An exit-flow model used in dynamic traffic assignment

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Abstract

We consider the behaviour of a link exit-flow model that has been used to model link flows in dynamic traffic assignment (DTA) on networks. In particular, we investigate how the model behaves when time and space (the link length) discretised and the discretisation is varied. We present numerical examples based on various inflow patterns and exit-flow functions and draw conclusions for applications of the model in discrete space and in discrete or continuous time. If inflows are always less than capacity and the link is homogeneous, with no obstructions at the exit, then if the discretisation is refined to the continuous limit the model goes to the solution of the well-known LWR model. However, we observe, somewhat counter intuitively, that the usual continuous-time model does not give as good an approximation to the LWR solution as does the discrete-time model: for a best approximation, the discretisation of space and time should be synchronised. We also investigate the ‘dip’ in outflows, or ‘jamming’ of outflows, that the model displays if inflows are permitted to exceed a capacity limit (as they sometimes do in published applications of the model) and the exit-flow function has a downward sloping part (which has usually been assumed away in DTA applications). In that case, if the number of spatial segments is increased, or the number of time intervals is reduced, then any dip in outflows occurs sooner and is more pronounced, and leads to earlier jamming. In the continuous limit, the jam occurs at the link entrance, preventing inflows in excess of capacity.

Introduction

We investigate the behaviour of a simple link flow model that has frequently been used in dynamic traffic assignment (DTA) models for road networks. The model is based on assuming that the outflow from each link at time t is a function of only the current number of vehicles on the link at time t, and also assuming with conservation of flow. We refer to this as the MNO model, since it was first used in DTA by Merchant and Nemhauser [1], [2] and has since been further investigated, applied and developed by others, including Ho [3], Carey [4], [5], [6], Carey and Srinivasan [7], Friesz et al. [8], Lasdon [9], Lam and Huang [10], Wie [11] and Wie et al. [12], [13], [14], Yang and Huang [15].

Though the MNO model has normally been applied to whole links, we here investigate dividing the link into segments and apply the MNO model sequentially to these. We investigate how the solution (the outflow profile) is affected by varying the level of discretisation of space (link length) and time, since in practice quite rough discretisations may be used. We illustrate this with numerical examples. The outflow function is also referred to as an exit-flow function or congestion function, and we will use these terms interchangeably.

For convenience, in the literature on the MNO model it is generally assumed that the exit-flow function not have a downward sloping part. In this paper we more realistically assume that it has. However, we also show that if the link (or link segment) is homogeneous, and the inflows do not exceed a certain link capacity, then the inflows and outflows will not move onto the downward sloping part of the exit-flow function. If the exit-flow function is nondecreasing, or only the nondecreasing part is utilised, then if the discretisation (of space and time) is refined to the continuous limit, the MNO solution converges to the solution of the well-known LWR traffic flow model [16], [17].

We also investigate what happens if the inflows exceed the link capacity, as is permitted by the standard MNO model. In that case the outflows can move onto the downward sloping part of the exit-flow function. This causes a fall or dip in outflows even if there is no fall in inflows. If inflows in excess of capacity persist long enough then the outflows can fall to zero, or ‘jam’ and remain jammed. This dipping or jamming of outflows in the MNO model could be removed by adopting the approach developed by Daganzo [18], [19] to obtain finite approximations to the LWR model. This would extend the MNO model by ignoring the downward sloping part of the flow–density curve when determining the flow available to exit from a segment (cell) and using the downward sloping part instead to determine the amount that the next segment (cell) can accept. This means that the amount available for outflow from a segment would never move onto the downward sloping part of the curve, hence never experience the dips or jams described above for the MNO model. However, in the present paper we do not adopt this approach, since we wish to investigate the properties of the MNO model as it has been presented and used in the DTA literature.

We address these issues in 2 The MNO and LWR models for congested links, 3 Some benchmark examples (without spatial or temporal discretisation), 4 Varying the discretisation of time (without discretising the link length), 5 Varying the discretisation of the link length (without discretising time), presenting a series of numerical examples. In Section 3 we consider the MNO model without temporal or spatial discretisation. In Section 4 we discretise time, refining the discretisation until we have effectively continuous time, for a whole undiscretised link. Taking this as the starting point in Section 5, we progressively refine the discretisation of the link length until arriving at a continuous treatment of both space and time. As this yields the same result as the LWR model, it provides a benchmark for the various discretisations. Throughout, we illustrate the behaviour of the MNO model for various exit-flow functions and inflow patterns.

To provide a more challenging test of the MNO model, we assumed patterns of inflow that are somewhat unusual, and are difficult for a continuous model to track accurately. For example, we assumed inflows jump suddenly from zero to a large inflow, or a steep ramp function that persists for a relatively short time. The model will perform at least as well, or much better, if inflows are assumed to rise and fall more smoothly, as is more usual.

As regards terminology, traffic flow is the number of vehicles passing a point per unit time and traffic density is the number of vehicles per unit distance. The following terminology used in this paper is common in the traffic assignment literature and in the operations research and economics literature, but unfortunately this differs from the terminology often used in the traffic flow literature. We refer to the number of vehicles on a link as the link volume, so that the exit-flow function can be referred to as the outflow–volume function. Traffic flow is said to be uncongested when its speed is not affected by the traffic density, in which case the flow–density (or flow–volume) curve in Fig. 1 is a straight line through the origin. This is also referred to as ‘free-flow’ traffic or light traffic. Traffic flow is said to be congested when its speed is affected by density (decreases with density), in which case the flow–density (or flow–volume) curve (Fig. 1) has a decreasing gradient. We refer to traffic flow as restricted when the traffic density is beyond the peak of the flow–density (or flow–volume) curve (Fig. 1), that is, when further increases in density cause the flow rate to fall.

Section snippets

The MNO model

Let vi be the volume of traffic (number of vehicles) on the link at the beginning of period i, let ui and qi be the inflow and outflow rate, respectively (in vehicles/unit time), for the link throughout period i, and Δt be the period length. The discrete-time MNO model for a link can be set out as follows. Treat the link outflow as a function of only the link volume, thusqi=g(vi)and to ensure conservation of traffic, let the volume on the link in the next period i+1 be the current volume plus

Some benchmark examples (without spatial or temporal discretisation)

In later sections we examine and illustrate how the MNO model behaves when we discretise link flows over time and space, and to do that we apply the model using three different q–v functions, namely quadratic, exponential and modified quadratic. Here we consider how the MNO model behaves for these q–v functions without any discretisation of time or of the link length, so that in later sections we can compare with the results obtained when using discretisation.

Varying the discretisation of time (without discretising the link length)

To see how the MNO model , is affected by the choice of time discretisation, we varied the discretisation, assuming various inflow patterns. The resulting outflow patterns are shown in Fig. 2, Fig. 3, Fig. 4, Fig. 5. As benchmarks with which to compare the results, in the figures we also show:

  • (a)

    The outflows that would occur if there was no congestion. In that case the outflows are simply the inflows shifted forward in time by the link trip time τ.

  • (b)

    The outflow profile that is obtained if we use

Varying the discretisation of the link length (without discretising time)

In this section we treat time as essentially continuous, by setting the period length Δt=0.001, and consider how the model behaves if we discretise the link into segments and refine the discretisation by increasing the number of segments.

We have already seen (Proposition 1) that if we take the MNO model , or , and divide the link into N spatial segments then the resulting MNO model converges to the LWR model as N→∞. While this is important, it does not tell us how the model behaves when N is

Concluding remarks

We set out an exit-flow model referred to as the MNO model and explore how this behaves if we discretise space (link length) and time, and vary the fineness of the discretisation. If the link inflow does not exceed capacity and the link is homogeneous with no exit controls, the MNO solution converges to the LWR solution as the discretisation is refined to the continuous limit. Since the latter is probably the most widely accepted traffic flow model it provides a useful benchmark for the MNO

Acknowledgements

This research was supported by a UK Engineering and Physical Science Research Council (EPSRC) grant number GR/R/70101, which is gratefully acknowledged. The authors also thank two anonymous referees.

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