focused on improving the existing routing algorithms or exploring new routing strategies but have failed in systematically discussing the inherent transport capacity of networks, especially the maximum throughput of urban street networks. Thus, we hypothesize that the network transport capacity cannot be greater than a constant on a connected network, regardless of applied routing strategies. We think that the throughput of a network is limited and mainly determined by the network s structure. In order to reveal the maximum throughput of street networks, we aim to further study the topological representation of street networks that is applicable to describe the transport process of traffic flow within the network. On the basis of this, we will analyze the relationships between the network throughput and the structural properties of the network, and then measure the maximum throughput of any connected network; finally, we try to estimate the maximum transport capacity of urban street networks. The remainder of this article is organized as follows. In Section 2, we introduce dual topological representations for urban street networks and the traffic flow model for arc arc topology. Section 3 analyzes the maximum transport capacity of a dual graph and the throughput of an urban street network. In Section 4, the experiments with and results of using the Chengdu street network as a case study are described. Section 5 discusses the significance of the maximum transport capacity of street networks. Section 6 concludes the article. 2 Topological Representations of Urban Street Networks 2.1 Dual Topological Representations of Urban Street Networks The topological representation of urban street networks is important for analyzing the structure and the functional characteristics of the street networks. In the GIS field, the most common method of topological representation is to describe the street map directly, based on classical graph theory (Jiang and Claramunt 2004), which constructs the street networks with street as arc and with crossing as junction (Figure 1a). In recent years, the complex network theory based on dual topology method has been applied to the GIS and traffic field, which is helpful for analyzing the complexity of the topological structure and traffic flow of the transport network (Jiang and Liu 2009; Barthelemy 2011; Liu et al. 2014). The scientists have studied the dual representation for the street-street relationship (Jiang and Liu 2009; Liu et al. 2014). In the dual graph for the street-street relationship shown in Figure 1b, a node corresponds to a street, and if

3 Transport Capacity Limit of Urban Street Networks 577 Figure 1 Topological representations of a street network: (a) Street network; (b) Street-street relationship (8 streets); (c) Arc network of streets; and (d) Arc-arc relationship (16 arcs) a link between two nodes exists, it means the two streets corresponding to these two nodes are interconnected. It has been concluded in previous work that the traditional street networks cannot fully reflect the overall morphological properties and complexity (Deng et al. 2010). From the point of view of geographic representation, the dual graph for street-street relationships represents the highest abstraction of street networks, which is helpful for further researching the topological properties of street networks, the importance of the street, the reliability of street networks, and the traffic behavior characteristics, etc. Previous study has shown that street street topologies tend to be better representations than the axial map; and vehicle flow is correlated to the morphological property of streets rather than to the axial map, suggesting the streetbased topological representations as an alternative GIS representation and topological analyses as a new analytical means for geographic knowledge discovery (Jiang and Liu 2009). However, although the dual graph for street-street relationships is more helpful for studying the structure and functional characteristics of the street network, it is not accurate in describing the local transport process of vehicle flow. Note that the drivers need to make a path choice only when they are arriving at an intersection, because they can just drive straight ahead before reaching the next intersection. Thus we can see that the dual graph for street-street

4 578 G Liu, P Gao and Y Li Figure 2 Geometric street network of Chengdu street network relationships cannot well describe this travel behavior at intersections. It needs to regard a local section as a research object, which is the section between two adjacent intersections; and it may be a part of an integrated street with a unique street name. A street should be divided into several sections which cannot cross the other streets, as shown in Figure 1c. Note that these sections are also called arcs in GIS. Therefore, the dual graph for the arc-arc relationship (Gutierrez and Medaglia 2008) (as shown in Figure 1d) is introduced to describe the transport process of vehicle flow. 2.2 Traffic Flow Model for Arc-arc Topologies To elucidate the thesis of this article, we use the Chengdu street network (Figure 2) for experiments, which contains 1,484 streets and 8,899 arcs. The dual topology method is adopted to extract the arcs and arc arc relationship and construct a dual graph for the aforementioned relationship, which contains 8,899 nodes and 23,026 edges. This dual topological network is then used to compare the transmission performance of the shortest-path routing algorithm and gravitational field routing algorithm. Assume that all nodes within the network have the functions of receiving, sending, and routing, and that the initial network load is equal to zero. The traffic model is described as follows. At each time step, p packets are generated and added into the system with randomly chosen sources and destinations. The capacity of node i is c i which

5 Transport Capacity Limit of Urban Street Networks 579 means the node can send at most c i packets at each time step. The queue length of each node is assumed unlimited, and the FIFO (first in first out) rule is applied at each queue. To navigate packets, packets are always delivered from the current node to one of its neighbors. If the packet s destination is located among the neighbors, the packet is directly sent to its target and then removed from the system. Otherwise, it is forwarded to one of its neighbors, in accordance with the routing algorithm. The traffic flow model suggests that at each time step, it is always creating new packets and destroying old packets. We can easily extract all the information (including travel time, route) about each packet and the traffic state of each node. Thus, we can easily analyze the dynamics of different routing strategies, including the network throughput, and distribution of traffic flow. 3 Transport Capacity Analysis of Urban Street Networks 3.1 Maximum Transport Capacity of Dual Graphs Basic ideas At each time step, there are always some packets created and some packets destroyed in the network. In a free-flow state, the accumulation (i.e. the number of packets on the network) in unit time is less than the network capacity so that no large queues exist on the network. In a steady state, inflow equals outflow over a long period of time. However, because the transmission of traffic flow is dynamic, the accumulation in the system could be changing. Therefore, whatever the system is in a free-flow or steady state, it indicates that inflow equals outflow and that there are no large queues. If the network flow sinks into a congestion phase, the number of created packets is greater than the number of destroyed packets so that more and more packets are stuck in the queue of nodes. In order to study the maximum transport capacity of networks, we should consider the following problems: 1. Obtain the number of packets remaining within the network at each time step. When the network is in a free-flow state, the number of packets remaining within the network should be small and the accumulation of packets is not remarkable. Consequently, if we can obtain the relation between the number of packets remaining within the network and time, we will further study the transport capacity of the network. 2. Study the relationship between the number of destroyed packets and the number of non-empty nodes in unit time. When the network is in a free-flow state, the number of created packets should be comparable to the number of destroyed packets. We believe that the number of non-empty packets is related to the number of packets remaining within the network and the number of created packets on average. 3. Analyze whether the number of created packets at each time step is related to node capacity and other topological measures (like the average path length), when the network flow is in a free-flow state. If the maximum transport capacity of networks does exist, it must be related to the structural and functional properties (like the average path length and node capacity, etc.) of networks. Therefore, we may obtain the maximum throughput of networks Argumentation 1. Estimate the number of packets remaining within the network at each time step. To evaluate the maximum transport capacity of dual graphs, it is helpful to analyze the throughput

6 580 G Liu, P Gao and Y Li of urban street networks. Recently, we discovered the relation between the throughput of a dual graph and its structural properties (Liu et al. 2015). In this article, we will first illustrate further the maximum transport capacity of a dual graph. It is certain that improving the routing strategy can enhance the throughput of the street network. However, when the network structure remains unchanged, the throughput will be limited. As p increases, the system undergoes a continuous phase transition from a free phase to a congestion phase. From analysis of the phase transition process, we can obtain the throughput of the network. In a free-flow state, no continual accumulation of packets occurs at any node in the network. In a congestion phase, however, the number of packets that accumulate in the residual queues of the nodes in a time step is greater than the capacity of the network. From a theoretical perspective, if the network structure and capacity of nodes do not change, an optimal path selection can improve the throughput of the network. However, with the consecutive development of the routing algorithm research, the improvement of network capacity has gradually become less significant. Determining the most suitable routing protocol and the maximum transport capacity of a network has become the primary issue. The free-flow state refers to the balance between the created and removed packets on average in unit time. In this study, we consider the maximum transport capacity of a network to be irrelevant to any routing strategy and to be determined by the structural properties of the network. Let WðtÞ be the total number of packets remaining within the network at time t and QðtÞ be the number of packets removed from the system at a time step. Note that WðtÞ is equal to the sum of the total number of packets within the network at time t21 (represented with Wðt21Þ) and the number of packets new created at time t minus QðtÞ. Therefore,we have: WðtÞ5Wðt21Þ1p2QðtÞ (1) 2. Computing the number of non-empty nodes at each time step. At each time step in the transport process, new packets are created and old packets are destroyed as they reach their destination. After p packets are created at each time step, each node whose queue is not empty (i.e. it contains at least one packet) would send a packet to one of its neighbors, if c i 51 for all i. At each time step, some packets reach their destinations and will be removed from the network. Other packets, however, do not reach their destinations and are still in the middle of their paths, therefore they will join the queue at the next node. Let XðtÞ be the number of non-empty nodes at time t. If the average path length between the sources and destinations on the network is assumed to be L, then the number of packets removed from the network can be estimated as QðtÞ5XðtÞ 1 L because a node can only send at most one packet to a neighbor at each time step (e.g. L51 indicates that a total of XðtÞ 1 L 5XðtÞ packets arrived at their destinations at one time step), if c i51 foralli. Then, we can see that, estimating XðtÞ is exactly the key to discover the dynamic mechanism of traffic balancing. We must note that when the network is in a free-flow state, the number of new created packets is less than the transport capacity of the network, and should be comparable to the number of destroyed packets on average at each time step. XðtÞ denotes the number of nodes which exist at least one packet prior to delivering the packets at time t. The minimum of XðtÞ may be equal to 1, whereas the maximum of XðtÞ cannot be greater than the network size N. In addition, XðtÞ can never be greater than Wðt21Þ1p. According to Equation (1), because the number of removed packets in a time step is estimated XðtÞ=L, then WðtÞ5Wðt21Þ1 p2xðtþ=l. Therefore, XðtÞ is defined as:

7 Transport Capacity Limit of Urban Street Networks 581 XðtÞ5LðWðt21Þ1p2WðtÞÞ (2) 3. Measuring the maximum throughput of networks. In the macro-perspective, the packets removed at each time step may be more or less than the new packets created. Consequently, the number of packets remaining within the network fluctuates. To ensure the load balancing of the network, the number of packet removed from the network should be comparable to the number of newly generated packets on average (i.e. when t!1, the average of removed packets at each time step is approximately equal to p) at each time step. On the basis of Equation (2), a mathematical formula is used to describe the load balancing as: lim t!1 ð t 1 XðsÞ 1 L ds t p (3) Equation (3) indicates that when t!1, the average number of packets removed from the network at each time step is equal to p. In the free phase, the traffic flow within a network can always maintain load balancing. According to the L Hospital rule (Pinelis 2004), Equation (3) is simplified as following: lim LWðt21Þ1p2WðtÞ ð Þ 1 t!1 L p (4) It may be concluded that when the system is in a free-flow state, it obtains lim t!1 WðtÞ5Wðt21Þ. Therefore, we can see that in a free-flow state, the number of removed packets in a time step is balanced with the number of generated packets in a time step, and the total number of packets remaining in the network is unchanged since WðtÞ5Wðt21Þ. According to Equation (4), the number of non-empty nodes is estimated to be pl, meaning XðtÞ pl. We consider that, in a free-flow state, the average lift time of a packet, that is, the step number within which the packet remains in the network, is estimated L, the average path length of a packet from a source node to a destination node, since the packet hops to another node in every time step. Therefore, the total number of packets remaining in the network can be estimated as pl, because p packets are generated in a time step and the total number of packets is unchanged. For a highly efficient routing strategy, it should balance the traffic load as much as possible. Therefore, if the traffic flow is in a free-flow state, and the node capacity c i 51 for all i, the number of non-empty nodes is approximately equal to the total number of packets remaining in the network, meaning XðtÞ WðtÞ. Note that the maximum number of packets sent in a time step within the network cannot be greater than X c i i. Therefore, XðtÞ < X c i i that means pl < X c i i.ifpincreases and pl approaches X c i i, the congestion state appears before pl5 X c i i at p5p c. Although the value of p c depends on the routing strategy, p c L cannot exceed X c i i for any routing strategy because the packets which are not delivered to other nodes certainly exist and the total number of packets in the network are sure to increase. Therefore, we have p c L < X c i i, then p c can be defined as: X i p c < c i (5) L Equation (5) means the network transport capacity is smaller than the ratio of the sum of all the node s capacities to the average path length of the network. This conclusion discovers a

8 582 G Liu, P Gao and Y Li fundamental theorem, which is universal and useful for the network transport of all dual graphs. Besides, Equation (5) indicates that, if the capacity of each node is equal to 1 that means c i 51 for all i, the maximum transport capacity of the network satisfies p c < N L. Therefore, we can see that if the network remains unchanged, the maximum transport capacity of the network is dependent on the size and average path length of the network. 3.2 Maximum Transport Capacity of Street Networks In Section 3.1, we present an important conclusion that the transport capacity of a dual graph is not greater than the ratio of the sum of all the node s capacities to the average path length of the graph. However, the urban street network is a geometric network in that the streets are of geometric length. In a dual graph, the distance between two nodes is the number of edges on the shortest path, which can be called hops. Moreover, the transmission capacity of streets is related to the number of lanes and average traffic speed of different street types. Therefore, the throughput obtained based dual graphs cannot describe the real transport capacity of the street network. We need to build up a mechanism between the throughputs of dual graphs and street networks. To obtain the real throughput of a street network, we should study the meanings of the distance and the capacity for dual graphs and street networks, respectively. For some street i, let V i (km/h) be the maximal average traffic velocity of street i, K i be the maximal density of vehicles for single lane and H i be the number of lanes on this street. Previous studies (Greenshields 1935; Hall et al. 1992) have discovered the relation between the traffic quantity, velocity and density, which fulfills Q i 5V i K i =4, where Q i is the maximal traffic quantity of street i. Therefore, we can obtain the maximal capacity of street i per minute as follows: c i 5 V i K i 4 H i V ik i H i 240 (6) Equation (6) indicates the capacity of street i in a minute. In a dual graph, the average travel time of a packet, T, that is, the step number within which the packet remains in the network, is estimated L, the average path length of a packet from a source node to a destination node, since the packet hops to another node in every time step. In the real traffic system, the traffic flow is dynamic and complex. In order to describe the time-variant of traffic flow, it is necessary to study real traffic using a reasonable time unit. Taking hour as the time unit cannot reflect the time-variant of traffic flow while setting the time by seconds is too detailed and is not necessary. Therefore, to describe the dynamics of real traffic flow in terms of minutes is reasonable, and we can make statistics of traffic flow by time in one-minute segments. According to Equations (5) and (6), the maximum transport capacity of a street network should fulfill: p c < X i V i K i H i 240, L (7) Therefore, based on Equation (7), if we extract the maximum speed of vehicles in a free-flow state (V i ), the vehicle density (K i ) and the number of lanes (H i ), then we can measure the maximum throughput of the street network. 4 Implementation Prototype In this section, we will use the Chengdu street network as a case study to illustrate that the transport capacity of a street network is never greater than the ratio of the sum of the capacities

9 Transport Capacity Limit of Urban Street Networks 583 of all the nodes to the average path length of the network. First, we will introduce a metric for measuring the transmission performance of the network. On the basis of this metric, some essential experimental data for both routing strategies are carried out. Then, an application of maximum throughput in street network analysis is presented. 4.1 Experiments and Results Throughput is measured using the critical generation rate p c, at which a phase transition from a free-flow state to a congestion state occurs. The order parameter proposed by Arenas (2001) has been introduced to describe the critical capacity accurately: 1 WðtÞ gðpþ5 lim t!1 p t (7) when p < p c, the traffic flow is in a free-flow state, so WðtÞ is far less than p t. Thus, gðpþ!0, which indicates that the system is in a free-flow state with no traffic congestion. When p > p c, WðtÞ and gðpþ abruptly increase. The system collapses with the continuous increase in p, gðpþ!1. The phase transition occurs at p5p c. p c is the critical generation rate under which the system maintains normal and efficient functioning. Thus, the transport capacity of the network is measured by p c. In order to prove our argument, it is necessary to compute the throughput for different routing strategies. The shortest path, traffic awareness (Wang et al. 2010), incremental (Jiang 2013) and gravitational field routing strategies (Liu et al. 2014) are chosen for simulation experiments to obtain the transport capacity of a network under different routing strategies. The shortest path routing strategy is a well-known method for network transport often used for simulation. The gravitational field routing strategy is a stable and highly efficient routing algorithm with a critical capacity several times higher than those of the shortest path routing strategy. The incremental and traffic awareness routing strategies are both highly efficient routing strategies as well. Combined with the Chengdu street network, suppose that in the free-flow state, the average maximal speeds of freeways, main streets, collector streets and branch streets are equal to 80, 60, 40 and 30 km/h, respectively; the numbers of lanes of these four street types are equal to 4, 3, 2 and 1 respectively. Because there is not sufficient traffic data for surveying the maximal vehicle densities of all streets of Chengdu city, here we suppose the average vehicle density is 100 cars/kilometer (K5100). Figure 3 shows the relation of the order parameter g change with the packet generation rate p under different routing strategies. The experimental results indicate that for each routing strategy, when p is very small, the traffic flow is in a free-flow state and there is no congestion, so the order parameter g is approximately equal to zero; when p reaches a certain value, g increases abruptly which means a phase transition occurs and the traffic network sinks into a congestion state. Therefore, the critical point for the phase transition is the throughput of the traffic network. For these four routing strategies, the corresponding throughput is different and the gravitational field routing strategy is more efficient. However, no matter what the routing strategy, the throughput of the street network is significantly less than the ratio of the sum of all the streets capacities to the average path length, X i c i=l511; 283. Consequently, the results indicate that the transport capacity of the street network is always less than X i c i=l, regardless of the adopted routing strategy. Therefore, we conclude that the maximum transport capacity of a street network is limited and is dependent on street capacity and average travel path. The street capacity can be obtained

10 584 G Liu, P Gao and Y Li Figure 3 Throughput of the Chengdu street network under different routing strategies according to the maximal average speed, vehicle density and the number of lanes of different street types. If we acquire the above data and the structure of the street network, we can evaluate the maximum throughput of the street network, which is useful for measuring the overall performance of street networks. Theoretically, reducing the average path length or improving node capacity can enhance the transport capacity of networks. To further illustrate that the transport capacity of a street network is never greater than the ratio of the sum of the capacities of all the nodes to the average path length of the network, the network capacity is obtained by adjusting the average path length with network size remaining unchanged. We introduce the BA scale-free network (Barabasi and Albert 1999) for experiments. Suppose that the node capacity c51 and 14 BA networks are created the same size but with different L. Figure 4 shows the variance of the transport capacity of BA networks with the increasing of X c i i=l. The results indicate that for the shortest path routing strategy, the throughput p c of the BA networks gradually grows with the increasing of X c i i=l, but always less than X c i i=l. We can see that for larger L, the value of X c i i=l is distinctly greater then p c. When L51, X c i i=l5100 which means that the BA network is a globally coupled network and X c i i=l reaches the maximum, the network capacity p c 599 < X c i i=l at this point. For the other three routing strategies, we can see that the transport capacity is improved greatly compared with the shortest path routing strategy, but is also less than X c i i=l whatever the value of X c i i=l. Theoretical analysis and numerical simulation demonstrate that, no matter what routing strategies, the maximal throughput of a network is less than the ratio of the sum of all the node s capacities to the average path length of the network. X c i i=l can be regarded as a natural characteristic of networks. For urban geometric street networks, the maximum transport capacity cannot be greater than N p Xi c i=l. Equation (5) indicates that improving node capacity can enhance the maximum throughput of the network. We can reduce the traffic congestion and improve the transmission performance of the street network through improving the street capacity. Therefore, some

11 Transport Capacity Limit of Urban Street Networks 585 Figure 4 Plot of p c versus R i c i =L for different L. Other parameters are network size N5100, growth index m5m 0 51; 2; 3; 4; 5; 7; 10; 15; 20; 25; 30; 40; 60; 100 for 14 BA networks with unchanged network size N5100, and c i 51 for all i statistical results of the maximum throughput and the network capacity under different routing methods are carried out as shown in Figure 5. The results indicate that for the four routing strategies, the network capacities all increase linearly with the increase of node capacity, but all are less than P i c i=l. Therefore, we can conclude that the maximum transport capacity of a network is a constant if the network remains unchanged; the maximum throughput can be improved by enhancing the node capacity or reducing the average path length; more importantly, the maximum throughput cannot be greater than the ratio of the sum of all the node s capacities to the average path length of the network, regardless of routing algorithms. Figure 5 Plot of p c versus R i c i =L for different c. Other parameters are the BA network size N5100 and m5m 0 54

12 586 G Liu, P Gao and Y Li Table 1 Maximum throughput of Chengdu street network for different speeds of freeways Maximal average speed of freeways (km/h) Maximum throughput 80 10, , ,283 The maximal average speeds of main streets, collector streets and branch streets are respectively equal to 60, 40 and 20 km/h. Table 2 Maximum throughput of Chengdu street network for different speeds of main streets Maximal average speed of main streets (km/h) Maximum throughput 60 10, , ,011 The maximal average speeds of freeways, collector streets and branch streets are respectively equal to 80, 40 and 20 km/h. 4.2 Effect of Different Ranks of Streets on Maximum Throughput of Street Networks In street network analysis, path selection is an important research subject, which can solve many traveling problems. However, for the street network planners or transport sector, it is important to study and improve the inherent throughput of street networks. The structural and functional properties of street networks are both related to the maximum throughput of street networks. For an urban street network, especially large cities, it is very difficult to change the topological structure of the network as a whole. Therefore, improving street capacity is one of the most important ways to enhance the network s throughput. Moreover, it is unrealistic to broaden the width of many streets, because this would destroy many buildings and is too costly. With this in mind, we study the maximum transport capacity of the Chengdu street network under different maximal average velocities of freeways, main streets, collector streets and branch streets, respectively, as shown in Tables 1 4. When the velocities of main streets, collector streets and branch streets remain unchanged, we can obtain the maximum throughput of the street network for different velocity of freeways. Table 1 shows that the maximum throughput is improved from 10,736 to 11,283, when the maximal average speed of freeways increases from 80 to 100 km/h. However, the analysis results indicate that the main streets and collector streets have greater impact on the maximum throughput of the street network. The reason for this is that the main streets and collector streets are crucial to the transmission of traffic flow and carry most of the traffic load. Therefore, in Table 3, the maximum throughput is improved from 10,736 to 12,549 when the maximal average speed of collector streets increases from 40 to 60 km/h.

13 Transport Capacity Limit of Urban Street Networks 587 Table 3 Maximum throughput of Chengdu street network for different speeds of collector streets Maximal average speed of collector streets (km/h) Maximum throughput 40 10, , ,549 The maximal average speeds of freeways, main streets and branch streets are respectively equal to 80, 60 and 20 km/h. Table 4 Maximum throughput of Chengdu street network for different speeds of branch streets Maximal average speed of branch streets (km/h) Maximum throughput 20 10, , ,830 The maximal average speeds of freeways, main streets and collector streets are respectively equal to 80, 60 and 40 km/h. Consequently, we can see that when it is difficult to improve the structure of street networks, street capacity can be enhanced by optimizing the maximal average speed in different classes of streets. Of course we cannot increase the speed limit blindly, because that might cause more traffic accidents. In order to improve the maximum throughput of a street network, we can add more pedestrian overpasses at main streets and collector streets that can separate pedestrians and vehicles. This will improve the maximal average traffic speed of streets and enhance the maximum transport capacity of the street network. 5 Discussion Generally, the users route choice behavior is overly dependent on the travel distance, which results in the serious congestion of some key streets and the low throughput of the street network. Therefore, exploring highly efficient routing strategies is one of the most important ways to enhance the transport capacity of the street network. However, the ultimate goal of routing optimization, that is to say whether the network throughput is limited, has not been studied systematically. The aforementioned theoretical analysis and experiments indicate that the transport capacity of a network is limited no matter what routing strategies are adopted and is directly related to the topological structure of the network. For a given street network, if the network remains unchanged, so that the capacity of streets, the topological structure and all the other properties of the network are fixed and

14 588 G Liu, P Gao and Y Li immovable, the network throughput is directly proportional to the sum of the capacity of all streets and inversely proportional to the average path length of the network. This suggests that improving the structural or functional properties of networks can enhance the network throughput, such as increasing street capacity or decreasing average path length. However, increasing street capacity (Figure 5) and decreasing average path length (Figure 4) both change the structural properties of the street network and therefore change the maximum transport capacity of the network which is irrelevant to routing strategies. For a street network, if the capacity of most of the streets is small and the average path length is great, then the maximum throughput of the street network must be very low. Our study indicates that road network planning is crucially important to the maximum transport capacity of the road network. The capacity of key streets at the core of network transport should be great and the average path length cannot be too large. Consequently, we can conclude that the maximum throughput of street networks does exist and is an inherent property of networks. We think that this result can help us to understand flow in real networks. 6 Conclusions We studied the phase transition from a free-flow state to a jammed phase and discovered a fundamental theorem on network transport. This article demonstrates an important phenomenon of traffic congestion and finds out that the transport capacity of street networks is limited and dependent on the topological and structural properties of networks. To summarize, this study contributes to the literature in the following ways: 1. We discuss the advantage of arc-arc topological relationships in network transport of real street networks; 2. We find that the maximum transport capacity of a network cannot be greater than the ratio of the sum of all the capacities of the nodes to the average path length of the network, regardless of the adopted routing algorithm; 3. This study reveals the relation among the street capacity, average travel time and average path length, and obtains the maximum throughput of street networks; and 4. Finally, this study provides a research prototype, which can be used for further research on network transport. To verify the correctness of our conclusion, the Chengdu urban street network and the BA networks are used as case studies, with experiments considering different routing strategies. The results show that for a dual graph, throughput is smaller than the ratio of the sum of all the capacities of the nodes to the average path length. For a real street network, the throughput is smaller than the maximum throughput for the corresponding dual graph multiplied by the ratio of the average travel time of packets within the dual graph to the average travel time of vehicles within the urban street network. The result is utilized to evaluate the inherent throughput of a network and discover various methods of improving the throughput (e.g. decreasing the average path length or enhancing the node capacity). After this study is completed, the network s maximum transport capacity is found to be a natural property of the network that is independent of routing strategies. Although the capacity of networks depends on the routing strategy, this conclusion is useful for evaluating the probable maximum capacity of networks. Our study reveals an important property of network transport, and presents the maximum transport capacity of a network, which is universal for any dual graphs. The future directions that our group intends to pursue concern controlling the congestion of traffic flow. The

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