Richard Mounce

Existence of Equilibrium in a Continuous Dynamic Queueing Model for Traffic Networks with Responsive Signal Control

Many real-life traffic systems incorporate responsive traffic signal control, i.e. where the green time assigned to a stage at a junction depends on the queue lengths on the various approaches. By making signals responsive one might expect the queueing pattern to approach equilibrium, i.e. a queueing pattern for which the responsive policy tells us to leave the signal settings unchanged. However, changing junction signal settings changes the costs of traversing the approaches to the junction and traffic may change route in the light of this. Hence, a responsive signal system is really at equilibrium only if it is at equilibrium with respect to its own rules and also with respect to the re-routing of traffic. The paper gives a framework for responsive signal control in the dynamic queueing model in terms of stage pressures. Three responsive signal policies are considered: delay-minimisation, equisaturation and P 0. A dynamical system is specified that describes both changes to signals due to the responsive signal policy and changes to route inflows due to the re-routing of traffic. An implicit function theorem is utilised in showing that the swap vector for the dynamical system is a continuous function of the route flow vector and green time vector. Then by Schauders fixed point theorem, there exists equilibrium of the dynamical system. Finally, the responsive policies are compared with fixed signals in network simulations.

Convergence to Equilibrium in Dynamic Traffic Networks when Route Cost Is Decay Monotone

This paper addresses the issue of convergence to equilibrium in dynamic traffic assignment. Within-day time is considered to be a continuous variable, so that traffic flows and costs are functions of within-day time. Flow propagates along routes connecting origin-destination (OD) pairs with the demand for travel between each OD pair considered to be rigid (fixed from day to day although it can vary within day). Day-to-day time is also modelled as continuous with the day-to-day dynamical system derived naturally from the usual dynamical user equilibrium (DUE) condition. This paper focuses on the bottleneck model, which has deterministic vertical queueing at bottleneck link exits when flow exceeds capacity. A new property called decay monotonicity is introduced. The link delay (and hence link cost) function is shown to be a decay monotone function of link flow provided that the link capacity is continuously differentiable and positive. In a restricted version of the single bottleneck per route case, it is shown that link cost decay monotonicity implies route cost decay monotonicity. Decay monotonicity of the route cost function is shown to be sufficient for convergence to equilibrium of the dynamical system.

Identifying Sampling Interval for Event Detection in Water Distribution Networks

It is a generally adopted policy, albeit unofficially, to sample flow and pressure data at a 15-min interval for water distribution system hydraulic measurements. Further, for flow, this is usually averaged, whereas pressure is instantaneous. This paper sets out the findings of studies into the potential benefits of a higher sampling rate and averaging for flow and pressure measurements in water distribution systems. A data set comprising sampling at 5xa0s (in the case of pressure), 1xa0min, 5xa0min, and 15xa0min, both instantaneous and averaged, for a set of flow and pressure sensors deployed within two DMAs has been used. Engineered events conducted by opening fire hydrants/wash outs were used to form a controlled baseline detection comparison with known event start times. A data analysis system using support vector regression (SVR) was used to analyze the flow and pressure time series data from the deployed sensors and hence, detect these abnormal events. Results are analyzed over different sensors and events....

Route Swap Processes and Convergence Measures in Dynamic Traffic Assignment

Dynamic traffic assignment algorithms typically proceed by iterating between route swapping and dynamic network loading. This chapter considers various route swap processes and investigated their convergence to equilibrium (assuming that the route cost vector is a monotone function of the route flow vector). Pairwise swapping (in which swapping can occur between each pair of routes) is shown to converge even with an arbitrary exponent on the cost differences (in the dynamic model and hence also in the steady state model). When swapping is only to the least costly route (for that origin destination (OD) pair), convergence is shown in the steady state model. Various forms of convergence measure were considered; this form is suggested by the route swap process. Equivalences were established between the different forms in that if one converged to zero (for a particular route swap process) then so would the other. These equivalences are independent of whether the swap process is continuous or discrete (i.e. the integral can be replaced with a sum).

Case-based reasoning to support decision making for managing drinking water quality events in distribution systems

In order to better leverage past experience of water quality incidents, and to tap into the unique incident database currently being maintained and required by regulatory authorities, a data mining approach is herein proposed. The quality of drinking water is paramount to protecting public health. However water quality failures do occur, with some of the hardest to understand and manage occurring within distribution systems. In the UK, a regulatory process is applied in which water service providers must report on significant water quality incidents, their causes, actions and outcomes. These reports form a valuable resource that can be explored for improved understanding, to help with future incident management and evaluate potential solutions. Case-based reasoning is a knowledge-based problem-solving technique that relies on the reuse of past experience. The WaterQualityCBR software system presented here was developed as such a decision support tool to more effectively manage water quality in distribution systems.

Stability of a dynamic model for traffic networks

The dynamic assignment model assumes flow moves towards cheaper routes at each time at a rate proportional to the product of the flow along the more expensive route and the cost difference. Therefore, it is important for the cost function to be monotone so that convergence to equilibrium will occur. Conditions on the bottleneck output function are given for the bottleneck delay function to be monotone, which will imply monotonicity of the route cost function in the single bottleneck per route case. It is shown that for reasonable bottleneck output functions, we have monotonicity of the product of link cost with a decaying exponential. This decay-monotonicity transfers to the route cost in certain given circumstances. This will in turn imply convergence of the dynamical system by applying Lyapunovs theorem using the appropriate Lyapunov function. It is then important to note that monotonicity of the route cost function implies decay-monotonicity of the route cost function and hence the convergence result is valid for the single bottleneck per route case with monotone link cost functions.

Optimisation through Control in Static and Dynamic Traffic Networks

There is clearly a need for optimising traffic systems in order to reduce congestion and improve network reliability. A system optimal assignment is a traffic flow pattern that minimises total network costs. In reality, travellers are not under any centralised control, but instead choose routes in order to minimise their own individual travel costs, which in general does not lead to a system optimal assignment. Travellers may be induced to choose routes that are closer to yielding a system optimal assignment through the use of tolls and signal control. The paper considers both approaches within a static traffic model (where flows and costs stay constant over time) and within a dynamic traffic model (where flows and costs vary over time).

Non-convergence in dynamic assignment networks?

In the steady-state assignment model where each link has a non-decreasing cost flow curve we have monotonicity not just at the link level but also at the route level. In our dynamical system we assume that the users swap to cheaper routes. Monotonicity of the route cost function is enough to guarantee that the given function V is in fact a Lyapunov function and hence that the system converges to equilibrium. In the dynamic assignment model, the route cost function is not a monotone function of route flow, as was shown previously by the author (2001). Therefore, convergence does not immediately follow, as it does in the steady-state case. This paper essentially shows that the dynamic counterpart of the steady-state Lyapunov function is in fact not a Lyapunov function. This does not at all imply non-convergence of the dynamical system simulating a swap to cheaper routes, but it does raise the question of convergence. Obviously, if another function could be found that satisfies the criteria of being a Lyapunov function this would be sufficient for convergence.