“…The bottleneck model has been widely recognized as an important tool for modeling dynamic congestion. 1 Researchers have proposed various tolling regimes to reduce the congestion externality in dynamic congestion, such as a fully time-varying toll (e.g., Arnott et al, 1993;Yang and Huang, 1997;Van den Berg and Verhoef, 2011a, b); a step toll (Arnott et al, 1990;Laih, 1994Laih, , 2004Lindsey et al, 2012;Van den Berg, 2012), and a time-independent flat toll (Xiao et al, 2012;Silva et al, 2014;Fu et al, 2018).…”