The classical Braess paradox problem refers to a user-equilibrium assignment model which all started with Braess's (Unternehmensforschung 12; 258-268, 1968) demonstrated example network. Some variants of Braess paradox and related theories were subsequently developed to detect this paradoxical phenomenon on a general network. In this paper, the authors are devoted to the classical Braess paradox problem involving situations whenever considering new links to be added to a network. Historical literature told us that existing theories for this problem were limited to networks which admit unique path flow solution. A generalized inverse approach is suggested to solve this problem without the assumption of unique path flow solution in this study. The change of equilibrium cost after link additions is derived as a generalized inverse formulation of which solution possesses the non-uniqueness and flow conservation over all perturbed paths. Based on this generalized inverse formulation of the change of equilibrium cost, the authors show that there exists at least one of the O/D pairs, connected by new added routes, such that Braess paradox doesn't (does) occur if the proposed test matrix is positive (negative) semi-definite. The derivations extend existing theories towards the situations when multiple routes are arbitrarily generated after link additions. These new theories deliver prior information to foresee Braess paradox taking place on a class of transportation networks which is more general than before and never reached by existing studies on the indicated classical Braess paradox problem.

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A theoretic formulation on how traffic time information distributed by ITS operations influences the trajectory of network flows is presented in this paper. The interactions between users and ITS operator are decomposed into three parts: (i) travel time induced path flow dynamics (PFDTT); (ii) demand induced path flow dynamics (PFDD); and (iii) predicted travel time dynamics for an origin-destination (OD) pair (PTTDOD). PFDTT describes the collective results of user’s daily route selection by pairwise comparison of path travel time provided by ITS services. The other two components, PTTDOD and PFDD, are concentrated on the evolutions of system variables which are predicted and observed, respectively, by ITS operators to act as a benchmark in guiding the target system towards an expected status faster. In addition to the delivered modelings, the stability theorem of the equilibrium solution in the sense of Lyapunov stability is also provided. A Lyapunov function is developed and employed to the proof of stability theorem to show the asymptotic behavior of the aimed system. The information of network flow dynamics plays a key role in traffic control policy-making. The evaluation of ITS-based strategies will not be reasonable without a well-established modeling of network flow evolutions.