Richard D. Connors

Network Equilibrium under Cumulative Prospect Theory and Endogenous Stochastic Demand and Supply

In this paper we consider a network whose travel demands and road capacities are endogenously considered to be random variables. With stochastic demand and supply the route travel times are also random variables. In this scenario travelers choose their routes under travel time uncertainties. Several evidences suggest that the decision making process under uncertainty is significantly different from that without uncertainty. Therefore, the paper applies the decision framework of cumulative prospect theory (CPT) to capture this difference. We first formulate a stochastic network model whose travel demands and link capacities follow lognormal distributions. The stochastic travel times can then be derived under a given route choice modeling framework. For the route choice, we consider a modeling framework where the perceived value and perceived probabilities of travel time outcomes are obtained via transformations following CPT. We then formulate an equilibrium condition similar to that of User Equilibrium in which travelers choose the routes that maximizes their perceived utility values in the face of transformed stochastic travel times. Conditions are established guaranteeing existence (but not uniqueness) of this equilibrium. The paper then proposes a solution algorithm for the proposed model which is then tested with a test network.

Evaluation and Design of Transport Network Capacity Under Demand Uncertainty

A flexible evaluation and design model for transport network capacity under demand variability is proposed. The future stochastic demand is assumed to follow a normal distribution. Traveler path choice behavior is assumed to follow the probit stochastic user equilibrium. The network reserve capacity is used to evaluate the performance of the network. Since the future demand is stochastic, the reserve capacity is measured by possible increases in both mean and standard deviation (SD) of the base demand distribution. The proposed model therefore represents the flexibility of the network in its robustness to origin-destination demand variation (i.e., high SD). The proposed model can also determine an optimal network design to maximize the reserve capacity of the network for both the mean and the SD of the increased demand distribution. The implicit programming approach is applied to solve the optimization problem. Sensitivity analysis is adopted to provide all necessary derivatives. The model and algorithm are tested with a hypothetical network to illustrate the merits of the proposed model.

On the existence and uniqueness of first best tolls in networks with multiple user classes and elastic demand

System optimal (SO) or first best pricing is examined in networks with multiple user classes and elastic demand, where different user classes have a different average value of value of time (VOT). Different flows (and first best tolls) are obtained depending on whether the SO characterisation is in units of generalised time or money. The standard first best tolls for time unit system optimum are unsatisfactory, due to the fact that link tolls are differentiated across users. The standard first best tolls for the money unit system optimum may seem to be practicable, but the objective function of the money unit system optimum is nonconvex, leading to possible multiple optima (and non-unique first best tolls). Since these standard first best tolls are unsatisfactory, we look to finding common money tolls which drive user equilibrium flows to time unit SO flows. Such tolls are known to exist in the fixed demand case, but we prove that such tolls do not exist in the elastic demand case. Although common money tolls do not exist which drive the solution to the exact time system optimal flows, tolls do exist which can push the system close to time system optimum (TSO) flows.

An Optimal Toll Design Problem with Improved Behavioural Equilibrium Model: The Case of the Probit Model

This paper considers the optimal toll design problem that uses the Probit model to determine travellers’ route-choices. Under probit, the route flow solution to the resulting stochastic user equilibrium (SUE) is unique and can be stated implicitly as a function of tolls. This reduces the toll design problem to an optimization problem with only nonnegativity constraints. Additionally, the gradient of the objective function can be approximated using the chain rule and the first order Taylor approximation of the equilibrium condition. To determine SUE, this paper considers two techniques. One uses Monte-Carlo simulation to estimate route choice probabilities and the method of successive averages with its prescribed step length. The other relies on the Clark approximation and computes an optimal step length. Although both are effective at solving the toll design problem, numerical experiments show that the technique with the Clark approximation is more robust on a small network.

Analytic approximations for computing probit choice probabilities

The multinomial probit model has long been used in transport applications as the basis for mode- and route-choice in computing network flows, and in other choice contexts when estimating preference parameters. It is well known that computation of the probit choice probabilities presents a significant computational burden, since they are based on multivariate normal integrals. Various methods exist for computing these choice probabilities, though standard Monte Carlo is most commonly used. In this article we compare two analytical approximation methods (Mendell–Elston and Solow–Joe) with three Monte Carlo approaches for computing probit choice probabilities. We systematically investigate a wide range of parameter settings and report on the accuracy and computational efficiency of each method. The findings suggest that the accuracy and efficiency of an optimally ordered Mendell–Elston analytic approximation method offers great potential for wider use.

Methodology for fitting and updating predictive accident models with trend

Reliable predictive accident models (PAMs) (also referred to as Safety Performance Functions (SPFs)) have a variety of important uses in traffic safety research and practice. They are used to help identify sites in need of remedial treatment, in the design of transport schemes to assess safety implications, and to estimate the effectiveness of remedial treatments. The PAMs currently in use in the UK are now quite old; the data used in their development was gathered up to 30 years ago. Many changes have occurred over that period in road and vehicle design, in road safety campaigns and legislation, and the national accident rate has fallen substantially. It seems unlikely that these ageing models can be relied upon to provide accurate and reliable predictions of accident frequencies on the roads today. This paper addresses a number of methodological issues that arise in seeking practical and efficient ways to update PAMs, whether by re-calibration or by re-fitting. Models for accidents on rural single carriageway roads have been chosen to illustrate these issues, including the choice of distributional assumption for overdispersion, the choice of goodness of fit measures, questions of independence between observations in different years, and between links on the same scheme, the estimation of trends in the models, the uncertainty of predictions, as well as considerations about the most efficient and convenient ways to fit the required models.

Updating outdated predictive accident models

Reliable predictive accident models (PAMs) (also referred to as safety performance functions (SPFs)) are essential to design and maintain safe road networks however, ongoing changes in road and vehicle design coupled with road safety initiatives, mean that these models can quickly become dated. Unfortunately, because the fitting of sophisticated PAMs including a wide range of explanatory variables is not a trivial task, available models tend to be based on data collected many years ago and seem unlikely to give reliable estimates of current accidents. Large, expensive studies to produce new models are likely to be, at best, only a temporary solution. This paper thus seeks to develop a practical and efficient methodology to allow currently available PAMs to be updated to give unbiased estimates of accident frequencies at any point in time. Two principal issues are examined: the extent to which the temporal transferability of predictive accident models varies with model complexity; and the practicality and efficiency of two alternative updating strategies. The models used to illustrate these issues are the suites of models developed for rural dual and single carriageway roads in the UK. These are widely used in several software packages in spite of being based on data collected during the 1980s and early 1990s. It was found that increased model complexity by no means ensures better temporal transferability and that calibration of the models using a scale factor can be a practical alternative to fitting new models.

Two-point spectral correlations for the square billiard

We investigate the two-point correlations in the quantum spectrum of the square billiard. This system is unusual in that the degeneracy of the energy levels increases in the semiclassical limit in such a way that the average level separation is not given by the inverse of the mean density of states. Hence, for example, the standard level spacings distribution does not tend to the Poissonian limit expected for integrable systems. In this paper we calculate the leading-order asymptotic form of a degeneracy-weighted two-point correlation function using a combination of probabilistic techniques and classical number theory. The result exhibits number-theoretical fluctuations about a mean which is a sum of two terms: one having the usual (constant) Poissonian form and the second representing a small correction which decays as the inverse of the correlation distance. This is confirmed by numerical computations.

A quasi-dynamic assignment model that guarantees unique network equilibrium

This article formulates a discrete-time dynamic traffic assignment (DTA) model and, under certain conditions, shows the existence and uniqueness of network equilibrium. Several theoretical issues need to be tackled. The inflow to a link in a particular discrete (time) period does not necessarily exit within the same period. We consider how flow is passed from one link and period to the next, and the corresponding costs. Under the proposed model, flow departing within a discrete period may experience different link travel times in different discrete periods, even if the flow chooses a single route. Route travel time must then be defined so that route and OD costs are meaningful. To this end, quasi-real route travel time is defined. Based on this definition, a quasi-equilibrium condition for DTA is proposed; a semi-dynamic analogue of user equilibrium. The existence and uniqueness of this equilibrium solution are proven.

Modeling Network Growth with Scaling Laws in a Linear Monocentric City

Scaling relationships have been observed across urban systems and appear to reveal global emergent properties that may help to characterize, understand, and predict urban growth. Empirical urban studies that show scaling laws relating the growth in population to that of the transportation network warrant theoretical investigation. Such scaling phenomena are considered in the context of a linear monocentric city. Commuters are distributed from the city boundary to the common destination (the central business district), and two modes (railway and car) are available along the corridor. Two scenarios of urban growth are considered: vertical and horizontal. Road capacity and corridor length both increase alongside the population growth. Within these scenarios, growth is constrained to follow a scaling law and the evolution of network performance is examined. The results suggest that different long-term evolution trajectories may exist depending on the scaling-law and network features. On this basis, theoretical limits need to be investigated on the possible evolutionary pathways for a city whose growth is dependent on its transport system.