## Extending Horizontal Queueing to Whole Routes of Moving Bottlenecks

### Authors

Nicholas B Taylor, TRL, UK

### Description

Paper develops mathematical model for efficient (progressive non-iterative) estimation of queues due to moving bottlenecks on varied routes, under various operational conditions, particularly where queuing spills back over several road sections

### Abstract

This paper is concerned with efficient estimation of queues caused by moving bottlenecks or obstructions along whole routes of varied length and composition, under various operational and demand conditions, particularly where queuing spills back over several road sections. Arising originally from a need to predict the impact of unusual vehicles, the practical requirement is to build and calibrate a route model section by section, using simple and readily available descriptions and typical traffic patterns, without needing information about the surrounding network, and without iterative calculations. The approach necessarily involves some idealisation and approximation, so internal consistency and robustness are important considerations, as well as accuracy in the most common situations. Estimation of passing capacities (references given), and details like acceleration, deceleration, lane differences, merging, and stop-start ?shock waves? are not considered explicitly. Mathematical analysis with diagrams and examples is given.

Queues caused by moving obstructions arise mainly from excess of demand over remaining capacity, swamping random effects, so a ?deterministic? macroscopic approach is considered adequate, and is efficient enough for embedding in larger simulations. Time-dependence is inherent, and the variable density and speed of moving queues mean that a ?horizontal? model taking account of physical extension is essential. Queues extending over several road sections effectively create interdependence between them, so in principle problem complexity can escalate as the number of sections ?linked? increases. Further complexity arises in the boundaries between different traffic flow states, which propagate upstream or downstream like wave-fronts. The model is most accurate for route segments of consistent road standard, but offers a practical approximation in more general cases. Queuing extending onto the surrounding network is not dealt with explicitly, but is allowed for by including entry flows in the total demand and by ascribing queues and delays to the road sections which cause them rather than those on which they are incurred.

The key to the approach is to anchor the model to feasible, if idealised, physical configurations and measurable, if simplified, values, while strictly observing conservation constraints. Horizontal queues are most conveniently modelled using space-time geometry. On a homogeneous road section under constant traffic conditions, the space-time diagram of a queue has triangular shape, bounded by the obstruction, the tail wave where arriving traffic joins the queue, and the discharge wave which propagates upstream from the point where the obstruction is removed. Queuing persists until the tail and discharge waves meet. Cumulative traffic is preserved, but suffers delay proportional to the extent of queuing and the excess travel time compared to free-flow. This simple model can fail where a queue spills back onto sections upstream. Once an obstruction has left a section where it has caused queuing, the section downstream is exposed to concentrated flow from the queue. For a time this increases the effective demand on the downstream section, compared to normal flow there. Thus, if a queue develops on the downstream section, its tail wave is generally in two parts with different speeds, giving rise to quadrilateral space-time geometry.

Quadrilateral queue segments, each ascribed to the passage of the obstruction along one road section, can be used to build up an extended queuing region affecting many sections, given certain simplifying assumptions. First, normal or ambient traffic volumes are assumed to be within road capacity, and turning proportions are assumed unaffected by queuing. This is not really restricting, since actual section volumes, as distinct from ultimate demands, cannot exceed capacity. Where there is pre-existing congestion and a moving obstacle could still have impact, this could be accommodated by adjusting ambient traffic parameters. Second, interdependence between sections is limited to description of the flow passing between the corresponding adjacent queuing regions, each of which is treated as homogeneous. This avoids long-range dependence, keeping problem complexity proportional to the number of route sections. Arguably, extended queues in this context are most likely to occur on motorways and major roads where road standard is consistent. If specific off-route impacts must be considered, additional section models can be set up to account for them.

The underlying model relies on relationships between traffic flow, speed and density. These satisfy the so-called ?fundamental relationship?, but a specific relationship between two of the variables is also needed. A speed/flow model is adopted for ?coupled? (non-free-flowing) traffic based on minimum distance and time headways, and calibrated to section speed and capacity characteristics. This form of model simplifies the description because it predicts that all discharge waves, and flow-change waves within queues, propagate upstream at the same speed (about 20 km/h). Queues are assumed to discharge at full carriageway capacity. There is some evidence that they discharge at a lower rate, but this could be addressed by separate research without affecting the basis of the model. The interface between section queues can take different forms depending on whether a gap of free-flowing traffic exists between them. The model automatically ensures the consistency of traffic flows and queuing conditions around the intersection.

Practical implementation of the route model embraces several additional mechanisms, including delay to vehicles while passing the obstruction, holding up of opposite direction traffic particularly on single-carriageway sections, modification of traffic volumes by diversion, and breaks in journey allowing queues to pass. These are relatively straightforward to calculate by simple vertical queue modelling or modification of section variables. The core queue model could potentially be extended to networks by including random queue modelling to allow equilibrium queues at junctions to be represented.

The sensitivity of results to the level of demand and turning proportions, and the effect of changes in road geometry, are illustrated by diagrams and calculated examples. Results are given for the whole route of a moving obstruction using motorway and other roads, showing that impact can be highly variable. Due to variations in ambient demand, the effect of journeys on different days and at different times is also highly variable, and not predictable by simple rules. Field measurements show there is no ?typical? result, confirming the need for disaggregate time-dependent modelling of individual cases.

#### Publisher

Association for European Transport