Position weighted backpressure intersection control for urban networks

Position weighted backpressure intersection control for urban networks Li Lia , Saif Eddin Jabaria,b,∗ arXiv:1810.11406v2 [cs.SY] 3 Feb 2019 a New b New York University Tandon School of Engineering, Brooklyn NY York University Abu Dhabi, Saadiyat Island, P.O. Box 129188, Abu Dhabi, U.A.E. Abstract This paper proposes a position weighted backpressure (PWBP) control policy for network traffic. This is a decentralized policy that considers the spatial distribution of vehicles along the length of network roads to account for spill-back dynamics. The approach can be implemented in real time, without a priori knowledge of network traffic demands, and can scale to large urban networks. The capacity region of the network is formally defined in the context of urban network traffic and it is proved, when exogenous arrival rates are within the capacity region, that PWBP control is network stabilizing. We demonstrate that PWBP can outperform adaptive signal control taken from a real-world setting. We perform comparisons against fixed signal timing, standard backpressure, and the capacityaware backpressure control policies using a calibrated microscopic simulation model of Abu Dhabi Island in the United Arab Emirates. We demonstrate the superiority of PWBP over the three other policies in terms of capacity region, network-wide delay, congestion propagation speed, recoverability from heavy congestion (outside of the capacity region), and response to incidents. Keywords: Decentralized control, backpressure, stochastic traffic flow, urban networks, intersection control 1. Introduction Various approaches have been proposed to optimize signal timing for isolated intersections, including mixed-integer linear models, rolling hori∗ Corresponding author, e-mail: sej7@nyu.edu Preprint submitted to Elsevier February 7, 2022 zon approaches, and store-and-forward models based on model predictive control; see (Yann et al., 2011; Gartner, 1983; Tettamanti and Varga, 2010; Mirchandani and Head, 2001; You et al., 2013; Ma et al., 2013) for examples. On the one hand, isolated intersection approaches fail to account for spillback from adjacent road segments, which can eventually lead to gridlock throughout a road network (Cervero, 1986). On the other hand, centralized techniques that include coordination between intersection controllers (Heung et al., 2005; Gettman et al., 2007) are not scalable and difficult to implement in real-world/real-time settings (Papageorgiou et al., 2003). For example, ACS-Lite (Gettman et al., 2007) can handle no more than 12 intersections in real-time. Recent articles in traffic control focus on connected-automated vehicles (CAVs). This is well-justified considering the vast opportunities and challenges that CAVs have to offer. The ability to control both trajectories and signals is one such opportunity that CAVs have to offer; we refer to (Li and Zhou, 2017; Yu et al., 2018; Feng et al., 2018) and references therein for recent examples. For more information, we refer to a recent review article that covers many aspects of intersection control: (Guo et al., 2019). Decentralized control techniques have been proposed to overcome the scalability issues associated with network control optimization. These techniques expect intersection controllers to be able to measure/estimate local traffic information in real-time. This information includes expected traffic demand at the intersection in the next cycle for heuristic approaches, e.g., (Smith, 1980; Lämmer and Helbing, 2008, 2010; Smith, 2011) or the queue sizes along the intersection arcs in max pressure based approaches (Wongpiromsarn et al., 2012; Varaiya, 2013; Xiao et al., 2014; Le et al., 2015). According to (De Gier et al., 2011), control strategies that use traffic conditions along both upstream and downstream arcs are more efficient and reliable than those that utilize upstream traffic conditions only. Among the decentralized techniques, backpressure (BP) based approaches serve as examples of techniques that utilize both upstream and downstream information. They were first independently proposed in (Wongpiromsarn et al., 2012) and (Varaiya, 2013) based on seminal work in communications networks (Tassiulas and Ephremides, 1992) (see (Neely et al., 2005; Georgiadis et al., 2006; Neely, 2010) for more details). In general, BP based approaches are scalable and come with theoretical guarantees of network stability. However, as it was originally developed for packet queueing in communications networks, the assumptions are not tailored to traffic problems and in some cases the assumptions are not suitable for traffic networks. Specifically, these assumptions include: (i) point 2 queues and (more critically) queue with infinite buffer capacities, (ii) separate queues for different commodities (corresponding to vehicles with different turning desires in traffic) and no interference between commodities. A consequence of the first assumption is that the models do not account for the spatial distribution of the queues, which has great impact on traffic control. For instance, Fig. 1 illustrates three different spatial distributions of vehicles with the same queue size. Clearly, signal control decisions at the downstream end should be very different for these three cases. A key point here is how vehicle flux out of road segments are affected by the vehicle distribution along the length of the road. While communications networks assume that such “transmission rates” will not be influenced by the distribution of packets along the channels, in vehicular traffic the situation is quite different. Point queueing techniques suffer this same drawback. For example, the flow rate over the course of a short time interval (e.g. 10 sec) at the downstream end of the road segment depicted in Fig. 1 should be very different in the three cases. A serious consequence of assumption (i) Queue  Build-­‐up  Dynamics Dense  Traffic Free-­‐flowing  Traffic Queue  Dissipation  Dynamics Free-­‐flowing  Traffic Dense  Traffic Queue  Build-­‐up  and  Dissipation  Dynamics Dense  Traffic Free-­‐flowing  Traffic Dense  Traffic Fig. 1: Three different spatial distributions of queues with same queue size. is loss of work conservation, in which flow is prohibited across the intersection despite the availability of (spatial) capacity in the outbound roads. Fig. 2 shows three cases in which BP control favors the eastbound approach (Q a to Qb ), despite the fact that flow rates will be zero along this approach if given priority. Recognizing the finite (spatial) capacity issue, (Gregoire et al., 2015) proposed an improvement, referred to as capacity aware back pressure (CABP) control. However, due to failure to account for the queue’s 3 spatial distribution, their approach can only avoid the case illustrated in Fig. 2a, but not the two depicted in Fig. 2b and Fig. 2c (in the former, the downstream queue is concentrated at the ingress of the road segment). As- (a) (b) (c) Fig. 2: Three non-work conserving cases (adopted from (Gregoire et al., 2015) and reproduced) sumption (ii) could be easily violated in traffic networks, e.g., shared lanes. Even when there are no shared lanes, road widening near the exits of inter4 section inbound roads, a very common geometrical features in urban networks, can create bottlenecks at the lane-branching point. Different turning movements (commodities) interact at the bottleneck, and one queue may block another if it gets too long as illustrated in Fig. 3. Work conservation may also be violated here, as traditional BP (and CABP) control would favor the through movement, despite the fact that no through vehicles can actually cross the intersection. This, in fact, serves as one physical mechanism that can lead to the scenario depicted in Fig. 2c. Loss of work conservation is a result of zero outflow if the through movement is given priority and the prime culprit is the fact that the spatial distribution of vehicles is not taken into consideration. Fig. 3: Bottleneck at the lane-branching point. This paper proposes decentralized intersection control techniques that apply macroscopic traffic theory to overcome the issues described above. We refer to this approach as position-weighted backpressure (PWBP). PWBP considers the spatial distribution of vehicles along the road, applying higher weights to queues that extend to the ingress of the road, thereby accounting for the possibility of spillback. Rates of flow across the intersection depend on both the control (signal status) and vehicle densities profiles (spatial distribution) along the inbound and outbound roads, capturing diminished flow rates at signal phase startups (startup lost times). We perform comparisons in isolated intersection settings against a real-world implementation of SCOOT as well as in a network setting against fixed intersection control, 5 standard BP, and CABP. We demonstrate superiority of PWBP in terms of capacity region, delay, congestion propagation speed, recoverability from heavy congestion and response to an incident. The type of control proposed is applied to intersection signal control (modern day traffic lights), but it can also be thought of as a prioritization scheme for connected vehicles at network intersections that can guarantee network stability. In both cases, when accurate measurement of the distribution of vehicles along the roads is not possible, one may employ a light-weight traffic state estimation technique. We refer to (Jabari and Liu, 2013; Seo et al., 2017; Zheng et al., 2018) for recent examples. The remainder of this paper is organized as follows: Sec. 2 describes the traffic dynamics model, macroscopic intersection control, and the proposed PWBP control policy. Sec. 3 rigorously demonstrates the network-wide stability properties of the PWBP approach using Lyapunov drift techniques. A comparison with adaptive control at the isolated intersection level and simulation experiments at the network level are provided in Sec. 4. Sec. 5 concludes the paper. 2. Problem formulation 2.1. Notation Consider an urban traffic network represented by the directed graph G = (N , A), where N is a set of network nodes, representing intersections and A ⊂ N × N is a set of network arcs, representing road segments. Each element of A is in one-to-one correspondence with an ordered pair of elements in N . For each node, n ∈ N , Πn and Σn denote, respectively, the set of (predecessor) arcs terminating in n and the set of (successor) arcs emanating from n. We also use Π( a) ⊆ A to denote the set of predecessor arcs to arc a ∈ A. That is, if n is the ingress node of arc a, then Π( a) = Πn . Similarly, Σ( a) is the set of successor arc to arc a. Fictitious source arcs are appended to the network to represent exogenous network arrivals. A new junction with indegree zero and outdegree one is created for each exogenous inflow and the new source arc connects this new node to the network boundary node; see Fig. 4. When exogenous inflows occur at the interior of the network (i.e., at a junction with non-zero in-degree) representing, for example, a parking ramp/lot, the associated arc can be broken into two arcs with a new node placed at the position of the merge; see Fig. 5. Source arcs will be assumed to have infinite jam densities (i.e., they serve as fictitious reservoirs), but the flow rates in and 6 Boundary   junction Fictitious   (source)  arc Fig. 4: Fictitious boundary source arcs Arc  with  exogenous   arrivals Fictitious   (source)  arc Fig. 5: Fictitious interior source arcs out of these arcs are assumed to be finite (i.e., finite capacities). They shall also be assumed to have zero physical length. Therefore, the traffic states associated with fictitious source arcs are point queues concentrated at the source node. We shall denote the set of (fictitious) source arcs by Asrc ⊂ A. Arcs in Asrc serve two purposes: the first (mentioned above) is to model exogenous network inflows. The second is to capture instabilities in the network: Roads have finite spatial capacities and traffic densities are always finite. Source arcs with infinite storage capacities are capable of capturing network instabilities. For example, a signal control policy that results in instabilities is one where congestion propagates to the source arcs and builds up there and exogenous arrivals can no longer be accommodated. 2.2. Stochastic arc dynamics We denote the length of each arc a ∈ A by la . With slight notation abuse, the upstream-most position (the entrance position) for each arc a in 7 the network is x = 0, while the downstream-most position (the arc exit position) is x = la (that these coordinates pertain to arc a only should be understood implicitly). We consider a multi-commodity framework, where ρba ( x, t) denotes the traffic density at position x along arc a that is destined to outbound arc b ∈ Σ( a) at time instant t. Similarly, qba ( x, t) denotes the flow rate at x along a that is destined to b at time t. We define the state of the system at time t as the vector of commodity-specific network traffic densities. This is denoted as1 ρ(t) ≡ {ρba (·, t)}(a,b)∈M . On the interiors of network arcs, we have the following conservation equation: for each a ∈ A and b ∈ Σ( a) ∂ρba ( x, t) ∂qb ( x, t) =− a x ∈ (0, la ), t ≥ 0. ∂t ∂x (1)
In a first order context, one sets qba ( x, t) ≡ φab ( x, t)Q a ρ a ( x, t) , where φab ( x, t) is the fraction of vehicles at position x along arc a that is destined to arc b at time t and Q a is a (stochastic) stationary flow-density relation pertaining to arc a. In a higher order context, qba ( x, t) = ρba ( x, t)v a ( x, t), where v is the macroscopic speed and ∂ dv a ( x, t) ∂  = + v a ( x, t) v a ( x, t) dt ∂t ∂x

∂ρ a ( x, t) ∂V a ρ a ( x, t)  loc , , = A a ρ a ( x, t), V a ρ a ( x, t) , ∂x ∂x (2) where Aloc a are ‘local’ macroscopic acceleration models (Treiber and Kesting, 2013, Chapter 9) and V a is a stationary stochastic speed-density relation. The stochasticity in Q a and V a is parametric, that is, they can be described as generalizations of equilibrium fundamental relations that capture heterogeneity in the driving population as described in (Jabari et al., 2014). For example, a generalization of Newell’s simplified relation (Newell, 2002):  Q a (ρ) = min v a ρ, wa (ρ − ρ a ) , (3) is one where the parameters v a , wa , and ρ a , denoting free-flow speed, backward wave speed, and jammed traffic density, respectively, are random 1 We use the ‘·’ notation as a function argument to represent the entire curve in the dimension in which it is used. In other words, ρba (·, t) denotes the traffic density curve along arc a destined to adjacent arc b at time instant t. 8 variables. We refer to (Jabari et al., 2018; Zheng et al., 2018) for the properties of the stochastic dynamics that arise as a result of a parametric treatment. Remark 1. We make no assumptions about the relationship between flux and density. The proposed approach is equally valid in first and second order contexts. The only assumptions we make are (i) flow conservation, (ii) probabilistic upper bounds on flux and density, and (iii) that arc parameters do not change along the length of the arc. The last assumption is easy to honor in a general network by splitting arcs with varying parameters into more than one arc. 2.3. Boundary dynamics and junction control At the arc boundaries, i.e., for x ∈ {0, la }, we employ a node model. Node models represent the coupling between adjacent arcs and are responsible for capturing queue spillback dynamics. Notation. For each node n ∈ N , let Mn denote the set of allowed movements between inbound and outbound road segments. The set Mn consists of ordered pairs ( a, b) such that a ∈ Πn and b ∈ Σn , i.e., Mn ⊆ Πn × Σn . The set of all network movements is denoted by M ≡ M1 ⊔ · · · ⊔ M|N | . A signal phase consists of junction movements that do not conflict with one another. We denote by Pn ⊆ 2Mn the set of allowable phases and by P ⊆ ⊗n∈N Pn the set of allowable network phasing schemes. Essentially, an allowable phase is one that does not allow crossing conflicts and only allows merging conflicts between a protected movement and a permitted movement. Example allowable phases are depicted in Fig. 6. Fig. 6: Example phases for a four-leg isolated intersection. Exogenous arrivals. For (fictitious) source arcs, we assume random arrivals; for commodity a ∈ Asrc and b ∈ Σ( a), let Aba (t) be a random (cumulative) arrival process with (instantaneous) rate λba (t) = E 9 dAba (t) dt . We, thus have that2 dAba (t) for a ∈ Asrc , b ∈ Σ( a). (4) dt Junction control. Let p a,in (t) and p a,out (t) denote the upstream and downstream control state at boundaries of arc a. The control state at time t is defined as the set of movements that are active at time t as implied by the active phases at the network junctions. The boundary flows are given by

b b b b q a (0−, t) = q a,in p a,in (t) and q a (la +, t) = q a,out p a,out (t) , where qba,in and qba,out are boundary flux functions, which depend on the (boundary) control variables and, implicitly, on the node dynamics (for instance, qba,in and qba,out cannot exceed local supplies
and demands at the arc boundaries).
We denote by q a,b pb,in (t) or equivalently q a,b p a,out (t) the rate of flow that departs arc a ∈ Π(b) into arc b at time t. These are related to the commodity flows at the arc boundaries as follows:

qc,a p a,in (t) (5) qba,in p a,in (t) = π a,b (t) ∑ qba (0−, t) = c∈Π( a):(c,a)∈M and

qba,out p a,out (t) = q a,b p a,out (t) , (6) where π a,b (t) is the percentage of flow into a at time t that is destined to adjacent arc b ∈ Σ( a). In the context of signalized urban networks, it was demonstrated in (Jabari, 2016) that the node coupling, represented by movement flows, is given uniquely by


 (7) q a,b p a,out (t) = 1 {(a,b)∈ pa,out (t)} min δa,b ρba (la +, t) , σb ρb (0, t) , where 1 {(a,b)∈ pa,out (t)} is an indicator function that returns 1 if the movement ( a, b) belongs to the phase p a,out (t) and returns 0 otherwise, δa,b is a commodity-specific (local) demand function that depends on the traffic density at the egress of arc a, σb is a (local) supply function that depends on the total traffic density at the ingress of arc b: ρb (0, t) = ∑ ρcb (0, t). (8) c:(b,c)∈M processes Aba may have jumps. To be more accurate in such situations, one defines Rt Rt Λba (t1 , t2 ) ≡ E t12 Aba (t)dt = t12 λba (t)dt. The boundary flux is then given by qba (0−, t−) = R t+∆t b ′ ′ Rt A a (t )dt . lim t−∆t Aba (t′ )dt′ and qba (0−, t+) = lim t 2 The ∆t↓0 ∆t↓0 10 Note that we adopt modified demand functions in order to account for startup lost times; see (Jabari, 2016) and references therein for more details. The local demand and supply functions are derived from the stationary
flowdensity relations Q a . Thus, the source of randomness in qba,in p a,in (t) and
qba,out p a,out (t) is also parametric (i.e., the stochasticity is driven by the random parameters). Finally, at the arc boundaries the conservation law (1) is given, for a ∈ A/Asrc , by
 b ∂qba ( x, t) ∂ρba ( x, t) q a,in p a,in (t) − qba (0, t)
x = 0 (9) =− = qba (la , t) − qba,out p a,out (t) x = la ∂t ∂x and for a ∈ Asrc by
dρba (t) dqb (t) dAba (t) =− a = − qba,out p a,out (t) . dt dx dt (10) 2.4. Network capacity region Under any network-wide phasing scheme, p ∈ P , the traffic network can “support” arrival processes with certain rates. Beyond these arrival rates, queues along the source arcs will grow indefinitely. For each p ∈ P , we say that the network can support an arrival rate vector λ( p) = [λ a ( p) · · · λ|A| ( p)]⊤ if lim T →∞ ∑ a∈A 1 T Z T 0  λ a ( p) + q a,in ( p) − q a,out ( p) dt = 0, (11) where with slight abuse of notation, q a,in ( p) and q a,out ( p) are the inflow and outflow rates obtained when the network phasing scheme p is active. This is interpreted as follows: the phasing scheme p is such that the total arc outflow exceeds the total arc inflow in the long run. In accord with (11), each p ∈ P defines a set of admissible arrival rates; denote these (convex) polytopes by Ω( p). Taking the union of these sets, ∪ p∈P Ω( p), we get the vectors of all possible arrival rates that the network can support under all p ∈ P . This is formally defined next. Definition 1 (Maximal throughput region). The maximal throughput region (a.k.a., capacity region) of the network, denoted by Λ, is the convex hull of all sets of admissible flows. That is,
Λ ≡ Conv ∪ Ω( p) . (12) p∈P 11 Arrival rates that lie in Λ but not in ∪ p∈P Ω( p) are interpreted as arrival rates that can be supported by switching between phasing schemes that lie in the latter (i.e., time-sharing). A control policy that can support all possible arrival rates in Λ is referred to as a throughput-maximal control policy. We denote a control policy by a vector of network control states: at time t the network control state is denoted by p(t) ≡ [· · · p a,in (t) p a,out (t) · · · ]⊤ , a policy is an entire curve p(·). We give two examples to illustrate the notion of capacity region. The first is the simple isolated intersection of two one-way streets depicted in Fig. 7a. If Phase p is active, the maximum arrival rate that can be accommodated is Ω( p). In the example below, each phase consists of only one movement as shown in Fig. 7b. Consequently, the maximal arrival rate that can be accommodated when Phase 1 (Phase 2) is active is a singleton denoted Ω( p1 ) (Ω( p2 )). Ω( p1 ) would consist of the saturation flow rate of the Arc1-Arc3 (eastbound) movement and Ω( p2 ) would consist of the saturation flow rate of the Arc2-Arc4 (northbound) movement. The capacity region, depicted in Fig. 7c, is the set of maximal arrival rates (λ1 , λ2 ) that can be accommodated by switching between phases p1 and p2 . λ2 <latexit sha1_base64="(null)">(null)</latexit> Arc 4 Ω(p2 ) <latexit sha1_base64="(null)">(null)</latexit> ! " Λ = Conv Ω(p1 ) ∪ Ω(p2 ) <latexit sha1_base64="(null)">(null)</latexit> Arc 3 Arc 1 Arc 2 Phase 1 (p1 ) Phase 2 (p2 ) <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> λ1 <latexit sha1_base64="(null)">(null)</latexit> Ω(p1 ) <latexit sha1_base64="(null)">(null)</latexit> (a) (b) (c) Fig. 7: Example isolated intersection and the associated capacity region, (a) intersection layout (arcs 1 and 2 are source arcs), (b) the two possible phases, (c) the capacity region. The second example is borrowed from (Varaiya, 2013, Example 3) and illustrates how an instability forms in the proposed model, namely given that we consider finite spatial arc capacities. In this example, depicted in Fig. 8, there is one source arc. Hence, the capacity region is one dimensional. The initial conditions depicted in the figure are such that the network is in a state of gridlock at time t = 0. Moreover, the turning desires shown in the figure prevent all vehicles from moving into their desired destination arcs. In this case, the capacity region consists of the singleton set 12 λ = 0. That is, the maximal arrival rate the network can accommodate is zero. Any other arrival rate is outside of the capacity region and cannot be accommodated by any control policy, not BP, not CABP, not PWBP, nor any signal timing optimization technique. In such cases, the only way to relieve gridlock is to re-route vehicles; the subject of control+routing is beyond the scope of the present paper, we leave it to future research. Arc 4 ρ24 (x, 0) = ρ4 <latexit sha1_base64="(null)">(null)</latexit> ρ42 (x, 0) = ρ2 <latexit sha1_base64="(null)">(null)</latexit> Arc 1 Arc 2 ρ32 (x, 0) = 0 <latexit sha1_base64="(null)">(null)</latexit> Arc 3 π2,3 (0) = π2,4 (0) = 0.5 <latexit sha1_base64="(null)">(null)</latexit> π4,2 (0) = π1,2 (0) = 1.0 <latexit sha1_base64="(null)">(null)</latexit> Fig. 8: A gridlock scenario (adopted from (Varaiya, 2013) and reproduced). 2.5. Position-weighted back-pressure (PWBP) For any intersection n ∈ N , we assume that controllers are capable of assessing the (average) movement fluxes associated with
any possible phase p ∈ Pn . That is, for any ( a, b) ∈ Mn , E q a,b ( p) ρ(t) ≡ Eρ(t) q a,b ( p) is known or can be estimated by the controller (locally). Omitting depenρ( t ) dence of δa,b and σb in (7) to simplify notation, define Pa,b ≡ P δa,b − σb ≤
0|ρ(t) . Then  Eρ(t) q a,b ( p) = 1 {(a,b)∈ p} Eρ(t) min δa,b , σb   ρ( t ) ρ( t )
= 1 {(a,b)∈ p} Pa,b Eρ(t) δa,b + 1 − Pa,b Eρ(t) σb . ρ( t ) (13) Note that Pa,b , Eρ(t) δa,b , and Eρ(t) σb are deterministic functions of ρ(t) that depend on the distributions of the parameters of δa,b and σb . These distributions can be established empirically using historical data (Jabari et al., 2014). The splits π a,b (t) are also treated as random quantities that are to be estimated or measured. In a fully automated system, these random variables may degenerate, that is, it is easy to imagine that they can be measured with high accuracy and become deterministic quantities. In present day settings they need to be estimated. The setting envisaged in this paper is one with mixed automated/connected and traditional vehicles. Connected 13 vehicles announce their turning desires upon entering arc a and may serve as probes to allow the controller to estimate traffic conditions along the arc and the split variables. Empirical techniques may also be employed for this purpose; we refer to (Zheng and Liu, 2017) for a recent approach and to (Rodriguez-Vega et al., 2019) for a recent article on reconstructing turning movements. The traffic state at time t, ρ(t), requires a traffic state estimation procedure that is capable of producing real-time estimates under present day instrumentation in the real world. We refer to (Seo et al., 2017; Zheng et al., 2018; van Erp et al., 2018) and references therein for recent research on traffic state estimation tools. For each n ∈ N and each ( a, b) ∈ Mn , we define the weight variable wa,b (t) = c a,b Z la x 0 la ρba ( x, t)dx Z lb lb − x − lb 0 ∑ cb,c πb,c (t)ρcb ( x, t)dx , c∈Σ(b): (b,c)∈M (14) which depends on the (commodity) density curves along arcs a and b. To interpret this, first note that Z la 0 ρba ( x, t)dx (15) is just the total traffic volume (queue size) along arc a that is destined to arc b. Then the first integral inside the square brackets in (14) can be interpreted as a weighted queue size, where traffic densities at the downstream end of arc a (at x = la ) have the (maximal) weight of one, while traffic densities at the upstream end of a (at x = 0) have a weight of zero. In between, the weights increase linearly with x. Similarly, the second integral inside the square brackets in (14) can also be interpreted as a weighted queue size, but with the weights decreasing linearly with x. Hence, the weight associated with movement ( a, b) decreases as the traffic densities at upstream end (ingress) of arc b increase and vice versa, it increases when the traffic densities are concentrated at the downstream end of arc a and vice versa. The movement constants c a,b in (14) allow for assigning higher weights to certain movements. The phase that is active at node n at time t under PWBP control, denoted pn,PWBP (t), is given by pn,PWBP (t) ∈ arg max p∈Pn ∑ ( a,b)∈Mn 14 wa,b (t)Eρ(t) q a,b ( p). (16) Since the number of possible phases at any intersection tend to be small (typically four-eight), (16) can be easily solved by direct enumeration. This allows for real-time distributed implementation of the proposed approach. When there exists more than one solution to (16), a randomization procedure that applies equal weight to all the maximizers is employed. This helps ensure work conservation as discussed below. Implementation of PWBP control for node n at time instant t is summarized in Algorithm 1. Algorithm 1: Position weighted backpressure phasing for node n at time t: PWBP(n, t) Input: Road geometry: {la } a∈Πn , {lb }b∈Σn ; movement constants: {c a,b }( a,b)∈Mn ; (estimated) traffic state at time t: ρ(t); (estimated) splits at time t: {π a,b (t)}(a,b)∈M ; the distributions of the parameters of {δa,b }(a,b)∈Mn and {σb }b∈Σn Iterate: 1: for ( a, b) ∈ Mn do Rl 2: wa,b (t) ←[ c a,b 0 a x la ρba ( x, t)dx − R lb 0 lb − x lb ∑ c∈Σ(b): cb,c πb,c (t)ρcb ( x, t)dx 3: for p ∈ Pn do  4: Eρ(t) q a,b ( p) ←[ 1 {( a,b)∈ p} Eρ(t) min δa,b , σb 5: end for 6: end for Output:  7: if arg max p∈Pn ∑( a,b)∈Mn wa,b (t)Eρ(t) q a,b ( p) 8: (b,c)∈M = 1 then pn,PWBP (t) ←[ arg max ∑( a,b)∈Mn wa,b (t)Eρ(t) q a,b ( p) p∈Pn 9: else 10: Select pn,PWBP (t) at random from the set n arg max p∈Pn ∑ wa,b (t)Eρ(t) q a,b ( p) ( a,b)∈Mn o assigning equal probabilities to each element in the set. 11: end if One of the advantages of a continuous time formulation is that Algorithm 1 can be implemented at pre-specified cadence. Moreover, the cadence can vary from one intersection to another in order to accommodate such constraints as minimum greens (to avoid aggressive oscillations in the control dynamics), pedestrian movements, and so on. To elaborate, let τn denote the minimum phase length for node n. The signal phasing sequence is given by pn,PWBP (kτn ) = PWBP(n, kτn ) where k is a positive integer. An15 other advantage of the proposed approach is that it is decentralized; that is, the calculations can be parallelized over the network nodes. PWBP and work conservation. Work conservation of PWBP control follows from two features of the proposed approach. The first feature is the node model used: It was demonstrated in (Jabari, 2016) that the node model produces holding-free solutions. Hence, for any chosen phase p ∈ P (which dictates the node supplies), as long as there exist supply along the outbound arcs, demands at the inbound are guaranteed to be served. The second feature is that the phase chosen by PWBP depends on both the movement weights wa,b and the expected movement fluxes, Eρ(t) q a,b ( p). Since the weights are non-negative, the phase chosen is guaranteed to result in (holding-free) flow across the node as long as at least one of the movements has a non-negative expected flux, Eρ(t) q a,b ( p). Holding only occurs when Eρ(t) q a,b ( p) = 0 for all movements ( a, b) ∈ Mn . However, this is gridlock scenario and no work is lost. In the case where the expected fluxes are positive but the weights are zero, if all other expected fluxes are zero, work may be lost. This corresponds to an alternative scenario where max p∈Pn ∑ wa,b (t)Eρ(t) q a,b ( p) = 0. (17) ( a,b)∈Mn In this case, the randomization procedure ensures with probability 1 that loss of work does not persist. 3. Network stability 3.1. Lyapunov functional and stability The traffic network is said to be strongly stable if (Neely, 2010, Definition 2.7): Z   Z la 1 T |ρba ( x, t)|dx dt < ∞. (18) E lim sup ∑ 0 T →∞ T 0 ( a,b)∈M Since the network traffic densities depend (implicitly) on the control decisions at the network nodes, strong stability implies that the control in place ensure that the network densities do not grow without bound in the long run. This section demonstrates that as long as such a control policy exists3 , the PWBP algorithm ensures strong stability. 3 Otherwise, there does not exist a control policy capable of stabilizing the network. Hence, this is a feasibility assumption. 16 Since the spatial capacities of non-source arcs in the network are naturally bounded, the stability condition in (18) can be restated in terms of source movements only, that is, the traffic network is said to be strongly stable if Z 1 T lim sup (19) ∑ src E|ρba (t)|dt < ∞. T 0 T →∞ ( a,b)∈M:a∈A Consider the network-wide energy functional V : D → R with domain D being an appropriately defined |A|-dimensional set of curves. V is defined as Z la Z la
la − x − x′ b ′ 1 V ρ( t ) ≡ c a,b ρ a ( x , t)ρba ( x, t)dx ′ dx, (20) ∑ 2 (a,b)∈M l 0 0 a where {c a,b }(a,b)∈M are non-negative finite constants. It can be easily shown
that V is a Lyapunov functional: (i) V ρ(t) ≥ 0 almost surely since traffic densities are non-negative (with probability 1) and (ii) V (ρ) = 0 if and only if ρba ( x, t) = 0 almost surely for all ( a, b) ∈ M and all x ∈ [0, la ]4 . Lemma 1 below provides a sufficient condition for strong stability using the definition of Lyapunov functionals. A notational convention used below is V̇ ≡ dV/dt. For source nodes, la = 0 (no physical length); consequently, source arc commodities (i.e., when a ∈ Asrc ) contribute ρba (t)2 only to the Lyapunov function, where dependence on position is dropped as an argument since these are point queues.
Lemma 1. For the Lyapunov functional (20), suppose EV ρ(0) < ∞. If there exist constants 0 < K < ∞ and 0 < ǫ < ∞ such that Z la
|ρba ( x, t)|dx (21) Eρ(t) V̇ ρ(t) ≤ K − ǫ ∑ E ( a,b)∈M 0 holds for all t ≥ 0 and all possible ρ(t), then the traffic network is strongly stable. P ROOF. We first integrate both sides of (21) over the interval [0, T ] and take expectation of both sides of the inequality to obtain E Z T 0 E ρ( t )
V̇ ρ(t) dt ≤ KT − ǫ Z T 0 ∑ ( a,b)∈M E Z la 0 |ρba ( x, t)|dxdt. (22) 4 One can construct pathological density curves with non-zero density spikes, where V = 0. However, such densities occur with probability zero. Technically, these are overcome by using equivalence classes of density curves, but we shall avoid this level of technicality to promote readability. 17 Reversing the order of the (outer) expectation and the integral, the lefthand side becomes Z T 0 Z
EEρ(t) V̇ ρ(t) dt = T 0 Z
EV̇ ρ(t) dt = E
V̇ ρ(t) dt 0

= EV ρ( T ) − EV ρ(0) . (23) T Combining this with (22), we get the inequality

EV ρ( T ) − EV ρ(0) ≤ KT − ǫ Z T 0 ∑ ( a,b)∈M E Z la 0  |ρba ( x, t)|dx dt. (24)
Dividing both sides by Tǫ and noting that EV ρ( T ) ≥ 0 (by definition of V), we get the inequality 1 T Z T 0 ∑ ( a,b)∈M E Z la 0 
K 1 |ρba ( x, t)|dx dt ≤ + EV ρ(0) . ǫ ǫT (25)
Noting that EV ρ(0) < ∞, the result follows immediately upon taking the limit on both sides as T → ∞. 3.2. Stability of PWBP According to Lemma 1, finding (finite) constants K and ǫ that satisfy the condition (21) will ensure strong stability of the dynamics at the network level. The constant K is established using the boundedness of the fluxes q a (·, ·), which is a property of traffic flow (i.e., a physical property that must be ensured by any model). On the other hand, ǫ depends on the intersection control polices. Lemma 2 provides a necessary ingredient that will be used later to establish the constant K. Lemma 2. Let a ∈ A/Asrc and suppose there exist constants 0 ≤ q a < ∞ and 0 ≤ ρ a < ∞ such that P (qba ( x, t) ≤ q a ) = 1 and P (ρba ( x, t) ≤ ρ a ) = 1 for any ( a, b) ∈ M, any x ∈ [0, la ], and any t ≥ 0. Then, there exist constants (1) (2) 0 ≤ Ka < ∞ and 0 ≤ Ka < ∞ such that, with probability 1, (i) for any ( x1 , x2 ) ⊆ [0, la ] Z  Eρ(t) qba (la , t) x2 x1 and 18  (1) ρba ( x, t)dx ≤ Ka (26) (ii) for any ( x1 , x2 ) ⊆ (0, la ) and any ( x3 , x4 ) ⊆ [0, la ] − E ρ( t ) Z x4 Z x2 x3 x1 ρba ( x, t) ∂qba ( x ′ , t) ′ (2) dx dx ≤ Ka . ∂x (27) P ROOF. First, note that − E ρ( t ) Z x2 ∂qba ( x, t) dx = Eρ(t) qba ( x1 , t) − Eρ(t) qba ( x2 , t) ≤ q a ∂x x1 (28) with probability 1. By the boundedness properties of ρba (·, ·) for a ∈ A/Asrc , it holds that Z x  4 b P (29) ρ a ( x, t)dx ≤ la ρ a = 1. x3 Hence, with probability 1, we have that −E ρ( t ) Z x4 Z x2 x3 x1 ∂q a ( x ′ , t) ′ dx dx ∂x Z   Z x4 b ρ( t ) ρ a ( x, t)dx E =− ρba ( x, t) x3 x2 x1 ∂qba ( x, t) dx ≤ la ρ a q a . (30) ∂x The bound in (27) follows immediately and (26) follows from (29) along with the boundedness of qba ( x, t) and then applying the Cauchy-Schwartz inequality. Corollary 1. Let a ∈ A/Asrc and assume the probabilistic bounds of Lemma 2. Then, there exists a constant 0 ≤ K < ∞ such that  Z la   x b ρ( t ) b c E q ( l , t ) ρ ( x, t ) dx a,b ∑ a a la a 0 ( a,b)∈M:a6∈Asrc   Z la Z la − l − x − x ′ ∂qba ( x ′ , t) ′  a ρ( t ) b ρ a ( x, t) −E dx dx ≤ K. (31) la ∂x 0+ 0 P ROOF. Since |(la − x − x ′ )/la | ≤ 1 for all ( x, x ′ ) ∈ [0, la ] × (0, la ), it follows (1) (2) from Lemma 2 that there exists constants 0 ≤ Ka < ∞ and 0 ≤ Ka < ∞ for each a ∈ A/Asrc that bind each of the terms in the sums in (31) from above. Defining (1) (2)
(32) c a,b Ka + c a,b Ka , K≡ ∑ ( a,b)∈M:a6∈Asrc the result follows immediately. 19 Theorem 3.1 (Stability of PWBP). Assume that the boundedness conditions of Lemma 2 hold for all a ∈ A/Asrc , assume that arrival rates lie in Λ, that is, there exists a control policy p∗ (·) that can stabilize the network in the sense defined in Lemma 1, and let wa,b (·) for each movement ( a, b) ∈ M be as defined in (14). Then the policy wa,b (t)Eρ(t) q a,b ( p) ∀n ∈ N , ∑ pn,PWBP (t) ≡ arg max p∈Pn (33) ( a,b)∈Mn ensures strong stability of the traffic network. P ROOF. Eρ(t) V̇ ρ(t)
d 1 = Eρ(t) ∑ c a,b 2 dt ( a,b)∈M = 1 c a,b Eρ(t) 2 (a,b∑ )∈M ∑ =− Z la Z la la − x − x′ ρba ( x ′ , t)ρba ( x, t)dx ′ dx la 0 0 Z la Z la la − x − x′ ∂ c a,b Eρ(t) ( a,b)∈M 0 la 0 ∂t Z la Z la la − x − x′ 0 0
ρba ( x ′ , t)ρba ( x, t) dx ′ dx ρba ( x, t) la ∂qba ( x ′ , t) ′ dx dx, ∂x (34) where the last equality follows from (1). For each a, the integrals inside the expectation can be decomposed as ∂qba (0, t) ∂x Z la la − x 0 la ρba ( x, t)dx + + Z la Z la − 0 0+ Z ∂qba (la , t) la x b ρ ( x, t)dx ∂x la a 0 ∂qb ( x ′ , t) ′ la − x − x′ b dx dx. (35) ρ a ( x, t) a la ∂x Then from (9), we have that
Eρ(t) V̇ ρ(t) = ∑ ( a,b)∈M:a∈Asrc c a,b Eρ(t)  dAb (t) a dt 
Z c a,b Eρ(t) qba,in p a,in (t) 
ρba (t) − qba,out p a,out (t) ρba (t)  la − x b ρ ( x, t ) dx ∑ a la 0 ( a,b)∈M:a6∈Asrc Z la   la − x b ρ( t ) b − c E q ( 0, t ) ρ ( x, t ) dx a,b ∑ a a la 0 ( a,b)∈M:a6∈Asrc  
Z la x b ρ( t ) b − c E q p ( t ) ρ ( x, t ) dx a,out a,b ∑ a,out la a 0 ( a,b)∈M:a6∈Asrc + 20 la ∑ + ( a,b)∈M:a6∈Asrc −E ρ( t )  Z  c a,b Eρ(t) qba (la , t) Z la 0 Z la − la − x − x′ la 0+ 0 la  x b ρ a ( x, t)dx la ∂qb ( x ′ , t) ′  dx dx ρba ( x, t) a ∂x  . (36) Appeal to Corollary 1 and noting that the third sum on the right-hand side is non-negative (and does not involve control variables), we have that there e < ∞ such that exists a constant 0 < K Z l a

x b ρ( t ) ρ( t ) e ρ a ( x, t)qba,out p a,out (t) dx E V̇ ρ(t) ≤ K − ∑ c a,b E la 0 ( a,b)∈M  Z la 
dA a (t)  la − x b b − dx , (37) ρ a ( x, t) q a,in p a,in (t) + la dt 0 where qba,in ( p a,in (t)) ≡ 0 for a ∈ Asrc and dAba (t)/dt ≡ 0 for a ∈ A/Asrc . Rl Also, for a ∈ Asrc the traffic density is concentrated at x = 0, i.e., 0 a |(la − x )/la |ρba ( x, t)dx = ρba (t). Upon re-arranging terms on the right-hand side of (37) and utilizing the properties of conditional expectation, we have that  Z la

x b ρ( t ) ρ( t ) e ρ a ( x, t)E q a,out p a,out (t) E V̇ ρ(t) ≤ K − ∑ c a,b la ( a,b)∈M 0 
la − x b la − x dAba (t) dx. (38) q a,in p a,in (t) − c a,b − c a,b la la dt By assumption, we have that there exist constants 0 < K ∗ < ∞ and ǫ∗ > 0 associated with the policy p∗ (·) such that E ρ( t )
∗ V̇ ρ(t) ≤ K − ǫ ∗ ∑ ( a,b)∈M E Z la 0 |ρba ( x, t)|dx. (39) By definition, we have for each t ≥ 0 that ǫ ∗ ∑ Z la ( a,b)∈M 0 ρba ( x, t)dx ≤ max − c a,b p∈P ∑ Z la ( a,b)∈M 0 la − x b q a,in ( p) − c a,b la  x b q ( p) la a,out  la − x dAba (t) dx. (40) la dt ρba ( x, t)Eρ(t) c a,b e), we have by appeal to Lemma 1 that the Hence, setting K ≡ max(K ∗ , K control policy, p(·) ∈ P , which maximizes the right-hand side of (40) for each t ≥ 0 is also network stabilizing. 21 It remains to show that is equivalent to (33). We have from (40) that   Z la la − x b x b b ρ( t ) q ( p) − q a,in ( p) dx ρ a ( x, t)E arg max ∑ c a,b la a,out la 0 p∈P ( a,b)∈M (41) for each t ≥ 0 is network stabilizing. (The last term on the right-hand side of (40) is dropped from the optimization problem since it constitutes an additive constant to the problem.) From (5) and (6), the objective function can be written as   Z la la − x x b ρ( t ) q ( p) − ∑ πa,b (t)qc,a ( p) dx, ∑ ca,b 0 ρa (x, t)E la a,b la ( a,b)∈M c∈Π( a): (c,a)∈M (42) which upon re-arranging terms and the orders of summation and integration becomes  Z la x ∑ ca,b 0 la ρba (x, t)dx ( a,b)∈M  Z lb lb − x c − cb,c πb,c (t)ρb ( x, t)dx Eρ(t) q a,b ( p). (43) ∑ l 0 b c∈Σ(b): (b,c)∈M The latter is bounded from above by ∑ ( a,b)∈M c a,b Z la x la 0 − ρba ( x, t)dx Z lb lb − x 0 lb ∑ cb,c πb,c (t)ρcb ( x, t)dx Eρ(t) q a,b ( p). (44) c∈Σ(b): (b,c)∈M Since intersection movements do not interact across nodes instantaneously, the optimization problem naturally decomposes by intersection. That is, maximizing (44) is equivalent to solving the |N | problems arg max p∈Pn ∑ wa,b (t)Eρ(t) q a,b ( p) ∀n ∈ N , ( a,b)∈Mn 22 (45) where wa,b (t) ≡ c a,b Z la x 0 la ρba ( x, t)dx − Z lb lb − x 0 lb ∑ cb,c πb,c (t)ρcb ( x, t)dx . c∈Σ(b): (b,c)∈M (46) This completes the proof. 4. Experiments 4.1. Real-world isolated intersection experiment In this section we will compare the SCOOT and PWBP using a calibrated intersection. The reason we only use a single intersection for the comparison between SCOOT and PWBP is that we want to make our simulation precisely reproduce the demand and corresponding SCOOT signal timing in real world, which is difficult to achieve in a network. Nevertheless, the comparison of network will be conducted in the following section among PWBP, BP, CABP and fixed timing using different virtual demands. Fig. 9 shows the layout of the calibrated single intersection: the Hamdan Bin Mohammed Street - Fatima Bint Mubarak Street intersection in Abu Dhabi in the United Arab Emirates (UAE). This intersection uses SCOOT to optimize the signal timing in real world. We chose December 6, 2017 as a typical working day and calibrated the traffic demand every 1 min for the whole day, using high-resolution detector data. Then we use the calibrated demand as the demand input in our simulation. The real time SCOOT signal timing was also extracted from historical data and imported to our simulator. In this way, our simulation reproduced what was happening in real world. In the second step, by keeping the demand unchanged, we used the PWBP to determine the signal timing in place of SCOOT. Fig. 10a shows the number of vehicles passing the intersection under both SCOOT and PWBP control over a 24 hour period, and Fig. 10b shows their average delay. The figures demonstrate that PWBP outperforms SCOOT substantially in terms of delay under both low and high demands. The average delay over the entire day for SCOOT is 95 seconds, while the average delay of PWBP is only 35 seconds. In addition, when demand is high, PWBP control has higher throughput. 4.2. Network experiments Network description. We utilize a microscopic traffic simulation network of a part of the city of Abu Dhabi consisting of eleven signalized intersections but also containing unsignalized intermediate junctions. The 23 Fig. 9: Layout of the calibrated single intersection in Abu Dhabi network layout is shown in Fig. 11. We compare PWBP control with three other control policies: fixed time, standard BP control, and CABP control. The fixed timing plans are optimized and include optimal offsets (i.e., signal coordination). BP, CABP, and PWBP are all implemented using a software interface. To simplify the experiments, we utilize a uniform demand at the boundaries, which we vary to gauge the capacity region of the network. Using a uniform (average) demand level allows us to use a single number (namely the demand) as a way to gauge the capacity region. Average network delay and network capacity region. Fig. 12 shows the total network delay under different demand scenarios (ranging from 500 to 1800 veh/h on average) for BP, CABP, and PWBP using two types of phasing schemes: one with four phases (“4-phase” scheme) and a scheme with eight phases (“8-phase scheme”). The 4-phase scheme includes phases 1-4 in Fig. 6, while the 8-phase scheme is all eight phases in Fig. 6. We observe that 40 s/veh is a threshold delay, beyond which the delay increases dramatically. We can hence treat 40 s/veh as indicative of reaching the boundary of the capacity region. From Fig. 12, with the 8-phase scheme, 24 Number of vehicles passing the intersection 1000 SCOOT PWBP 900 800 700 600 500 400 300 200 100 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Time (h) (a) 200 SCOOT PWBP 180 Average delay (s/veh) 160 140 120 100 80 60 40 20 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Time (h) (b) Fig. 10: Comparison between SCOOT and PWBP in (a)flow, (b)delay. we see that delays begin to increase rapidly at a higher average demand levels for the PWBP: 1620 veh/h for the 8-phase scheme vs. 1580 veh/h for the 4-phase scheme. However, this is not the case for BP and CABP control, since they do not distinguish left-turning and through queues, which results in blocking at the points where roads widen (left-turn lane addition). This indicates that BP and CABP have a wider capacity region using a 4-phase scheme compared to the 8-phase scheme. All subsequent experiments use an 8-phase scheme with PWBP and 4-phase schemes with 25 150 140 130 120 110 100 90 80 70 60 50 40 30 20 10 0 PWBP_8 Phase PWBP_4 Phase Delay (s/veh) Delay (s/veh) Fig. 11: Simulation network in Abu Dhabi. 1580 1620 150 140 130 120 110 100 90 80 70 60 50 40 30 20 10 0 BP_8 Phase BP_4 Phase 1370 (b) CABP_8 Phase CABP_4 Phase Delay (s/veh) Delay (s/veh) (a) 150 140 130 120 110 100 90 80 70 60 50 40 30 20 10 0 1555 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800 Demand (veh/h) 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800 Demand (veh/h) 1400 1570 150 140 130 120 110 100 90 80 70 60 50 40 30 20 10 0 Fixed timing 1225 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800 Demand (veh/h) 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800 Demand (veh/h) (c) (d) Fig. 12: Delay patterns at varying demand levels for different control policies. BP and CABP. The demands at which delays begin to increase quickly for fixed signal timing, BP, CABP, and PWBP are 1225, 1555, 1570, and 1620 26 veh/h, respectively. Fig. 13 shows a comparison of network delays for the four control policies under varying demands. 160 Fixed timing BP CABP PWBP 140 Delay (s/veh) 120 100 80 60 40 20 0 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 Demand (veh/h) Fig. 13: Network delays associated with different control policies. Congestion propagation. In the following experiments, we set the demand levels to the deterioration bounds of the control policies and compare how congestion levels propagate over time. Since the deterioration bounds for BP and CABP are close, we just use CABP’s bound (1570 veh/h); we, hence, compare three demand scenarios. Fig. 14 – 15 show how the speeds of all vehicles within the network are distributed under demand levels 1225, 1570 and 1620 veh/h. The horizontal axes in these figures are time and the vertical axes are percentage of vehicles traveling at or below the color-coded speeds. Under the different demand levels, the network eventually becomes grid-locked (at different levels for the different control policies). Specifically, it takes about four hours until total network gridlock under a fixing timing plan when the demand reaches 1225 veh/h, under BP and CABP it takes approximately six hours (at 1570 veh/h) until gridlock, and for PWBP, it takes approximately seven hours. This indicates that PWBP is more resilient than the other policies. Fig. 16 shows how the total number of vehicles (stuck) in the network evolves with time. Recoverability from congestion. Fig. 17 shows how different control policies recover from congestion. The total simulation time is eight hours, the time interval from t = 120 min to t = 240 min is set as a congested period, during which demand levels are set to the deterioration bounds. We set a demand of 1000 veh/h for the remainder of the eight-hour simulation 27 100% 20 40 km/h 10 30 km/h 0.1 40% 60 60 50 km/h 40 km/h 60% 50 50 40 30 20 10 0.1 40 30 20 10 0.1 20 km/h 20% 10 km/h 0 50 100 150 200 Time (min) 250 0 km/h 0 50 100 100% 50 km/h 200 250 40 km/h 60% 50 30 km/h 60 60 0 km/h 60 km/h 50 km/h 80% Speed Percentile Speed Percentile 100% 60 km/h 80% 50 10 km/h (b) BP@1225vph 60 60 150 20 km/h Time (min) (a) FT@1225vph 40% 40 km/h 60% 50 50 40 40 30 20 10 30 km/h 40% 50 100 150 200 250 30 20 10 0.1 20% 10 km/h 0.1 0.1 0 20 km/h 40 30 20 10 40 30 20 10 20% 0 km/h 0 50 100 150 200 250 0 km/h (d) PWBP@1225vph 100% 60 km/h 50 km/h 40 km/h 0.1 30 km/h 40% 60 km/h 50 40 30 80% Speed Percentile 80% 60 60 50 km/h 20 50 10 40 km/h 60% 4 300 20 10 50040 20 3 0.1 2 10 0 10 km/h Time (min) (c) CABP@1225vph 100% 20 km/h 0.1 Time (min) Speed Percentile 30 km/h 40% 20% 60% 60 km/h 80% Speed Percentile 60% 100% 50 km/h 10 1 40 30 40 30 20 0. 80% Speed Percentile 60 km/h 50 50 10 0.1 0.1 30 km/h 40% 0.1 20 km/h 20% 10 km/h 0 50 100 150 200 250 300 350 20 km/h 20% 0 km/h 10 km/h 0 Time (min) 50 100 150 200 250 300 350 0 km/h Time (min) (e) FT@1570vph (f) BP@1570vph Fig. 14: Network speed evolution, (a) fixed timing under a demand level of 1225 veh/h, (b) BP under a demand level of 1225 veh/h, (c) CABP under a demand level of 1225 veh/h, and (d) PWBP under a demand level of 1225 veh/h, (e) fixed timing under a demand level of 1570 veh/h, (f) BP under a demand level of 1570 veh/h. 28 60 60 50 km/h 80% 0.1 40 km/h 40 30 km/h 0.1 0.1 100 150 200 250 300 350 0.1 0 km/h 0 50 100 0.1 150 250 300 20% 60 60 km/h 150 200 250 300 350 400 20 10 30 km/h 10 0.1 20 km/h 0 km/h 10 km/h 0 50 100 150 Time (min) 200 250 300 350 100% 0.1 60 km/h 60 60 50 0. 50 60% 40 0 3 20 40 km/h 10 30 km/h 0.1 20 km/h 20 km/h 20% 20% 10 km/h 0 40 0 3 20 40% 0.1 50 km/h 50 10 Speed Percentile 20 30 km/h 40% 80% 0.1 Speed Percentile 40 km/h 0.1 10 50 100 150 200 250 60 km/h 1 3 12000 50 40 50 km/h 40 30 0 km/h (d) BP@1620vph 50 60% 400 Time (min) (c) FT@1620vph 60 40 km/h 0.1 60% 20% 10 km/h 100 0 km/h 50 km/h 40 0 3 20 km/h 80% 350 50 40% 100% 20 km/h 2400350 0 0.1 30 km/h 40% 50 200 60 80% Speed Percentile 40 km/h 10 0.1 Speed Percentile 100% 60 km/h 50 km/h 0 0.1 (b) PWBP@1570vph 80% 60% 30 km/h Time (min) (a) CABP@1570vph 4300500 2 40 30 20 10 10 km/h Time (min) 100% 40 km/h 40 30 20 10 20% 10 km/h 50 50 50 60% 20 km/h 20% 0 60 km/h 50 km/h 40% 10 20 60 40 30 20 10 0.1 40% Speed Percentile 3040 20 0 1 60% 60 80% 50 30 Speed Percentile 100% 60 km/h 0.1 50 4 103020 100% 300 350 400 0 km/h 10 km/h 0 Time (min) 50 100 150 200 250 300 350 400 0 km/h Time (min) (e) CABP@1620vph (f) PWBP@1620vph Fig. 15: Network speed evolution, (a) CABP under a demand level of 1570 veh/h, (b) PWBP under a demand level of 1570 veh/h, (c) fixed timing under a demand level of 1620 veh/h, and (d) BP under a demand level of 1620 veh/h, (e) CABP under a demand level of 1620 veh/h, (f) PWBP under a demand level of 1620 veh/h. 29 Number of vehicles in the network 7000 Fixed timing BP CABP PWBP 6000 5000 4000 3000 2000 1000 0 0 50 100 150 200 250 300 350 400 450 300 350 400 450 300 350 400 450 Time (min) (a) Number of vehicles in the network 7000 Fixed timing BP CABP PWBP 6000 5000 4000 3000 2000 1000 0 0 50 100 150 200 250 Time (min) (b) Number of vehicles in the network 7000 Fixed timing BP CABP PWBP 6000 5000 4000 3000 2000 1000 0 0 50 100 150 200 250 Time (min) (c) Fig. 16: Evolution of total numbers vehicles in the network under different control policies and demand levels of (a) 1225 veh/h, (b) 1570 veh/h, and (c) 1620 veh/h. 30 60 Fixed timing BP CABP PWBP Delay (s/veh) 50 40 30 20 10 0 0 30 60 90 120 150 180 210 240 270 300 330 360 390 420 450 480 Time (min) (a) demand@1225vph 200 Fixed timing BP CABP PWBP 180 160 Delay (s/veh) 140 120 100 80 60 40 20 0 0 30 60 90 120 150 180 210 240 270 300 330 360 390 420 450 480 Time (min) (b) demand@1570vph 200 Fixed timing BP CABP PWBP 180 160 Delay (s/veh) 140 120 100 80 60 40 20 0 0 30 60 90 120 150 180 210 240 270 300 330 360 390 420 450 480 Time (min) (c) demand@1620vph Fig. 17: Average network delay under varying peak period demands. time. Fig. 17a, b and c only differ in the demand levels during the congested period. The congested period demand levels are 1225, 1570 and 1620 veh/h 31 in Fig.17a, b and c, respectively. According to Fig. 17, for all tested scenarios, PWBP outperforms the other three control policies in terms of both delay and recovery time. Even when the peak demand reaches 1620 veh/h, PWBP only needs 30 min to recover from the congestion, while fixed timing needs about 90 min to recover with a peak demand of 1225 veh/h. Note that when the peak demand reaches 1570 and 1620 veh/h, the delay levels under fixed timing becomes too high and hence cannot be shown in Fig. 17b and c. We also see that using fixed timing, the network does not eventually recover from congestion. Response to an incident. We investigate the performance of PWBP in the presence of an incident located at the yellow spot in Fig. 11. The incident is located half-way between intersections A and B, along a 3-lanes arc. We test scenarios where one lane and two lanes are blocked for a duration of one and two hours, and under different demand levels. Fig. 18 shows the results for one-lane blocked cases when demand is 1500 veh/h. Fixed 140 120 BP_NI CABP_NI BP_1h CABP_1h BP_2h CABP_2h PWBP_NI PWBP_1h PWBP_2h Delay (s/veh) 100 80 60 40 20 0 0 30 60 90 120 150 180 210 240 270 300 330 360 Time (min) Fig. 18: Delays associated with different policies with one lane blocked by the incident under a demand level of 1500 veh/h. timing is not included here since 1500 veh/h is beyond its capacity region and the delays will only increase without bound. Dotted lines represent the non-incident cases, while dashed and solid lines represent the incident cases with one and two hour durations, respectively. The incident starts at the 60th min in both cases. When the incident duration is one hour, we see that the network recovers within 30 minutes after the incident is cleared 32 under BP, CABP and PWBP. However, when the incident duration is two hours, PWBP only needs one hour to completely recover, while congestion in the network persists for significantly longer under BP and CABP: the effects of the incident are still felt in the network three hours after the incident is cleared (compared to the no-incident scenarios). Fig. 19 shows the two-lanes-blocked cases when demand is 1200 veh/h. The network fails to recover under fixed timing, BP and CABP control 90 Fixed timing_NI Fixed timing_1h 80 BP_NI CABP_NI PWBP_NI BP_1h CABP_1h PWBP_1h 70 Delay (s/veh) 60 50 40 30 20 10 0 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 Time (min) Fig. 19: Delays associated with different policies with two lanes blocked by the incident under a demand level of 1200 veh/h. when the incident blocks two of the three lanes. The delays increase sharply and the whole network becomes gridlocked. In contrast, using PWBP control the incident hardly has any impact at all on network delay. The reason of the performance difference between BP, CABP and PWBP originates from how the model deals with scenarios in Fig. 2b and Fig. 2c. With an incident located half-way between intersections A to B, the incident results in congested conditions (queueing) between the incident location and intersection A and low volume traffic between incident location and intersection B. When the queue spills back to intersection A (similar to Fig. 2b), PWBP will forbid the movements from A to B, while BP and CABP fail to capture the spillback dynamics. In addition, PWBP does not allocate green time at intersection B to the movement from A when there are actually no vehicle near the stop line (similar to Fig. 2c), while BP and CABP may still allocate green time to this movement. 33 5. Conclusion and outlook Backpressure (BP) based intersection control is a control policy that was originally developed for communications networks. Many of the assumptions made in the original theory were adopted in the BP applications to traffic networks despite them not being applicable to vehicular traffic. Specifically, infinite arc capacities, point queues, independence of commodities (turning movements), and there being no analogue for start-up lost times in communications networks. These are critical features in intersection control. To accommodate these features, we develop a backpressure control technique that is based on macroscopic traffic flow, which we refer to as position-weighted backpressure (PWBP). PWBP considers the spatial distribution of vehicles when calculating the backpressure weights. The proposed PWBP control policy is tested using a microscopic traffic simulation model of an eleven-intersection network in Abu Dhabi. Comparisons against coordinated and optimized fixed signal timing, standard BP, and a capacity-aware variant of BP (CABP) were carried out. The results indicate that PWBP can accommodate higher demand levels than the other three control policies and outperforms them in terms of total network delay, congestion propagation speed, recoverability from heavy congestion, and response to an incident. This paper has focused on prioritization of movements at network intersections. As a possible future research direction, this can be extended to include real-time route guidance. Another possible avenue for future research is a combined perimeter/interior control policy. Perimeter control (Yang et al., 2017; Chiabaut et al., 2018; Ortigosa et al., 2014; Ambühl et al., 2018; H. et al., 2018; Yang et al., 2018) is emerging as a useful tool for network control at a macroscopic level. A study of the trade-offs between the capacity region of an intersection control policy and perimeter control could serve as a powerful network-wide control tool. 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