Title: Increasing the stability and robustness of simulation-based network assignment models for largescale applications Author: Michael Mahut, INRO Consultants Inc. Larger-scale dynamic network models based on traffic flow principles are becoming increasingly popular, from wide-area micro-simulation to sub-regional or even regional dynamic traffic assignment (DTA), which may also be a coupled with an activity-based demand model in an integrated advanced travel demand model framework. Although the specifics of the traffic flow models may vary, from highly detailed lane-level micro-simulation to hydrodynamic or other mesoscopic models, the promise of these approaches lies largely in their ability to provide greater realism over traditional network models for investigating complex, time-dependent mechanisms, such as congestion-based HOT lane tolls, dynamic parking pricing and departure-time choice modeling. A fundamental issue that is commonly encountered, particularly in larger-scale applications, is the response of the network model to a scenario in which the travel demand is not sufficiently commensurate with the network supply (flow capacities). A key feature shared by the types of network models mentioned above is that they strictly respect flow capacities, i.e., of links, turning movements. These models also respect basic properties of traffic flow: traffic congestion is represented as physical queues which spill back upstream against the direction of traffic, and as they do they reduce the capacities of upstream links and turning movements. When this occurs, the reduction in capacity of the upstream elements leads to even more queueing, and thus even more spillback, which results in even more queueing, and so on In this sense, the actual or effective capacities (throughputs) are not exogenously specified capacities, but are in fact a result of the demand itself, or more specifically, the degree to which the demand exceeds the network supply. In cases where the demand and supply are not well matched, these cascading queues can lead to gridlock, and can often result in severe congestion over large swaths of the network. These extremely congested conditions tend to render the model results unusable. The question is not primarily about the realism of the model results, but rather about their emergent properties, which demonstrate highly nonlinear behavior, making the model unstable or even chaotic. This has several negative consequences. First of all, the network assignment may take a very long time to converge, or may never converge at all. Secondly, even if it does, the results may not be reliable: due to the nonlinear behaviour of the model, small changes to model inputs can lead to large changes in model outputs, and thus a model in this state may be overly sensitive to errors in the model inputs. Any effort to remedy this situation must address the basic question of the trade-off between the mathematical properties of a model and its correctness or realism. If the primary objective is the practical usability of the model, the unavoidable conclusion is that as a model becomes sufficiently unstable, the results become unusable, even if the model is based on realistic assumptions. For example, if the network supply is artificially low due to network coding errors, the excessive congestion makes it virtually impossible to identify the critical bottlenecks that initiated the traffic breakdown. If the demand is unrealistically high, such as might be the case in the context of a demand model within
which the network assignment runs iteratively as a sub-model, the instability of the network assignment may prevent the demand model from converging. Although matrix adjustment methods can be useful for addressing certain situations where demand exceeds supply, they are not applicable or relevant in all contexts. Due to the inherent trade-off between stability and realism for these types of models, the only way to address the instability is to allow the models to be less realistic somehow in certain situations. However, this must be done in a well structured way that (1) aims to maintain the overall value of the model outputs (maximize the realism that can be achieved), and (2) provides a meaningful quantitative measure of the degree to which the underlying traffic flow properties have been relaxed or stretched in order to generate a more stable solution. This paper proposes such a mechanism, describes the associated measure, and provides numerical results obtained with a detailed traffic simulation model on a relatively large real-world network. Some attempts have been made in the past, and which are in current use today, to address this issue via built-in rules, such as removing vehicles entirely from the simulation or allowing vehicles to change paths simply in order to reduce the congestion building up behind them (in some cases this may result in the vehicle being unable to reach its intended destination). These ad-hoc approaches are unsatisfying because the outputs of the model are no longer based on the entirety of the inputs (the model violates conservation of flow) but also because the rules are arbitrary, are not parameterized, and tend to be model specific rather than based on the underlying theory that is common across all traffic flow models. As the potential applications of simulation-based network models continues to grow, along with the accessibility of the computational power that makes them feasible to use, there is a need to address fundamental methodological issues such as this in a more satisfying way. The approach taken here to address this issue is inspired from a well-known idea in optimization which involves formulating the dual problem of a given problem in which constraints of the original problem are relaxed (the well known gap measure used in network models is in fact the duality gap which is the difference between the primal and dual solutions). The specifics of the approach proposed here are motivated by the question of which constraint(s) in a traffic flow model can be relaxed without losing the overall realism which is essential to the value of the model. The approach taken here is based on several observations about the behaviour of capacity-constrained traffic models (and real traffic, for that matter). First of all, the property of strictly respecting flow capacities is essential as it is key to the realism and value offered by this type of model. Secondly, the nonlinear behaviour of these models and resulting instabilities are tied to the nonlinear growth of queues, which is highly dependent on the degree to which the demand exceeds the capacity at key bottlenecks. Thirdly, the congestion spill-back effect, which reduces upstream throughput and thus drives the nonlinear/cascading queue process, is less pronounced on multi-lane facilities due to the potential for vehicles to overtake one another (e.g. when congestion is mainly restricted to some of the lanes of a roadway but not all).
The mitigating effect of overtaking, which can be referred to equivalently as non-link-fifo (first-in-firstout) behaviour, merits a more detailed description. As a queue grows beyond the immediate link upstream of the bottleneck, it will, depending on the specifics of the local network and demand patterns, incur delays on drivers whose paths do not in fact take them through the bottleneck itself. These drivers thus do not influence the demand/capacity ratio of the bottleneck, their presence is unrelated to the initial growth of the queue, but they will be delayed by the queue and contribute to (i.e. accelerate) the rate of growth of the queue. The more of these drivers (vehicles) that are trapped in a queue, the faster it grows and the more it traps (delays) other vehicles, both destined and not destined for the bottleneck in question. The delays incurred by these drivers, which can be thought of as indirect delays with respect to the bottleneck, are essentially what drives the extremely rapid type of area wide queue formation that is characteristic of unstable micro- and meso-scopic network models. The proposed mechanism for restraining queue formation thus focuses on reducing only the indirect delays, and this is done by partially relaxing the link-fifo constraint as it applies to these vehicles. In concrete terms, relaxing FIFO implies that overtaking is permitted in a situation where it is not physically possible to do so. In order to only reduce (rather than eliminate) indirect delays, and more generally to prevent the mechanism from overly impacting the model results, the relaxation mechanism is triggered though an exogenous parameter which defines the amount of delay that can be passed upstream from one vehicle to the next, if the two vehicles are exiting the link by different turning movements. Thus, the mechanism enforces FIFO by turning movement ( turn FIFO ) while relaxing link FIFO. The maximum amount of delay that is allowed to be transferred in this way is referred to as the delay threshold. In more general terms, the delay threshold parameter controls how much overtaking can take place as a function of the amount of delay that is propagating upstream from the vehicle at the stop line. The mechanism was designed to be relatively subtle in order to avoid overly relaxing and thus modifying the results. In particular, in terms of the total amount of indirect delay (vehicle-seconds) in a queue that is reduced as a function of the initial delay propagating back from the stop line, the mechanism can be characterized as sub-linear in general, and linear in the limiting case where all of the propagated delay is indirect delay. For example, considering a queue of 10 vehicles, all of which are leaving the link by the left turn, behind the first vehicle at the stop line, which is leaving by the right turn, and a delay threshold of 1 minute: the total delay in queue is reduced by zero if the first vehicle experiences one minute of delay only (or less), 5% if the delay is 2 minutes, 10% for 3 minutes,..., 45% for 10 minutes, etc, meaning that there is more intervention with increasing delay, but the intervention strictly follows a (sub) linear rule in order to avoid undesirable nonlinear effects. The proposed link FIFO relaxation (LFR) mechanism was implemented in a simulation-based dynamic traffic assignment model which simulates the movement of individual vehicles on discrete lanes of the roadway using vehicle-interaction models such as car-following and gap-acceptance (Dynameq software developed by INRO Consultants). It should be noted that link FIFO is not respected in an absolute way by real traffic, in particular on multi-lane facilities, and this aspect of real traffic can be represented quite realistically using a lane-based traffic model. The LFR mechanism implemented here allows a model of this kind to further decrease link FIFO in heavily congested conditions.
A quantitative measure of the impact of the LFR mechanism was implemented and is referred to as the virtual queue. The virtual queue refers to the number of vehicles that have been allowed to execute the overtaking action but have not yet left the link as they are still delayed by preceding vehicles destined for the same exiting turn. The virtual queues were reported for each lane and turning movement of each roadway link. The virtual queue has a useful intuitive interpretation as it is a direct measure of the number of vehicles by which the queue has been shortened, relative to the case where no LFR is permitted. A brief overview of numerical results is provided here focusing on a single test network. The test case is partially hypothetical as it was important to test the method under severe conditions. The test program also investigated the relative changes in stability with different values of the delay threshold, whereas the results reported here will focus only on one test, with a delay threshold of 30 seconds, referred to as LFR-30. The test case was intentionally set up in such a way that the standard model (no LFR) could not generate an acceptably converged solution in a reasonable number of iterations. This situation is easy to create for any model that strictly respects link and turn capacities (and the resulting spill-back effects) by simply increasing the demand by a sufficient amount. A comparison of the relative gaps (per departure time interval) for the first 50 iterations of the two model runs is presented in figure 1. The LFR-30 model, shown the left, was easily able to generate a stable solution within the first 50 iterations, which is a typical number for a scenario with a well balanced demand and supply. In fact, though not shown here, even the LFR-60 model was able to converge reasonably well within 50 iterations. This was found to be a very positive result as it only required a relatively gentle intervention within the normal functioning of the traffic model. Figure 2 shows link flows scaled to bar widths and colored by link speeds, and virtual queues summed over turning movements at each node, scaled to node circle diameter and colored by delay, during the most congested 30-minute period. The LFR-30 model (left side) clearly shows significantly higher traffic volumes than the standard model, which can be seen very clearly along a major artery in the middle of the network heading into the CBD. The virtual queues, even when summed at each node, were not nearly as prevalent as might have been expected. The orange circle that is clearly visible on the left side represents 70 vehicles. In all, during the worst 30 minutes of congestion, there were only 57 nodes in the network with total virtual queues (summed over all turning movements) greater than 10 vehicles. The LFR-60 model had only 27 nodes with total virtual queue over 10 during the same half hour, with a maximum value of 32. In conclusion, the proposed link FIFO relaxation (LFR) method provides a very effective way to address the issue of instability due to severe congestion in simulation-based dynamic assignment models. The idea behind the mechanism is to reduce the nonlinear growth of queues in severely congested conditions, while respecting bottleneck capacities and turn FIFO discipline, by allowing a certain relaxation of link FIFO. In this way, the mechanism reduces indirect delay, which is defined as delay that is propagated upstream between two vehicles which are exiting the current link by different turning movements. The mechanism reduces total indirect delay in a queue in a sub-linear fashion relative to the initial (stop line) delay being propagated. This adaptive property of the mechanism is key to ensuring
that the mechanism does not over-react and that in general it has little or no effect in traffic conditions that are only moderately congested. The LFR virtual queue length outputs indicated that the mechanism was able to drastically improve the convergence properties of extremely congested test scenarios with only a modest relaxation of the physical constraints. The LFR mechanism is motivated by the need to trade-off model stability against realism (fidelity) for simulation-based network models that strictly respect flow capacities and traffic flow properties such as congestion spill-back, when the inputs to the model result in unusable results typified by gridlock and non-convergence of the network assignment. For cases where these conditions are due to network coding, the more fluid traffic flows combined with the virtual queue output yield far more usable outputs that provide insight into the underlying demand/supply discrepancies rather than simply trying to mimic the resulting traffic conditions as they might occur in reality. For situations where the modeling context necessarily requires the network model to handle severe congestion, a mechanism of this kind can play a critical role in helping the model achieve convergence, without necessarily being invoked in the converged solution. Examples of this include the early iterations of an assignment run, or when an assignment is nested within a demand model and is expected to provide meaningful O-D impedances in response to non-converged demands that are necessarily unrealistic. FIGURES Figure 1 Relative gaps by departure time interval: LFR-30 (left), Standard model (right)
Figure 2 Link flows scaled to bar widths colored by speed, and virtual queue lengths scaled to node diameter and colored by delay: LFR-30 (left), Standard model (right) BIBLIOGRAPHY Dynamic Traffic Assignment, a Primer. Transportation Research Circular E-C153. Transportation Research Board, June, 2011. http://onlinepubs.trb.org/onlinepubs/circulars/ec153.pdf Guidebook on the Utilization of Dynamic Traffic Assignment in Modeling, Traffic Analysis Toolbox vol. 14. Federal Highway Administration, November 2012. http://ops.fhwa.dot.gov/trafficanalysistools/