whitehou govC ONGESTION M ODELLING* C. Robin Lindy1 and Erik T. Verhoef2*
1Department of Economics
University of Alberta
Edmonton
Alberta
Canada
Phone: +1-780-4927642
Fax: +1-780-4923300
E-mail: robin.lindy@ualberta.ca
2Department of Spatial Economics
Free University Amsterdam
De Boelelaan 1105新东方英语论坛
1081 HV Amsterdam
The Netherlands
Phone: +31-20-4446094
Fax: +31-20-4446004
E-mail: everhoef@econ.vu.nl This version: 05/11/99
Key words: congestion, road pricing, networks
JEL codes: R41, R48, D62
格鲁吉亚语
Abstract
Transportation rearchers have long struggled to find satisfactory ways of describing and analysing traffic congestion, as evident from the large number of often competing approaches and models that have been developed. This paper aims to provide a review of the literature on this topic. The paper s
tarts with the modelling of homogeneous traffic flow and congestion on an isolated road under stationary conditions. We t up the supply-demand framework ud to characterize equilibrium and optimal travel volumes. Next, an overview of macroscopic and microscopic models of nonstationary traffic flow is given. We then describe how trip timing can be modelled, and discuss the esnce of dynamic equilibrium. The paper next reviews the principles of static and dynamic equilibrium on a road network in a deterministic environment, and then identifies equilibrium concepts that account for stochasticity in demand and capacity. Finally, conceptual and practical issues regarding congestion pricing and investment on a network will be addresd.
*The authors would like to thank Ken Small, Richard Arnott and André de Palma for stimulating comments on an earlier version of this paper. Any remaining errors, however, are the authors’ responsibility alone.
**Erik Verhoef is affiliated as a rearch fellow to the Tinbergen Institute.
1I NTRODUCTION
Traffic congestion is one of the major liabilities of modern life. It is a price that people pay for the various benefits derived from agglomeration of population and economic activity. Becau land is sc
arce and road capacity is expensive to construct, it would be uneconomical to invest in so much capacity that travel were congestion-free. Indeed, becau demand for travel depends on the cost, improvements in travel conditions induce people to take more trips, and it would probably be impossible to eliminate congestion.hape
Transportation rearchers have long struggled to find satisfactory ways of describing and analysing congestion, as evident from the large number of often competing approaches and models that have been developed. Early rearchers hoped to develop models bad on fluid dynamics that would not only be accurate, but also universally applicable. But congestion, unlike fluid flow, is not a purely physical phenomenon but rather the result of peoples' trip-making decisions and minute-by-minute driving behaviour. One should therefore expect the quantitative — if not also the qualitative — characteristics of congestion to vary with automobile and road design, rules of the road, pace of life, and other factors. Models calibrated in a developed country during the 1960s, for example, may not fit well a developing country at the beginning of the twenty-first century.
Congestion in transportation is of cour not limited to roads: it is also a problem at airports and in the airways, at harbours, on railways, and for travellers on bus and subway networks. For modelling purpos uful parallels can often be drawn between traffic congestion and congestion at other facil
ities. But given space constraints, and in the interest of maintaining focus, attention is limited in this paper to road traffic congestion. Broadly speaking, traffic congestion occurs when the cost of travel is incread by the prence of other vehicles, either becau speeds fall or becau greater attention is required to drive safely. Traffic engineering is largely concerned with traffic congestion and safety, and it should therefore be no surpri that traffic flow theory will feature prominently in this paper.
Traffic congestion can be studied either at a microscopic level where the motion of individual vehicles is tracked, or at a macroscopic level where vehicles are treated as a fluid-like continuum. Queuing theory is a form of microscopic analysis. But most of the literature on queuing is of limited relevance becau it focus on steady-state conditions — which rarely prevail in traffic, and on stochastic aspects of individual customer/traveller arrival and rvice times — which are arguably of condary importance (except at junctions) for traffic flows heavy enough to cau congestion (Hurdle, 1991). Queuing theory thus will not be treated here. Car-following theory is another form of microscopic analysis that will be mentioned. Macroscopic analysis will nevertheless occupy the bulk of attention.武汉理工大学网络教育学院
The paper is organized as follows. Section 2 concerns the modelling of homogeneous traffic flow an
d congestion on an isolated road under stationary conditions. It also ts up the supply-demand framework ud to characterize equilibrium and optimal travel volumes. Section 3 provides an overview of macroscopic and microscopic models of nonstationary traffic flow. It then describes how trip timing can be modelled, and discuss the esnce of dynamic equilibrium. Section 4 reviews the principles of static and dynamic equilibrium on a road network in a deterministic environment, and then identifies equilibrium concepts that account for stochasticity in demand and capacity. Section 5 address conceptual and practical issues regarding congestion pricing and investment on a network. Finally, Section 6 concludes.
Congestion Modelling
22 T IME-I NDEPENDENT M ODELS
Time-independent models of traffic congestion rve as a stepping stone toward the development and understanding of more complicated and realistic time-dependent models. They may also provide a reasonable description of traffic conditions that evolve only slowly. Such traffic is sometimes called 'stationary', although a preci definition of 'stationary' is rather delicate (Daganzo, 1997).
Traffic streams are described by three variables: density (k ), speed (v ), and flow (q ), measured res
pectively in vehicles per lane per km., km. per hour, and vehicles per lane per hour. At the macroscopic level the variables are defined under stationary conditions at each point in space and time,and are related by the identity q k v =⋅. Driver behaviour creates a cond functional relationship between the three variables that can be shown by plotting any one variable against another. Figure 1(a)depicts a speed-density curve, dubbed the fundamental diagram of traffic flow (Haight, 1963). Though studied for decades (e May, 1990 for a literature review), understanding about the shape of this curve continues to evolve, as evidenced by changes to the third edition of the venerable U.S. Highway Capacity Manual (Transportation Rearch Board, 1992). The preci shape on a given road gment depends on various factors (Roess et al ., 1998, Chs. 10, 21). The include the number and width of traffic lanes,grade, road curvature, speed limit, location vis-à-vis entrance and exit ramps, weather, mix of vehicle types, proportion of drivers who are familiar with the road, and idiosyncrasies of the local driving population.
Figure 1. (a) speed-density curve, (b): speed-flow curve, (c): flow-density curve
For safety reasons speed usually declines as density increas. Nevertheless, on highways speeds t
end to remain clo to the free-flow speed, v f , up to flows of 1,000-1,300 vehicles per lane per hour. At higher densities the speed-density curve drops more rapidly, passing through the point (,)k v 00 at which flow reaches a maximum, q k v 000=, and reaching zero at the jam density, k j , where speed and flow are一月 英文
Congestion Modelling 3
both zero. Speed-flow and flow-density curves corresponding to the speed-density curve in Fig. 1(a) are shown in Fig. 1 (b) and (c) respectively. Note that any flow q q '<0 can be realized at either a low density and high speed, (,)k v L L , or at a high density and low speed (,)k v H H . Economists refer to the upper branch of the speed-flow curve as congested , and to the lower branch as hypercongested . In the engineering literature the upper branch is variously referred to as uncongested , unrestricted or free flow ,and the lower branch as congested , restricted or queued . The term queued is apposite for the hypercongested branch in that queuing usually occurs in this state; e Section 3. But congestion also occurs on the upper branch whenever speed is below the free-flow speed. For this reason, the economics terminology will be ud here.
Following Walters (1961) the speed-flow curve can be ud for economic analysis by interpreting flo
w as the quantity of trips supplied by the road per unit of time. A trip cost curve can be generated of the form C q c L v q ()/()=+0α, where αis the unit cost of travel time, L is trip distance, v q () is speed expresd in terms of flow, and c 0 denotes trip costs other than in-vehicle travel time, such as monetized walk access time and fuel costs (if the costs do not depend on congestion). The trip cost curve, shown in Fig. 2, has a positively-sloped portion corresponding to the congested branch of the speed-flow curve,and a negatively-sloped backward-bending portion corresponding to the hypercongested branch. A flow of q ' can be realized at a cost C L on the positively-sloped portion, as well as at a higher cost C H on the negatively-sloped portion. Becau the same number of trips is accomplished, the latter outcome is inefficient.
Figure 2. Backward-bending travel cost curve, C q (), and travel demand curve, p q ()
If flow is also interpreted to be the quantity of trips "demanded" per unit of time, then a demand curve p q ()can be combined with C q () to obtain a supply-demand diagram. Candidate equilibria occur where p q () and C q () interct. In Fig. 2 there are three interction points: x , y and z , with flow congested at x , and hypercongested at y and z . There has been a heated debate in the literature (e Chu and Small, 1997; Verhoef, 1999; and McDonald et. al for reviews and recent contributions to this debate)about whether hypercongested equilibria are stable, and also whether it
makes n to define the supply Flow
C L q 'q 0z
y
C H x
p q ()
C q ()
Congestion Modelling
4and demand for trips in terms of flow. The emerging view ems to be that hypercongestion is a transient phenomenon that can be properly studied only with dynamic models; e Section 3.
四级查分网站For economic analysis (e.g . Button, 1992), it is common to ignore the hypercongested branch of the speed-flow curve and to specify a functional form for C q () on the congested branch directly, rather than beginning with a speed-density function and then deriving C q (). Given C q (), the socially opti
mal usage of the road, and the congestion toll that supports it, can be derived as shown in Fig. 3. As in Fig. 2, the unregulated equilibrium flow, q E , occurs at point E , the interction of C q () and p q (). Now, since 'external benefits' of road u are not likely to be significant (benefits are normally either purely internal or pecuniary in nature), p q () specifies both the private and the marginal social benefit of travel. Total social benefits can thus be measured by the area under p q (). Analogously, C q () measures the cost to the traveller of taking a trip. If external travel costs other than congestion, such as air pollution or accidents,are ignored, then C q () measures the average social cost of a trip. The total social cost of q trips is then TC q C q q ()()=⋅, and the marginal social cost of an additional trip is MSC q TC q q ()()=∂ =+⋅C q q C q q ()()∂ .
The socially optimal number of trips, q *, occurs in Fig. 3 at point F where MSC q () and p q ()interct. The optimum can be supported as an equilibrium if travellers are forced to pay a total price of p MSC q **()=. Becau the price of a trip is the sum of the individual’s physical travel cost and the toll,p C q =+()τ, the requisite toll is τ***()()=−MSC q C q =⋅q C q q **()/∂∂ , where q C q q **()/⋅∂∂ is the marginal congestion cost impod by a traveller on others. This toll is known as a 'Pigouvian' tax, after its spiritual father Pigou (1920).
英国出国留学Figure 3. Equilibrium road usage, q E , optimal road usage, q *, and optimal congestion toll, τ*
Imposition of the toll rais social surplus by an amount equal to the shaded area FGE in Fig. 3.Nevertheless, travellers end up wor off if the toll revenues are not ud to benefit them. The q *individuals who continue to drive each suffer a loss per trip of p p E *−, resulting in a collective loss equal to area HIFJ . And the q q E −* individuals who are priced off the road, either becau they switch to another mode or give up travelling, suffer a collective loss equal to area JFE . The loss
are the root of the longstanding opposition to congestion tolling in road transport. Transportation analysts and planners are now trying to devi ways of spending toll revenues so as to improve the acceptability of pricing (Small, 1992b).
Flow
p MSC q **(
=p C q E E =()C q ()*C(q)E
Congestion Modelling 5geeta
3 T IME-D EPENDENT M ODELS
Time-dependent or dynamic traffic models allow for changes in flow over time as well as over space.The most widely ud dynamic macroscopic model is the hydrodynamic model developed by Lighthill and Whitham (1955) and Richards (1956) — henceforth the LWR model; e Daganzo (1997) for a review. The esntial assumption of the LWR model is that the relationship in stationary traffic between speed and density, shown in Fig. 1, also holds under nonstationary conditions. The model is completed by imposing the condition that vehicles are neither created nor destroyed along t
he road. If x denotes location and t time, and if the requisite derivatives exist, this results in a partial differential equation,∂∂∂ q t x x k t x t (,)(,)+=0, known as the conrvation equation . In cas where q and k are discontinuous, and therefore not differentiable, a discrete version of the conrvation equation still applies, as will be shown in the following example.
Figure 4. (a): Transition from A to B on flow-density curve, (b): trajectories in time-space diagram.To illustrate how the LWR model behaves, suppo that traffic on a roadway is initially in a congested stationary state, A , with density k A , speed v A and flow q A , as shown in Fig. 4(a) which is adapted from May (1990, Fig. 11.1). Inflow at the entrance then falls abruptly from q A to q B , moving traffic to a new state, B , at another point on the same flow-density curve. State B will propagate downstream as a shock wave with some speed, w AB . Vehicles upstream in state B catch up to the shock wave at a speed v w B AB −, and thus leave state B at a flow rate ()v w k B AB B −. Given conrvation of vehicles, this must match the rate at which they enter state A : ()v w k A AB A −. Equating the two rates, and recalling that q k v i A B i i i ==,, , one obtains w q q k k AB A B A B =−−()(). This wave speed corresponds to the slope of a line joining states A and B on the flow-density curve in Fig. 4(a). The wave speed is slower than the speed of vehicles in either state, v A and v B .q B
q A
Flow v B v A
w AB A B
k k Density Time 0
lighters eminemC
(a)(b)
