On the existence of stationary states in general road networks |
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Affiliation: | 1. School of Economics and Management, Beihang University, Beijing 100191, China;2. Key Lab of Complex System Analysis and Management Decision, Ministry of Education, China;3. Department of Civil and Environmental Engineering, The Hong Kong University of Science and Technology, Kowloon, Hong Kong, China;1. Department of Economics, University of California, Riverside, USA;2. Department of Mathematics, University of California, Riverside, CA 92521, USA;1. KTH Royal Institute of Technology, Department of Transport Science, 11428 Stockholm, Sweden;2. Massachusetts Institute of Technology (MIT), Department of Civil & Environmental Engineering, Cambridge, MA 02139, USA;1. School of Transportation Engineering, Hefei University of Technology, Hefei 230009, China;2. Department of Civil Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong;3. School of Traffic and Transportation, Beijing Jiaotong University, Beijing 100044, China;4. School of Economics and Management, Beihang University, Beijing 100191, China |
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Abstract: | Our daily driving experience and empirical observations suggest that traffic patterns in a road network are relatively stationary during peak periods. In numerous transportation network studies, there has been an implicit conjecture that stationary states exist in a network when origin demands, route choice proportions, and destination supplies are constant. In this study, we first rigorously formulate the conjecture within the framework of a network kinematic wave theory with an invariant junction model. After defining stationary states, we derive a system of algebraic equations in 3-tuples of stationary link flow-rates, demands, and supplies. We then introduce a new definition of junction critical demand levels based on effective demands and supplies. With a map in critical demand levels, we show that its fixed points and, therefore, stationary states exist with the help of Brouwer’s fixed point theorem. For two simple road networks, we show that the map is well-defined and can be used to solve stationary states with a brute-force method. Finally we summarize the study and present some future extensions and applications. |
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Keywords: | Network kinematic wave model Stationary states Demand and supply Critical demand levels Fixed point |
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