To appear in Proceedings of the 14th IEEE ITS Conference (ITSC 2011), Washington DC, USA, October 2011.
Dynamic Lane Reversal in Traffic Management Matthew Hausknecht, Tsz-Chiu Au, Peter Stone School of Civil and Environmental Engineering {mhauskn,chiu,pstone} {davidfajardo2,s.travis.waller} Abstract— Contraflow lane reversal—the reversal of lanes in order to temporarily increase the capacity of congested roads—can effectively mitigate traffic congestion during rush hourand emergency evacuation. However, contraflow lane reversaldeployed in several cities are designed for specific trafficpatterns at specific hours, and do not adapt to fluctuationsin actual traffic. Motivated by recent advances in autonomous An illustration of contraflow lane reversal (cars are driving on vehicle technology, we propose a framework for dynamic lane the right side of the road). The total capacity of the road is increased by reversal in which the lane directionality is updated quickly and approximately 50% by reversing the directionality of a middle lane.
automatically in response to instantaneous traffic conditionsrecorded by traffic sensors. We analyze the conditions under systems, more aggressive contraflow lane reversal strategies which dynamic lane reversal is effective and propose an integer can be implemented to improve traffic flow of a city without linear programming formulation and a bi-level programming increasing the amount of land dedicated to transportation.
formulation to compute the optimal lane reversal configuration An important component of implementing dynamic lane that maximizes the traffic flow. In our experiments, activecontraflow increases network efficiency by 72%.
reversal is fully understanding the systemwide impact ofincreasing capacity on an individual link. We define the objective of contraflow as follows: given a road network,a specification of vehicles’ locations and destinations, and Traffic congestion is a major issue in today’s transportation a method for determining network efficiency (such as an systems. Contraflow lane reversal, the reversal of traffic flow objective function), assign a direction of flow to each lane along a lane to temporarily increase the capacity of congested such that network efficiency is maximized. To study the roads at the expense of under-utilized ones, is a method to network effects of dynamically repurposing lanes, we cast the increase traffic flow without adding additional roads or lanes.
problem as a maximum multi-commodity flow problem— On the left of Fig. 1, the top lanes are being more heavily a version of the maximum flow problem in graph theory utilized than the bottom ones. On the right, by temporarily with multiple commodities (or goods) flowing through the converting a lane to flow in the opposite direction, the network. Then we propose an integer programming formula- instantaneous capacity in the left-to-right direction of the tion and a bi-level programming formulation to compute the road is increased by 50%. Contraflow lane reversal has been maximum flow in the network. We evaluate our approaches used routinely in several cities in order to alleviate traffic in grid-like transportation networks representative of many during rush hours as well as to reroute traffic around certain downtown metropolitan areas where it will have the most areas such as construction sites or stadiums.
Today, contraflow lane reversal is used at a macro time The rest of the paper is organized as follows. In Section II, scale at rush hour or for quick evacuations from an area. In we discuss the hardware needed for implementing a dynamic both cases however, the change in flow must be carefully lane reversal scheme. In Section III and IV, we analyze under planned before the event, with little or no room for dynamic what conditions dynamic lane reversal will be useful for an changes. Today’s hardware for traffic monitoring is good individual road and intersection. In Section V and VI, we enough to gather real-time traffic data. With the help of introduce both the macroscopic ILP traffic model as well modern computerized traffic control systems, it is possible as the bi-level formulation, and investigate the performance to quickly and dynamically open and close lanes or entire gains imparted by dynamically reconfiguring lanes.
roads, or even change the directionality of lanes based onreal-time usage statistics, such that effective capacity of a road can be dynamically changed based on the demand.
A reversible lane (or contraflow lane) is a lane in which Rapid changes of lane directions, however, may confuse traffic may travel in either direction. The common hard- human drivers. To fully utilize the potential of dynamic lane ware for creating reversible lanes is overhead traffic lights reversal, we will need to rely on the upcoming availability (Fig. 2(a)). In many cities, barrier transfer machines, also of computer-aided driving systems and fully autonomous known as zipper machine, are used to relocate the moveable vehicles that will help vehicles to adjust to the rapid changes barriers such that the road in one direction can be dynami- of lane directions. With the help of computerized driving cally widened at the expense of the other (Fig. 2(b)).
Hardware for controlling contraflow lane.
By definition, the throughput of the road increases after the The basic requirements to support dynamic lane reversal are that the reversal has to be done quickly and safely, andthat the drivers must be notified about the change immedi- λ(L1,2) + λ(L2,1) < λ (L1,2) + λ (L2,1) ately. While it’s conceivable to devise a system that satisfies where λ (L1,2) = min(I1),c(L1,2) − c(l)}, and λ (L2,1) = these requirements using the hardware in Fig. 2, there is min(I2), c(L2,1) + c(l)}.
likely to be significant cost and risk of driver confusion.
In general, lane reversal is beneficial only when one of the However once most cars are controlled by computer, these directions is oversaturated while the other is undersaturated, costs and risks may be significantly reduced by real time up- as shown in Fig. 1. Formally, we have the following theorem: dates of lane direction over wireless network communication, Theorem 1: The throughput of the road R increases after as computerized driving systems (i.e., autonomous vehicles) the reversal of a lane l ∈ La,b if and only if La,b is un- can react to the changes of lane directions much quicker than dersaturated by δa while Lb,a is oversaturated by δb, where human drivers. For example, Dresner and Stone proposed an max{c(l) − δa, 0.0} < δb.
intersection control mechanism called Autonomous Intersec- Proof Sketch. Due to space limitations, we only consider tion Management (AIM) that uses a wireless communication the case in which c(l) > δa and δb < c(l). The reversal of l protocol to enable fine-grained interleaving of vehicle routes reduces the effective traffic rate of Lb,a by x = c(l) − δa > 0 through an intersection [1]. With some modifications to the while the effective traffic rate of La,b increases by δb. Thus AIM protocol, autonomous vehicles can be informed about the throughput of the road increases if and only if δb > x = the current lane directions as well.
max{c(l) − δa, 0.0}.
We begin by considering, from a theoretical perspective, Analyzing the change of the intersection throughput is the effects of lane reversal on a single road. Consider a road necessary because intersections may potentially be the bot- between intersections I1 and I2. Let R be the road between tlenecks of the traffic flow, preventing the adjacent roads I1 and I2, L1,2 be the set of lanes from I1 to I2, and L2,1 from achieving their maximum throughput as predicted by be the set of lanes from I2 to I1. As an example, in Fig. 3, Theorem 1. Estimating the effects on intersection throughput L1,2 = {l1,l2} and L2,1 = {l3,l4}. The capacity of a lane l, theoretically, however, can be a challenging task, especially denoted by c(l), is the maximum rate at which vehicles enter when vehicles from different roads can enter the intersection the lane and is measured by the number of vehicles per hour.
at the same time. Therefore, we use empirical methods to see We assume the capacity of a set L of lanes, denoted by c(L), whether intersections can handle the increase of the incoming is the sum of the capacities of all lanes (c(L) = ∑l∈L c(l)).
traffic when the directions of adjacent lanes reverse.
For simplicity, we ignore the effect of lane changing which We experiment with the intersection in Fig. 4, which has potentially reduces the capacity of L.
six lanes on each incident road. Initially, 3 lanes are incoming Assume both I1 and I2 are sources at which vehicles are lanes and 3 lanes are outgoing lanes. We set the target traffic “generated” to travel along R at the target traffic rates β (I1) rate of the eastbound road be 5500 vehicles per hour, the and β (I2) respectively. But the effective traffic rates λ (L1,2) target traffic rate of westbound road be 1100 vehicles per and λ (L2,1) at which vehicles actually enter the road are lim- hour, and the traget traffic rates of both northbound and ited by the capacity of the lanes. More precisely, λ (L1,2) = southbound roads are 1650 vehicles per hour. Thus, the traffic min(I1), c(L1,2)} and λ (L2,1) = min(I2),c(L2,1)}. If on the eastbound road is several times higher than other λ(L1,2) = c(λ(L1,2)), we say L1,2 is saturated. If β(I1) > roads, causing traffic congestion on the eastbound road. We c(L1,2), L1,2 is oversaturated by an amount of β (I1)−c(L1,2).
check whether reversing the direction of two lanes on the Clearly, if L1,2 is oversaturated, L1,2 is saturated. L1,2 is westbound road can help to increase the throughput of the undersaturated by an amount of c(L1,2) − β (I1) if β (I1) < eastbound road as well as the intersection throughput (the c(L1,2). Clearly, if L1,2 is undersaturated, L1,2 is not saturated.
number of vehicles entering the intersection per hour). The The saturation of L2,1 is defined in the same manner.
new lane configuration is shown on the right side in Fig. 4.
The throughput of the road R is the sum of the effective We repeated the experiment 30 times and in each run traffic rates of the lanes (i.e., λ (L1,2) + λ (L2,1)). Now con- we measured 1) the total number of vehicles entering the sider what happens if the direction of l ∈ L1,2 is reversed.
intersection during the 1-hour period, and 2) the number of The multi-commodity flow problem is a generalization of the well-known max flow problem in which multiplecommodities or goods flow through the network, each withdifferent source and sink nodes. Modeling a road networkat the macroscopic level allows us to map the well-studiedproblem of multi-commodity flow directly onto our problem Fig. 4. The reversal of two lanes on the westbound road of an intersection.
of dynamically reconfiguring lanes. In order to solve this problem we utilize the the mathematical machinery of linear A Linear Program contains a linear function to be max- imized over a set of variables, subject to constraints. Innormal linear programs these variables are allowed to as- vehicles entering the intersection from each road during a 1- sume fractional values, but since all of our flow demands hour period. The average of the number of vehicles and the are required to be integer-valued, we must approach this 95% confidence intervals are shown in Table I. The results problem as an Integer Linear Program. Unfortunately, the show that the throughput of the intersection increased by 6%, multicommodity flow problem has long been known to be and this is mainly due to the increase of incoming traffic NP-complete when dealing with integer flows, even for only from the eastbound road whose throughput is increased by 13%. Note that both increases are statistically significant.
We define the following Integer Linear Program: Given a After lane reversal, the eastbound road’s traffic rate is much graph G = {V, E}, each edge (u, v) has some integer capacity closer to the target traffic rate, and this means that the c(u, v) representing the total number of lanes present on intersection successfully handled the increase of the traffic that road. There are k distinct commodities (traffic flows) coming from the eastbound road. The lane reversal has only K1, ., Kk where each commodity Ki = (si,ti, di) has an as- minor detrimental effects on other roads, because they are sociated source si, destination ti, and demand di. Flow of undersaturated and the lane reversal does not reduce the commodity i over edge (u,v) is denoted fi(u, v).
capacity of these roads below their target traffic rate.
Our objective is to find an assignment of flows which satisfy the following three constraints: The capacity con- straint, shown in Equation 2, specifies that the total amountof flow (in both directions) over a given (u, v) edge must While the ability to improve throughput on individual not exceed the capacity of that edge (note c(u, v) = c(v, u) in roads and at individual intersections are important proofs the undirected case). The conservation constraint, Equation of concept, the true question is whether (and how much) 3, ensures that for all non sink/source vertices the amount of dynamic lane reversal can help on a full road network. To inflow of a given commodity equals the amount of outflow.
address this question we model a road network as a graph Finally Equation 4 specifies that the flow of each commodity consisting of vertices and edges. Each node of the graph must meet or exceed the demand for that commodity.
represents an intersection and each edge represents a road between intersections. Additionally, each u, v edge has an (u,v)∈E ∑(fi(u,v)+ fi(v,u)) ≤ c(u,v) associated capacity c(u, v) which constrains the maximum amount of traffic that road can handle (in this section we ∀i∈1.k,v∈V−{s,t}[ ∑ fi(u,v) = ∑ fi(v,w)] model intersections as having infinite capacity).
Traffic in the network is modeled in terms of aggregate ∀i∈1.k fi(si,w) = ∑ fi(w,ti) ≥ di demand. Specifically we consider a finite number of flows, where each flow has an associated source vertex, destination The goal of the ILP solver is to find an assignment of vertex, and integer-valued demand. For example, a parking directionality to each lane which maximizes the objective garage at a mall could be a source, and a bridge at the function (Equation 5) subject to the constraints specified edge of the network could be a destination. The demand above. This is done by assigning integer values to individual represents the instantaneous number of vehicles that want to flows. We choose to use the maximum multi-commodity travel between these two points. Vehicles may take any path objective function in which the objective is to reconfigure between the source and destination so long as they do not the network to maximize the sum of all commodity flows: violate capacity constraints of roads. We seek to determine 1) whether or not the lanes of a given road network can be maximize ∑ ∑ fi(si,w) dynamically reconfigured in order to accommodate a given set of traffic flows and 2) what is the maximum demand a While multiple possible objective functions could meet our criterion of finding a lane configuration capable of handling a given set of flows, we choose the maximum multicommodityobjective because it forces the ILP solver not only to satisfy the flow demands, but also to find the absolute maximum amount of traffic a network can handle. In contrast, thefollowing section explores an alternative, least-cost objectivefunction.
C. Bi-Level Programming Formulation The multicommodity flow formulation of our problem is convenient in that its solution—the maximum flow—is unique and independent of vehicles’ behavior. However it ignores the fact that drivers are self-interested—they are A example generated graph with two incident S, T flows. Thick concerned about their own travel times and have no incentive lines represents highways, medium lines represent arterial roads, and thin to cooperate to achieve the maximum flow of the network.
Therefore, we consider an alternative formulation for the adaptive capacity problem using a bi-level approach, where ROAD TYPES COMPOSING THE RANDOMLY GENERATED NETWORKS.
the objective is to set link capacities such that, as flows are determined by User Equilibrium behavior, the total system travel time, i.e. the sum of the travel times of all users, is minimized. By modeling the route choice user behavior as a travel cost minimization, we can more accuratelycharacterize the behavior of users in a traffic context. The allocated to a link (i, j) that is taken from the reverse link mathematical formulation is shown in Equations 6–8. The ( j,i). The fitness function used is the total system travel upper level problem includes the allocation of capacity x to time in the underlying UE problem given a decision vector each of the links, while the lower level problem is the classic User Equilibrium model presented by Wardrop [3]: min f (x) s.t. − ci j < xi j < c ji In this section we use the ILP solver to empirically compare different traffic management systems – those which can reverse lane directions quickly and those which can reverse slowly or not at all. We hypothesize that traffic ∑vodji bod(i) ∀i,od.vij ≥ 0 ∀i, j (8) managers which have the ability to quickly reverse the flow of traffic along lanes will achieve higher throughputs than where c is the capacity vector, vod is the flow vector for each OD, x is the dynamic lane capacity allocation vector, t fis the free flow speed vector, and α and β are parameters, and bod(i) is equal to the node supply/demand for each OD pair We automatically generate graphs qualitatively similar to od. As bi-level problems such as this are difficult to solve downtown regions of many cities. Each graph takes the form exactly, we present a Genetic Algorithms based solution of a connected planar grid. To determine the capacities for each road, we randomly select from one of the three roadtypes shown in Table II with the associated probabilities.
D. Genetic Algorithm Solution Method Flows are generated by selecting a random source, sink Genetic algorithms (GAs) is a global search heuristic that vertex pair from the graph. An example road network is uses techniques inspired by evolutionary biology ([4], [5]).
GAs are based on the assumption that the best solution isfound in regions of solution space having a high proportion of good solutions. GAs explore the solution domain to To measure the performance difference between traffic identify the promising region and then search the promising control systems with the ability to quickly reverse lanes (such regions more intensely. GAs start with a population of ran- as the AIM protocol) and those that can only slowly reverse domly regenerated individuals that evolves with generations lanes (such as zipper machines), we evaluate each traffic based on the principle of survival of the fittest. Unlike management system for 10 hours. Each hour a new set of classical methods, GAs work with a population of points.
random flows is spawned, and it is the job of the traffic Therefore, the chances of getting trapped at local optima manager to accommodate this traffic as well as possible by are reduced. Moreover, many variations of GAs are suitable reversing lane directionality. However not all traffic managers for handling complex problems involving discontinuities, will be able to reverse lanes every hour. Traffic management disjoint feasible spaces, and noisy function evaluation [6], systems differ in their reconfiguration period – the amount [7], [8]. In our formulation, a gene represents the capacity of time that must elapse between lane reconfigurations. For example, a system with reconfiguration period of 2 will reconfigure lane directions every other hour while a system with reconfiguration period of 5 will only reconfigure everyfifth hour. If a traffic manager is unable to reconfigure lanes for a given hour, the network throughput is computed by finding the maximum multicommodity flow over the cur- rent lane configuration (e.g. directed multicommodity flow) without allowing the directions of any lanes to change. On the other hand, if a traffic manager is able to reconfigure lanes for a given hour, we compute throughput by finding the maximum multicommodity flow over the network consistingentirely of undirected edges. This gives managers with lowreconfiguration period the ability to more fully adapt to An example run proceeds as follows: we first randomly Total network throughput (vehicles) as a function of DLR period.
generate a road network using the procedure described above.
A reconfiguration period of 1 means that the network was reconfiguredeach timestep; 2 means every other timestep; ∞ means the network was Initially this network is configured in a balanced manner never reconfigured (always in balanced, directed configuration). Error bars in which the capacity of each road is divided as evenly as denote 95% confidence intervals. Period 1,2,3 are statistically significant possible between lanes flowing in one direction and those with respect to period 0 as well as each other.
in the opposite. Next, random flows are generated and the technology which can adapt quickly to changing traffic flows ILP solver is used to compute the network throughput for that hour either on the current directed configuration (if themanager cannot reconfigure this hour) or on the undirected configuration (if the manager can reconfigure this hour). Any This section shows preliminary results of the implemen- changes made to the directionality of lanes carry over into tation of the bi-level formulation presented in Section V-C.
the next hour when new flows are generated.
The objective of these numerical results is to show that the In our experiments we evaluated networks of size 100 algorithm is implementable for modest sized problems.
(10x10) for 10 hours. Each hour contained a set of 4 We solve a problem on a 10x10 grid network, where the randomly generated flows. Demands for these flows were number of lanes varies for each link, the capacity is 1800 set to 0 to ensure that the ILP solver would both reach a vehicles/hour/lane, α = 0.1, β = 4, and t f is 25 mph, 35 mph valid solution and maximize the achievable throughput.1 The and 55 mph depending on the number of available lanes.
throughput of each traffic manager was evaluated at each For the GA, the size of the population and the number of timestep and the total throughput for a traffic manager with a generations were both set to 30, the mutation probability was given reconfiguration period was the sum of the throughputs set to 0.002, and the cross-over probability was set to 0.75.
it achieved over all 10 hours. We evaluated the performance Based on these parameters, the model was solved for each of each traffic manager over 34 different networks (34 trials) of 10 time periods. In order to compare the behavior of the GA based solution method with the ILP solver, we examine Figure 6 shows the total network throughput achieved by the resulting lane configurations after each time period and traffic managers with different reconfiguration periods. The calculate the correlation coefficient of the solution vectors results show a significant increase in performance of traffic for the ILP and GA. The correlations in lane configuration managers who have some form of lane reconfiguration in for each of the 10 hours are shown in Table III. As shown by comparison to no lane reconfiguration. This is not surpris- the resulting correlation coefficients, the two solution vectors ing since the benefits of contraflow are well established.
are quite different, which conforms to the idea that increased However, we seek to address the question of how much accountability in the driver route choice response process performance increase is bestowed by frequent rather than will lead to significantly different solution strategies.
infrequent lane reconfiguration. Infrequent reconfiguration While the bi-level and ILP solutions show significant (reconfiguration period = 3,4,5) shows only modest im- differences, we expect that both solutions are valid under provements over the static configuration, approximately 11% the different assumptions each model makes. Because of the throughput gain. However, decreasing our reconfiguration pe- highly redundant grid based network topology, we expect that riod to 2, we see a 32% performance gain over the static case.
(source, destination) demands may be satisfied by multiple Finally, the fully dynamic reconfiguration period 1 traffic possible paths. We hypothesize that the discrepancy in the manager provides a 72% increase in throughput compared solutions results from different paths being utilized for the to the static network. This trend suggests that the large same set of demands. However, while both models are find- gains in traffic efficiency are achievable with reconfiguration ing valid solutions given their constraints, we should expectthe bi-level formulation’s solution to be more realistic in an 1Incorporating non-zero demands is straightforward to implement but could have resulted in the ILP solver being unable to find valid solutions actual traffic network because the bi-level model incorporates many aspects of real traffic such as congestion, equilibrium agenda includes designing car control and intersection con- CORRELATION COEFFICIENTS OF ILP AND GA SOLUTION VECTORS trol policies for dynamic lane reversal.
Acknowledgments. This work has taken place in part in theLearning Agents Research Group (LARG) at UT Austin. LARG and speed limits. On the other hand, the bi-level model can research is supported in part by NSF (IIS-0917122), ONR (N00014- guarantee only approximate solutions while the ILP solver 09-1-0658), and the FHWA (DTFH61-07-H-00030).
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