A comparative study of Macroscopic Fundamental Diagrams of arterial road networks governed by adaptive traffic signal systems
aa r X i v : . [ n li n . C G ] D ec A comparative study ofMacroscopic Fundamental Diagrams of arterial roadnetworks governed by adaptive traffic signal systems
Lele Zhang a,b , Timothy M Garoni b, ∗ , Jan de Gier c a ARC Centre of Excellence for Mathematics and Statistics of Complex Systems, Departmentof Mathematics and Statistics, University of Melbourne, Victoria 3010, Australia b School of Mathematical Sciences, Monash University, Clayton, Victoria 3800, Australia c Department of Mathematics and Statistics, The University of Melbourne, Victoria 3010,Australia
Abstract
Using a stochastic cellular automaton model for urban traffic flow, we study andcompare Macroscopic Fundamental Diagrams (MFDs) of arterial road networksgoverned by different types of adaptive traffic signal systems, under variousboundary conditions. In particular, we simulate realistic signal systems thatinclude signal linking and adaptive cycle times, and compare their performanceagainst a highly adaptive system of self-organizing traffic signals which is de-signed to uniformly distribute the network density. We find that for networkswith time-independent boundary conditions, well-defined stationary MFDs areobserved, whose shape depends on the particular signal system used, and alsoon the level of heterogeneity in the system. We find that the spatial hetero-geneity of both density and flow provide important indicators of network per-formance. We also study networks with time-dependent boundary conditions,containing morning and afternoon peaks. In this case, intricate hysteresis loopsare observed in the MFDs which are strongly correlated with the density het-erogeneity. Our results show that the MFD of the self-organizing traffic signalslies above the MFD for the realistic systems, suggesting that by adaptively ho-mogenizing the network density, overall better performance and higher capacitycan be achieved.
Keywords: macroscopic fundamental diagram, traffic signal system,simulation ∗ Corresponding author
Email addresses: [email protected] (Lele Zhang), [email protected] (Timothy M Garoni), [email protected] (Jan de Gier)
Preprint submitted to Elsevier December 12, 2012 . Introduction
A central goal of traffic science is the formulation of appropriate macro-scopic variables characterizing and relating demand and performance of roadinfrastructure. On the level of a single street (or freeway), the fundamentaldiagram (FD), introduced by Greenshields (1935), expresses flow as a functionof density. Fundamental diagrams for a single link are generically unimodal,describing a free-flow regime at low densities and a congested regime at highdensities . It is far from clear, however, to what extent such simple relationsshould extend to more complex systems such as urban road networks. Earlystudies in this direction date back at least to Godfrey (1969). The first con-vincing empirical evidence that congested urban networks can display simplerelationships between network-aggregated demand and performance was pre-sented in Geroliminis and Daganzo (2007, 2008). These works clearly indicatethe existence of a Macroscopic Fundamental Diagram (MFD) in the city of Yoko-hama, relating network-aggregated production and accumulation . Analyticaltheories attempting to explain the existence of MFDs have been developed byDaganzo and Geroliminis (2008) and Helbing (2009), and match the Yokohamadata quite well. The discussion in Daganzo and Geroliminis (2008) analyzes theeffects on MFDs of applying different signal timings, by treating traffic signalsas exogenous capacity constraints. One of the main aims of the current work isto study and compare MFDs in networks governed by different types of adaptive traffic signal systems.Given the existence of MFDs, it is natural to ask under what conditionsshould they be observed? In previous work on MFDs, Daganzo and Geroliminis(2008) postulate sufficient regularity conditions under which MFDs should beexpected to exist, including slow-varying and distributed demand, and ho-mogeneous network infrastructure. Helbing (2009) argues that the details ofMFD curves should be expected to depend not only on the aggregated den-sity, but also on the spatial density distribution. Taking these observationsfurther, Mazloumian et al. (2010) argue that the aggregated flow should in factbe a function of both the aggregated density and also its spatial variation.Geroliminis and Sun (2011b) demonstrate using empirical data that while stricthomogeneity of traffic states is not necessary to observe a well-defined MFD, thespatial distribution of density is indeed a key quantity. In this work we studythe spatial heterogeneity of both density and flow, and demonstrate how theirbehavior can be used as predictors of network performance.In particular, we discuss the relationship between the time evolution ofspatial heterogeneity and hysteresis . Hysteresis has been observed and stud-ied in freeway networks in Geroliminis and Sun (2011a). While hysteresis wasnot observed in Geroliminis and Daganzo (2008), a careful empirical study of Even in this simplest of cases, however, it is typically found experimentally that significantscatter is observed in flow-density relations of congested links; see Kerner (1998). The production and accumulation are surrogates for the flow and density, and are morereadily measured in empirical trials. commuter corridors , i.e. portions ofan arterial network wedged between a strong source (e.g. a residential area)and a strong sink (e.g. the central business district), the input/output rateson the boundary may be significantly higher than in the bulk of the network.Other situations where such behavior might arise include arterial networks inthe presence of perimeter control. We observe from our simulations that forsuch systems both clockwise and anticlockwise hysteresis loops can genericallyappear, and we show that these observations can be explained using a modifiedversion of the model presented in Gayah and Daganzo (2011).Our simulations utilize a stochastic cellular automaton (CA) model, intro-duced by de Gier et al. (2011). This model is mesoscopic, in the sense that al-though individual vehicles are modeled, fine-grained details of individual driverbehavior are deliberately treated in a course-grained, statistical, manner. Whiledetails of vehicular motion through intersections are deliberately ignored, real-istic signal phasing at intersections is included in the model. In fact, the modelwas specifically designed to provide a simple and fast way to study arbitrarytraffic signal systems, on arbitrary networks. Using this CA model we studythe existence and shape of MFDs for three specific traffic signal systems, usingboth time-dependent and time-independent boundary conditions. In particu-lar, we simulate variants of the SCATS traffic signal system, which is cur-rently employed by numerous road authorities worldwide, including in Sydneyand Melbourne. In order to study the effect of increased adaptivity on MFDs,we then compare these results for SCATS with the highly-adaptive (idealized) self-organizing traffic lights (SOTL) system, originally introduced for Manhat-tan lattices by Gershenson (2005), and then generalized to arbitrary networksin de Gier et al. (2011). In particular, the version of SOTL that we study isspecifically designed to minimize the spatial heterogeneity of the density withinthe network.Ji and Geroliminis (2012) have recently addressed the question of how todecompose road networks into subnetworks so that each subnetwork has a well-defined MFD, while Daganzo (2007) and Haddad and Geroliminis (2012) havestudied how to optimize the control of flows between such subnetworks. SOTLcan be viewed as a possible mechanism for adaptively homogenizing the densitydistributions within subnetworks.The remainder of this paper is organized as follows. In Section 2, we de- Sydney Coordinated Adaptive Traffic System
2. Cellular Automata Model
We briefly outline the cellular automata model used in our simulations, whichwe refer to as the
NetNaSch model. For a comprehensive description of themodel see de Gier et al. (2011).Cellular automata (CA) are models which are discrete in time, space andstate variables, whose dynamical rules are local. The NetNaSch model rep-resents a road network by a directed graph, in which the nodes represent in-tersections and the links represent streets. With each link is associated anordered list of lanes, and each lane is a simple one-dimensional stochastic CAobeying a (slight generalization of) the Nagel-Schreckenberg (NaSch) dynamics;see Nagel and Schreckenberg (1992). In addition, vehicles may move betweenneighboring lanes via simple lane-changing rules. Thus, the dynamics alongeach given link is essentially a standard CA freeway model, albeit with inputand output rates that are determined dynamically by the rest of the network.The NetNaSch model intentionally avoids modeling the detailed motion of vehi-cles as they move through intersections; the underlying assumption being thatthe actual time a vehicle physically spends in an intersection is unimportantcompared to the time spent on the inbound link waiting to traverse the inter-section. This course-grained approach allows the model to be easily applied tonetworks of arbitrary topology, using any choice of desired signal phasing.In order to mimic origin-destination behavior, the NetNaSch model demandsthat each vehicle makes a random decision about which link it wants to turninto at the approaching intersection. More precisely, for each node n , we assignto each ordered pair ( l, l ′ ), where l is an inlink and l ′ an outlink of n , theprobability p T ( l → l ′ ) that a vehicle on l wants to turn into l ′ when it reaches n . The turning decision is made when the vehicle first enters l , since its choiceof which link to turn into at the approaching intersection should influence itsdynamics as it travels along l . In particular, it influences the vehicle’s choiceof when to change lanes. In order to guarantee the robustness of the model,however, we do allow for drivers to adaptively change their turning decisionswhen faced with very high levels of congestion. This enables the model to avoidbecoming frozen in pathological states of gridlock caused by drivers adheringstrictly to their turning decisions. Specifically, suppose that a vehicle is queuedat an intersection in a lane that is consistent with its current turning decision,but that the vehicle has waited through more than n G green signals without4eing able to clear the intersection, due to spillback on the link onto whichit wishes to turn. In this instance, the NetNaSch model allows the vehicle toremake its random turning decision. We set n G = 6 in the simulations in thiswork.The NetNaSch model can be used with a variety of boundary conditions. Inthis paper we use open boundary conditions, and so the density in the networkis not controlled directly. Instead, at each time step, vehicles enter and exitthe network stochastically, according to prescribed input/output rates. Wecall in- and output at the boundary of the network exogenous , while internalsources/sinks are called endogenous , for example representing parking garages.In general, both exogenous and endogenous input and output are allowed.A boundary link is a link which has one of its two endpoints within thenetwork, and one external to the network. Boundary links are classified aseither boundary inlinks , if their to-node belongs to the network, or boundary outlinks , if their from-node belongs to the network. A bulk link is a link whoseendpoints are both contained in the chosen network. In the NetNaSch model,each lane λ of each boundary inlink is assigned an input probability α λ : ateach discrete time step a new vehicle is inserted into the first cell of lane λ with probability α λ . Likewise, each lane λ of each boundary outlink is assignedoutput probability β λ , which determines the probability that a vehicle wishingto exit the network from the last cell of lane λ at a given time step actually beallowed to do so.The collections { α λ } , { β λ } therefore specify the exogenous input/output ofthe network, i.e. they describe the level of demand imposed on the network byits environment. Intuitively, one can view α λ as being the density of an externalreservoir of vehicles being fed into boundary lane λ . And likewise, one can view(1 − β λ ) as being the density of an external sink being fed vehicles by boundaryout-lane λ . Indeed, for the 1-lane NaSch freeway model, where the reservoirand sink can be thought of as on- and off-ramps, these interpretations are quiterealistic.As discussed in Section 3, the boundary links are not considered to be partof our network, in the sense that we do not include their densities and flows inour network aggregated values. Instead, the boundary links are simply viewedas buffers allowing a realistic way to couple the bulk network to its external en-vironment. A practical issue that must be decided upon is how long to make theboundary links. There is no unique best answer to this question; a detailed dis-cussion of the pros and cons of different possibilities is presented in de Gier et al.(2011). For the simulations discussed in the present work, we simply set thelength of the boundary links equal to the length of the bulk links. One advan-tage of this approach is that spillback caused by over-saturation on boundaryoutlinks is modelled dynamically, in a realistic way.In addition to these exogenous inputs/outputs, each lane of each bulk link isassigned an input probability γ λ , and an output probability δ λ , which determinethe rate of input and output from internal sources and sinks. Intuitively, onecan view these internal sources and sinks as parking garages, for example. Then γ λ is the probability that at a given time step, a vehicle will leave the parking5arage and enter the network, while δ λ is the probability that a vehicle passingsuch a parking garage will leave the network to enter the parking garage. Forthe simulations discussed in the present work, the internal sources and sinkswere located near the middle of the lane, with the sink occurring before thesource.For simplicity, we refer to the collection of all exogenous and endogenousinflow and outflow rates as the “boundary conditions” for the network, de-spite the fact that the endogenous rates are actually properties of the bulk.In principle then, the boundary conditions for a network are specified by thecollections { α λ : λ is a lane of an inlink } , { β λ : λ is a lane of an outlink } , { γ λ : λ is a lane of a bulk link } and { δ λ : λ is a lane of a bulk link } . For a given net-work, one could conceive of varying all of these parameters independently, from0 to 1, and studying the resulting distributions of flow and density. In order tomeaningfully investigate MFDs however, we instead vary the α λ , β λ , γ λ and δ λ in a given systematic manner, corresponding to a reasonable demand scenariofor an arterial network. We discuss several such scenarios in Section 2.4. Weemphasize that the values of α λ , β λ , γ λ and δ λ can vary with time.We now summarize the details of the specific network and input parameterssimulated in the present study. According to the NaSch model, the speed v of each vehicle can take one of v max + 1 allowed integer values v = 0 , , , . . . , v max . Taking the length of a cellto be 7.5m, corresponding to the typical space occupied by each vehicle in a jam,and the duration of each time step to be 1s, suggests v max = 3 is a reasonablechoice for an urban network. I.e., each vehicle can move 0, 1, 2 or 3 cells per timestep in such a CA model, depending on local traffic conditions. These are thevalues used in our simulations. In addition, the NaSch model (and consequentlythe NetNaSch model) includes, at each time step and for each vehicle, a randomunit deceleration which is applied with probability p noise . By setting p noise sothat it is 0 . v max , and 0 . × . Fig. 1 shows a typical intersection in detail. With the exceptionof Section 5.1.5, the length of each bulk and boundary link was set to 750m,corresponding to 100 cells, and the length of each turning lane was set to 120m.This choice of link length corresponds to the distance between signalized in-tersections in an arterial network, and is typical of arterial road networks inpredominantly suburban cities such as Melbourne. Vehicles drive on the left side of the road in Australia. igure 1: Illustration of a typical node in the simulated network.(a) (b) (c) (d)Figure 2: The four phases used at each node of the simulated network. Each node was given the same four phases: a north/south phase, an east/westturning phase, an east/west phase and a north/south turning phase. See Fig. 2.This fixed ordering of phases was applied to our simulations of SCATS. Note thatthe phase in Fig. 2-(a) is not necessarily the first phase of the cycle; for SCATSwith signal linking this is determined by the linking protocol as described inSection 4.1.
In all our simulations, each link was assigned the same turning probability of p T for left and right turns, implying a probability 1 − p T of continuing straightahead. The probability p T was set to 0 . p T = 0 . , . We consider a number of representative scenarios for the boundary condi-tions, which we summarize as follows.I.
Time-independent.
All input and output rates are constant in time. Twomain variations were studied. 7a)
Isotropic.
The same value of α is applied to all inlinks, the samevalue of β to all outlinks, and the same values of γ and δ to all bulklinks. To obtain MFDs, the values of α and β were varied while thevalues of γ and δ were held constant.(b) Anisotropic . The value of α on the west boundary is twice as largeas on the other three boundaries, and likewise β is only half as largeon the east boundary. This sets up a west-to-east anisotropy in thedemand imposed on the network. The endogenous sources and sinkswere ignored in this case, ( γ, δ ) = (0 , Time-dependent . In this case the boundary conditions are isotropic, butthe values of α , β , γ and δ are time-dependent.The values of α and β were changed every 30 minutes, and the networkwas simulated for 20 hours. For each of the three signal systems, theprofiles of α and β were reverse-engineered to produce time series for thedensity profiles that resemble the empirical data for Yokohama presentedin Geroliminis and Daganzo (2008). In each case, the peak densities werechosen so as to drive the MFD into the high-density regime during loading,and then allow it to relax back to the stationary low-density curve duringrecovery. Two extreme cases were studied.(a) Boundary Loading . Vehicles can enter and exit the network only viaboundary links. I.e. γ and δ are strictly zero.(b) Uniform Loading . We allow vehicles to be loaded and unloaded uni-formly throughout the network by setting γ = α and δ = β .In the time-independent cases we compute MFDs by varying the exogenousdemand, for a number of fixed choices of endogenous demand. This corre-sponds in some sense to viewing the endogenous demand as an inherent partof the network, comparable to the choice of signal system etc, and studyinghow the network responds to different levels of external demand. Systems withlow endogenous demand correspond to “commuter corridors”; portions of thearterial network wedged between a strong source (e.g. a residential area) anda strong sink (e.g. the central business district). Higher values of ( γ, δ ) corre-spond to introducing shopping centers and parking garages etc into the bulk ofthe network.In the time-dependent case, in addition to the two extreme cases of boundaryloading and uniform loading, we also studied a number of intermediate scenarios,in which the strength of the endogenous demand was non-zero but less than theexogenous demand. The resulting behavior of these systems was intermediatebetween the two extremes (II.a) and (II.b) and so for the purposes of illustrationit suffices to focus on boundary and bulk loading.Finally, we note that Buisson and Ladier (2009) present a careful discussionof possible sources of network heterogeneity in empirical studies of MFDs, in-cluding the position of detectors, types of roads, and traffic signals. We deliber-8tely study symmetric square-lattice networks for which all links are equivalentand all nodes are equivalent.
3. Observables
We define the density, ρ l ( k ), of link l at the k th time step of a simulationto be the fraction of all cells on l which are occupied at that instant. The flow, J λ ( k ), of lane λ during the k th time step is simply the indicator for the eventthat a vehicle crosses the boundary between a fixed pair of neighboring cellsduring the k th update . The flow J l ( k ) on link l at the k th time step is thensimply the sum of the J λ ( k ) over all lanes λ in link l . We emphasize that sinceour model is stochastic, the observables ρ l ( k ) and J l ( k ) are random variables .Since we are interested in the dynamics on the order of traffic cycles, ratherthan iterations of our model, we bin the instantaneous link flow and densityinto bins of size b , using b = 5 minutes in our simulations. In a slight abuse ofnotation, we define ρ l ( t ) := 1 b kb X k =( t − b +1 ρ l ( k ) and J l ( t ) = 1 b kb X k =( t − b +1 J l ( k ) , (1)where the physical time t is measured in intervals of b = 5 minutes.Let us denote the set of all bulk links in the network by Λ. We emphasizethat the boundary links in the NetNaSch model are not considered to be partof the network, and serve only as an effective means of connecting the bulk tothe external environment. From the link observables (1), we then define thefollowing macroscopic network-aggregated observables: ρ ( t ) := 1 | Λ | X l ∈ Λ ρ l ( t ) ,h ρ ( t ) := s | Λ | X l ∈ Λ [ ρ l ( t ) − ρ ( t )] ,J ( t ) := 1 | Λ | X l ∈ Λ J l ( t ) ,h J ( t ) := s | Λ | X l ∈ Λ [ J l ( t ) − J ( t )] . (2)Again, we emphasize that these macroscopic observables are random variablesin our model, although by aggregating the data over both time and space thefluctuations of these macroscopic observables will be significantly suppressedrelative to the original instantaneous link observables. In our simulations, the flow is measured 2 v max cells from the upstream node of the link. J ( t ) and ρ ( t ) are the network-aggregated flow and density.We refer to the quantities h ρ ( t ) and h J ( t ) as the heterogeneity (spatial vari-ability) of the density and flow respectively, since they give a measure of theextent to which the spatial distribution of the link-level observables differ fromthe corresponding network-aggregated values. Note that the heterogeneitiesachieve their lower bound of 0 only when the corresponding link observables areall equal.A fundamental question to be studied via our simulations is the extent towhich ρ ( t ) and/or h ρ ( t ) determine the value of J ( t ). The statement that aninvariant MFD exists implies that J ( t ) should be a function of ρ ( t ) alone. How-ever, recent work by Mazloumian et al. (2010) and Geroliminis and Sun (2011b)suggest that h ρ ( t ) is also an important indicator of network performance. Oursimulations confirm this. In fact, we find that both h ρ ( t ) and h J ( t ) providevaluable indicators of network performance. For each distinct choice of traffic signal system and boundary conditions,we performed n independent simulations (with n ranging between 10 and 30),in order to estimate the expected values of the network-aggregated quantitiesdefined in (2). For a given observable X ( t ), if we denote its realization in the i th run by X ( i ) ( t ) then we compute X ( t ) = 1 n n X i =1 X ( i ) ( t ) , (3)err( X ( t )) = vuut n ( n − n X i =1 [ X ( i ) ( t ) − X ( t )] , (4)where X ( t ) is the natural estimator for the expected value of X ( t ) and err( X ( t ))is its standard error. The density, ρ l , gives the fractional occupation of the link l , relative to themaximum physically possible density, ̺ max . Consequently, ρ l is a dimensionlessquantity lying in the interval [0 , ̺ max = 400 / ≈ ρ l can be multiplied by ̺ max = 133. Thenetwork aggregated density, ρ , and its heterogeneity, h ρ , have the same dimen-sions as ρ l and the same conversions apply.Taking the duration of each time step in the NetNaSch model to be 1s, asdiscussed in Section 2.1, implies that the flow J l on link l is measured in unitsof veh/s. We emphasize that the link flows we measure are not normalized bythe number of lanes, which in all cases is 2. The network aggregated flow, J ,and its heterogeneity, h J , have the same dimensions as J l .10 . Traffic Signal Systems We simulate and study the existence of MFDs in networks using three dis-tinct traffic signal systems:
SCATS-L:
A model of SCATS with linking and adaptive cycle lengths.
SCATS-F:
A “free” version of SCATS-L, with no signal linking.
SOTL:
Self-organizing traffic lights.The SCATS traffic signal system, which controls the traffic signals in nu-merous cities around the world, uses knowledge of the recent state of trafficto choose appropriate values of three key signal parameters: cycle length, splittime, and linking offset. At each intersection it can adaptively adjust both thetotal cycle length, and the fraction ( split ) of the cycle given to each particularphase. In addition, it can coordinate ( link ) the traffic signals of several consec-utive nodes along a predetermined route by introducing fixed offsets betweenthe starting times of specific phases, thereby creating a green wave. Both theSCATS-L and SCATS-F models are special cases of our general SCATS model,which we outline in Section 4.1As a benchmark with which to compare the SCATS-like traffic signal sys-tems, we also considered the highly-adaptive self-organizing traffic lights (SOTL)system (see de Gier et al. (2011)). The SOTL system is based on the simpleprinciple that each node (intersection) should choose its current phase to bethe phase which currently has the highest demand. Unlike SCATS, no directcoordination is enforced between the signals at neighboring nodes, however suchcoordination is often seen to arise via self-organization , since neighboring nodesdo indirectly communicate with each other via the levels of traffic that theyaccept and release. The particular version of SOTL that we study here usesdensity as the demand metric, and it therefore strives to adaptively minimizethe network’s density heterogeneity. The details of the SOTL system are sum-marized in Section 4.2.
The SCATS control system adaptively controls three key signal parameters:linking offset, cycle length and split time. We discuss below how our model ofSCATS chooses each of these parameters. A subsystem in a SCATS network is a group of nodes which all share acommon cycle length. If a node does not belong to any subsystem, we call ita non-subsystem node . Within each subsystem, we appoint a unique master node m and a number of slave nodes s . Fig. 3 illustrates an 8 by 8 networkwith eight subsystems, each consisting of one master node and six slave nodes.To implement linking, each node is assigned a special phase P ∗ , which is its linked phase . If the linked phase P ∗ m of the master node m starts at time t then11 igure 3: An 8 by 8 lattice network with 8 linearly linked subsystems. P ∗ s of the slave node starts at time t + T s , where T s is the offset . Ideally, thelinking offset should be chosen based on the distance L between m and s , andthe instantaneous local space-mean speed. In practice, actual implementationsof SCATS tend to operate with fixed offsets during a given period of the day(for example morning peak hour). In our simulations we therefore use a fixed linking speed v = 54km/hr, which is just slightly less than the average free-flowspeed of about 60km/hr, see Section 2.1. SCATS chooses the unique cycle length within a subsystem based on thelocal traffic conditions in the neighborhood of the master node, as quantified bythe
Degree of Saturation (DoS). Every time a master node is about to restart itscycle, the cycle length is adjusted adaptively based on recent measured valuesof the DoS. In our model of SCATS, the cycle length is selected based on the volume ratio . For an inlink l and phase P the volume ratio is defined to be R ( l, P ) = 1 N ( l, P ) V ( l, P ) S ( P ) , (5)where V ( l, P ) is the measured traffic volume out of inlink l during phase P , and S ( P ) is P ’s current split time. The quantity N ( l, P ) denotes a fixed benchmarkvolume for the link and phase, measured in vehicles per second . If the volume In our simulations N ( l, P ) was set to 1 veh/sec for all phases. . , . Based on the model of SCATS outlined above, we considered two variants:SCATS-L and SCATS-F. The first variant, SCATS-L, operates on the linkednetwork shown in Fig. 3. By contrast, SCATS-F operates on the network whereno subsystems or linking are imposed, so that each node chooses its own cyclelength and split time plan according to its local traffic state, independently ofits neighbors.
The third signal system we study is the self-organizing traffic lights (SOTL)system described in de Gier et al. (2011). While SCATS-L and SCATS-F areadaptive in their cycle length and split time selections, they both maintain afixed cyclic ordering of each node’s phases. By contrast, the SOTL system isacyclic, and is designed so that at each phase change, the phase which currentlyhas highest demand is selected. This procedure is applied independently to eachnode.Suppose we agree on a suitable demand function d ( P ) which quantifies thedemand of each phase P of each given node. Phases with large values of d ( P )should be candidates for being the next choice of the active phase. However, oneshould also keep track of the time τ ( P ) that each phase has been idle, since wedo not want a given phase to remain idle for too long, unless it has strictly zerodemand. The key idea behind SOTL is to compute a threshold function, κ ( P ),for each phase P , which depends on both the phase’s idle time and demandfunction, and when κ ( P ) reaches a predetermined threshold value, κ ( P ) > θ, we consider making P the active phase. For a detailed general discussion of theSOTL methodology, see de Gier et al. (2011).In the simulations performed in the current work, the demand d ( P ) of phase P was simply chosen to be the total number of vehicles over all its inlinks, andthe threshold function was κ ( P ) = d ( P ) τ ( P ) P P ′ d ( P ′ ) , (6)This particular choice for the demand function implies that SOTL attempts, ateach instant of time, to adaptively minimize the network’s density heterogeneity.We used a threshold value of θ = 5.A precise algorithmic description of SOTL is given in Algorithm 2 in Appendix A.2.13 . Simulations: Time-independent boundary conditions We simulated the network described in Section 2, using the three differenttraffic signal systems described in Section 4, and measured the macroscopicobservables defined in (2). In this section, we present the results for the time-independent boundary conditions defined in Section 2.4. We simulated eachsystem for 10 hours, which ensured that stationarity was always reached.
In this section, we present our results for simulations using the isotropicBoundary Condition (I.a). In this case, we have two free parameters, α and β , where α is the input probability on each boundary inlink, and β the outputprobability on each boundary outlink. Fixed values for the bulk input and out-put probabilities γ and δ were applied to all bulk links. Three different choicesfor the fixed values of ( γ, δ ) were simulated: ( γ, δ ) = (0 , , (0 . , . , (0 . , . γ and δ Fixing γ = δ = 0, we simulated the network using a number of differentvalues of α and β , in order to obtain a range of values of the aggregated networkdensity, ranging from very low to very high. We observed that, in all cases, theflow and density reach approximately stationary values by hours 5 or 6. Fora given choice of traffic signals, the longest relaxation times were observed forflows close to capacity. For SOTL, stationarity was always achieved much fasterthan for SCATS (at comparable densities), typically by hours 3 or 4, whichimplies that the relaxation time of the network using SOTL is much smallerthan when using SCATS.In Figs. 4(a), (b) and (c) we plot J against ρ for SCATS-F, SCATS-L, andSOTL respectively, at hours 1 , , . . . , . , . J vs ρ curves are converging to a well-defined stationary MFD as time increases. Afterapproximately 6 hours of simulation, the J vs ρ curves at all later times areessentially indistinguishable. We note that Mazloumian et al. (2010) observedsimilar time-dependent behavior during three-hour simulations of their model,and they concluded that such time-dependence would likely persist at all latertimes. Fig. 4 would suggest however that such time-dependence is in fact tran-sient, and, for all practical purposes, ceases to be observable after some finitetime. It is conceivable that the behavior observed in Mazloumian et al. (2010)is an consequence of their use of periodic boundary conditions.14 ne t w o r k f l o w SCATS−F: isotropic and time independent rates ( γ =0, δ =0), p T = 0.1 (a) Time evolution of MFDs: SCATS-F ne t w o r k f l o w SCATS−L: isotropic and time independent rates ( γ =0, δ =0), p T = 0.1 (b) Time evolution of MFDs: SCATS-L ne t w o r k f l o w SOTL: isotropic and time independent rates ( γ =0, δ =0), p T = 0.1 (c) Time evolution of MFDs: SOTL ne t w o r k f l o w Isotropic and time independent rates ( γ =0, δ =0) , p T = 0.1 at 6 hr SOTLSCATS−LSCATS−F (d) Stationary MFDsFigure 4: Figs. (a), (b), and (c) show MFDs of SCATS-F, SCATS-L, and SOTL, at hours1,2,. . . 6, for a network with isotropic and time-independent boundary conditions, and nointernal sources/sinks. Fig. (d) shows a comparison of the stationary MFDs for the threesignal systems. Error bars corresponding to one standard deviation are shown, but are oftensmaller than the symbol size of the data point. .
43 at a density around 0 .
34, while SCATS-F and SCATS-Lobtain maxima of approximately 0 .
39 at densities around 0 .
19. This representsa 10% increase in network capacity by using SOTL, compared with SCATS. Wenote that SCATS-F performs better than SCATS-L in the high density regime.This is in fact to be expected, since for an isotropic network, linking shouldat best be merely unhelpful, and at worst it will be counterproductive becauseit will reduce the system’s adaptivity. We return to the comparison betweenSCATS-F and SCATS-L in Section 5.2.
In order to gain insight into the underlying cause of the transient behaviordisplayed by the MFDs, we consider how the corresponding heterogeneity curvesevolve with time. We begin by considering the density heterogeneity.In Fig. 5(a) we plot h ρ for SCATS-L at hours 1,2,. . . 6. Geroliminis and Sun(2011b) find empirically that for a given network density, the flow should belower when the density heterogeneity is higher. Comparison of Fig. 4(b) withFig. 5(a) confirms this; if one observes the time evolutions of the MFD curveand h ρ vs ρ curve in the neighborhood of a given value of ρ , there is a clearanticorrelation between the flow and the density heterogeneity. At moderatevalues of the density, h ρ starts at low values early in the simulation, and thenincreases to a stationary curve at around hour 5 or 6. Conversely, the flow startsat relatively high values and decreases to its stationary value, again reachingstationarity at around hour 5 or 6. Similar behavior is observed for SCATS-F.If one considers the analogous plots for SOTL, Figs. 4(c) and 5(c), at hours3,. . . , 6, precisely the same behavior can be observed, although it is less pro-nounced. The most obvious feature in Figs. 4(c) and 5(c), however, is thebehavior of the instantaneous curves at hour 1 (and to a lesser extent, hour 2):for moderate values of density, the flow is below its stationary limit, while h ρ isabove its stationary limit. This again demonstrates the anticorrelation betweenthe network flow and the density heterogeneity. In fact, the same type of be-havior displayed by SOTL at hour 1 is also present in SCATS-L and SCATS-F,however it largely occurs before hour 1 in these cases. This type of behav-ior is presumably related to the fact that the simulations are started from acompletely empty network, with vehicles only entering via the boundary.The general conclusion to be drawn from these observations is that whilethe transient behavior of the flow at a given density might be subtle (e.g. non-monotonic in time), at all times there is a strong anticorrelation between flow16 ne t w o r k den s i t y he t e r ogene i t y SCATS−L: isotropic and time independent rates ( γ =0, δ =0), p T = 0.1 (a) Density heterogeneity: SCATS-L ne t w o r k f l o w he t e r ogene i t y SCATS−L: isotropic and time independent rates ( γ =0, δ =0), p T = 0.1 (b) Flow heterogeneity: SCATS-L ne t w o r k den s i t y he t e r ogene i t y SOTL: isotropic and time independent rates ( γ =0, δ =0), p T = 0.1 (c) Density heterogeneity: SOTL ne t w o r k f l o w he t e r ogene i t y SOTL: isotropic and time independent rates ( γ =0, δ =0), p T = 0.1 (d) Flow heterogeneity: SOTLFigure 5: Heterogeneity versus density at hours 1,2,. . . ,6 for network governed by SCATS-Land SOTL, with isotropic and time-independent boundary conditions. Error bars correspond-ing to one standard deviation are shown, but are often smaller than the symbol size of thedata point. J and h ρ plays an important role in understand-ing hysteresis in networks with time-dependent boundary conditions.As a practical observation, we note that h ρ is considerably smaller whenthe network is governed by SOTL than when governed by SCATS-L, which isunsurprising given that SOTL is specifically designed to adaptively reduce thedensity heterogeneity. Given the previous observation that the SOTL MFD liesstrictly above the SCATS-L MFD, this suggests that a methodology of adaptivedensity homogenization can provide a highly effective means of network control.The relationship between the transient behaviors of the flow and the flowheterogeneity is more complicated, however. We shall present a careful timeseries analysis of the cross correlations between h J and J elsewhere. However,the stationary behavior of the h J vs ρ curves appear to contain a considerableamount of information. In particular, for all three signal systems studied, thereappears to be a point of inflection in the stationary h J vs ρ curve either at, orslightly before, the density ρ c at which the network reaches capacity. Figs. 5(b)and 5(d) show h J vs ρ for SCATS-L and SOTL, respectively. For the case ofSOTL there is in fact a long plateau, implying that for the majority of the free-flow regime the flow heterogeneity is independent of the network density in thiscase. We now consider the effect of introducing non-zero values of the internalinput and output probabilities, γ and δ , which model endogenous sources/sinkssuch as parking garages. Figs. 6(a) and 6(b) show the MFDs for SCATS-L andSOTL produced by varying α and β for two distinct, non-zero, fixed values of( γ, δ ), and compares them with the ( γ, δ ) = (0 ,
0) case already discussed. Theresults for SCATS-F are intermediate between those of SCATS-L and SOTL.The first observation is simply that different values of ( γ, δ ) clearly producedifferent MFD curves as α and β are varied. This is to be expected, since thestronger the internal sources and sinks, the more homogeneous is the network;for sufficiently strong internal sources and sinks the effects of the boundarybecome essentially irrelevant.The second observation is that for both SCATS-L and SOTL, the MFDs aretranslated to the right (higher densities) as γ and δ increase. This is intuitivelyreasonable; for ( γ, δ ) strictly zero the links deep in the bulk of the network wouldbe expected to have lower density than those near the boundary, implying thatthe aggregated network density should be lower. This scenario explains whythe translation of the MFD is more pronounced for SCATS-L than for SOTL,since we have already seen in Fig. 5 that the heterogeneity of SCATS-L is muchhigher than that for SOTL.A final observation is that for SOTL the capacity is also marginally higherwith non-zero γ, δ than with zero γ, δ . For SCATS-L, there is considerably morestatistical noise in the high density branch when simulating with non-zero γ, δ ,and it is not clear to what extent the capacities change, if at all.18 ne t w o r k f l o w SCATS−L: isotropic rates, p T = 0.1 at 6hr γ =0, δ =0 γ =0.05, δ =0.1 γ =0.1, δ =0.2 (a) MFDs for SCATS-L ne t w o r k f l o w SOTL: isotropic rates, p T = 0.1 at 6hr γ =0, δ =0 γ =0.05, δ =0.1 γ =0.1, δ =0.2 (b) MFDs for SOTLFigure 6: MFDs for SCATS-L and SOTL with γ = 0 , . , . δ = 0 , . , .
2. Error barscorresponding to one standard deviation are shown.
Apart from these translations and rescalings, Figs. 6(a) and 6(b) show thatno qualitative change in the shapes of the stationary MFDs is introduced byapplying non-zero internal sources/sinks. By contrast, we will see in Section 6that when using time-dependent boundary conditions, the introduction of suf-ficiently strong internal sources and sinks can qualitatively affect the behaviorof the network.
We now briefly turn our attention to the impact on MFDs of varying theturning probabilities, again using isotropic boundary conditions (Boundary Con-dition (I.a)). Figs. 7(a), 7(b) and 7(c) compare the MFDs produced for net-works with turning probabilities p T = 0 . , . , .
2, for SCATS-F, SCATS-L,and SOTL, respectively. While the low density branches are invariant with p T ,it is clear that the high density branches become more rapidly decaying as p T increases. For both p T = 0 .
15 and p T = 0 .
2, there is a critical density ρ j suchthat the flow remains constant for all ρ > ρ j . This value of ρ j decreases as p T increases. Although it is not observable for p T = 0 .
1, it is presumably presentin principle but at a density higher than any we simulated. We note that evenfor ρ > ρ j , the flow is non-zero, so the network is not locked into a rigid grid-lock. The value of this plateaued high-density flow can be seen to decrease as p T increases.On a practical level, we note that for p T = 0 .
15 and p T = 0 . p T = 0 .
2. The transient behavior is qualitativelythe same as that shown in Fig. 4(a) for p T = 0 .
1. The low density branchis stationary already by hour 1, while the high density branch decreases to its19 ne t w o r k f l o w SCATS−F: isotropic and time independent rates ( γ = 0, δ = 0), time 6hr p T =0.1p T =0.15p T =0.2 (a) MFDs for SCATS-F with various p T ne t w o r k f l o w SCATS−L: isotropic and time independent rates ( γ = 0, δ = 0), time 6hr p T =0.1p T =0.15p T =0.2 (b) MFDs for SCATS-L with various p T ne t w o r k f l o w SOTL: isotropic and time independent rates ( γ = 0, δ = 0), time 6hr p T =0.1p T =0.15p T =0.2 (c) MFDs for SOTL with various p T ne t w o r k f l o w SCATS−F: isotropic and time independent rates ( γ = 0, δ = 0), p T = 0.2 (d) Time evolution of MFDs for SCATS-Fwith p T = 0 . p T = 0 . , . , .
2, for a network with isotropic and time-independent boundaryconditions. Fig. (d) shows MFDs for SCATS-F at hours 1,2, . . . ,6 with p T = 0 .
2. Error barscorresponding to one standard deviation are shown, but are often smaller than the symbolsize of the data point. stationary limit, which is reached by hour 4 or 5. The extent to which the curvesat hours 1 and 2 differ from their stationary limits is clearly much larger for p T = 0 . p T = 0 . As discussed in Section 2.1, in general, the networks we simulate have bulklinks of length 750m, with right-turning lanes of length 120m. In this sectionwe study the effect of varying the length of both the main lanes and the turninglanes.We begin by considering the effect of doubling the length of the turninglanes. In Fig. 8(a) we compare the MFDs produced by a system governed by20 ne t w o r k f l o w SOTL: isotropic and time independent rates ( γ = 0, δ = 0), time 6hr p T =0.15, long TLp T =0.2, long TLp T =0.15, short TLp T =0.2, short TL (a) MFDs using different turning lane lengths ne t w o r k f l o w Isotropic and time independent rates ( γ =0, δ =0) , p T = 0.1 at 6 hr SOTLSCATS−LSCATS−F (b) MFDs using different bulk link lengthsFigure 8: Fig. (a) shows MFDs of SOTL at hour 6 with turning probabilities p T = 0 . , . p T = 0 .
1, fora network with short (210m) bulk links. In both cases, time-independent, isotropic boundaryconditions are applied, with zero sources and sinks. Error bars corresponding to one standarddeviation are shown, but are often smaller than the symbol size of the data point.
SOTL with turning probabilities p T = 0 . , .
20, with turning lanes of length120m (denoted short TL) and 240m (denoted long TL). All other parametersare set as in Section 5.1.4. The data for the short TL system is in fact identicalto that plotted in Fig. 7(c); we repeat it here for ease of comparison with thelong TL case. For both p T = 0 .
15 and p T = 0 .
2, we see that the systems withshort and long turning lanes have identical behavior at very-low and very-highdensities. This is to be expected; at very low densities both turning lanes willbe relatively unoccupied, whereas at very high densities both turning lanes willsuffer from spillback. At moderately high densities however, the MFD of thelong TL system lies strictly above that of the short TL system. We concludethat for sufficiently large turning probabilities, increasing the length of turninglanes can produce a non-trivial increase in the network’s MFD at moderatelyhigh densities.An analogous comparison of the long and short TL systems was also madewith p T = 0 .
1, however there was no observable difference in this case. Thisis intuitively reasonable; for fixed lane lengths, there should be a value of theturning probabilities below which the effect of congestion on the turning lanes isnegligible compared to congestion on the main lanes. In such a case, increasingthe length of the turning lane further does not modify the network’s MFD. Ourresults show that this is already the case for p T = 0 . downtown networks. In Fig. 8(b) we show the results ofrepeating the simulations presented in Fig. 4 using 210m bulk links. To facilitatecomparison with Fig. 4(d), we left the lengths of the turning lanes fixed. Acomparison of Figs. 8(b) and 4(d) shows a number of differences. Firstly, for21ach signal system, we observe that the value of the maximum flow is lowerfor the network with shorter links. For example, the maximum flow for SOTLdrops from around 0 .
43 for the network with long bulk links to around 0 . ρ ≈ .
5. Beyond that density,the network becomes highly congested and the flow plateaus at a very smallvalue, less than 0 .
05. Comparing the different signal systems, we see that againSOTL outperforms the SCATS systems. Furthermore, the relative improvementof SOTL over SCATS in the low-density branch of the MFD is significantly morepronounced for the system with short links.Finally, we note that in reality, networks with short links would likely alsohave shorter turning lanes. We therefore also repeated the simulations presentedin Fig. 8(b) using turning lanes of length 45m. The relationship between theresulting MFDs and Fig. 8(b) are found to be similar to the scenario illustratedin Fig. 8(a). In this case however, we observe that the MFD of the systemwith the shorter turning lanes lies strictly below that of the system with longerturning lanes, even at p T = 0 .
1. This illustrates the converse of our observationabove: for a given value of p T , the high-density branch of the MFD can bedecreased by shortening the length of the turning lanes sufficiently. In this section we present our results for simulations using Boundary Con-dition (I.b). In this case, we again have two free parameters, α and β , where α is the input probability on each inlink on the west boundary, and β the outputprobability on each outlink on the east boundary. All other boundary inlinkshave input probability α/ β . No internal sources or sinks are present ( γ = δ = 0) and the turningprobability is p T = 0 .
1. These boundary conditions imply that the demand inthe west-to-east direction is twice that of other directions. In the presence ofsuch anisotropy, applying linking with SCATS-L is a very natural thing to do,and our simulations of SCATS-L are linked in the west-to-east direction.In Fig. 9(a) we plot a comparison of the stationary MFDs of the networksusing SCATS-F, SCATS-L, and SOTL, corresponding to hour 6 of our simula-tions, by which time each system had reached approximate stationarity. Thetransient behavior prior to stationarity is qualitatively the same as described forthe case of isotropic boundary conditions discussed in Section 5.1. The first ob-servation is simply that for each signal system, well-defined stationary MFDs doexist in the presence of anisotropic boundary conditions. However, the shapesof the stationary curves are clearly quite different to those produced by isotropicboundary conditions as shown in Fig. 4. While in the isotropic case the MFDsare smooth curves which qualitatively resemble a typical single-link FD, theMFDs shown in Fig. 9(a) have a more intricate structure. In particular, for22 ne t w o r k f l o w Anisotropic and time independent rates ( γ =0, δ =0) , p T = 0.1 at 6 hr SOTLSCATS−LSCATS−F (a) Stationary MFDs ne t w o r k den s i t y he t e r ogene i t y Anisotropic and time independent rates ( γ =0, δ =0) , p T = 0.1 at 6 hr SOTLSCATS−LSCATS−F (b) Stationary density heterogeneityFigure 9: Comparison of SCATS-L, SCATS-F, and SOTL at stationarity, on a network withanisotropic α and β in the west-to-east direction. Linking in SCATS-L is applied along thehigh-demand direction. Fig. (a) shows the MFDs while Fig. (b) shows density heterogeneity.Error bars corresponding to one standard deviation are shown, but are often smaller than thesymbol size of the data point. all three signal systems, the stationary MFD displays two steep drops in flow.The first of these occurs for ρ ≈ .
4, and is most pronounced for SCATS. ForSOTL, the location of this first drop is close to the density ρ c which producesmaximum flow, while for SCATS it occurs well beyond ρ c .As observed in the isotropic case for p T = 0 .
2, for densities above a criticalvalue, ρ j , an oversaturated regime with constant flow is obtained. Again, evenfor ρ > ρ j the flow remains non-zero, although it is very low. For all threesignal systems, a second steep drop can be observed for ρ ≈ ρ j . This drop isvery sharp, even for SOTL. Some insight into this behavior can be obtainedby comparing with the corresponding density heterogeneities in Fig. 9(b). Forall three signal systems, the location of the drop in the MFD near ρ j coincidesexactly with a cusp in the density heterogeneity. In addition, for SCATS-L andSCATS-F, the sharp drop in the MFD near ρ ≈ .
6. Simulations: Time-dependent boundary conditions
In the previous section, we applied time-independent boundary conditions,and observed that the system took up to 6 hours to reach stationarity. Inreal-world scenarios, however, traffic demand typically varies with time, andin practice a network may never actually reach stationarity. Understandingtransient behavior of the network dynamics is therefore of significant practicalimportance.In this section we present our results for simulations using Boundary Con-dition II. of Section 2.4. We select appropriate values of α and β so that wecan simulate the network traffic variation during a typical weekday. Specifically,we consider a 20 hour period, and enforce two peaks in the demand; one corre-sponding to the morning, the other to the afternoon. At each instant, the samevalue of α (resp. β ) is applied to each boundary inlink (resp. outlink), so theboundary conditions are isotropic. We consider two scenarios for the internalsources and sinks: γ = δ = 0, in which case the demand is entirely driven bythe boundary; and γ = α , δ = β , in which case the bulk input (output) oc-curs at the same rate as the boundary input (output). We refer to these twocases as boundary loading and uniform loading, respectively. Unlike the time-independent case discussed in the previous section, in the time-dependent casewe find that varying the relative strength of the internal sources/sinks comparedwith the boundary demand can produce rather different qualitative behavior.For each of the three signal systems, and for both boundary and uniformloading, the peak densities were chosen so as to drive the MFD into the high-density regime during loading, and then allow it to relax back to the stationarylow-density curve during recovery. The time series of the resulting networkdensity and flow for SCATS-L are shown in Fig. 10; the corresponding profilesfor SCATS-F and SOTL were very similar, although SOTL produced higherand sharper peaks in the density.Figs. 11(a), 11(c), and 11(e) show the relationship between the density andflow for each of the three signal control systems studied, in the case of uniformloading, while Figs. 12(a), 12(c) and 12(e), show the analogous plots in the caseof boundary loading. Each plot is in fact a parametric plot in the density-flowplane, parameterized by time. In each case, the density-flow curve obtained dur-ing the first 4 hours of the simulation coincides exactly with the corresponding24 ne t w o r k ob s e r v ab l e s SCATS−L : time dependent rates, p T = 0.1( γ = 0, δ = 0) densityflow (a) Boundary loading ne t w o r k ob s e r v ab l e s SCATS−L : time dependent rates, p T = 0.1( γ = α , δ = β ) densityflow (b) Uniform loadingFigure 10: Time series of network-aggregated flow and density for the SCATS-L traffic signalsystem under time-dependent isotropic boundary conditions. Error bars corresponding to onestandard deviation are shown. stationary MFD. During this period, the network is initially empty, and as thedensity increases the flow increases until maximum flow is obtained. Through-out this period, the network remains uncongested, and no transient effects areobserved. However, as the morning peak in demand is approached, which occursat around hour 6 of the simulations, we see that the density-flow curves developnon-trivial time dependences, and hysteresis effects emerge. Let us analyze the observed hysteresis patterns in more detail. We begin withthe case of uniform loading. Fig. 11(e) shows the time evolution of the MFD forSOTL. The system reaches capacity at approximately hour 4, after which timeflow decreases as density is further increased. This continues until around hour6, when the first peak in demand is reached. After this point, density dropsand flow begins to increase again until it again reaches capacity, at around hour8. However the curve followed during this “recovery” process (hours 6 to 8)lies below the curve followed in the original “loading” process (hours 4 to 6).The MFD curve from hour 4 to hour 8 therefore defines a clockwise hysteresisloop. Similar behavior occurs at later times also. Indeed, we observe a furthertwo clockwise hysteresis loops: from capacity, to low density, back to capacity(hours 8 to 12); from capacity, to high density, back to capacity (hours 12 to 16).Finally, the system recovers from capacity to the very low density regime (hours16 to 20); this final recovery curve is initially below the initial loading curve(hours 0 to 4), but coincides with it at sufficiently low densities. Qualitativelysimilar behavior is also observed for SCATS-L and SCATS-F, although the sizeof the loops tend to be somewhat larger.Clockwise hysteresis loops were recently observed in an empirical study ofMFDs in Toulouse, presented in Buisson and Ladier (2009), where it was ar-gued that hysteresis is caused by spatial heterogeneity in network density. In25 ne t w o r k f l o w SCATS−F : time dependent rates, p T = 0.1( γ = α , δ = β ) (a) SCATS-F: MFD den s i t y he t e r ogene i t y SCATS−F: time dependent rates ( γ = α , δ = β ), p T = 0.1 (b) SCATS-F: Density heterogeneity ne t w o r k f l o w SCATS−L: time dependent rates, p T = 0.1( γ = α , δ = β ) (c) SCATS-L: MFD den s i t y he t e r ogene i t y SCATS−L: time dependent rates ( γ = α , δ = β ), p T = 0.1 (d) SCATS-L: Density heterogeneity ne t w o r k f l o w SOTL: time dependent rates, p T = 0.1( γ = α , δ = β ) (e) SOTL: MFD den s i t y he t e r ogene i t y SOTL: time dependent rates ( γ = α , δ = β ), p T = 0.1 (f) SOTL: Density heterogeneityFigure 11: Performance of network under time-dependent isotropic boundary conditions, whenusing SCATS-F, SCATS-L, and SOTL traffic signal systems with uniform loading. Left col-umn: Instantaneous MFDs. Right column: Density heterogeneities. Error bars correspondingto one standard deviation are shown. ne t w o r k f l o w SCATS−F : time dependent rates, p T = 0.1( γ = 0, δ = 0) (a) SCATS-F: MFD den s i t y he t e r ogene i t y SCATS−F: time dependent rates ( γ =0, δ =0), p T = 0.1 (b) SCATS-F: Density heterogeneity ne t w o r k f l o w SCATS−L: time dependent rates, p T = 0.1( γ = 0, δ = 0) (c) SCATS-L: MFD den s i t y he t e r ogene i t y SCATS−L: time dependent rates ( γ = 0, δ = 0), p T = 0.1 (d) SCATS-L: Density heterogeneity ne t w o r k f l o w SOTL: time dependent rates, p T = 0.1( γ = 0, δ = 0) (e) SOTL: MFD den s i t y he t e r ogene i t y SOTL: time dependent rates ( γ =0, δ =0), p T = 0.1 (f) SOTL: Density heterogeneityFigure 12: Performance of network under time-dependent isotropic boundary conditions, whenusing SCATS-F, SCATS-L, and SOTL traffic signal systems with boundary loading. Left col-umn: Instantaneous MFDs. Right column: Density heterogeneities. Error bars correspondingto one standard deviation are shown. two-bin model presented in Gayah and Daganzo (2011), under the assumptionof perfectly symmetric loading, provides a simple analytical explanation of thisbehavior.If the constraint of perfectly uniform loading is relaxed however, differentbehaviour can arise. If loading is sufficiently non-uniform then density may infact become more heterogeneously distributed in loading than in recovery, whichwould produce anticlockwise hysteresis loops in the MFD, rather than clockwiseloops. To study this possibility, we have therefore simulated networks in whichthe boundary sources/sinks are stronger than the internal sources/sinks. Situ-ations where such behavior might arise include commuter corridors as well asarterial networks in the presence of perimeter control, or gating . The most ex-treme example of such loading is when the strength of the internal sources andsinks is set identically to zero, corresponding to the profiles in Fig.12.Consider Fig. 12(e) for the SOTL signal system. During the first 10 hours ofthe simulation corresponding to the lead-up to, and recovery from, the morningpeak hour, we observe two distinct hysteresis loops. The first of these loops,which occurs at moderately high density, is traced out by the flow-density curvein an anticlockwise direction, as time evolves, while the second loop, occuring atlow density, is traced out in a clockwise direction. As the density then increasesagain due to the afternoon peak, a second, larger, anticlockwise hysteresis loopis traced out at high densities. The low-density behavior is qualitatively similarto that observed for uniform loading, and the clockwise loops in the MFD againcoincide with anticlockwise loops in the density heterogeneity, Fig. 12(f); c.f.Figs. 11(a) and 11(b) for example. The behavior at moderately high density isthe opposite of that observed for uniform loading, but it can again be understoodfrom Fig. 12(f) as follows. An initially empty network is loaded just aboveits capacity, in a non-uniform manner which leaves some links very congested,with others still uncongested. After the morning peak (around hour 6), asthe inflow drops and outflow increases, vehicles disperse through the network,and the density becomes more evenly distributed. Fig. 13 illustrates this effectby comparing the spatial density distibrutions during loading and recovery, at afixed value of the network density, during the afternoon peak. We clearly observethat the centre of the network is less congested than the boundary region during28 OTL: Hr 13.25, boundary loading (a) Loading
SOTL: Hr 15.4167, boundary loading (b) RecoveryFigure 13: Link densities in a network governed by SOTL with boundary-loading. On eachlink, the length of the superimposed solid line is proportional to the magnitude of that link’smean density during a specific 5-minute time interval. The densities on boundary links arenot shown, since these are not considered part of the network, and are not included in ournetwork-aggregated observables. Fig. (a) is measured during loading, at time 13:25, and has ρ = 0 . ± . J = 0 . ± .
001 and h ρ = 0 . ± . ρ = 0 . ± . J = 0 . ± .
002 and h ρ = 0 . ± . loading, whereas the distribution during recovery is reasonably uniform. Theseheterogeneity differences cause the anticlockwise loops seen in the MFD at highdensity. In Section 6.3 we show that this behavior can be predicted using aheterogeneous version of the two-bin model discussed in Gayah and Daganzo(2011).The two scenarios, boundary loading and uniform loading, can be consideredopposite extremes; in practice, one would likely expect that the strength of theinternal sources and sinks would be non-zero but smaller than the boundaryrates. We therefore also simulated a scenario with γ = α/ δ = β/
3. Theresults are intermediate between the uniform loading and boundary loadingcases discussed above. In particular, for the SOTL system, we observe bothanticlockwise and clockwise hysteresis at high density: anticlockwise hysteresisduring hours 4 to 8 (as observed for boundary loading); and clockwise hysteresisduring hours 12 to 16 (as observed for uniform loading).Finally, let us compare the behavior of the SOTL simulations with the corre-sponding simulations of SCATS. For both boundary loading and uniform load-ing, we again find that SOTL has lower values of density heterogeneity andhigher capacities than both the SCATS systems. For the specific case of bound-ary loading, we note that while Figs. 12(a) and 12(c) display clockwise loops inthe MFDs of SCATS-L and SCATS-F, as observed for SOTL, the anticlockwiseloops appear to be absent. By zooming in on Fig. 12(d), one finds that thereis, in fact, a small clockwise hysteresis loop in the density heterogeneity plot forSCATS-L, at the start of the final recovery process, and similarly zooming in onFig. 12(c) one observes a corresponding anticlockwise loop in the MFD. However29he loops are comparable to the size of the error bars in the simulations and sotheir existence cannot be firmly established. The presence of strong anticlock-wise loops for SOTL but not for SCATS can be understood as a consequence ofSOTL’s generally stronger ability to homogenize the network density.
The two-bin model was introduced by Daganzo et al. (2011). It representsthe interaction between two subnetworks, or bins , in a larger road network.A state of the system consists of the pair ( ρ , ρ ), and the expression for theflow J ( · ) in each bin is assumed to be given by the same triangular MFD, withcapacity ( ρ c , J c ) and jam density ρ j . The dynamical evolution of the system isdefined by the following system of ordinary differential equationsd ρ d t = a − b J ( ρ ) + p J ( ρ ) − p J ( ρ ) L , d ρ d t = a − b J ( ρ ) + p J ( ρ ) − p J ( ρ ) L , (7a)for ρ , ρ < ρ j , and dρ dt = dρ dt = 0 if ρ or ρ = ρ j . (7b)The model as stated in (7) has 8 free parameters: a i , the rate of inflow intobin i ; b i , the proportion of traffic flowing out of the network from bin i ; p i , theproportion of traffic turning out of bin i into the other bin; L i , the total networklength of bin i .The perfectly symmetric case, in which a = a , b = b , p = p , and L = L , was studied in detail by Gayah and Daganzo (2011). It was foundthat if hysteresis was observed in the aggregated MFD, it was typically orientedclockwise. The results for the uniform loading scenario studied in Sections 6.1and 6.2 are in perfect agreement with these results.In this section, we consider instead a highly asymmetric case, in which a = b = 0, and p and p are not necessarily equal. This can be viewed as a two-binmodel of the boundary loading scenario studied in Sections 6.1 and 6.2. In thisinterpretation, bin 1 corresponds to the links adjacent to the boundary, whilebin 2 corresponds to the remainder of the bulk links, see Fig. 3. This impliesthat L is slightly smaller than L . The aggregated density and flow are then ρ = L ρ + L ρ L + L , J = L J ( ρ ) + L J ( ρ ) L + L . (8)Figs. 14(a) and 14(b) show phase plots for the system (7) during loading andrecovery, respectively, for a typical choice of the model parameters. The loadingprocess has a = 0 . b = 0, while the recovery process has a = 0 and b = 0 .
1, so that no vehicles exit the network during loading, and no vehiclesenter the network during recovery. In both cases p = 0 .
08 and p = 0 .
02. Forthe perfectly symmetric case studied by Gayah and Daganzo (2011), trajectories30 ρ ρ a = 0.1, b = 0, p = 0.08, p = 0.02 ρ j ρ j ρ c ρ c (a) Loading: a = 0 . b = 0, p = 0 . ρ ρ a = 0, b = 0.1, p = 0.08, p = 0.02 ρ j ρ j ρ c ρ c (b) Recovery: a = 0, b = 0 . p = 0 .
00 a = 0.1, b = 0.1, p = 0.08, p = 0.02J c ρ j ρ c (c) MFD corresponding to (a) and (b). ρ ρ a = 0.1, b = 0, p = 0.12, p = 0.02 ρ j ρ j ρ c ρ c (d) Loading: a = 0 . b = 0, p = 0 . a = b = 0. The loading paths have b = 0 while the recoverypaths have a = 0. Fig. (c) shows the trajectory in the ρ − J plane corresponding to a typicalloading/recovery process, given by the black paths in (a) and (b). The star represents themaximum value of density obtained during this loading/recovery process, and the diamond(triangle) represents the maximum flow attained during loading (recovery). Fig. (d) showsthe effect on Fig. (a) of increasing p . ρ = ρ , remain on that line. We note that for theasymmetric system studied here, however, this is not the case.To illustrate the effect on the MFD of combining such loading and recov-ery processes, let us now consider the black trajectories shown in Figs. 14(a)and 14(b). In Fig. 14(a), the black trajectory starts with ρ = 0 and ρ = 0,which corresponds to the initial state of the boundary loading scenario simulatedin Section 6.1. Suppose we stop the loading process at a finite time, denoted bythe star in Fig. 14(a), and then begin a recovery process. The resulting recoverytrajectory is shown in black in Fig. 14(b). The diamond (triangle) in Fig. 14(a)(Fig. 14(b)) shows the location at which maximum flow was attained on theblack loading (recovery) trajectory. In Fig. 14(c) we show the trajectory inthe ρ − J plane resulting from combining these loading and recovery processes,superimposed on the underlying MFD of the model. There is a clearly visibleanticlockwise hysteresis loop as the system recovers from high density, preciselyas observed for the boundary loading scenario simulated in Section 6.1.Since the recovery path in Fig. 14(b) initially moves towards more balancedstates, i.e. moves toward the diagonal, the higher flow during the initial stagesof recovery can be seen as a consequence of lower heterogeneity, as observedin Section 6.2. Note that the location of the maximum flow during recoveryoccurs precisely at zero heterogeneity, ρ = ρ . As the recovery trajectorycontinues further, Fig. 14(b) shows that the heterogeneity increases again, andthe corresponding trajectory in the MFD then drops below the original loadingcurve and a clockwise hysteresis loop results. This combination of anticlockwisehysteresis loops at high density and clockwise loops at low density agrees exactlywith the behavior observed in Fig. 12(e). This behavior is quite generic, andsimilar MFD hysteresis is observed for any similar pairs of loading and recoverytrajectories, whenever loading begins with ρ ≪ ρ c and ends with ρ ≫ ρ c .As a final observation, Fig. 14(d) shows an alternative loading process, forwhich all parameters are the same as Fig. 14(a) except that p = 0 .
12 is slightlyhigher. While the trajectories are qualitatively similar, it is apparent that evena small variation in the rate that vehicles from the boundary can enter thenetwork can produce quite different loading trajectories, starting from a giveninitial state. One significant factor that would contribute to the effective valueof p in an actual arterial network is the type of traffic signal system used. Fromthe perspective of the two-bin model, it is therefore unsurprising that differenttypes of signal systems can have rather different levels of hysteresis in theirMFDs.
7. Discussion
We have studied macroscopic fundamental diagrams (MFD) on a square-lattice traffic network for a variety of traffic scenarios. In particular we havestudied networks as a function of both anisotropy (directional bias) and unifor-mity (presence of bulk sinks and sources) in demand. We furthermore studiedthe impact of various turning probabilities, and we have studied these networkswith three different traffic signal systems, SCATS-F, SCATS-L and SOTL.32ur main findings include the following: • For time-independent demand, MFDs do exist even when demand is notuniform, but their shapes depend on the nature of the non-uniformity: – Systems with more uniformly distributed demand achieve similar ca-pacities, but capacities occur at higher densities; – Systems subject to anisotropic exogenous demand display a steepdrop in the flow just beyond the maximum of the MFD; – As turning rates increase, capacity and jamming occur at lower den-sities, capacity decreases, and the congested branch of the MFD de-creases; • For time-dependent demand, MFDs show clear hysteresis which is stronglycorrelated with the spatial heterogeneity of the density. The qualitativebehaviour of this hysteresis is strongly dependent on the level of uniformityof loading and unloading; • The choice of traffic signal system plays a crucial role in determining anetwork’s performance. The idealized control scheme SOTL, which isdesigned to uniformize the network density distribution, always results ina higher MFD compared to the commonly used SCATS system. SOTLincreases the network capacity and produces higher flows in the congestedregime.We finally remark that given the strong dependence of the choice of trafficsignal system on the shape of the MFD, one can in principle use MFDs as ametric for comparing the performance of different traffic signal systems.
Acknowledgments
We gratefully acknowledge the financial support of the Roads Corporationof Victoria (VicRoads), and we thank VicRoads staff, in particular AdrianGeorge, Andrew Wall and Hoan Ngo, for numerous valuable discussions. Wealso thank Carlos Daganzo, Vikash Gayah, Yibing Wang and John Gaffney foruseful discussions, and we thank two anonymous referees for their invaluablecomments. This work was supported under the Australian Research Coun-cil’s Linkage Projects funding scheme (project number LP120100258), and T.G.is the recipient of an Australian Research Council Future Fellowship (projectnumber FT100100494). This research was undertaken with the assistance ofresources provided at the NCI National Facility through the National Compu-tational Merit Allocation Scheme supported by the Australian Government. Wealso gratefully acknowledge access to the computational facilities provided bythe Monash Sun Grid. 33 eferences
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Appendix A. Details of Traffic Signal Control Systems
Appendix A.1. SCATS
The strategy for adapting the cycle length C based on the volume ratio R ,defined in (5), of a master node is presented as below. Algorithm 1.
SCATS cycle length decision
Case 1: if C = MIN & R > . , then C = STOPPER
Case 2: if C = STOPPER & R < . , then C = MIN
Case 3: if
R > . , then C = min { C + STEP , MAX } Case 4: if
R < . , then C = max { C − STEP , STOPPER } Otherwise: C remains unchanged. The parameters used in Algorithm 1 are set as follows:
MIN : minimum cycle length 44 seconds;
STOPPER : stopper cycle length 64 seconds;
MAX : maximum cycle length 130 seconds;
STEP : fixed amount of increment/decrement 6 seconds.Fig. A.15 illustrates the cycle length decision process implemented by Algo-rithm 1. The main strategy underlying the above algorithm is to attempt tokeep the volume ratio within the range [0 . , . cyc l e l eng t h ( s e c ) MIN STOPPER MAX
Case 1 Case 3Case 4Case 2
Figure A.15: Cycle length selection by SCATS-like systems. during the previous cycle (over 0 .
95) the cycle length is increased by a fixedamount. Otherwise if green time was wasted on the previous cycle, signaledby
R < .
85, the cycle length is decreased. The
STOPPER is included to allowa steep increase in the cycle length due to a sudden increase in traffic volume,when the cycle length is at its minimum.The volume ratio of a master node, m , is given by R ( m ) = R ( l ∗ , P ∗ ) , (A.1)where l ∗ is the unique inlink flowing in the linked direction, and P ∗ is the linkedphase. The cycle length of each slave node is equal to that of its master.For a non-subsystem node, n , the adaptive cycle length strategy remainsvalid, except that the volume ratio is defined by maximizing R over all inlinksand phases, R ( n ) = max P max l R ( l, P ) . (A.2)Given the cycle length C , the split time of phase P for a master or non-subsystem node is given by S = d ( P ) P P d ( P ) [ C − number of phases × S min − total amber time] + S min , (A.3)with the demand function d ( P ) = max l V ( l, P ) where l is an inlink of phase P .We impose a fixed delay T wait to the nodes each time there is a phase change andthe next phase does not share any path with the current one. During that time,only right-turning vehicles that have been giving way to others may traversethe intersection. This delay mimics the amber time for phase change. We set T wait to 2 seconds in our simulations, however the precise value does not impactgreatly on the simulation results provided that the split times are not too small.For SCATS, the total amber time in a cycle is 4 second. The minimum splittime in our simulations was set to S min = 5 seconds.36or slave nodes, we demand that the split time of the linked phase must bethe same as that of its master node. The remaining portion of the cycle is thenshared between the other phases according to their maximum inlink volumes.Initially, at the beginning of each simulation, the cycle length of each node isset to the minimum value and the split time plan is chosen based on the turningprobabilities. We note that since split times are adaptive, the initial conditionfor the splits is unimportant. Appendix A.2. SOTL
SOTL is an acyclic signal system, in the sense that no fixed ordering of thephases is imposed. The following algorithm provides a precise description ofhow SOTL operates at each time step and node. The observable τ ( n ) acts as aclock for node n , recording how long the current phase has been activate for. Algorithm 2.
SOTLIncrement τ ( n ) for each phase P 6 = P active do Increment τ ( P ) end forif τ ( n ) > S min then Let Π ′ = {P ∈ Π = {P , P , . . . } : κ ( P ) > θ } if Π ′ = ∅ then Let Π ′′ = {P ∈ Π ′ : κ ( P ) = max P ′ ∈ Π ′ κ ( P ′ ) } Let Π ′′′ = {P ∈ Π ′′ : τ ( P ) = max P ′′ ∈ Π ′′ τ ( P ′′ ) } Uniformly at random choose
P ∈ Π ′′′ and set P active = P Set τ ( P active ) = 0 Set τ ( n ) = 0 end ifend if When the idle time of node n is greater than the minimal split time, S min ,SOTL chooses the phases for which the threshold functions κ are greater thanthe threshold θ , and among those it selects the phases with the maximal κ , thenamong those it selects the phases with the longest idle time. If there is morethan one element in this latter set, then the next active phase will be chosen atrandom from it, however in practice there is typically only one such phase tochoose from. The fixed delay T wait is also applied to SOTL. In our simulations,the minimal split time was set to S minmin