Integrated daily commuting patterns and optimal road tolls and parking fees in a linear city

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Abstract

For decades, the dynamic traffic patterns of morning and evening commutes have been investigated separately, and it is often assumed that they are simple mirror symmetries. In this paper, we use a two-stage differential method to establish a daily traffic pattern that links the morning and evening commutes as an integrated one. Based on a bi-direction bottleneck network with a spatial pattern of parking, we use analytical models to describe travelers’ behavior in choosing departure times in their morning and evening trips, where a commuter’s morning and evening decisions are joined by a parking location. Given fixed parking locations of commuters, we firstly derive the evening commute pattern, which is a Nash equilibrium in the sense that no one can reduce her/his travel cost given other commuters’ decisions. Then the individual evening commute costs are allocated to different parking locations in modeling the morning commuting behavior, and the morning travel pattern is a user equilibrium in the sense that everyone has equal daily travel cost and no one can reduce private travel cost by unilaterally changing travel decisions. Then we propose a time-varying road toll regime to eliminate queuing delay and reduce schedule delay penalty. Furthermore, a time-varying road toll and location-dependent parking fee regime is developed to achieve a system optimum where the morning schedule delay cost is further reduced to the minimum by reversing the spatial order of parking. In view of the fact that road pricing is hard to implement, we propose a location-dependent parking fee regime with no road tolls to optimize the morning commute pattern, without improving the evening commute pattern.

Introduction

In most urban areas parking space in downtowns is of high demand and limited supply, and thus the management of parking could have significant impact on the performance of the transportation system in which parking is an integral part. Although various aspects of parking, such as parking policy (Verhoef et al., 1995, Bifulco, 1993), parking behavior (Hester et al., 2002), parking demand (Lam et al., 1998), and parking price (Arnott et al., 1991, Glazer and Niskanen, 1992, Miller and Everett, 1982, Shoup, 1982, Arnott and Rowse, 1999) have been considered in the literature, an integral analysis of downtown parking considering commuting behavior, bottleneck dynamics and parking/congestion fees is still lacking. In this paper, we examine the interplays among commuters’ travel behavior, congestion pricing policies, parking demand and the resulting dynamic traffic patterns in a simplified linear city, and analyze how parking fees and road tolls are fairly charged to improve system efficiency.

The basic modeling approach used in our study is dynamic traffic assignment. Dynamic user equilibrium has been studied for more than two decades, which has accumulated a vast literature. A typical and analytically solvable dynamic user equilibrium (DUE) problem is the morning commute problem with a bottleneck model originated by Vickrey (1969): commuters to the central business district (CBD) travel through a bottleneck of limited capacity, and experience queuing delay at the bottleneck if traffic arrival rate exceeds bottleneck capacity. Travelers therefore choose different departure times so as to minimize their combined cost of congestion delay and late/early arrival penalty, and in equilibrium no one can (strictly) reduce his/her trip cost by changing his/her departure time unilaterally.

Past research paid more attention to the modeling of morning home-to-work commute than that of evening work-to-home commute. It was generally believed that the evening trip-timing pattern is a mirror image of the morning one, since every morning commute trip is matched by a reverse trip in the evening. Some researchers examined the differences between morning and evening travel patterns. The distinction between arrival-time-determined scheduling preferences in the morning commute and the departure-time-determined scheduling preferences in the evening commute is frequently noted in the literature. In line with this, Hurdle (1981) identified some differences between the equilibrium patterns of morning and evening commutes in terms of the departure time pattern and magnitude of congestion. Moreover, when travelers are not identical, the symmetry between morning and evening commutes breaks down. Vickrey, 1969, Fargier, 1983 demonstrated this in their insightful articles, and recently de Palma and Lindsey (2002) gave a more thorough analysis to this.

Arnott et al. (1991) constructed a model that incorporates both the spatial distribution of parking and the temporal peaking of travel pattern on a highway with a single-bottleneck. In their model, such cost components as queuing time, walking time, schedule delay, road toll and parking fee are considered, but only morning home-to-work commute is modeled. They investigated and compared the efficiency of various road toll and parking fee regimes, and showed that when parking is incorporated, the rush-hour traffic pattern is different from that generated by the traditional bottleneck model. Specifically, queuing dead-weight cost can be eliminated and schedule delay can be minimized with the implementation of time-varying road tolls and location-dependent parking fees. As the authors pointed out, neglected in their paper are several important real-world features of parking. First, the return commute in the evening is ignored, which can be an important limitation if the traffic pattern in the evening commute is not a mirror image of the morning commute when parking is considered in the analysis. Second, in practice the commuting traffic demand is not fixed, as mentioned by the authors. Rather, it is elastic and is generally believed to be a decreasing function of trip cost. Moreover, the travel demand should be a function of the daily commuting cost rather than that of the single morning or evening trip cost.

Motivated by Arnott et al. (1991), in this paper we attempt to give a more thorough and complete investigation of the daily commuting problem with consideration of parking. First, the evening commuting pattern is firstly established given fixed parking locations of commuters. Second, the parking location-dependent commuting cost in the evening is considered as part of the daily commuting cost in deriving the morning commute pattern. Third, several road toll and parking fee regimes are proposed, and their mechanisms and efficiencies are discussed.

As is commonly adopted in similar studies, the simple network depicted in Fig. 1 is used in our study. It has a single origin–destination pair connected by a highway corridor of two routes. The origin represents a residential area and the destination a city business center. Among the residents living in the origin, some would go to the city center for work in daytime and come back home after work. For simplicity, we suppose all these travelers are behaviorally homogenous. Moreover, each of the two parallel paths connecting the origin and the destination has a bottleneck with limited capacities, and it is further assumed that the constant travel times on the line haul parts of these two paths are zero. The same network is used in Zhang et al. (2005) to study the dynamic scheduling of work trips using stochastic models with no consideration to spatial parking.

In Fig. 1, the service capacity of the bottleneck in the home-to-work direction is s,1 and that in the work-to-home direction is s. Assuming N commuters travel from the origin to the destination for work everyday, a queue will develop if the arrival rate exceeds the capacity.2 There is a parking lot around the CBD, where the parking spots that are treated as continuous variables are indexed by n in order of increasing distance from the CBD. Walking time to the workplace from location n is taken to be W+wn, where W is the walking time from the parking lot to the workplace and w is the time taken for passing one parking spot on foot. Since the cost related to W is the same to everyone, and thus assumed to be zero for simplicity. The in-vehicle travel time within the parking area is ignored, so the interference between vehicles being parked, and between parkers and through traffic is assumed away.

Individuals are assumed to have a common preferred arrival time at work (e.g. their official work starting time) t, and a common preferred leaving time from work (their official off work time) t. The cost of unit early arrival time in the morning is taken to be β, and late arrival for work is not allowed. In the evening, early departure from work is not allowed, and the cost of unit late departure time is γ. The unit cost of in-vehicle travel time is α (for both morning and evening commutes), and the unit cost of walking time is λ. To ensure the existence of a deterministic equilibrium it is assumed that α>β and α>γ. Since in general people prefer to drive rather than to walk, λ > α.

In modeling the departure time choice behavior, we use two equilibrium assumptions, namely Nash equilibrium and user equilibrium. In a Nash equilibrium, no one can reduce private travel cost by changing departure time when all other commuters’ departure times are given. In a user equilibrium, every one has the same travel cost and no one can reduce her/his travel cost by unilaterally changing departure time. Clearly, Nash equilibrium is a more general assumption, of which user equilibrium is a special case.

The paper is organized as follows. In Section 2, we establish a Nash-equilibrium traffic pattern in the evening commute without pricing for a given set of assigned parking spaces, and then the user-equilibrium morning traffic pattern is derived considering daily travel cost by adding the parking location-dependent evening travel cost into the morning trip. In Section 3, we optimize the evening and morning travel patterns with time-varying road tolls. In Section 4, we derive the system optimal commuting pattern under time-varying road tolls and location-dependent parking fees. In Section 5, we propose a location-dependent parking fee and no road toll regime to optimize the morning commuting pattern. Section 6 provides some numerical examples. Finally, Section 7 concludes the paper.

Section snippets

Nash-equilibrium travel pattern in the evening commute with fixed parking locations

The parking locations of commuters in the evening were decided in their morning commute and are considered given when we model their evening commute. Since all commuters leave work after t, the travel cost of an individual who leaves the bottleneck at time t with a parking spot n isCf(t,n)=αD(t)s+λwn+γt-D(t)s-wn-t,where D(t) is the length of the queue upon the arrival of a commuter who leaves the bottleneck at time t. On the right hand side of the above equation, the first

Morning and evening commuting patterns with time-varying road tolls (regime r)

In this section, we investigate the situation that time-varying tolls are charged in both morning and evening commutes, in order to eliminate the efficiency loss in regime f.

Time-varying road tolls and location-dependent parking fees for a system optimum (regime o)

In the morning commute of regime f, commuters park their cars outwards due to the competition for a convenient parking spot. Actually parking inwards is more efficient since it can decrease the total unpunctuality penalty cost. If commuters park inwards, arrival time to the workplace of the first commuter can be postponed by 2wN, and hence the average early arrival time is reduced by wN. The system optimal morning traffic pattern with a reversed parking order is shown in Fig. 7, where AB is the

Optimal location-dependent parking fees without road tolls (regime p)

As discussed in Arnott et al. (1991), road pricing has disadvantages in implementation, such as high operating cost, interfering traffic and political opposition etc. Parking fees are relatively more favorable than road tolls, since the transaction entailed in making payment generally does not impede traffic flow. That’s why road pricing has been implemented only in few cities, and almost all cities are collecting parking fees in urban area. In this section, we discuss the location-dependent

Numerical examples

In this section, we present some numerical examples to clearly demonstrate the analytical results obtained in the above sections. In the network shown in Fig. 1, the service rate of each bottleneck is assumed to be s=s=1.0×102veh/min, and totally N = 1.0 × 104 people are assumed to live in the residential area. Each person drives his/her own car to work in the morning and returns home in the evening. The official work start time is t=9:00am and the end time is t=17:00pm. The unit cost of

Conclusions

In this paper, we investigated the joint morning and evening rush-hour commutes with a two-stage differential method, based on a linear city with one end of residential area and another end of workplace. In the first stage, the evening work-to-home commuting pattern was developed, and travel costs have been derived for individual commuters with different parking locations. Then in the second stage, by attaching the evening travel costs to different parking locations, the morning home-to-work

Acknowledgements

The authors wish to thank Prof. Kenneth A. Small of University of California at Irvine and an anonymous referee for their helpful comments in improving the paper. This study has been substantially supported by the National Natural Science Foundation Council of China through three projects (#70401016, #70571058 and #70521001).

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