## 1 INTRODUCTION

Congestion pricing has been advocated to reduce traffic congestion since 1920 (see, [1] for a recent review). Furthermore, the advanced electronic tolling system makes congestion pricing more practical and successful implementations (e.g., Singapore and London). However, getting the public to accept congestion pricing is still a major obstacle. Among reasons of public opposition, social inequality is an important one. Many have mentioned the social inequality issue of congestion pricing [2, 3]; Levinson [4]; Ecola and Light [5].

It has been suggested [6] that the strategy proposed as “hybrid between rationing and pricing” has the potential to reduce traffic congestion without penalizing anyone. In [7], Daganzo and Garcia have explored the similar strategy under a dynamic setting with single bottleneck congestion. Guo and Yang [8] investigate congestion pricing and revenue refunding schemes that reduce total travel time and make every user better off. Nie and Liu [9] examine the impacts of traveler's value of time on the existence of a pricing-refunding scheme that is both self-financing and Pareto-improving. In [10], Lawphongpanich and Yin have investigated the Pareto-improving tolls on general transportation networks that do not require revenue distribution. In Wu *et al*. [11], Pareto improvement is achieved by appropriately charging tolls on highway links and adjusting the fares of transit lines.

After Daganzo proposed the rationing and pricing (RP) scheme in 1995, Nakamura and Kockelman [12] presented an empirical application of the “hybrid between rationing and pricing” strategy to San Francisco Bay Bridge corridor. Han *et al*. [13] has analyzed the efficiency of the plate number-based traffic rationing in general networks. Wang *et al*. [14] have investigated the effects of road space rationing from both short-term and long-term perspectives. (long-term equilibrium under road rationing takes into account the public's self-adjustment activities such as car consumption and car disposal) This paper follows a similar strategy with Daganzo [6] and considers the Pareto-improving RP scheme without revenue refunding. Under the RP scheme, every day, each commuter is classified as either “free” or “rationed”. “Free” commuters are allowed to use the highway without paying the toll, whereas “rationed” commuters have to pay the toll if they drive. Each day a fraction of commuters are rationed in their free use of the highway, and the rationing fractions are determined systematically so that everyone is equally rationed in a given period. Different from Daganzo's single bottleneck model, we take a continuum modeling approach and conduct a general bi-modal analysis in a linear monocentric city with a competitive railway and highway system. Continuum modeling approach has been applied to explore the general tendencies and patterns of commuters' behaviors and their responses to policy changes in the transportation system at a macroscopic level [15, 16]. Ho and Wong [17] provide a recent review of the development and applications of the two-dimensional continuum traffic equilibrium modeling approach.

Unfortunately, the complexity of the two-dimensional space often makes it difficult to obtain analytical properties. By considering a simplified one-dimensional continuum corridor, analytical solution and properties of the spatial traffic equilibrium with multimodal choices can be obtained. Haring *et al*. [18] are the first to examine a simple solution of the user equilibrium on a traffic corridor with several congested modes. Jehiel [19] proves that the simple solution would exist if capacities of two congested modes are constant. In recent years, many have conducted analysis of various problems [e.g., road pricing and park-and-ride (P&R) facility] with linear monocentric city model [20-28]. In [29], Arnott and de Palma have considered the no toll equilibrium on a traffic corridor that connects a continuum of residential locations to a point central business district (CBD). De Palma and Arnott [30] examined a single-lane road with Lighthill-Whitham-Richards (LWR) flow congestion and Greenshields' relation.

The remainder of this paper is organized as follows. Section 2 makes some assumptions and formulates the continuum user equilibrium model without RP in a linear city. Section 3 proposes the RP scheme and explores some properties of the user equilibrium under this scheme. Extension to RP with cordon and P&R service has been made in Section 4. Numerical results are presented in Section 5, and the concluding remarks and suggestions for further researches are provided in Section 6.