Analysis of a Markovian Queuing Model for Autonomous Signal-Free Intersection
AAnalysis of a Markovian Queuing Model forAutonomous Signal-Free Intersections
Hanwen Dai and Li Jin ∗† September 29, 2020
Abstract
We consider a novel, analytical queuing model for vehicle coordination at signal-free intersections. Vehicles arrive at an intersection according to Poisson processes,and the crossing times are constants dependent of vehicle types. We use this modelto quantitatively relate key operational parameters (vehicle speed/acceleration, inter-vehicle headway) to key performance metrics (throughput and delay) under the first-come-first-serve rule. We use the Foster-Lyapunov drift condition to obtain stabilitycriteria and an upper bound for average time delay. Based on these results, we comparethe efficiency of signal free intersections with conventional vehicles and with connectedand autonomous vehicles. We also validate our results in Simulation of Urban Mobility(SUMO).
Keywords:
Queuing systems, connected and autonomous vehicles, signal-free intersections
Modern connected and autonomous vehicle (CAV) technology has a strong potential forimproving urban traffic efficiency. A connected and fully autonomous vehicle is capable ofsensing the environment, communicating with other vehicles and infrastructures, and usingthe advanced control system to process the gathered information and decide its movementand navigation path [5, 8]. In particular, the CAV technology enables high-speed signal-freeintersections, where a centralized controller coordinates the movement of CAVs crossing anintersection with point-to-point guidance [13]. An efficient coordinate algorithm of intersec-tion control can reduce time delay and save energy for autonomous vehicles [10].In this paper, we study the performance of signal-free intersections in an analytical man-ner. We propose a Markovian queuing model for automated intersections. Our model is ∗ This work was in part supported by the C2SMART University Transportation Center, and the NYUTandon School of Engineering. † The authors are with the Tandon School of Engineering, New York University, Brooklyn, New York,USA. H. Dai is also with the Department of Industrial Engineering, Tsinghua University, Beijing, China.Emails: [email protected], [email protected] a r X i v : . [ ee ss . S Y ] S e p nalogous to M/D/ ∞ models but has two important and peculiar features for signal-freeintersections. First, we allow delayed parallel service. Second, we impose a switching delay.Vehicles cross the intersection on a first-come-first-serve (FCFS) basis. Using the queuingmodel, we provide a system stability criteria and an upper bound of the average vehicle timedelay. To validate our theoretical findings, we use Simulation of Urban Mobility (SUMO [7])to run experiments and make comparison. We find that our theoretical results match thesimulation results well.Figure 1: A signal-free intersection with CAVs.The merging of multiple traffic streams is a major component of traffic flow problems.Early in 1969, a unified approach was established to solve the problem of merging two ormore strings of high-speed vehicles into a single lane [1]; Dresner and Stone [4] proposeda reservation-based system for coordinating vehicles at intersections in 2004. A n -pathmerging problem can be regarded as a scheduling problem [2], which involves determiningthe merging sequence and vehicle trajectory with collision avoidance. Many researches usekinematic models that track vehicle position, velocity and acceleration/deceleration [2, 6,11, 12, 13, 14, 15, 16, 18]; at the more macroscopic level, Zhu and Ukkusuri proposed alane based traffic flow model to reduce control complexity at intersections [17]. Our queuingmodel stands between the kinematic model and the traffic flow model; our model capturesboth microscopic vehicle heterogeneity/interaction and macroscopic performance metrics.Related to Miculescu and Karaman [10]’s research work of extending the polling system tointersection control, our work gives the theoretical analysis of the two-queue polling systemwith constructing Lyapunov function and referring to Foster-Lyapunov criteria.We formulate the intersection control problem as a continuous time two-queue systemwith Poisson arrival process, constant crossing time depending on the vehicle class, and acooldown time between two consecutive vehicles. Unlike conventional queuing models, wedefine the system state as a tuple of the residual system time, the class, and the constantcrossing time of the last entered vehicle. Thus, our model is continuous-time and continuous-state, which cannot be analyzed using classical queuing-theoretic approaches. To addressthis, we proceed with an infinitesimal generator for the queuing processes, which is derivedbased on the theory of piecewise-deterministic Markov processes [3].We analyze the system stability with the help of the Foster-Lyapunov drift condition [9].2o achieve system stability, the proposed quadratic Lyapunov function should drift in thenegative direction as the residual system time of the last entered vehicle increases, whichcomes to our stability criteria. The drift condition also leads to an upper bound of theaverage time delay. We use the upper bound as a proxy for the exact delay, which is veryhard to compute.Finally, we run simulation by using SUMO to validate our theoretical findings. Sinceour queuing model makes some simplification and disregard some details, such as vehiclefollowing model and vehicle acceleration profile. SUMO is able to provide a more realisticintersection environment and vehicle interaction. Given vehicle flow input and running thesimulation, SUMO will return the average time delay to be compared with the theoreticalresults.The main contributions of this paper are as follows. First, we develop an analyticalqueuing model integrating the advantages from macroscopic traffic flow model and micro-scopic vehicle dynamics model. Second, based on our model, we provide the system stabilitycriteria and an upper bound of the average time delay. Third, we validate our modeling andanalysis approaches via microscopic simulation.The rest of this paper is organized as follows. In section 2, we formulate the signal-freeintersection as a polling system with two queues. We provide theoretical analysis for systemstability in section 3. In section 4, we validate our theoretical findings through simulationexperiments. We conclude the paper with remarks in section 5. We model a two-direction intersection as a system of two queues. We label the classes ofvehicles in these two directions as 1 and 2. Vehicles join each queue as Poisson processesof rate λ and λ , respectively. Every vehicle will experience two stages as it go throughFigure 2: The queuing model for a two-way intersection.the intersection, viz. queuing and crossing; see Fig. 2. The queuing occurs outside thecrossing zone, where vehicles queue and wait for discharge instruction from the rode-sideunit (RSU). After the RSU instructs a vehicle to enter the crossing zone, the vehicle will3raverse the crossing zone with a time S and at speed u . To capture the heterogeneity intraffic mixture, we assume that S are independent and identically distributed (IID) randomvariables with a discrete probability mass function (PMF) { p s ; s ∈ S} , where S is the set ofvehicle type-specific (e.g. cars, trucks, buses, etc.) crossing times.We model the crossing zone as ∞ servers with vehicle type-dependent crossing (service)times. Note that, although the crossing time is heterogeneous due to the heterogeneity ofvehicle types, it is known to the RSU before the crossing starts. However, these serversare not independent, and we can not let infinite vehicles enter the crossing zone; instead,the system of servers is subject to a cooldown time, and the number of vehicles crossingthe intersection will be constrained by the cooldown time. The cooldown time captures thesafety constraint on inter-vehicle headways. That is, if one server starts serving a vehicle(the leading vehicle) at time t , then any other server cannot start serving the next vehicle(the following vehicle) until the time t + c ; the cooldown time c is equal to θ if the leadingand following vehicles are from the same direction and is equal to θ if they are from distinctdirections. In general, θ ≥ θ ; this is similar to the switch-over time in [10]. In this paper,we assume that the cooldown time is smaller than the minimal crossing time.Note that the queuing model as described above would be not Markovian if we were touse the number of vehicles as the state variable. The reason is that the crossing time, whichresembles the service time in classical queuing models, is not exponentially distributed. Toresolve this, we consider the state ( X ( t ) , Y ( t ) , S ( t )), where X ( t ) ∈ R ≥ is the residual systemtime for the vehicle that last entered the system up to time t , and Y ( t ) ∈ { , } is the classof the vehicle that last entered the system up to time t and S ( t ) ∈ S is the crossing time ofthe vehicle that last entered the system up to time t .We assume that vehicles are scheduled to cross the intersection in a first-come-first-serve(FCFS) manner. Note that our modeling and analysis approach apply to alternative scheduleschemes as well. When a vehicle arrives at time t , Y ( t ) and S ( t ) are updated according tothe class of this vehicle. Then, X ( t ) is updated as follows: X ( t ) = X ( t − ) + θ + S ( t ) − S ( t − ) Y ( t − ) = Y ( t ) , X ( t − ) ≥ S ( t − ) − θ ,X ( t − ) + θ + S ( t ) − S ( t − ) Y ( t − ) (cid:54) = Y ( t ) , X ( t − ) ≥ S ( t − ) − θ ,S ( t ) otherwise, where S is a random variable with PMF p S ( s ). Between two vehicle arrivals, X ( t ) evolvesas follows: ddt X ( t ) = (cid:40) X ( t ) = 0 , − X ( t ) > . We can also compactly express the queuing dynamics with an infinitesimal generator L suchthat for any function g : R ≥ × { , } × S → R ≥ that is smooth in the continuous argument.If x ≥ s − θ , which means that when the current vehicle arrives, the cool-down timebetween this vehicle and the previous one is not over, then the arrived vehicle needs to waitfor (cid:0) x − ( s − θ ) (cid:1) , where θ = θ , if these two vehicles are in the same direction; θ = θ , if thesetwo vehicles are in distinct directions. By summing up the waiting time and the crossing4ime s (cid:48) , we can update x (cid:48) as x (cid:48) = x + s (cid:48) + θ − s : L g ( x, y, s ) = − g (cid:48) ( x, y, s )+ λ y (cid:16) (cid:88) s (cid:48) ∈S p s (cid:48) g ( x + s (cid:48) + θ − s, y, s (cid:48) ) − g ( x, y, s ) (cid:17) + λ − y (cid:16) (cid:88) s (cid:48) ∈S p s (cid:48) g ( x + s (cid:48) + θ − s, − y, s (cid:48) ) − g ( x, y, s ) (cid:17) . If s − θ ≤ x < s − θ , which means that if the arrived vehicle is in the same direction asthe previous one, then it can cross the intersection without waiting, otherwise it still need towait for (cid:0) x − ( s − θ ) (cid:1) . By summing up the waiting time and crossing time, we can update x (cid:48) : L g ( x, y, s ) = − g (cid:48) ( x, y, s )+ λ y (cid:16) (cid:88) s (cid:48) ∈S p s (cid:48) g ( s (cid:48) , y, s (cid:48) ) − g ( x, y, s ) (cid:17) + λ − y (cid:16) (cid:88) s (cid:48) ∈S p s (cid:48) g ( x + s (cid:48) + θ − s, − y, s (cid:48) ) − g ( x, y, s ) (cid:17) . If 0 ≤ x < s − θ , which means that when the current vehicle arrives, the cool-downtime between this vehicle and the previous one is over, and the arrived vehicle can cross theintersection without waiting. Therefore x (cid:48) is updated as x (cid:48) = s (cid:48) : L g ( x, y, s ) = − I x> g (cid:48) ( x, y, s )+ λ y (cid:16) (cid:88) s (cid:48) ∈S p s (cid:48) g ( s (cid:48) , y, s (cid:48) ) − g ( x, y, s ) (cid:17) + λ − y (cid:16) (cid:88) s (cid:48) ∈S p s (cid:48) g ( s (cid:48) , − y, s (cid:48) ) − g ( x, y, s ) (cid:17) . For the above equations, λ − y denotes λ k such that k ∈ { , } and k (cid:54) = y , and s denotesthe crossing time of the last arrived vehicle, while s (cid:48) denotes the crossing time of the currentarriving vehicle. In this section, we quantify the throughput of signal-free intersections by studying stabilityof the queuing model. We define throughput based on the notion of stability. We saythat the queuing system is stable if there exists
Z < ∞ such that for any initial condition( x, y, s ) ∈ R ≥ × { , } × S , D ( x, y, z ) ≤ Z , where D ( x, y, z ) denotes the average time delay.The main result of this paper states when such an upper bound Z exists and if yes, howto quantify it. Theorem 1
Consider an intersection with arrival rates λ , λ in two directions, mean cross-ing time ¯ S , offset time θ , and switch-over time θ . Then, the queues are stable under the CFS sequencing policy if max { λ , λ } ( θ − θ ) + ( λ + λ )( θ + s − s min ) < . (1) Furthermore, if the above holds, then the average time delay ¯ D , which is the actual averagesystem time minus the theoretical minimum system time, is upper-bounded by ¯ D ≤ ( λ + λ ) (cid:80) s (cid:48) ∈S p s (cid:48) s (cid:48) − ( λ + λ )( θ + s − s min ) − max { λ , λ } ( θ − θ ) . (2)We can present the stability criteria (1) by plotting the border lines, where max { λ , λ } ( θ − θ )+( λ + λ )( θ + s − s min ) = 1. As shown in figure 3, the red lines in the heat maps representthe border lines, and the area under the red lines satisfies our stability criteria. The upperbound of average time delay can be presented in a 2-D line plot by letting λ = λ . As shownin figure 4, the red dashed line represents the theoretical upper bound of the conventionalsignal-free intersections, and the gray dashed line represents that of the CAV signal-freeintersections, with equal arrival rate in each direction. Since all these figures combines thetheoretical results with the experiment results, we will illustrate them in details in section 4.The main proof idea is that we first construct the function g ( x, y, s ) as a quadraticLyapunov function V ( x, y, s ) = | x | . The infinitesimal generator L is piece-wise accordingto x , and it becomes linear for x ≥ s max − θ . Hence a negative slope − c < ≤ x < s max − θ , we prove the piece-wise function is bounded by d . In all, we prove that L V ( x, y ) ≤ − cx + d, ∀ ( x, y, s ) ∈ X × Y × S , and according to[9, Theorem 4.3] the average time delay D is upper-bounded by d/c . The detailed proof isattached below. Proof.
Consider the Lyapunov function V ( x, y, s ) = 12 | x | . Then we have:If x ≥ s − θ : L V ( x, y, s ) = (cid:16) − λ y ( θ + s − s ) + λ − y ( θ + s − s ) (cid:17) x + 12 λ y (cid:88) s (cid:48) ∈S p s ( θ + s − s ) + 12 λ − y (cid:88) s (cid:48) ∈S p s ( θ + s − s ) , If s − θ ≤ x < s − θ : L V ( x, y, s ) = (cid:16) − λ − y ( θ + s − s ) (cid:17) x − λ y x + 12 λ y (cid:88) s (cid:48) ∈S p s (cid:48) s (cid:48) + 12 λ − y (cid:88) s (cid:48) ∈S p s (cid:48) ( θ + s (cid:48) − s ) , If 0 ≤ x < s − θ : L V ( x, y, s ) = − x −
12 ( λ y + λ − y ) (cid:16) x − (cid:88) s (cid:48) ∈S p s (cid:48) s (cid:48) (cid:17) . a) CAV intersections with 100 time steps(b) conventional intersections with 100 timesteps Figure 3: heatmaps of CAV signal-free intersections and conventional signal-free intersections7igure 4: line plot of average time delayFor x ≥ s max − θ , to ensure the stability, the function L V ( x, y, s ) should decrease as x increases. Therefore, we have λ y ( θ + s − s ) + λ − y ( θ + s − s ) < , ∀ s ∈ S . For s − θ ≤ x < s − θ , we can compute the vertex x ∗ : x ∗ = − λ − y ( θ + s − s ) λ y < − λ y ( θ + s − s ) λ y = s − θ − s < s − θ . Therefore, L V ( x, y, s ) is still decreasing for s − θ ≤ x < s − θ .For 0 ≤ x < s − θ , similarly, we can compute the vertex x ∗ : x ∗ = − λ y + λ − y < . Therefore, L V ( x, y, s ) is still decreasing for 0 ≤ x < s − θ . For x < s max − θ , L V ( x, y, s )is decreasing, and the maximum is obtained at x = 0: L V max = 12 ( λ y + λ − y ) (cid:88) s (cid:48) ∈S p s (cid:48) s (cid:48) . Therefore we have λ y ( θ + s − s min ) + λ − y ( θ + s − s min ) < , L V ( x, y ) ≤ − cx + d, ∀ ( x, y, s ) ∈ X × Y × S , where c = 1 − λ y ( θ + s − s min ) − λ − y ( θ + s − s min ) , d = ( λ y + λ − y ) (cid:80) s (cid:48) ∈S p s (cid:48) s (cid:48) . Hence, by[9, Theorem 4.3] the average time delay D is upper-bounded by (2). (cid:3) D8 Simulation-based validation
To validate the above results, we use Simulation of Urban Mobility (SUMO [7]) to simulatethe time delay for each vehicle due to the congestion. The network consists of a two-laneeast/west road crossing a two-lane north/south road in a single intersection. Vehicles appearon the inbound roads with given arrival rates, pass through the central intersection, and thenexit the system on the outbound roads. The vehicle flow from west to east merges with theflow from south to north with no turning allowed, and the merging sequence is determinedaccording to the FCFS rule. The vehicle attributes, such as length and width, are presentedin TABLE 1. Based on the same network and vehicle attributes, two types of intersections,conventional signal-free intersections and CAV signal-free intersections, are used to validateour queuing model and stability criteria under distinct pairs of arrival rates. According tothe simulation results, we find that 100 time steps of vehicle generation is able to distinguisha stable system and an unstable one. For each pair of arrival rates and intersection type, 20replications can give a good estimation to the average time delay for each vehicle. And thetime step size of our simulation is 0.1 second, which means the system state will be updatedevery 0.1 second. Table 1: Vehicle attributes and experiment settings.Length Width Maximalspeed Maximalacceleration Maximaldecelera-tionVehicle 5m 1.8m 7m/s 0.8m/s Crossing distanceIntersection 14.4mNumberof replica-tions Number of stepsgenerating vehi-cles Step sizeExperiment 20 100 0.1sFor ease of presentation, we assume that for the same intersection type, each vehicle hasthe equal intersection crossing time s , with ignoring the velocity fluctuation when vehiclescross the intersections. For conventional signal-free intersections, vehicles have to come to afull stop before crossing. Hence, the crossing time is given by s = (cid:114) b + l ) a , where a is the maximal acceleration, b is the vehicle length, and l is the crossing distance ofthe intersection. In addition, large headways are needed for human-driven vehicles, whichare 7.5 meters in our simulations. For CAV signal-free intersections, vehicles can maintainthe maximal speed as they cross the intersection. Hence, the crossing time is given by s = l + b ¯ v , v is the maximal speed allowed for crossing. Smaller headways are allowed for CAVs.In this section, we assume that inter-CAV headways are 5.5 meters. Note that to avoidpotential longitudinal collision, we need to ensure that v / d ≤ h , where h denotes thesafety headways, and d denotes the maximal deceleration. And we assume that the offsettime and switch-over time between two consecutive human-driven vehicles are twice as longas those between two CAVs. The following table summarizes the difference of parametersettings between theses two types of intersections.Table 2: Parameters for conventional vehicles and CAVsMinimalheadway Crossingtime Offsettime Switch-over timeConventional 7.5m 6.96s 2s 4sCAV 5.5m 2.77s 1s 2sWe visualize our simulation results of the conventional signal-free intersections and theCAV signal-free intersections on two separated heat maps in Fig. 3, where the shade of thecell represents the average time delay. We also plot the derived stable criteria 1 in the graphas the red lines, and the area under the lines satisfies our stable criteria. We believe if theaverage time delay exceeds 120 seconds, the system is not stable, which corresponds to theblack area in the heat maps.According to the heat maps in Fig. 3, most of the area bounded by our stable criteria 1has the light color, which represents system stability. Intuitively, we can see that the white10rea of the CAV intersections is larger than that of the conventional intersections. And underthe same pair of arrival rates, the average system time of the CAV signal-free intersectionsis less than that of the conventional signal-free intersections, since the CAV has the smallercooldown time and headways for two consecutive vehicles. Moreover, the average systemtime for the conventional signal-free intersections expands faster with the increase of thearrival rates.The upper bound of average time delay under the same arrival rate is presented in Fig. 4,with red lines related with the conventional intersections and black lines related with theCAV intersections. In the line plot, the dashed lines represent the theoretical upper boundof the average time delay, the solid lines represent the actual average time delay according tothe simulation results, and the gray vertical dotted lines represent the stability criteria 1. Wecan observe that for the conventional intersections, the theoretical upper bound holds well.However, for the CAV intersections, the actual average time delay will exceed the theoreticalupper bound with the increase of the arrival rate, which does not match our expectation.With tracking the vehicle crossing speed, we find that CAVs can not maintain the maximalspeed when crossing the intersection under a dense arrival, due to the vehicle interaction inSUMO. Therefore, we modify the original theoretical upper bound of the CAV intersectionsby using the average crossing time under 0.20 vehicle arrival rate, and the average crossingtime is given by the simulation results. The modified upper bound is presented in the plotby the black dash-dotted line. After modification, the theoretical line performs better. In this paper, we formulate a two-queue polling system with Poisson arrival process, constantcrossing time depending on the vehicle class, and the cooldown time between two consecutivevehicles. Vehicles cross the intersection following FCFS rules. Unlike common queuingmodel, we define the system state as a tuple of the residual system time, vehicle class, andcrossing time of the last entered vehicle. With the state transition functions, we expressthe queuing dynamics with an infinitesimal generator. We further provide the analyticalsystem stability criteria and an upper bound of the average time delay by using Foster-Lyapunov criteria. To validate our theoretical results, we use SUMO to run simulations oftwo signal-free intersection types. According to the simulation results, the stability criteriaholds for both types of intersections, but the theoretical upper bound does not matches ourexpectation for the CAV intersection, which is due to the crossing time fluctuation underintersection congestion.We provide a novel modeling method and an efficient analytical framework for vehiclesequencing at signal-free intersections. One major future work is integrating more coordina-tion strategies into our model, with more complex state transition functions and infinitesimalgenerators, and we will be able to find the optimal control to minimize the average systemdelay. 11 eferences [1] Michael Athans. a unified approach to the vehicle-merging problem.
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