Connectivity-Aware Traffic Phase Scheduling for Heterogeneously Connected Vehicles
aa r X i v : . [ c s . S Y ] A ug Connectivity-Aware Traffic Phase Scheduling forHeterogeneously Connected Vehicles
Shanyu Zhou
University of Illinois at Chicago [email protected]
Hulya Seferoglu
University of Illinois at Chicago [email protected]
ABSTRACT
We consider a transportation system of heterogeneously con-nected vehicles, where not all vehicles are able to communi-cate. Heterogeneous connectivity in transportation systemsis coupled to practical constraints such that (i) not all ve-hicles may be equipped with devices having communicationinterfaces, (ii) some vehicles may not prefer to communicatedue to privacy and security reasons, and (iii) communica-tion links are not perfect and packet losses and delay occurin practice. In this context, it is crucial to develop controlalgorithms by taking into account the heterogeneity. In thispaper, we particularly focus on making traffic phase schedul-ing decisions. We develop a connectivity-aware traffic phasescheduling algorithm for heterogeneously connected vehiclesthat increases the intersection efficiency (in terms of the av-erage number of vehicles that are allowed to pass the inter-section) by taking into account the heterogeneity. The simu-lation results show that our algorithm significantly improvesthe efficiency of intersections as compared to the baselines.
Keywords
Cyber-physical systems, transportation systems, connectedvehicles, heterogeneous communication.
1. INTRODUCTION
The increasing population and growing cities introduceseveral challenges in metropolitan areas, and one of the mostchallenging areas is transportation systems. In particular,the rapidly increasing number of vehicles in metropolitantransportation systems, has introduced several challengesincluding higher traffic congestion, delay, accidents, energyconsumption, and air pollution. For example, the averageof yearly delay per auto commuter due to congestion was 38hours, and it was as high as 60 hours in large metropolitanareas in 2011 [21]. The congestion caused 2.9 billion gal-lons of wasted fuel in 2011, and this figure keeps increasingyearly [21], e.g., the increase was 3.8% in Illinois betweenyears 2011 and 2012 [1]. This trend poses a challenge for
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CarSys’16, October 03-07, 2016, New York City, NY, USA c (cid:13) http://dx.doi.org/10.1145/2980100.2980105 (a) Phase I ( φ = 1) (b) Phase II ( φ = 2)(c) Phase III ( φ = 3) (d) Phase IV ( φ = 4) Figure 1: An example intersection with four possibletraffic phases. efficient transportation systems, so new traffic managementmechanisms are needed to address the ever increasing trans-portation challenges.Fortunately, advances in communication and networkingtheories offer vast amount of opportunities to address everincreasing challenges in transportation systems. In partic-ular, connected vehicles, i.e., vehicles that are connectedto the Internet via cellular connections and to each othervia device-to-device (D2D) connections such as Bluetoothor WiFi-Direct [2], are able to transmit and receive infor-mation to improve the control and management of traffic,which has potential of reducing congestion, delay, energy,and improving reliability. In this context, it is crucial tounderstand how heterogeneous communication affects theperformance of transportation systems.Heterogeneity in transportation systems is coupled to prac-tical constraints such that (i) not all vehicles may be equippedwith devices having communication interfaces, (ii) some ve-hicles may not prefer to communicate due to privacy and se-curity reasons, and (iii) communication links are not perfectand packet losses and delay occur in practice. It is crucial todevelop control algorithms by taking into account the het-erogeneity. In this paper, we particularly focus on makingtraffic phase scheduling decisions. The next two examples il-lustrate the traffic phase scheduling problem and the impactof heterogeneous communications on the scheduling. (a) Only the first vehiclecommunicates L (b) Only the second vehiclecommunicates Figure 2: An example single-lane intersection,where vehicles are going straight, turning left andturning right respectively.
Example Let us consider Fig. 1, which shows an iso-lated intersection, and all four possible traffic light phases.Traffic lights could be configured in four different phases:Phases I, II, III, and IV.
E.g.,
Phase I corresponds to thecase that only north-south and south-north bounds are al-lowed to pass through the intersection. The traffic light schedul-ing determines the phase that should be activated. Note thatonly one phase could be activated at a time. It is clear thatscheduling decisions should be made based on the congestionlevels of different directions (or traffic bounds). For example,selecting either Phase I or Phase III in the specific exampleof Fig. 1 looks a better decision as compared to Phase IIor Phase IV, because Phase I and Phase III have a largernumber of vehicles in their corresponding queues.
Example 1 is a widely known problem in network con-trol and optimization theory, and the optimal solution tothis problem is the popular max-weight algorithm [22]. Thebroader idea behind max-weight algorithm is to prioritizethe scheduling decisions with larger weights, which corre-sponds to congestion level, loss probabilities, and link qual-ities. The max-weight idea is applied to transportation sys-tems as well in previous work [7, 24, 25, 27] that schedulestraffic phases according to congestion levels, which has po-tential of allowing more vehicles to pass and reduce waitingtimes at intersections. This approach works well in a sce-nario that the directions of all vehicles are known a-priori.For example, if all devices communicate with the trafficlight in terms of their intentions about their directions ( e.g., turn right, go straight, etc.), the traffic light determineswhich phase to activate using the max-weight scheduling al-gorithm. However, due to heterogeneity of communicationin connected vehicles, only a percentage of vehicles commu-nicate their intentions. In this heterogeneous setup, newconnectivity-aware traffic phase scheduling algorithms areneeded as illustrated in the next example.
Example Let us consider Fig. 2, which shows one ofthe four incoming traffic lanes in an intersection. This is aone-way single-lane road, where we call the first vehicle atthe intersection as the head-of-line (HoL) vehicle. In Fig.2(a), the HoL vehicle has communication ability, and thevehicles are going straight, turning left, and turning right,respectively. In this case, the traffic light knows that theHoL vehicle is going straight (because the HoL vehicle com-municates), so it arranges its phase accordingly.Now let us consider Fig. 2(b), where the directions of ve-hicles are the same; i.e., straight, left, and right. Yet, in thisscenario HoL vehicle does not communicate, but only thevehicle behind HoL communicates. In this case, although the traffic light knows that the second vehicle is going to theleft, it has no idea of the HoL vehicle’s intention. If thetraffic phase, possibly determined as a solution to the max-weight algorithm, does not match the intention of the HoLvehicle, then the HoL vehicle blocks the other vehicles at theintersection, and no vehicles can pass. Similarly, HoL block-ing can be observed in more involved multiple-lane scenarios[28]. As seen, the max-weight algorithm may not be optimalin some scenarios due to heterogeneous connectivity, whichmakes the development of new scheduling algorithms, by tak-ing into account heterogeneity, crucial. ✷ In this paper, we develop a connectivity-aware traffic phasescheduling algorithm by taking into account heterogeneouscommunications of connected vehicles. Our approach fol-lows a similar idea to the max-weight scheduling algorithm,which makes scheduling decisions based on congestion levelsat intersections. However, our algorithm, which we nameConnectivity-Aware Max-Weight (CAMW), is fundamen-tally different from the max-weight as we take into accountheterogeneous communications while determining conges-tion levels. In particular, CAMW has two critical compo-nents to determine congestion: (i) Expectation: This com-ponent calculates the expected number of vehicles that canpass through the intersection at every phase based on thenumber of vehicles, and the percentage of communicatingvehicles at the intersection. (ii) Learning: This componentlearns the directions of vehicles even if the vehicles do notdirectly communicate with the traffic light. The expectationand learning components of our algorithm operate togetherin harmony to make better decision on traffic phase schedul-ing. The simulation results demonstrate that CAMW algo-rithm significantly improves the intersection efficiency (interms of the average number of vehicles that are allowedto pass the intersection) over the baseline algorithm; max-weight. The following are the key contributions of this work: • We investigate the impact of heterogeneous commu-nication on traffic phase scheduling problem in trans-portation networks. We develop a connectivity-awaretraffic scheduling algorithm, which we name Connectivity-Aware Max-Weight (CAMW), by taking into accountthe congestion levels at intersections and the hetero-geneous communications. • The crucial parts of CAMW are expectation and learn-ing components. In the expectation component, wecharacterize the expected number of vehicles that canpass through the intersections by taking into accountthe heterogeneous connectivity. In the learning compo-nent, we infer the directions of vehicles even if they donot directly communicate. The expectation and learn-ing components collectively determine the number ofvehicles that can pass through the intersections. • We evaluate CAMW via simulations, which confirmour analysis, and show that our algorithm significantlyimproves intersection efficiency as compared to the base-line; the max-weight algorithm.The structure of the rest of this paper is as follows. Section2 presents the related work. Section 3 introduces the sys-tem model. Section 4 develops our connectivity-aware trafficphase scheduling algorithm by taking into account hetero-geneous communications. Section 5 presents the simulationresults. Section 6 concludes the paper. . RELATED WORK
This work combines ideas from traffic phase scheduling,queuing theory, and network optimization. In this section,we discuss the most relevant literature from these areas.
Traffic phase scheduling:
Design and development of traf-fic phase scheduling algorithms have a long history; morethan 50 years [14]. Thus, there is huge literature in the area,especially on the design of optimal pre-timed policies [14, 6,3], which activate traffic phases according to a time-periodicpre-defined schedule. These policies do not meet expecta-tions under changing arrival times, which require adaptivecontrol [15]. The adaptive control mechanisms including [4],[6], [10], [11], [13] and [16], optimize control variables, suchas traffic phases, based on traffic measures, and apply themon short term.
Queueing theory:
Using queuing theory to analyze trans-portation systems has also very long history [26].
E.g., [14,19, 9] considered one-lane queues and calculated the ex-pected queue length and arrivals using probability gener-ation functions. Other modeling strategies are also studied;such as the queuing network model [20], cell transmissionmodel [12], store-and-forward [2], and petri-nets [5].
Network optimization and its applications to transporta-tion systems:
Max-weight scheduling algorithm and back-pressure routing and scheduling algorithms [22] arising fromnetwork optimization area has triggered significant researchin wireless networks [17, 18]. This topic has also inspiredresearch in transportation systems [7, 24, 25, 27]. Feed-back control algorithms to ensure maximum stability areproposed both under deterministic arrivals [25] and stochas-tic arrivals [23, 27] following backpressure idea. The infinitebuffer assumption of backpressure framework is studied bycapacity aware back-pressure algorithm in [8].
Our work in perspective:
As compared to the previouswork briefly summarized above, our work focuses on con-nected vehicles and investigates the scenario where vehiclescommunicate heterogeneously. In this scenario, we developan efficient connectivity-aware traffic phase scheduling algo-rithm by employing expectation and learning of congestionlevels at intersections.Our previous work [28] investigates the impact of theblocking problem at intersections, characterizes the wait-ing times, and develops a shortest delay routing algorithmin transportation systems. As compared to this work, inthis paper, we develop a connectivity-aware traffic phasescheduling algorithm by taking into account heterogeneouscommunications.
3. SYSTEM MODEL
In this section, we present our system model includingtraffic lights and phases as well as our queuing models ofthe traffic.
Traffic lights and phases:
In our system model, we focuson an intersection controlled by a traffic light. The fourtraffic phases we consider in this paper are shown in Fig. 1.We define φ as a phase decision, e.g., φ = 1 corresponds toPhase I in Fig. 1. The set of phases is Φ, and φ ∈ Φ.We consider that time is slotted, and at each time slot t ,a phase decision is made. Each traffic phase lasts for n timeslots. Vehicles have a chance to pass the intersection onlywhen the corresponding traffic phase is active, i.e., ON. Forinstance, vehicles in the south-north bound lanes may passthe intersection only when phase φ = 1 is ON in Fig. 1. L λ λ RSRL S (a) Queue I λ λ S LS L RSL (b)
Queue II
Figure 3: Two queuing models considered in this pa-per, where λ and λ are the arrival rates of straight-going and left-turning traffic, respectively.(a) Single-lane traffic model. (b) One+two lane model. Modeling intersections with queues:
We model the iso-lated intersection as a set of queues following [28]. Typically,there are four queues for each direction (for south-north,north-south, west-east, and east-west bound) at an intersec-tion. We specifically focus on one direction and model itusing two models:
Queue I , which is one-lane model shownin Fig. 3(a) and
Queue II ; which is a one+two lane modelshown in Fig. 3(b).Note that for both of
Queue I and
Queue II , we canconsider straight-continuing and right-turning traffic as thesame traffic, since they share the similar right of way. Thus,to demonstrate the analysis in a simple way, we simply con-sider that the right-turning and straight-continuing trafficsare combined together, and we call both right-turning andstraight-continuing vehicles as straight-going vehicles.At each slot, vehicles arrive into intersections, where λ and λ are the average arrival rates of straight-going andleft-turning vehicles, respectively. In our analysis, the ar-rivals can follow any i.i.d. distribution. In this setup, whena vehicle enters the intersection, it can connect to the trafficlight either using cellular or vehicle-to-vehicle communica-tions. Thus, it can communicate its intention with the trafficlight about its destination, i.e., turning left, going straight,etc. The probability of communication for each vehicle is ρ .If a vehicle does not communicate, we model their in-tentions probabilistically, where p is the probability that avehicle (which does not communicate its intention) will gostraight, while p is the probability that it will turn left.
4. CONNECTIVITY-AWARE TRAFFIC PHASESCHEDULING4.1 CAMW: Connectivity-Aware Max-Weight
In this section, we develop our connectivity-aware trafficphase scheduling algorithm by taking into account hetero-geneous communications. We consider the setup shown inFig. 1 for phases. Our scheduling algorithm, which we callConnectivity-Aware Max-Weight (CAMW), determines thephase φ by optimizingmax φ X i ∈{ ,... } Q i ( t ) ˜ E ( K φi ( t ))s.t. φ ∈ Φ . (1)where Q i ( t ) is the number of vehicles in the i th incomingqueue at time slot t , and ˜ E ( K φi ( t )) is the estimated num-ber of vehicles that can pass the intersection from the i th v v v T n-2n-1n Figure 4: An illustrative example of communicatingvehicles in a queue at a time slot. Communicatingvehicles are at labeled locations; v , v , · · · , v T . incoming queue under traffic phase φ ∈ Φ. Note that oneactive phase lasts for n time slots and it takes one time slotfor a vehicle to pass the intersection. In other words, atmost n vehicles in a queue can pass the intersection duringone green light phase. The optimization problem in (1) ap-plies to all queuing models ( i.e., includes both Queue I and
Queue II ).Note that (1) determines the phase by taking into account Q i ( t ) and ˜ E ( K φi ( t )). The queue size information Q i ( t ) canbe easily determined by traffic lights using sensors that countthe number of approaching vehicles. In other words, (1) pri-oritizes phases with larger Q i ( t ) values. This is an approachfollowed by the classical max-weight algorithm. However, aswe discussed earlier, using Q i ( t ) alone is not sufficient whenvehicles heterogeneously communicate with traffic lights. Inthis case, since each device has different destinations, block-ing can occur. I.e., even if Q i ( t ) is large, the number of ve-hicles that can pass through the intersection could be smalldue to blocking. Thus, to reflect this fact, we include theterm ˜ E ( K φi ( t )) in the optimization problem.˜ E ( K φi ( t )) is the estimated number of vehicles that canpass the intersection from the i th incoming queue undertraffic phase φ ∈ Φ. ˜ E ( K φi ( t )) is found using two steps:expectation and learning. The key idea behind expectationpart is to calculate the expected number of vehicles, whichis E ( K φi ( t )), that can pass the intersection at phase φ , whilethe key idea of the learning part is to fine tune E ( K φi ( t ))and find ˜ E ( K φi ( t )) by learning the directions of vehicles thatdo not communicate. In the next two sections, we presentthe expectation and learning components of CAMW. E ( K φi ( t )) for Queue I
Let us focus on phase φ ∈ Φ and the i th queue, where i ∈ { , , , } . In this setup, T ( t ) ( T ( t ) ≤ n ) denotes thenumber of vehicles that have communication abilities at timeslot t , and v l ( t ) ( l = 1 , , · · · , T ) denotes the location ofthe l th communicating vehicle in the queue. For example, v ( t ) = 3 means that the second communicating vehicle inthe queue is actually the third vehicle in the queue. Fig. 4illustrates an example locations of communicating vehicles.Note that the vehicles that do not communicate are notassigned any location labels.Now, let us define two conditions; C and C . The firstcondition C requires that all communicating vehicles wouldlike to go to the same direction and aligned with the trafficphase, while the second condition C corresponds to the casethat the first communicating vehicle that is not aligned withthe traffic phase is in the location of v L ( t ) ( L = 1 , , · · · , T ).Note that the conditions C and C are complementary. Thenext theorem characterizes the expected number of vehiclesthat would leave queue i at phase φ . Theorem Assume that all the queues in an intersec-
LE L (a) Conf. I L S L (b) Conf. II
LS S (c) Conf. III E S S (d) Conf. IV
Figure 5: Four possible configurations (Conf. I toConf IV) for the first three vehicles in
Queue II ,where L and S denote that the intention of the vehi-cle is to turn left or go straight, respectively, while E denotes that the location is empty (due to previousblocking). tion follow Queue I . The expected number of vehicles thatwould leave the i th queue and pass the intersection at trafficphase φ ∈ Φ is characterized by E ( K φi ( t )) = P T ( t ) l =0 p − l p (( p + p v l ( t )) p v l ( t ) − +(1 − p − p v l +1 ( t )) p v l +1 ( t ) − )+ np n − T ( t )1 , if C holds P L − l =0 p − l p (( p + p v l ( t )) p v l ( t ) − +(1 − p − p v l +1 ( t )) p v l +1 ( t ) − )+( v L ( t ) − p v L ( t ) − L , if C holds.(2) Proof.
The proof is provided in Appendix A. E ( K φi ( t )) for Queue IIQueue II assumes that there are dedicated lanes for left-turning and straight-going vehicles, which makes it funda-mentally different than
Queue I . In this setup, we considerthat traffic lights can sense whether the HoL location of eachdedicated lane is empty or not. Thus, in
Queue II , the firsttwo vehicles in the queue will indirectly communicate theirintentions to the traffic light. Fig. 5 demonstrates four pos-sible configurations for HoL vehicles. For example, in Fig.5(a), HoL position of the straight going lane is empty (shownwith E ), the traffic light will know that two vehicles in thequeue will turn left. On the other hand, in Fig. 5(b), thetraffic light knows that in the dedicated lanes, one vehiclewill go straight, and the other will turn left, but it does notknow the intentions of the other vehicles as long as they donot explicitly communicate with the traffic light.The crucial observation with Queue II is that if the ve-hicles that indirectly communicate with the traffic light areseparated from the queue, the rest of the vehicles form a sub-queue . For example, all the vehicles other than (i) thefirst two left-turning vehicles in Fig. 5(a), and (ii) the twovehicles that are going straight and turning left in Fig. 5(b),form a sub-queue . The important property of the sub-queue is that it follows
Queue I , and can be modeled using thelocation labels as shown in Fig. 4. Thus, we can calculate E ( K φi ( t )) of Queue II using the similar analysis we have inSection 4.2.1. Next, we provide the details of our E ( K φi ( t ))calculation.et T ( t ) denotes the number of communicating vehicles inthe sub-queue at time t , C is the condition that all commu-nicating vehicles in the sub-queue go to the same directionaligned with the traffic phase, and C denotes the conditionthat the first communicating vehicle in the sub-queue thatgoes to a different direction than what the traffic phase al-lows is at location v L ( t ) ( L = 1 , , · · · , T ). The next theoremcharacterizes the expected number of vehicles that wouldleave queue i at phase φ for model Queue II . Theorem Assume that all the queues in an intersec-tion follow
Queue II . Then, if the first three vehicles of the i th incoming queue are in the form of Fig. 5(a) or Fig.5(d), the expected number of transmittable vehicles is char-acterized by E ( K φi ( t )) = P T ( t ) l =0 p − l p (( p + p v l ( t )) p v l ( t ) − + (1 − p − p v l +1 ( t )) p v l +1 ( t ) − ) + ( n − p n − − T ( t )1 , if C holds2 + P L − l =0 p − l p (( p + p v l ( t )) p v l ( t ) − + (1 − p − p v l +1 ( t )) p v l +1 ( t ) − ) + ( v L ( t ) − p v L ( t ) − L , if C holds(3) where T ( t ) ≤ n − .And if the first three vehicles of the i th incoming queue arein the form of Fig. 5(b) or Fig. 5(c), the expected numberof transmittable vehicles is characterized by E ( K φi ( t )) = P T ( t ) l =0 p − l p (( p + p v l ( t )) p v l ( t ) − + (1 − p − p v l +1 ( t )) p v l +1 ( t ) − ) + ( n − p n − − T ( t )1 , if C holds1 + P L − l =0 p − l p (( p + p v l ( t )) p v l ( t ) − + (1 − p − p v l +1 ( t )) p v l +1 ( t ) − ) + ( v L ( t ) − p v L ( t ) − L , if C holds(4) where T ( t ) ≤ n − . Proof.
The number of vehicles that can be guaranteedto pass the intersection under certain traffic phase dependson the configuration of the first three vehicles in the queue.First, we consider the case that the first three vehicles arein the form of Fig. 5(a) or Fig. 5(d). In this case, at leasttwo vehicles can pass the intersection for the correspondingtraffic phase, so we need to consider the rest of the vehicles, i.e., n − n is the queue size. Not-ing that n − sub-queue in this setup, andassuming that T ( t ) ( T ( t ) ≤ n −
2) vehicles communicate the sub-queue , it is clear that the sub-queue is represented by
Queue I . Thus, (3) is obtained by adding two to (2).On the other hand, if the first three vehicles are in theform of Fig. 5(b) or Fig. 5(c), at least one vehicle canpass the intersection at any traffic phase configuration. Inthis scenario, one vehicle is considered as guaranteed to betransmitted, and the rest of the vehicles ( n − sub-queue . Similar to above discussion, the sub-queue follows Queue I , so (4) is obtained by adding one to (2). Thisconcludes the proof.
In the previous section, we characterized the expectednumber of vehicles E ( K φi ( t )) that can pass an intersectionat phase φ from queue i . However, in our CAMW algorithm,which solves (1), we do not use E ( K φi ( t )). The reason is that E ( K φi ( t )) is an expected value and its granularity is poor. Inother words, if we use E ( K φi ( t )) in (1), we may end up withchoosing a traffic phase that allows no vehicles passing theintersection. In this case, the intersection is blocked . Moreimportantly, once the intersection is blocked , if we keep using E ( K φi ( t )) in (1), we may end up with choosing the wrongtraffic phase next time with high probability, which leads toa deadlock. To address this issue, we introduce the learningmechanism, which works in the following way.We assume that traffic lights can infer if blocking occursat intersections, and use this information in future decisions.For example, assume that the selected traffic phase at time t − φ = 1 (as shown in Fig. 1(a)), and ˜ E ( K φ =1 i ( t − E ( K φ =1 i ( t − E ( K φ =1 i ( t ))is set to zero at time t so that φ = 1 is not selected again.˜ E ( K φ =1 i ( t +∆)) is set to E ( K φ =1 i ( t +∆)) again immediatelyafter some vehicles are transmitted from the queues. Thismay take ∆ time slots. This learning mechanism appliesto both Queue I and
Queue II , but in
Queue II , separatelanes for each direction makes the learning process by de-fault.
I.e., in Queue II , ˜ E ( K φi ( t )) = E ( K φi ( t )), ∀ t .
5. PERFORMANCE EVALUATION
In this section, we consider an intersection controlled bya traffic light. Each arriving vehicle to the intersection cancommunicate with probability ρ . Each green phase lasts forone or more time slots. We assume that the arrival rateto each queue in the intersection is the same; i.e., λ and λ are the same ∀ i ∈ { , , , } . We present the simulationresults of our Connectivity-Aware Max-Weight (CAMW) al-gorithm for both of Queue I and
Queue II , as compared tothe baseline, the max-weight algorithm, which is briefly de-scribed next.
The max-weight scheduling algorithm determines a trafficphase as a solution tomax ρ X i =1 Q i ( t ) K φi ( t )s.t. φ ∈ Φ , (5)where K φi ( t ) is the weight of queue i for phase φ . Thevalue of K φi ( t ) depends on the intersection type and thecorresponding queuing models, which is explained next.First, let us consider Queue I . If the HoL vehicle in the i th queue can communicate, then K φi ( t ) = 1 for the phasethat is aligned with the direction of the HoL vehicle and K φi ( t ) = 0 for the other three phases. If the HoL vehicle Note that K φi ( t ) = 1 in the original max-weight algorithm,while it varies in (5) as explained in this section. Thus,although we call this baseline the max-weight algorithm, itis actually the improved version of the classical max-weightalgorithm. ime0 2000 4000 6000 8000 10000 A v e r age queue s i z e ρ =0.4 ρ =0.7 ρ =1.0 (a) Max-weight Time A v e r age queue s i z e ρ =0.4 ρ =0.7 ρ =1.0 (b) CAMW Figure 6: The average queue size versus time for
Queue I . Each green phase lasts for two time slots.The arrival rate is λ = 0 . and λ = 0 . to each ofthe queue in the intersection. cannot communicate, max-weight considers K φi ( t ) = 1 forthe phases that control the i th queue if the queue length islarger than zero.Second, we assume that all the queues in the intersectionfollow Queue II . In this setup, we take into account the firsttwo vehicles in the dedicated lanes. For example, if the firsttwo vehicles from the i th incoming queue are in the form ofFig. 5(a), then K φi ( t ) = 1 for the left turning phase, and K φi ( t ) = 0 for the other phases. On the other hand, if thefirst two vehicles are in the form of Fig. 5(b), then K φi ( t ) = 1for both the left-turning and straight-going phases. Queue I
We first assume all the queues in the intersection follow
Queue I , and evaluate our CAMW algorithm as comparedto the baseline; max-weight. The evolution of the averagequeue size of the intersection for different scheduling algo-rithms is presented in Fig. 6. Each green phase lasts fortwo time slots. The arrival rate is λ = 0 .
18 and λ = 0 . ρ = 1 .
0, bothof the algorithms have the similar performance. This is be-cause every vehicle can communicate, so the max-weight al-gorithm, since the traffic light can communicate with theHoL vehicle, can align the phases with the direction of HoLvehicle. However, when the communication probability re-duces to ρ = 0 .
7, max-weight cannot stabilize the queues,while CAMW stabilizes. When ρ = 0 .
4, neither CAMWnor max-weight can stabilize the queues, because the arrivalrates fall out of the stability region. As can be seen CAMWsupports higher traffic rates than the max-weight algorithmthanks to exploiting connectivity of vehicles.Fig. 7 presents the intersection efficiency versus total ar-rival rate to each queue for different communication proba-bility ρ . The intersection efficiency is defined as the ratio ofdeparting traffic to arrival traffic. In this setup, each greenphase lasts for two time slots. Each queue has the samearrival rate, and λ = 1 . λ . It can be observed that when ρ = 1 .
0, both of the algorithms can achieve very similarintersection efficiency. However, if ρ = 1, the intersectionefficiency of max-weight scheduling algorithm drops almostto zero, while CAMW can still achieve satisfying intersec-tion efficiency thanks to taking into account heterogeneouscommunication probabilities. Queue II
In this section, we assume all the queues in the intersec-tion follow
Queue II . The evolution of the average queuesize of the intersection using CAMW and max-weight algo-
Total arrival rate to each queue I n t e r s e c t i on e ff i c i en cy CAMWMax−weight (a) ρ = 0 . Total arrival rate to each queue I n t e r s e c t i on e ff i c i en cy CAMWMax−weight (b) ρ = 0 . Total arrival rate to each queue I n t e r s e c t i on e ff i c i en cy CAMWMax−weight (c) ρ = 0 . Total arrival rate to each queue I n t e r s e c t i on e ff i c i en cy CAMWMax−weight (d) ρ = 1 . Figure 7: Intersection efficiency versus total arrivalrate to each queue with different communicationprobability ρ for Queue I . Each green phase lasts fortwo time slots and each queue has the same arrivalrate and λ = 1 . λ . A v e r age queue s i z e CAMWMax−weight (a) ρ = 0 . A v e r age queue s i z e CAMWMax−weight (b) ρ = 0 . Figure 8: The evolution of the average queue size ofthe intersection using our algorithm and max-weightalgorithm for different communication probability ρ for Queue II . The arrival rate to each queue is λ = λ = 0 . and each green phase lasts for two time slots. rithm for different communication probability ρ is presentedin Fig. 8. The arrival rate to each queue is λ = λ = 0 . ρ is small, CAMW is slightly better than the max-weightalgorithm, which is because both of the two algorithm selecttraffic phases in a similar way when ρ is small. The averagequeue sizes over 10,000 time slots when ρ = 0 . ρ is large, our algorithm improves much overmax-weight algorithm. This is because the estimation accu-racy in our algorithm improves as ρ increases, which allowsmore vehicles to pass at each green phase. When ρ = 0 . ρ in-creases in Queue II , which is against the observation we hadin
Queue I . The reason is that while ρ affects max-weight’sdecision about HoL vehicles as explained in (5) in Queue I ,it does not have any effect in
Queue II .Fig. 9 presents the intersection efficiency versus total ar-rival rate to each queue for different communication prob- I n t e r s e c t i on e ff i c i en cy CAMWMax−weight (a) ρ = 0 . I n t e r s e c t i on e ff i c i en cy CAMWMax−weight (b) ρ = 0 . I n t e r s e c t i on e ff i c i en cy CAMWMax−weight (c) ρ = 0 . I n t e r s e c t i on e ff i c i en cy CAMWMax−weight (d) ρ = 1 . Figure 9: Intersection efficiency versus total arrivalrate to each queue with different communicationprobability ρ for Queue II . Each queue has the sametotal arrival rate and λ = λ , and each green phaselasts for two time slots. abilities ρ . Each queue has the same total arrival rate and λ = λ . Each green phase lasts for two time slots. It canbe observed that the performance of our algorithm improvesas the communicating probability ρ increases, while max-weight has the same performance as ρ changes. The reasonis that the estimation accuracy in our algorithm improves as ρ increases, so CAMW performs better than the max-weightalgorithm as ρ increases. Note that CAMW improves overmax-weight by 14%, which is significant.
6. CONCLUSION
In this paper, we considered a transportation system ofheterogeneously connected vehicles, where not all vehiclesare able to communicate. For this setup, we developeda connectivity-aware max-weight scheduling (CAMW) al-gorithm by taking into account the connectivity of vehi-cles. The crucial components of CAMW are expectationand learning components, which determine the estimatednumber of vehicles that can pass through the intersectionsby taking into account the heterogeneous communications.The simulations results show that CAMW algorithm signifi-cantly improves the intersection efficiency over max-weight.
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APPENDIXA. PROOF OF THEOREM 1
In this section, we specifically focus on the calculationof E ( K φ =1 i ( t )), where φ = 1 corresponds to the phase inFig. 1(a) to explain our the proof in an easier way. Notethat E ( K φ =1 i ( t )) calculation can be directly generalized to E ( K φi ( t )), ∀ φ ∈ Φ.We first derive the calculation of E ( K φ =1 i ( t )) when allcommunicating vehicles are going straight. The calculationof E ( K φ =1 i ( t )) for other cases will be obtained based on thisderivation. If all communicating vehicles are going straightat time slot t , we can consider the queue as divided into( T + 1) blocks by the T communicating vehicles. (Note that T is the number of communicating vehicles in a queue).Let a random variable J denote the number of vehiclesthat can pass the intersection. The probability that j vehi-cles pass the intersection is P [ J = j ], and it behaves simi-larly to the geometric distribution. However, the probabilitydistribution is different when j falls into different blocks dueto the communicating vehicles that go straight. To be moreprecise, we have P [ J = j ] = p j p , ≤ j ≤ v − p j − p , v ≤ j ≤ v − p j − T p , v T ≤ j ≤ n − p n − T , j = n (6)Note that P [ J = v − P [ J = v − · · · , P [ J = v T − v , v , · · · , v T are all going straight, and if v l − v l vehicles can passthe intersection for sure ( l = 1 , , · · · , T ). Using (6), we can obtain the expected number of vehi-cles that can pass the intersection as E ( K φ =1 i ( t )) when allcommunicating vehicles are going straight. I.e., E ( K φ =1 i ( t )) = v − X j =1 jp j p + v − X j = v jp j − p + · · · + n − X j = v T jp j − T p + np n − T (7)In (7), P v l +1 − j = v l jp j − l p can be expressed as p − l p P v l +1 − j = v l jp j − = p − l p ∂ ( P vl +1 − j = vl p j ) ∂p = p − l p (( p + p v l ) p v l − + (1 − p − p v l +1 ) p v l +1 − ). Thus, we can obtain E ( K φ =1 i ( t ))when all communicating vehicles are going straight as E ( K φ =1 i ( t )) = T X l =0 p − l p (( p + p v l ) p v l − +(1 − p − p v l +1 ) p v l +1 − ) + np n − T (8)Note that we have v = 1 , v T +1 = n + 1 in (8) to make itconsistent with (7).When there are some communicating vehicles going left,let v L ( t ) be the location of the first communicating vehiclethat goes left. There are ( L −
1) communicating vehiclesin front of v L ( t ) that going straight and ( T − L ) communi-cating vehicles behind v L ( t ) which will be blocked for sure.Now, we only focus on the vehicles between the location 1to ( v L ( t ) − L −
1) communicating vehiclesamong them, and all of the communicating vehicles are go-ing straight. Thus, we can use the similar analysis as used in(7) except that now the maximum number of vehicles thatcan pass the intersection is ( v L ( t ) −
1) instead of n . There-fore, we have the expected number of vehicles that can passthe intersection E ( K φ =1 i ( t )) when the first communicatingvehicle that turns left is at location v L ( t ). Thus, E ( K φ =1 i ( t )) = L − X l =0 p − l p (( p + p v l ) p v l − +(1 − p − p v l +1 ) p v l +1 − ) + ( v L ( t ) − p v L ( t ) − L (9)By taking into account all the ( T + 1) situations, we con-clude that E ( K φi ( t )) = P T ( t ) l =0 p − l p (( p + p v l ( t )) p v l ( t ) − +(1 − p − p v l +1 ( t )) p v l +1 ( t ) − )+ np n − T ( t )1 , if C holds P L − l =0 p − l p (( p + p v l ( t )) p v l ( t ) − +(1 − p − p v l +1 ( t )) p v l +1 ( t ) − )+( v L ( t ) − p v L ( t ) − L , if C holds.(10)By following the same analysis, we can obtain E ( K φi ( t ))for φ = 2 ,,
1) instead of n . There-fore, we have the expected number of vehicles that can passthe intersection E ( K φ =1 i ( t )) when the first communicatingvehicle that turns left is at location v L ( t ). Thus, E ( K φ =1 i ( t )) = L − X l =0 p − l p (( p + p v l ) p v l − +(1 − p − p v l +1 ) p v l +1 − ) + ( v L ( t ) − p v L ( t ) − L (9)By taking into account all the ( T + 1) situations, we con-clude that E ( K φi ( t )) = P T ( t ) l =0 p − l p (( p + p v l ( t )) p v l ( t ) − +(1 − p − p v l +1 ( t )) p v l +1 ( t ) − )+ np n − T ( t )1 , if C holds P L − l =0 p − l p (( p + p v l ( t )) p v l ( t ) − +(1 − p − p v l +1 ( t )) p v l +1 ( t ) − )+( v L ( t ) − p v L ( t ) − L , if C holds.(10)By following the same analysis, we can obtain E ( K φi ( t ))for φ = 2 ,, ,,