Optimal Traffic Flow Distributions on Dynamic Networks

Abstract

In a parallel network, the Wardrop equilibrium is the optimal distribution of the given total one unit flow across alternative parallel links that minimizes the effective costs of the links which are defined as the sum of the latency at the given flow and the price of the link. Meanwhile, the system optimum is the optimal distribution of the given total one unit flow for which the average effective cost is minimal. In this paper, we study the so-called Wardrop optimal flow that is the Wardrop equilibrium as well as the system optimum of the network. We propose a discrete-time replicator equation on a Wardrop optimal network for which the Nash equilibrium, the Wardrop equilibrium and the system optimum are the same flow distribution in the dynamic network. We also describe the conceptual and functional model of intelligent information system for dynamic traffic flow assignment in transportation networks.

A Appendix

Definition 4

A sequence $\left\{ \textbf{x}^{(n)}\right\} _{n\in \mathbb {N}}$ with $\textbf{x}^{(1)}:=\textbf{x}$ is called an orbit of the point $\textbf{x}\in \mathbb {S}^{m-1}$.

Let $\mathcal {E}_{\varepsilon \textbf{L}_m}(\textbf{y},\textbf{x}):=\varepsilon (\textbf{y},{\textbf{L}_m}(\textbf{x}))=\varepsilon \sum _{i=1}^{m}y_i\ell _i(x_i)$ and $\mathcal {E}_{\varepsilon \textbf{L}_m}(\textbf{x},\textbf{x}):=\varepsilon (\textbf{x},{\textbf{L}_m}(\textbf{x}))=\varepsilon \sum _{i=1}^{m}x_i\ell _i(x_i)$ for any $\textbf{x},\textbf{y}\in \mathbb {S}^{m-1}$ and $\varepsilon \in (-1,0)$.

Definition 5

A flow $\textbf{x}$ is called a Nash equilibrium if one has $\mathcal {E}_{\varepsilon \textbf{L}_n}(\textbf{x},\textbf{x})\ge \mathcal {E}_{\varepsilon \textbf{L}_n}(\textbf{y},\textbf{x})$ for any $\textbf{y}\in \mathbb {S}^{m-1}$. A flow $\textbf{x}$ is called a strictly Nash equilibrium if one has $\mathcal {E}_{\varepsilon \textbf{L}_n}(\textbf{x},\textbf{x})> \mathcal {E}_{\varepsilon \textbf{L}_n}(\textbf{y},\textbf{x})$ for any $\textbf{y}\in \mathbb {S}^{m-1}$ with $\textbf{y}\ne \textbf{x}$.

Definition 6

A point $\textbf{x}\in \mathbb {S}^{m-1}$ is called a common fixed point of the sequence of the replicator equations $\{\mathcal {R}_n\}_{n\in \mathbb {N}}$ if one has $\mathcal {R}_n\left( \textbf{x}\right) =\textbf{x}$ for any $n\in \mathbb {N}$.

Definition 7

A continuous function $\varphi :\mathbb {S}^{m-1}\rightarrow \mathbb {R}$ is called a Lyapunov function if the number sequence $\left\{ \varphi \left( \textbf{x}^{(n)}\right) \right\} _{n\in \mathbb {N}}$ is a bounded monotone sequence for any initial point $\textbf{x}^{(1)}:=\textbf{x}\in \mathbb {S}^{m-1}$.

Definition 8

A common fixed point $\textbf{x}\in \mathbb {S}^{m-1}$ is called stable if for every neighborhood $U(\textbf{x})\subset \mathbb {S}^{m-1}$ of $\textbf{x}$ there exists a neighborhood $V(\textbf{x})\subset U(\textbf{x})\subset \mathbb {S}^{m-1}$ of $\textbf{x}$ such that an orbit $\left\{ \textbf{y}^{(n)}\right\} _{n\in \mathbb {N}}$ with $\textbf{y}^{(1)}:=\textbf{y}$ of any initial point $\textbf{y}\in V(\textbf{x})$ remains inside of the neighborhood $U(\textbf{x})$.

Definition 9

A common fixed point $\textbf{x}\in \mathbb {S}^{m-1}$ is called attracting if there exists a neighborhood $V(\textbf{x})\subset \mathbb {S}^{m-1}$ of $\textbf{x}$ such that an orbit an orbit $\left\{ \textbf{y}^{(n)}\right\} _{n\in \mathbb {N}}$ with $\textbf{y}^{(1)}:=\textbf{y}$ of any initial point $\textbf{y}\in V(\textbf{x})$ converges to $\textbf{x}$. A fixed point $\textbf{y}\in \mathbb {S}^{m-1}$ is called asymptotically stable if it is both stable and attracting.

Definition 10

A common fixed point $\textbf{x}\in \mathbb {S}^{m-1}$ is called asymptotically stable if it is both stable and attracting.