Back-pressure traffic signal control with unknown routing rates
Jean Gregoire, Emilio Frazzoli, Arnaud de La Fortelle, Tichakorn Wongpiromsarn
BBack-pressure traffic signal controlwith unknown routing rates
Jean Gregoire ∗ Emilio Frazzoli ∗∗ Arnaud de La Fortelle ∗∗∗
Tichakorn Wongpiromsarn ∗∗∗∗∗
Mines ParisTech, Paris, [email protected]. ∗∗ Massachusetts Institute of Technology, Boston, [email protected] ∗∗∗
Mines ParisTech, Paris, France; Inria Paris-Rocquencourt, Francearnaud.de la [email protected] ∗∗∗∗
Thailand Center of Excellence for Life Sciences, [email protected]
Abstract:
The control of a network of signalized intersections is considered. Previous worksproposed a feedback control belonging to the family of the so-called back-pressure controls thatensures provably maximum stability given pre-specified routing probabilities. However, thisoptimal back-pressure controller (BP*) requires routing rates and a measure of the number ofvehicles queuing at a node for each possible routing decision. It is an idealistic assumption forour application since vehicles (going straight, turning left/right) are all gathered in the samelane apart from the proximity of the intersection and cameras can only give estimations ofthe aggregated queue length. In this paper, we present a back-pressure traffic signal controller(BP) that does not require routing rates, it requires only aggregated queue lengths estimation(without direction information) and loop detectors at the stop line for each possible direction. Atheoretical result on the Lyapunov drift in heavy load conditions under BP control is providedand tends to indicate that BP should have good stability properties. Simulations confirm thisand show that BP stabilizes the queuing network in a significant part of the capacity region.
Keywords: road traffic, traffic lights, traffic control, transportation control, queuing theory,back-pressure, network control.1. INTRODUCTIONIn today’s metropolitan transportation networks, trafficis regulated by traffic light signals which alternate theright-of-way of users (e.g., cars, public transport, pedes-trians). Congestion is a major problem resulting in a lossof utility for all users due to delayed travel times overthe network Shepherd (1992). That is why it is of highinterest to find a control policy that can stabilize a net-work of signalized intersections under the largest possiblearrival rates. Under traffic light control, a particular setof feasible simultaneous movements, called a phase, isdecided for a period of time Papageorgiou et al. (2003).Controlling a traffic light consists of designing rules todecide which phase to apply over time. Pre-timed policiesactivate phases according to a time-periodic pre-definedschedule, and the signal settings can be fixed by optimiza-tion, assuming within-day static demand Cascetta et al.(2006); Miller (1963); Gartner et al. (1975). They are notefficient under changing arrival rates which require adap-tive control. Many major cities currently employ adap-tive traffic signal control systems including SCOOT Huntet al. (1982), SCATS Lowrie (1990), PRODYN Henryet al. (1984), RHODES Mirchandani and Head (2001),OPAC Gartner (1983) or TUC Diakaki et al. (2002). Thesesystems update some control variables of a configurable pre-timed policy on middle term, based on traffic mea-sures. Control variables may include phases, splits, cycletimes and offsets Papageorgiou et al. (2003). More re-cently, feedback control algorithms that ensure maximumstability have been proposed both under deterministicarrivals Varaiya (2013), and stochastic arrivals Varaiya(2009); Wongpiromsarn et al. (2012). These algorithmsare based on the so-called back-pressure control presentedin the seminal paper Tassiulas and Ephremides (1992)for applications in wireless communication networks andrequire real-time queues estimation. An optimal back-pressure traffic signal controller (BP*) is presented inWongpiromsarn et al. (2012) and Varaiya (2009). They aredefined under different modelling assumptions but theyare algorithmically equivalent. The key benefit of back-pressure control is that it can be completely distributedover intersections, i.e., it requires only local informationand it is of O (1) complexity. However, the strong assump-tions of the model in Varaiya (2009) (and also implicitlyin Wongpiromsarn et al. (2012)) is that controllers requirerouting rates and a measure of the number of vehiclesqueuing at every node of the network for each possiblerouting decision. However, in reality, apart from the prox-imity of the intersection, vehicles (going straight, turningleft, turning right, etc.) are all gathered, and it is difficultto estimate the number of vehicles queuing for each di- a r X i v : . [ c s . S Y ] M a r ection (see Figure 1). Cameras can give good estimationsof the total number of vehicles queuing at a given node,but not the direction of vehicles. However, it is feasible todetect if there are some vehicles (or no vehicle) that wantto go to a given destination, if we assume the existenceof dedicated lanes from the proximity of the intersectionwith loop detectors at the stop line. Dedicated lanes indicated by road markings
Fig. 1. Dedicated lanes for turning vehicles. The dedicatedlanes are indicated by road markings when vehiclesapproach the intersection. Apart from the proximityof the intersection, vehicles are all gathered.The back-pressure control proposed in this paper(BP)requires such loop detectors and an estimation of the totalnumber of vehicles queuing at each node (gathering allpossible directions). It does not assume any knowledge ofrouting rates. We evaluate the performance of BP withregards to the optimal BP* control. The contribution ofthe paper is to provide a back-pressure traffic signal con-troller based on more realistic assumptions on the availablemeasurements than state-of-the-art back-pressure trafficsignal control and to show in simulations that stability isconserved in a significant part of the capacity region.The paper is organized as follows. Section 2 describes thequeuing network model. Sections 3 is mainly expository:it describes BP* highlighting its stability-optimality. Thecontributions of the paper are presented in Section 4 and5. Section 4 exhibits BP and a theoretical result on theLyapunov drift that tends to indicate that it should havegood stability properties. The simulations of Section 5confirm this and show that BP stabilizes the network in asignificant part of the capacity region. Section 6 concludesthe paper and opens perspectives.2. MODELAs standard in queuing network control, time is slotted,and each time slot maps to a certain period of time duringwhich a control is applied. It is convenient to use a fixedpre-defined time slot length, whose size corresponds tothe minimal duration of a phase. When the time slotsize is fixed, the traffic signal control problem consists ofcomputing at the beginning of each time slot t the phaseto apply during slot t . The network of intersections ismodelled as a directed graph of nodes ( N a ) a ∈N and links( L j ) j ∈L . Nodes represent lanes with queuing vehicles, andlinks enable transfers from node to node: this is a standardqueuing network model.It is a multiple queues one server queuing network. Everysignalized intersection is modelled as a server managing a N N Q Q C N C Q Q N C N C Q Q N C N C Q Q J i λ λ λ λ λ λ λ λ N C C Fig. 2. A junction with 4 incoming nodes and 4 outgoingnodes which corresponds to the intersection depictedin Figure 3.junction which consists of set of links. Junctions ( J i ) i ∈J are supposed to form a partition of links. For everyjunction J , I ( J ) and O ( J ) denote respectively the inputsand the outputs of J . Inputs (resp. outputs) of junction J are nodes N such that there exists a link L ∈ J pointing from (resp. to) N . The reader should consider theintroduction of junctions in the model as an overlay of thequeuing network model. For the sake of simplicity, we donot represent links in the queuing network representationof Figure 2.Every server maintains an internal queue for every in-put/output, and server work enables to transfer vehiclesfrom an input to an output of the junction. The internalqueue at node N a is a vector Q a and Q ab ( t ) denotesthe number of vehicles in the queue of node N a enter-ing N b upon leaving N a . The aggregated queue length Q a ( t ) = (cid:80) b Q ab ( t ) denotes the total number of vehiclesat node N a considering all possible routings after exiting N a . In this paper, queues are supposed to have infinitecapacities: there is no blocking (see Gregoire et al. (2013a)for an adaptation of back-pressure traffic signal control inthe context of finite capacities).At every time slot t , servers work, resulting in vehiclestransfers. Under phase-based control, the transmissionrate offered by servers are set by activating a given signalphase p i at each junction J i from a predefined finite set offeasible phases P i at every time slot t . Let P = (cid:81) i ∈J P i denote the set of feasible global phases. Each global phase p = ( p i ) i ∈J ∈ P results in a different service matrix µ ( p ) where µ ab ( p ) represents the transmission rate offeredby servers to transfer vehicles from N a to N b in a timeslot when phase p is activated. The transmission rate isassumed to be binary, µ ab ( p ) ∈ { , s ab } : it is zero or itequals the saturation rate s ab . Only the vehicles whichare at a node at the beginning of time slot t can betransferred from that node to another node during slot t . Figure 3 depicts the 4 typical phases of a 4 inputs/4outputs junction. N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N (a) (b)(c) (d) Fig. 3. A typical set of feasible phases at a junction. Forexample, supposing that service rates equal 0 or 1, thenon zero service rates for phase (a) are µ , µ , µ and µ Exogenous arrivals occur at every node of the network.Let A a ( t ) denote the number of vehicles that exogenouslyarrive at node N a during slot t . The arrival process A a ( t )is assumed to be rate convergent with long-term arrivalrate λ a ≥
0. When a quantity of vehicles arrives at node N a ∈ I ( J i ) during slot t , exogenously and endogenously, itis split and added into queues Q ab , b ∈ O ( J i ). The routingprocess is exogenous and assumed to be rate convergentwith ratios r ab with (cid:80) b r ab ≤ − (cid:80) b r ab represents the exit rate ofvehicles entering node N a , it is the ratio of vehicles directlyremoved from the network when entering node N a , i.e.not added to any queue Q ab . Note that the only variablethat is controlled is the activated phase at every time slot t , denoted by p ( t ), and yielding a service matrix µ ( p ( t ))during slot t . 3. BP* CONTROLLER In the following, we expose BP* signal control. It is an ex-tension of the algorithm proposed in Varaiya (2009) whereinternal/exit links are not differentiated, because exits mayoccur at any link of the network. It is quite equivalentto the back-pressure controller of Wongpiromsarn et al.(2012), assuming the nodes carry direction information.Loosely speaking, the idea of back-pressure control is tocompute pressure at every node based on node occupancyand to open flows which have a high upstream pressureand a low downstream pressure, like opening a tap.Algorithm 1 defines BP* control. At every junction i , foreach phase p ∈ P i , the weighted sum (cid:80) a,b W ab ( t ) µ ab ( p )is computed. W ab ( t ), the weight associated to transfersfrom N a to N b , is the difference between the upstream Algorithm 1
BP* control
Require:
Queues lengths matrix Q ( t ),Pressure functions P ab ( Q ab ) for all a, b ∈ N ,Routing matrix r . function BP* for i ∈ J dofor a ∈ I ( J i ) , b ∈ O ( J i ) do Π ab ( t ) ← P ab [ Q ab ( t )] end forfor a ∈ I ( J i ) , b ∈ O ( J i ) do W ab ( t ) ← max (Π ab ( t ) − (cid:80) c r bc Π bc ( t ) , end for p (cid:63)i ( t ) ← arg max p i ∈P i (cid:80) a ∈I ( J i ) ,b ∈O ( J i ) W ab ( t ) µ ab ( p i ) end forreturn Phase p (cid:63) ( t ) to apply in time slot t end function pressure Π ab ( t ) and the weighted downstream pressure (cid:80) c r bc Π bc ( t ). BP* consists of selecting the phase thatmaximizes the weighted sum. Moreover, we assume thatin case of equality the selected phase p ∗ ( t ) always satisfies µ ab ( p ∗ ( t )) = 0 if W ab ( t ) = 0. The following theorem states that under linear pressurefunctions with strictly positive slope, BP* as defined byAlgorithm 1 is optimal in terms of stability, i.e. stabilizesthe network for all arrivals rates that can be stabilizedconsidering all possible control strategies. It is an exten-sion of the results of Varaiya (2009), because vehicles canenter/exit the network at any node, there is no distinctionbetween exit nodes and internal nodes. Moreover, in con-trast with Varaiya (2009); Wongpiromsarn et al. (2012),pressure functions are just assumed to be linear withstrictly positive slope in this paper: P ab ( Q ab ) = θ ab Q ab , θ ab > Theorem 1. (Back-pressure optimality). Assuming thatpressure functions are linear with strictly positive slopes,BP* as defined by Algorithm 1 is stability-optimal.
Proof.
Due to space limitations, the full proof is not pro-vided in this paper and is available in the supplementarymaterial Gregoire et al. (2013b). Stability is proved usingthe Lyapunov function V ( t ) = V ( Q ( t )) = (cid:80) a,b θ ab Q ab ( t ) .The existence of B, η > E { V ( t + 1) − V ( t ) | Q ( t ) } ≤ B − η (cid:88) a,b Q ab ( t ), (1)enables to conclude stability for the queuing network usingthe sufficient condition proved in Neely (2003).4. BP CONTROLLER Back-pressure control proposed in Section 3 requires com-plete knowledge of the queues lengths matrix Q ( t ) and therouting rates. For our application, a complete knowledgeof Q ( t ) is not realistic because dedicated lanes for turningehicles are only from the proximity of the junction. Far-ther, all vehicles are gathered and the controller does nothave access to the direction of every vehicle in the absenceof vehicle-to-infrastructure communications. That is whywe propose in the present paper a controller that usesonly the aggregated queues lengths Q a ( t ) = (cid:80) b Q ab ( t ),i.e. a queue length without direction information. It isdefined by Algorithm 2. It computes the phase to ap-ply at every time slot without requiring neither routingrates nor complete knowledge of queues lengths matrix Q ( t ) and takes as inputs the aggregated queues lengths Q a ( t ) = (cid:80) b Q ab ( t ). However, it still requires vehicle de-tectors variables d ab ( t ) ∈ [0 ,
1] defined below: d ab ( t ) = min( Q ab ( t ) /s ab ,
1) (2)The variable d ab ( t ) can be measured by loop detectorspositioned at dedicated lanes. Algorithm 2
BP control
Require:
Queues lengths Q a ( t ),Pressure functions P a ( Q a ),Loop detectors variables d ab ( t ). function BP for i ∈ J dofor a ∈ I ( J i ) ∪ O ( J i ) do Π a ( t ) ← P a [ Q a ( t )] end forfor a ∈ I ( J i ) , b ∈ O ( J i ) do W ab ( t ) ← d ab ( t ) max (Π a ( t ) − Π b ( t ) , end for p (cid:63)i ( t ) ← arg max p i ∈P i (cid:80) a ∈I ( J i ) ,b ∈O ( J i ) W ab ( t ) µ ab ( p i ) end forreturn Phase p (cid:63) ( t ) to apply in time slot t end function Algorithm 2 defines BP control. Note that for transfersfrom N a to N b , the upstream pressure is now Π a ( t ) and thedownstream pressure is Π b ( t ): individual queue pressuresΠ ab ( t ) are not required. Moreover, the difference betweenthe upstream pressure and the downstream pressure ismultiplied by d ab ( t ) to form W ab ( t ). Hence, if at time slot t , there is no vehicle at N a going to N b , the weight W ab ( t )associated to transfers from N a to N b vanishes. Let us consider the Lyapunov function V ( Q ) and itsevolution through time V ( t ) defined below: V ( t ) = V ( Q ( t )) = (cid:88) a θ a Q a ( t ) = (cid:88) a θ a ( (cid:88) b Q ab ( t )) (3)Let us define heavy load conditions at time slot t as statesof the network such that if the right-of-way is given to anyindividual queue, it can be emptied at saturation flow, i.e.there are enough vehicles in the individual queue to ensuresaturation: ∀ a, b ∈ N , Q ab ( t ) ≥ s ab (4)The following theorem proves that under heavy load con-ditions the Lyapunov drift respects the sufficient conditionfor network stability if λ + (cid:15) ∈ Λ r , for sufficiently large (cid:15) . Theorem 2. (Lyapunov drift under heavy load conditions).Assume λ + (cid:15) ∈ Λ r , BP control as defined in Algorithm2 is applied and the network is in heavy load conditions,then there exists B, η > E { V ( t + 1) − V ( t ) | Q ( t ) } ≤ B − η (cid:88) a Q a ( t ) (5)for sufficiently large (cid:15) . Proof.
Due to space limitations, the full proof is not pro-vided in this paper and is available in the supplementarymaterial Gregoire et al. (2013b).The above theorem tends to indicate that the networkshould have good stability properties because the condi-tion for stability is verified in heavy load conditions for λ sufficiently interior to the capacity region. Unfortunately itdoes not enable to conclude that the network is stable in asignificant part of the capacity region. Indeed, heavy loadconditions can not be guaranteed at all time, and when anindividual queue Q ab is below the saturation flow s ab , it isa constraint for the emptying of Q a , that can unstabilizethe queuing network. Hence, the characterization of thestability region of the queuing network under BP controlwith the modelling assumptions presented in Section 2is still a challenging problem. That is why we proposeto implement the two back-pressure controllers and tocompare their behaviour. The results of the simulationsare presented in the next section.5. SIMULATIONS The model and the algorithms presented in this paperhave been implemented into a simulator coded in Java.It simulates a grid network and every junction of the gridhas 4 inputs, 4 outputs, and 4 feasible phases as depicted inFigure 3. The height and the width are parameters. Everyindividual flow allowed by phases of Figure 3 equals 10(it is the saturation rate). Vehicles are generated at eachnode N a at an arrival rate λ a that can be set as desired.The arrival process generates individual arrivals as well asbatches of 10 vehicles. The routing ratios are fixed at thebeginning of the simulation. Simulations have been carried out for a 21 ×
21 square gridnetwork (see Figure 4). First of all, we present simulationsresults in the case of a network that has been configuredwith the same arrival rates and routing rates at every nodeof the network.
Simulation results for a particular network and particulararrival/routing rates
The numerical results of Figure 5correspond to the following parameters. Turn left probabil-ity when a vehicle enters a node: 0 .
2; turn right probabilityig. 4. The 21 ×
21 grid network used for the presentedsimulations.when a vehicle enters a node: 0 .
2; go straight probabilitywhen a vehicle enters a node: 0 .
5; exit probability when avehicle enters a node: 0 .
1; probability of a batch: 0 . P a ( Q a ) = Q a and P ab ( Q ab ) = Q ab ( θ a = θ ab = 1); vehicles are generated at every node withthe same arrival rate λ > λ = 0.4, 0.5, 0.6, 0.65, 0.7, 0.75, 0.8 and 0.9 vehiclesper time slot. Figure 5 depicts the global queue of thenetwork over time, i.e. (cid:80) a Q a ( t ) = (cid:80) a,b Q ab ( t ), for theheight arrival rates, under BP* control and under BPcontrol. One can observe in Figure 5 that under BP*control, the queuing network is stabilized for λ ≤ . λ = 0 .
75. Under BP control, it isstabilized for λ ≤ .
65 and gets unstable from λ = 0 . . / . (cid:39) Evaluation of BP with regards to BP* on several samplesof parameters
In the following simulations, the rout-ing/arrival process parameters are not uniform over nodesany more. 10 samples of parameters have been generated.For each sample, the routing/arrival rates are generatedas follows. For each direction (straight, left, right), (uni-formly) random values between 0 and 1 are generated,say y s , y l , y r ; a (uniformly) random value between 0 and0.1 is generated for exits, say y ω ; and the routing ratesare set by normalization of the generated real values,i.e. for the left direction for example, the routing rateis y l / ( y s + y l + y r + y ω ). The arrivals rates are set bygenerating a (uniformly) random value between 0 and 1for every node, say λ a . At the beginning of the simulation,a parametrizable scaling value x enables to fix the actualarrival rate of the current simulation: λ a = xλ a , where x has the same value over nodes. The value of 0.1 for thescale of exits is quite arbitrary and, loosely speaking, fixes Fig. 5. Evolution of the global queue of the networkover time for height arrival rates. Comparison of thebehaviour of the network under BP/BP* control.the averaged number of travelled nodes before exiting thenetwork.Note that the routing rates and the values λ a are fixedfor a given sample. However, the value of λ a depends onthe value of x set at the beginning of the simulation. Theparameter x enables to define a performance for BP withregards to BP* for a given sample. We let x vary and weobserve the maximum value of x such that the network isstable under BP versus BP* (say x ∗ max for BP* and x BPmax for BP). We define the performance of BP with regardsto BP*, or more shortly the performance of BP (becauseBP* is optimal), as follows:performance(BP) = x BPmax /x ∗ max (6)As for previously presented simulations, the probabilityof a batch is 0.05 and the pressure functions are linearwith slope 1. Figure 6 depicts the performance obtained forthe 10 samples, the average performance and the standarddeviation. The average performance is around 80%, i.e.the optimality gap is about 20%. The simulation resultsprove that the performance of BP is affected by the rout-ing/arrival rates. Hence, the distribution (over samples) ofthe performance would be different for a different distribu-tion of routing/arrival rates. Nevertheless, in the particularsetting of the experiment, the average optimality gap of20% seems again a low price to pay with regards to theuch more realistic assumptions on the measurementsavailable to compute the control.Fig. 6. Performance distribution for ten samples. The pointabove the axis represents the average performanceover samples and the horizontal bar is the standarddeviation.However, these promising results can not be extended toany kind of network of intersections and further simula-tions with a more general structure of network shouldbe carried out to confirm the closeness of performance.We are currently implementing our algorithms in a trafficsimulator in order to test the performance of BP controlwith real traffic data of the city of Singapore.6. CONCLUSION AND PERSPECTIVESThe simulation results of this paper prove that BP isnot optimal but tend to indicate that it stabilizes thequeuing network in a significant part of the capacityregion. The benefits of BP originate from the more realisticassumptions on queues measurements. Computing thephase to apply only requires aggregated queues lengthsestimation that can be provided by cameras, and loopdetectors at dedicated lanes. The optimality gap, around20% in the particular setting of the experiments, seemsa low price to pay for the benefits of relaxed assumptionson the available measurements. However, simulations havebeen conducted in a grid network, which is a particularstructure, and with synthetic data which can stronglydiffer from real traffic data. To confirm the closeness ofperformance, simulations should be carried out in a moreadvanced traffic network simulator.Finally, the emergence of vehicle-to-infrastructure commu-nications opens avenues to enhance traffic signal control.The traffic signal controllers can have access, in particular,to the destination node of every vehicle. As a result, back-pressure control with a multiple-commodity queuing net-work model, as proposed in Neely (2003) in the context ofwireless communication networks, should be investigated.ACKNOWLEDGEMENTSThis work was supported in part by the Singapore Na-tional Research Foundation through the Future Urban Mo-bility Interdisciplinary Research Group at the Singapore-MIT Alliance for Research and Technology.REFERENCESCascetta, E., Gallo, M., and Montella, B. (2006). Modelsand algorithms for the optimization of signal settingson urban networks with stochastic assignment models. Annals of Operations Research , 144(1), 301–328.Diakaki, C., Papageorgiou, M., and Aboudolas, K. (2002).A multivariable regulator approach to traffic-responsivenetwork-wide signal control.
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