Abstract
This study proposes a nonlinear min-cost-pursued swapping dynamic (NMSD) system to model the evolution of selfish routing games on traffic network where travelers only swap from previous costly routes to the least costly ones. NMSD is a rational behavior adjustment process with stationary link flow pattern being the Wardrop user equilibrium. NMSD is able to prevent two behavioral deficiencies suffered by the existing min-cost-oriented models and keep solution invariance. NMSD relaxes the homogeneous user assumption, and the continuous-time NMSD (CNMSD) and discrete-time NMSD (DNMSD) share the same revision protocol. Moreover, CNMSD is Lyapunov-stable. Two numerical examples are conducted. The first one is designed to characterize the NMSD-conducted network traffic evolution and test the stability of day-to-day NMSD. The second one aims to explore the impacts of network scale on the stability of route-swaps conducted by pairwise and min-cost-pursed swapping behaviors.
Highlights
Smith [1] proposed the classical proportional-switch adjustment process (PAP) to describe the evolution of network traffic
PAP is viewed as the most natural route-swapping process [2]. It has a simple mathematic formulation and an intuitive behavior basis; in addition, PAP explicitly addresses the original micromechanism of network traffic evolution, that is, how and how much traffic will swap from the beingused routes to the other ones
Recalling (1), we can conclude that ∑w ∑p∈K−wt (C∗wt − Cpwt)fpwtρpwt ≤ 0, which proves that continuous-time NMSD (CNMSD) is a rational behavior adjustment process
Summary
Smith [1] proposed the classical proportional-switch adjustment process (PAP) to describe the evolution of network traffic. PAP assumes traffic swaps from the previous costly routes to less costly ones with swapping rate being proportional to the pairwise absolute cost differences. Different from PAP and NPSD, this study investigates another route-swapping behavior; that is, travelers only swap from the previous costly routes to the least costly alternatives. This kind of greedy route-swapping behavior (compared with the pair-wise swapping behavior) was studied by Mounce and Carey [2] In their dynamic model, the swapping traffic from a costly route to one of the least costly ones was proportional to the absolute difference, just analogous to PAP. The swapping traffic from a costly route to one of the least costly ones was proportional to the absolute difference, just analogous to PAP It still suffered from the PAP shortcomings.
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