Malachy Carey

Optimal Time-Varying Flows on Congested Networks

This paper develops a well-behaved convex programming model for least-cost flows on a general congested network on which flows vary over time, as for example during peak/off-peak demand cycles. The model differs from static network models and from most work on multiperiod network models because it treats the time taken to traverse each arc as varying with the flow rate on the arc. We develop extensions of the model to handle multiple destinations and multiple commodities, though not all of these extensions yield convex programs. As part of its solution, the model yields a set of nonnegative time-varying optimal flow controls for each arc. We determine and discuss sufficient conditions under which some or all of these optimal flow controls will be zero-valued. These conditions are consistent with computational experience. Finally, we indicate directions for further research.

A MODEL AND STRATEGY FOR TRAIN PATHING WITH CHOICE OF LINES, PLATFORMS, AND ROUTES

Train pathing is concerned with assigning trains and train times for a set of rail links, stations stops, etc., so as to meet a system of constraints on headways, trip times, dwell times, etc. while minimizing delays or costs and meeting travel demands. In a previous paper we presented a model, algorithms, and strategy for pathing trains of different speeds and stopping patterns for a double track rail line dedicated to trains in one direction. Here we extend this to more general more complex rail networks, with choice of lines, station platforms, etc, as is more typical of the high density scheduled passenger railways in Britain and Europe. We apply the model to a small network and find acceptable solution times. Applying addition search strategies from the previous paper should reduce solution times by further orders of magnitude.

Scheduling and platforming trains at busy complex stations

Abstract We consider the problem of train planning or scheduling for large, busy, complex train stations, which are common in Europe and elsewhere, though not in North America. We develop the constraints and objectives for this problem, but these are too computationally complex to solve by standard combinatorial search or integer programming methods. Also, the problem is somewhat political in nature, that is, it does not have a clear objective function because it involves multiple train operators with conflicting interests. We therefore develop scheduling heuristics analogous to those successfully adopted by train planners using “manual” methods. We tested the model and algorithms by applying to a typical large station that exhibits most of the complexities found in practice. The results compare well with those found by traditional methods, and take account of cost and preference trade-offs not handled by those methods. With successive refinements, the algorithm eventually took only a few seconds to run, the time depending on the version of the algorithm and the scheduling problem. The scheduling models and algorithms developed and tested here can be used on their own, or as key components for a more general system for train scheduling for a rail line or network. Train scheduling for a busy station includes ensuring that there are no conflicts between several hundred trains per day going in and out of the station on intersecting paths from multiple in-lines and out-lines to multiple platforms, while ensuring that each train is allowed at least its minimum required headways, dwell time, turnaround time and trip time. This has to be done while minimizing (costs of) deviations from desired times, platforms or lines, allowing for conflicts due to through-platforms, dead-end platforms, multiple sub-platforms, and possible constraints due to infrastructure, safety or business policy.

A CONSTRAINT QUALIFICATION FOR A DYNAMIC TRAFFIC ASSIGNMENT MODEL

This note resolves a hitherto open question as to whether a dynamic traffic assignment model, which was developed and analyzed in earlier issues of this journal, satisfies a “constraint qualification.” It is shown that the model does in fact satisfy a constraint qualification, which establishes the validity of the optimality analysis presented by previous authors.

Ex ante heuristic measures of schedule reliability

Measures of reliability and punctuality of scheduled public transport services are important in planning, management, operating and marketing of these services. Various methods can be used to measure reliability. Analytic methods are usually practical for only very simple structured systems. Simulation methods are very time consuming and require data which may not be available. As a result, the most widely used measures are ad hoc or heuristic. However, the assumptions and properties of these measures, and the relationships between them, are seldom discussed, hence we discuss them here. We consider existing measures, extensions of these, and new measures. For specificity, we use the example of train arrivals and departures at a train station: stations with several hundred trains per day and multiple platforms are common in many countries, for example throughout Europe. Some measures of reliability are based on the observed delays, hence can be used only after the event. However, we here focus on measures which can be used in advance, for example for estimating the reliability of proposed schedules or changes in schedules at the design stage. In this we distinguish between measures which require some information about probabilities of delay and those which do not. We also distinguish between exogenous delays, which are beyond the influence of the scheduler (delays due to problems in engineering or operations), and knock-on delays which are affected by schedule design: both types are of interest for schedule reliability, but the latter are of more interest for measuring schedule robustness.

Extending a train pathing model from one-way to two-way track

A train pathing model and algorithms for a rail network was presented and tested in a previous paper. There, it was assumed that each rail line has two or more tracks and each was dedicated to traffic in one direction as is usual in Europe. Here, we show how to adapt and extend that model and algorithms so as to handle trains on single line two-way track, as is usual in North America. Though introducing two-way track makes the formulation appear more complex, I show that somewhat surprisingly, (a) it does not increase the number of constraints or variables, including 0-1 variables, and (b) the model with two-way track should generally be easier to solve.

Externalities, average and marginal costs, and tolls on congested networks with time-varying flows

For congested networks on which flows vary over time, we derive system marginal costs, user perceived costs and user externality costs, for each arc and path. We also obtain a set of optimal congestion tolls and flow controls which may be used to shift the user determined flows toward a socially preferred pattern. An important way in which our results differ from the usual static analysis is that the social cost externality depends not only on the level of congestion, but also on the rate of increase or decrease of congestion. This is intuitively explicable as follows. Consider users delayed on an arc. Their delays will be further compounded or multiplied if congestion has increased during the time they are delayed. On the other hand, their delays will be reduced if congestion has declined during the time they are delayed. This multiplier effect is such that the resultant dynamic externalities can easily be a few times larger, or smaller, than the externalities derived in the usual static analysis. As a r...

The existence, uniqueness and computation of an arc-based dynamic network user equilibrium formulation

Pad devices disposed between a lift sling and a load and permitting relative sliding motion between the load and the sling without the load and sling being in direct engagement and thereby avoid and/or minimize damage to the sling and to the load.

Testing schedule performance and reliability for train stations

On busy congested rail networks, randomdelays of trains are prevalent, and these delays have knock-on effects which result in a significant or substantial proportion of scheduled services being delayed or rescheduled. Here we develop and experiment with a simulation model to predict the probability distributions of these knock-on delays at stations, when faced with typical patterns of on-the-day exogenous delays. These methods can be used to test and compare the reliability of proposed schedules, or schedule changes, before adopting them. They can also be used to explore how schedule reliability may be affected by proposed changes in operating policies, for example, changes in minimum headways or dwell times, or changes in the infrastructure such as, layout of lines, platforms or signals. This model generates a reliability analysis for each train type, line and platform. We can also use the model to explore some policy issues, and to show how punctuality and reliability are affected by changes in the distributions of exogenous delays.

Optimizing scheduled times, allowing for behavioural response

We consider transport activities for which time has to be allocated or scheduled in advance. When the schedule is implemented, the time actually taken by each activity is subject to random variation, hence can exceed the scheduled time. To reduce such over-runs or lateness, and improve reliability and costs, some extra time is usually allowed for some, or all, activities in the schedule. However, it is well known that if more time is allocated for an activity then the activity often tends to take longer. Because of this behavioural response, some or all of the benefits (in reliability, costs, etc.) of the extra time allowance are lost. To compensate for this, should even more time be allowed for each activity, or should less be allowed, and if so, how much? We consider this question here and in particular we discuss the effect of such behavioural response on expected costs and on the optimal time to allow for an activity. We find that the optimal time to allow depends, in very simple ways, on a behavioural response ratio, and on the ratio of scheduled time costs to lateness costs. The model is applicable to computing optimal times for public transport timetables, for buses, trains or airlines. It is also relevant to choosing how much time to allow for each of a set of operations in production scheduling or service scheduling.