Poster: Resource Allocation with Conflict Resolution for Vehicular Sidelink Broadcast Communications

P oster: Resource Allo cation with Conflict Resolution fo r V ehicula r Sidelink Broadcast Communications Luis F. Aban to-Leon Eindho v en Univ ersit y of T ec hnology Eindho v en, Netherlands l.f.aban to@tue.nl Arie K opp elaar NXP Semiconductors Eindho v en, Netherlands arie.k opp elaar@nxp.com Sonia Heemstra de Gro ot Eindho v en Univ ersit y of T ec hnology Eindho v en, Netherlands sheemstradegro ot@tue.nl ABSTRA CT In this pap er w e presen t a graph-based resource allo cation sc heme for sidelink broadcast V2V comm unications. Harness- ing a v ailable information on geographical position of vehicles and spectrum resources utilization, eNo deBs are capable of allotting the same set of sidelink resources to differen t vehicles distributed among several communications clusters. Within a comm unications cluster, it is crucial to preven t time-domain allo cation conflicts since v ehicles cannot transmit and receive sim ultaneously , i.e., they must transmit in orthogonal time resources. In this research, w e present a solution based on a bipartite graph, where v ehicles and sp ectrum resources are represen ted b y v ertices whereas the edges represen t the ac hiev able rate in each resource based on the SINR that eac h vehicle p erceives. The aforementioned time orthogonal- it y constrain t can be approached b y aggregating conflicting v ertices in to macro-v ertices which, in addition, reduces the searc h complexit y . W e show mathematically and through sim ulations that the proposed approach yields an optimal so- lution. In addition, we provide sim ulations showing that the prop osed method outp erforms other comp eting approaches, sp ecially in scenarios with high v ehicular density . KEYW ORDS resource allo cation; v ehicular communications; sidelink 1 INTR ODUCTION V ehicle–to–vehicle (V2V) communications is one of the nov el use cases in 5G and has attracted m uch interest, sp ecially for safet y applications, since connected vehicles may hav e the p oten tial to prev ent accidents [1]. In V2V Mo de 3 , eNo deBs assign resources to v ehicles for them to p erio dically broadcast CAM messages [ 2 ]. Once the allocation has b een accomplished, data is disseminated directly b etw een vehicles. Con versely , in conv entional cel- lular comm unications, data tra verse the eNo deB via up- link/do wnlink b efore it can be forwarded. Thus, since data Permission to make digital or hard copies of part or all of this work for p ersonal or classro om use is granted without fee provided that copies are not made or distributed for profit or commercial adv antage and that copies b ear this notice and the full citation on the first page. Copyrigh ts for third-party components of this work m ust be honored. F or all other uses, con tact the owner /author(s). MobiCom ’17, Snowbird, UT, USA © 2017 Copyright held by the owner/author(s). 978-1-4503-4916- 1/17/10. DOI: 10.1145/3117811.3131260 Figure 1: V2V Broadcast Comm unications via Sidelink traffic is controlled b y the eNo deB, users can b e allo cated in the same time resource but different frequency sub c hannels. Nev ertheless, in V2V Mo de 3 the flo w of data is not con- trolled, th us it is fundamen tal to guarantee that vehicles will transmit in orthogonal time resources to prev ent conflicts. On the other hand, assignment problems can b e represented as weigh ted bipartite graphs, where the ob jective is to find a maximal matching. This classical problem is called herein unc onstr aine d weighte d gr aph matching . Resource allo cation for V2V comm unications has a time orthogonality constraint whic h cannot b e handled by the men tioned metho d. W e hav e therefore envisaged a solution called c onstr aine d weighted gr aph matching , which incorporates the men tioned constraint. The ob jectiv e of this pap er is t w o-fold:  i  pro ve that an optimal solution for the c onstr aine d weighted gr aph matching problem exists and  ii  discuss the suitability of suc h an ap- proac h for av oiding resource allo cation conflicts in broadcast v ehicular communications. 2 MOTIV A TION AND CONTRIBUTIONS Our motiv ation is to develop an approac h capable of  a  prev enting allo cation conflicts—by enforcing constraints— and  b  maximizing the sum-rate capacity of the system. F or instance, in one of the clusters of Fig. 1, a resource conflict can b e observed betw een vehicles V 8 and V 10 since they hav e b een allotted resources in the same time subframe. The con tributions of our work are summarized: • Kuhn-Munkres [ 3 ] is a computationally efficien t metho d for solving matching problems in bipartite graphs. Ho wev er, due to additional time orthogonalit y con- strain ts, the resultan t problem is not directly ap- proac hable b y it. In our solution, vertices conflicting v 1 v 2 . . . v N r 1 r 2 . . . r K r K + 1 r K + 2 . . . r 2 K . . . r K N − 1 + 1 r K N − 1 + 2 . . . r K N macro- macro- macro- v ertex R 1 v ertex R 2 v ertex R N V V ehicles R R esour c es Figure 2: Constrained W eighted Bipartite Graph among each other hav e b een aggregated into macro- v ertices yielding a resultant graph whic h is solv able b y Kuhn-Munkres. • V ertex aggregation cuts down the num b er of effec- tiv e v ertices and therefore narrows the amount of p oten tial solutions, without affecting optimality . • W e sho w through simulations that our approac h is capable of providing fairness among all vehicles, esp ecially in scenarios with high v ehicle density . 3 PR OPOSED APPR OA CH Let G =  V , R , E  b e a bipartite graph such that |R| = K |V | = K N , as depicted in Fig. 2. In this scheme, the K N v ertices in R are clustered in to N disjoin t groups {R α } N α = 1 called macro-vertices, suc h that R = ∪ N α = 1 R α , R α ∩ R α 0 = ∅ , ∀ α , α 0 . Eac h macro-vertex R α is an aggregation of K v ertices, i.e., |R α | = K . The target is to find a vertex–to– v ertex (or vehicle–to–resource) matc hing with maxim um sum of w eights suc h that no tw o vertices in V are matc hed to any t wo v ertices in the same macro-vertex R α . This condition m ust be satisfied as it depicts the time orthogonality require- men t that preven ts allo cation conflicts. W e will show that the optimal solution is tantamoun t to finding the maximum v ertex–to–macro-vertex matching. In Fig. 2, vertices v i represen t the vehicles in cluster V whereas v ertices r j represen t the allotable resources managed b y the eNo deB. K represen ts the num b er of resources p er subframe. N represen ts the amoun t of av ailable subframes in which the resource allo cation task can b e accomplished. The problem is formulated by (1) max c T x sub ject to  I N × N ⊗ 1 1 × N 1 1 × N ⊗ I N × N  | {z } A ⊗ 1 1 × K x = 1 (1) where ⊗ represen ts the tensor pro duct operator, c ∈ R M , x ∈ B M with M = K N 2 . The solution and weigh t vectors are x =  x 1 , 1 , . . . , x N ,N  T and c =  c 1 , 1 , . . . , c N ,N  T , resp ec- tiv ely . Also, c ij = B log 2  1 + SINR ij  . Because x exists on the binary subspace, the cost function can b e equiv a- len tly expressed as c T x = x T diag  c  x without affecting optimalit y . Also, we can add zero-v alued terms c ij x ij x ik (with r j , r k ∈ R α ) to the cost function without affecting the solution. A more generalized expression is given by x T  I M × M ⊗ 1 K × K − I K × K  diag  c  x = 0 . No w, the augmented cost function can b e recast as c T x = x T diag  c  x + x T  I M × M ⊗  1 − I  K × K  diag  c  x = x T  I M × M ⊗ 1 K × K  diag  c  x = x T  I M × M I M × M ⊗ 1 K × 1 1 1 × K  diag  c  x = x T  I M × M ⊗ 1 K × 1  | {z } y T  I M × M ⊗ 1 1 × K  diag  c  x | {z } d (2) F rom (2), we obtain that d =  I M × M ⊗ 1 1 × K  diag  c  x and y =  I M × M ⊗ 1 1 × K  x . Similarly , we obtain that x =  I M × M ⊗ 1 1 × K  † y . In the following, w e use the previous relations to simplify the constraint in (1),  I N × N ⊗ 1 1 × N 1 1 × N ⊗ I N × N  ⊗ 1 1 × K   I M × M ⊗ 1 † 1 × K  y = 1 =  I N × N ⊗ 1 1 × N 1 1 × N ⊗ I N × N  I M × M  ⊗  1 1 × K 1 † 1 × K  | {z } 1 y = 1 =  I N × N ⊗ 1 1 × N 1 1 × N ⊗ I N × N  y = 1 (3) Th us, the problem in (3) can b e recast as (4) max d T y sub ject to  I N × N ⊗ 1 1 × N 1 1 × N ⊗ I N × N  | {z } A y = 1 . (4) Fig.3 shows the transformation pro cess from (1) to (4). W e notice that d dep ends on x whic h is not desirable. In order to eliminate this dep endency , we state without a pro of—due I M × M ⊗ 1 1 × K I M × M ⊗ 1 1 × K × diag · x c y d Figure 3: T ransformation Pro cess to space limitations—that d = lim β →∞ 1 β ◦ log { I M × M ⊗ 1 1 × K e ◦ β c } (5) where ◦ log {·} and e ◦{·} are the element-wise natural logarithm and Hadamard exp onen tial [4], resp ectiv ely . 4 SIMULA TIONS W e consider a 10 MHz channel which is divided into sev- eral resource ch unks, eac h with an extent of 1ms in time and 1.26 MHz in frequency . T o wit, 1.26 MHz corresp onds to 7 resource blo cks (RBs), each consisting of data (5RBs) and control information (2RBs) [ 5 ]. In our mo del, we con- sider that clusters are indep enden t from each other. Thus, resources used in a certain cluster can be repurp osed by ve- hicles in other clusters. Since w e consider a message rate of 10 Hz, the resource allo cation task is carried out every 0.1 s and therefore the maximum num b er of allotable sub- frames is 100. W e also assume that the sidelink channel conditions of each vehicle are rep orted to the eNo deB via uplink. The resource allo cation is broadcasted to v ehicles via do wnlink. Notice that sidelink resources are exclusively used for comm unications. In Fig. 4, we compare 4 differen t algorithms in base of the av erage ov er 1000 sim ulations. W e hav e considered N = 100 vehicles in each cluster. Through sim ulations, we sho w that our sc heme is optimal since it ac hieves the same p erformance as exhaustive search. The greedy algorithm p erforms as goo d as the prop osed approac h when w e examine the highest-rate v ehicle only . This is logical as the premise of the greedy algorithm is to assign the b est resources on first-come first-served basis. Considering the system av erage rate, our proposed approac h has an adv antage o ver the greedy algorithm. Also, when considering the w orst-rate vehicle, our prop osal excels as it is capable of providing a higher lev el of fairness. In all cases, the random allo cation algorithm is outp erformed by the other approaches. Fig. 5 shows Highest- Rate V ehicle W orst-Rate V ehicle System A verage Rate System Rate Standard Deviation 0 2 4 6 8 10 8 . 97 7 . 12 8 . 22 1 . 16 8 . 97 7 . 12 8 . 22 1 . 16 8 . 91 5 . 85 8 . 02 1 . 25 7 . 63 1 . 76 4 . 52 1 . 67 Rate [Mbits / s / resource] Exhaustive Search Graph-based Algorithm Greedy Algoritm Random Algorithm Figure 4: V ehicles Data Rate 20 40 60 80 100 2 3 4 5 6 7 8 Number of V ehicles Rate [Mbits / s / resource] Graph-based Algorithm Exhaustive Search Greedy Algoritm Random Algorithm Figure 5: W orst-rate V ehicle 2 4 6 8 10 0 . 1 0 . 4 0 . 7 1 Rate x [bits / s / Hz] Pr  Rate < Rate x  Graph-based Algorithm Exhaustive Search Greedy Algoritm Random Algorithm Exhaustive Search w/o constraints Figure 6: Cumulativ e Distribution F unction the achiev able rate for the worst-rate v ehicle. The prop osed graph-based algorithm attains the same p erformance as the exhaustiv e search. W e observe that when the vehicle density p er cluster is lo w, the greedy approach attains near optimal solutions as there are far more resources than vehicles to serv e. How ever, as the density increases, esp ecially near the o verload state, its p erformance drops. The random allo cation algorithm p erforms w orse than the other approaches. Fig. 6 sho ws the CDF of the achiev able rates. W e observe that the prop osed approach outperforms the other tw o ap- proac hes. F or the sake of comparison, we ha ve included the results of the unconstrained system, which do es not takes in to account conflict a voidance constrain ts. This is of course not desirable but it serves as a comparison b ound. 5 CONCLUSION W e hav e presented a nov el resource allo cation algorithm for V2V communications considering conflict constraints. W e w ere able to transform the original problem into a simplified form. In our future work, w e will consider ( i ) p o w er control and ( ii ) the assumption that a subset of v ehicles ma y b elong to more than one cluster simultaneously . REFERENCES [1] Crash data analyses for vehicle-to-infrastructure communications for safety applications. T echnical Report FHW A-HR T-11-040, U.S. Department of T ransp ortation, Nov ember 2012. [2] Intelligen t transp ort systems (its); stdma recommended parameters and settings for co op erative its; access la yer part. T echnical Report ETSI TR 102 861, ETSI, January 2012. [3] J. Munkres. Algorithms for the assignment and transportation problems. SIAM J Appl Math , 5(1):32–38, March 1957. [4] F umio Hiai. Monotonicity for entrywise functions of matrices. Journal of Line ar Algebr a and its Applic ations , 431(8):1125–1146, September 2009. [5] T echnical sp ecification group radio access network; evolv ed uni- versal terrestrial radio access (e-utra); physical layer pro cedures; (release 14) v14.2.0. 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