1 Throughput Optimizing Localized Link Scheduling for Multihop Wireless Networks Under Physical Interference Model Yaqin Zhou, Xiang-Yang Li, Senior Member, IEEE, Min Liu, Member, IEEE, Xufei Mao, Member, IEEE, Shaojie Tang, Member, IEEE, Zhongcheng Li, Member, IEEE 3 Abstract—We study throughput-optimum localized link Thelinkschedulingproblemhasbeenstudiedwithdifferent 1 0 scheduling in wireless networks. The majority of results on optimizationobjectives, e.g., throughput-optimumscheduling, link scheduling assume binary interference models that simplify 2 minimum length scheduling. Our study mainly focuses on interferenceconstraintsinactualwirelesscommunication.While maximizing throughput in multihop wireless networks. Nev- n the physical interference model reflects the physical reality a more precisely, the problem becomes notoriously harder under ertheless, a common primary issue here is to find a set of J the physical interference model. There have been just a few conflict-free links with regard to the interference models. 4 existingresultsonlinkschedulingunderthephysicalinterference Interference models matter significantly in scheduling al- 2 model,andevenfeweronmorepracticaldistributedorlocalized gorithm design and analysis. It is well known that the time scheduling. In this paper, we tackle the challenges of localized complexity of a throughput-optimum scheduling that tries ] link scheduling posed by the complex physical interference I constraints. By cooperating the partition and shifting strategies to find a maximum weighted independent set of links, is N into the pick-and-compare scheme, we present a class of local- polynomial under the 1-hop interference model in switched s. ized scheduling algorithms with provable throughput guarantee wired networks, whereas it becomes NP-hard under more c subject to physical interference constraints. The algorithm in general interference models in wireless networks [4]. Here [ the linear power setting is the first localized algorithm that achieves at least a constant fraction of the optimal capacity the weight associated with every link is its corresponding 2 regionsubjecttophysicalinterferenceconstraints.Thealgorithm queue length. Motivated by providing efficient solutions for v in the uniform power setting is the first localized algorithm the NP-hard problem, numerous low complexity scheduling 8 with a logarithmic approximation ratio to the optimal solution. algorithmswith suboptimalthroughputguarantee[5], [4], [6], 3 Our extensive simulation results demonstrate correctness and [7], [8], [9], [10], [11], [12], [13], have been proposed in 7 performance efficiency of our algorithms. 4 literature. . Index Terms—localized link scheduling, physical interference Despite of the numerous significant results gained for the 1 model, maximum weighted independent set of links (MWISL), problem, most of them assume simple binary interference 0 capacity region. 3 models,e.g.,hop-based,range-based,andprotocolinterference 1 models [14]. Under this category of interference model, a v: I. INTRODUCTION set of links are conflict-free if they are pairwise conflict-free. i AS a fundamental problem in wireless networks, link Conflictoftransmissionsontwodistinctlinksispredetermined X scheduling is crucial for improving networking perfor- independently of the concurrent transmissions of other links. r mances through maximizing throughput and fairness. It has Thus, interferencerelationshipsbased on these modelscan be a recently regained much interest from networking research represented by conflict graphs, and we can leverage classic community because of wide deployment of multihop wire- graph-theoreticaltools for solutions. However, in actual wireless networks, e.g., wireless sensor networks for monitoring less communication, interference constraints among concur- physical or environment[1] [2] through collection of sensing rent transmissions are not local and pairwise, but global and data [3]. Generally, link scheduling involves determination of additive. Conflict of distinct transmissions is determined by which links should transmit at what times, what modulation the cumulativeinterferencefrom all concurrenttransmissions, and coding schemes to use, and at what transmission power which is often depicted by the physical interference model, levels should communication take place [4]. In addition to e.g.,the Signal-to-Interference-plus-NoiseRatio (SINR) inter- its great significance in wireless networks, developing an ference model. But due to the simplification of binary inter- efficient scheduling algorithm is extremely difficult due to ference models on physical reality, the corresponding results the intrinsically complex interference among simultaneously no longer hold under the more realistic physical interference transmitting links in the network. model. Recent results [15], [16] have shown that throughput differssignificantlyunderthe two kindsofmodels.Moreover, Yaqin Zhou, Min Liu, and Zhongcheng Li are with Research Center of the global and additive nature of the physical interference Network Technology, Institute ofComputing Technology, Chinese Academy ofSciences, Beijing, China. modeldrivesprevioustraditionaltechniquesbased on conflict Xiang-YangLiandShaojieTangarewithDepartmentofComputerScience, graphsinapplicableortrivial.Consequently,designingandan- Illinois Institute ofTechnology, Chicago, IL,USA. alyzing scheduling algorithms under the physical interference Xufei Mao is with TNLIST, School of Software, Tsinghua University, Beijing, China. model becomes especially challenging. 2 Some recent research results [17], [14], [18], [19], [20], is also applicable. We prove that our revised algorithm [21], [22], [23], [24] have addressed a few challenges related canachieveO(log V )fractionofcapacityregionwhere | | tothelinkschedulingproblemunderthephysicalinterference V is the number of nodes. | | model. To the best of our knowledge, however, all of these, 3) We conduct extensive simulations to evaluate our al- withthroughputmaximizationorotheroptimizationobjectives gorithms. The simulation results demonstrate the cor- suchasaminimumlengthschedule[14],justfocusoncentral- rectness of our algorithms and corroborate our theo- ized implementation.Distributed or evenlocalized scheduling retical analysis. The comparisons with two centralized under the physicalinterference model is more demandingout algorithms in terms of average throughput performance of practical relevance. further show performance efficiency of our algorithms. Though[25],[26],[27]considerdistributedimplementation The remainder of the paper is organized as follows. In of centralized algorithms, they fail to provide an effective SectionII,wedefinetheexactsystemmodelsforourproblem. localized scheduling algorithm with satisfactory theoretical InSectionIII,wedescribethebasicideasofoursolutions,and guarantee [25] [26] and require global propagation of mes- the proposed localized scheduling algorithm with theoretical sages. A scheduling algorithm without theoretical guarantee performance analysis. In Section IV, we revise our solutions may cause arbitrarilybad throughputperformance,and global for the uniform power setting, and propose a localized algo- propagation of messages throughout network is inefficient in rithmwitha logarithmicapproximationratio.We evaluateour terms of time complexity [28], especially for large-scale net- solutions for both power settings in Section V. In Section VI, works. This motivatesus to developlocalized link scheduling we review related works on link scheduling. In Section VII algorithms with provable throughput performance. Here by a we discuss some related issues on proposed algorithms. We localized scheduling algorithm we mean that each node only conclude this paper in Section VIII. needsinformationwithinconstantdistancetomakescheduling decisions, while a distributed algorithm may inexplicitly need informationfar away. Since just local informationis available II. MODELSANDASSUMPTIONS for each node to collaborate on schedulings with globally A. Network Communication Model coupledinterferenceconstraint,it posessignificant challenges to designing efficient localized scheduling algorithms with We modela wireless networkbya two-tuple (V,E), where theoretical throughput guarantee. V denotes the set of nodes and E denotes the set of links. In this paper we tackle these challenges of practical lo- Eachdirectedlinkl=(u,v) E representsacommunication ∈ calized scheduling for throughput maximization under the request from a sender u to a receiver v. Let l or uv k k k k realistic physical interference model with the commonly- denote the length of link l. We assume each node knows its used linear and uniform power assignment. For the primary own location, which is necessary in the partition and shifting challenge of decoupling the global interference constraints, process. on observing that distance dominates the interference, we partition the links into disjoint local link sets with a certain B. Interference model distance away from each other. By this way, we bound the Under the physical interference model, a feasible schedule interference from all the other local link sets with a constant isdefinedasanindependentsetoflinks(ISL),eachsatisfying so that independent scheduling inside each local link set is possible. Then we shift the partitions to ensure every link P η uv −κ getscheduled.Usingthisweimplementlocalizedschedulings. SINRuv =∆ uP· ·ηk wkv −κ+ξ ≥σ, Furthermore,wenovellycombinethesetwostrategiesintothe w∈Tu w· ·k k P pick-and-compare scheme by which we can prove that our where ξ denotesthe ambientnoise, σ denotescertain thresh- algorithms obtain a theoretical capacity guarantee. old, and T denotes the set of simultaneous transmitters with u We summarize our main contributions as follows. u. It assumes path gain η uv −κ 1, where the constant ·k k ≤ 1) We proposea localized link schedulingalgorithmunder κ>2ispath-lossexponent,andη isthereferencelossfactor. the linear power setting with provable throughputguar- We considerthe followingtwo transmissionpowersettings. antee. We successfully decouple the global interference 1) Linearpowersetting:asenderutransmitstoareceiver constraints forthe implementationof localized schedul- v always at the power P = c uv β P where c u ing, by proving that there exists a constant distance and β are both constant satisfyi·nkg ck> 0≤,0 < β < κ, between any two disjoint local link sets to ensure inde- and P is the maximum transmission power. pendentlink schedulinginside a locallinkset underthe 2) Uniform powersetting: alllinksalwaystransmitatthe linear power setting. We then prove that our algorithm same power P =P. u can achieve a constant fraction of capacity region. We use P and P alternatively to denote the transmitting 2) We further revise the algorithm to work under the l u power of link l=(u,v). uniform power setting, and give a detailed analysis on We also assume all links having a length less than the theachievablecapacityregion.Whileundertheuniform maximum transmission radius κ ηP. The distance between powersetting,weprovethatifthesizeofthescheduling σξ setforlocallinksetsisupperboundedbyaconstant,the u and v satisfies r uv Rq, where r and R respectively ≤k k≤ solution to decouple the global interference constraints denotes the shortest link length and the longest link length. We suppose that r and R are known by each node. 3 C. Traffic models and scheduling In the rest of the paper we look for solutions to generate localized link schedulings of approximately optimal weight, Themaximumthroughputschedulingisoftenstudiedinthe following models. It assumes time-slotted wireless systems, with a certain constant probability at least. and single-hop flows with stationary stochastic packet arrival process at an average arrival rate λ. The vector A(t) = III. THEALGORITHM IN THELINEARPOWERSETTING l Al(t) denotes the number of packets arriving at each link Inthissection,wefocusonthedesignoflocalizedschedul- { } in time slot t. Every packet arrival process A (t) is assumed ing algorithm for the linear power setting. We firstly demon- l tobei.i.dovertime.We alsoassumeallpacketarrivalprocess strate the basic ideas of the algorithm design before involved Al(t) have bounded second moments and they are bound by in the details of implementation. A , i.e., A (t) A , l E. Let a vector 0,1 E max l max | | A. Basic idea ≤ ∀ ∈ { } denotea schedule S(t) at each time slot t, where S (t)=1 if l The basis of our idea is to create a set of disjoint local link l is active in time slot t and S (t)=0 otherwise. Packets l link sets in which the scheduling can be done independently departure transmitters of activated links at the end of time without violating the global interference constraint. The de- slots. Then, the queue length (it is also referred to as weight coupling of the global interference constraint is based on or backlog) vector Q(t) = Q(t) evolves according to the l { } the fact that distance dominates the interference between two following dynamics: distinctlinks. Thatis, if a transmitting link is placeda certain Q(t+1)=max 0,Q(t) S(t) +A(t+1). distance away from all the other transmitting links, the total { − } interference it receives may get bounded. We prove later that The throughput performance of link scheduling algorithms thisassumptionholdsactually.Based onthis, thenwe employ ismeasuredbyasetofsupportablearrivalratevectors,named thepartitionstrategiestodividethenetworkgraphintodisjoint capacity region or throughputcapacity. Thatis, a scheduling local areas such that each local area is separated away by policy is stable, if for any arrival rate vector in its capacity a certain distance to enable independent local computation region [12], of schedulings inside every local area. Links lying outside lim E[Q(t)]< . t ∞ localareaswillkeepsilentto ensureseparationoflocalareas. →∞ A throughput-optimalschedulingalgorithmcan achievethe As links lying outside local areas cannot remain unscheduled optimal capacity region. It is well known that the policy all the time or it will induce network instability, we use the of finding a maximum independent set of links to schedule shifting strategy to change partitions at every time slot to regardingtotheunderlyinginterferencemodelsisthroughput- makesurethateverybackloggedlinkwillbescheduled.These optimal[29].Unfortunately,theproblemof findinga MWISL locally computed scheduling link sets compose a new global itself is NP-hard generally [30]. Thus we have to rely on schedule (t) at every time slot t. X approximation or heuristic methods to develop suboptimal Inlightofthepick-and-comparescheme,wechooseamore schedulingalgorithmsrunningin polynomialtime.A subopti- weightedschedule,denotedasS(t), betweena newgenerated malschedulingpolicycanjustachieveafractionoftheoptimal schedule (t)andthelast-timescheduleS(t 1)usingQ(t). X − capacityregion.Thefractionallyguaranteedcapacityregionis Meanwhile, by Proposition 1, if we guarantee that depicted by efficiency ratio γ [4]. P(S(t) Q(t) γS (t) Q(t)) δ (1) ∗ Asuboptimalschedulingpolicywithefficiencyratioγ must · ≥ · ≥ find a γ-approximation scheduling at every time slot t to for some constant γ > 0, δ > 0, the queue length vector achieveγ timesoftheoptimalcapacityregion[31].Itremains Q(t) will eventually converge to a stable state. Then we can difficult to achieve in a decentralized manner. The pick-and- get a constant approximation ratio scheduling policy for the compare approach proposed in [5] enables that we just need optimal. to find a γ-approximationscheduling with a constant positive B. Detailed description probability.Thebasic pick-and-compare[5] worksasfollows. At every time slot, it generates a feasible schedule that has a Now we describe the details of our method. constant probability of achieving the optimal capacity region. We firstgivea briefdescriptionofthepartitionandshifting If the weight of this new solution is greater than the current strategies [11], as illustrated in Fig. 1. We partition the plane solution, it replaces the current one. Using this approach intocellswithside lengthd=R,usinghorizontallinesx=i achieves the optimal capacity region. The proposition below andverticallinesy =j forallintegersiandj.Averticalstrip further extends this approach to suboptimal cases. withindexiis (x,y)i0ofgeneratingan verticalstripiandahorizontalstripj.Asuper-subSquare(i,j) independentset (t) of links with weight at least γ times the is the set of cells: cell(x,y)x [i K +a ,(i+1) K + t X { | ∈ ∗ ∗ weightoftheoptimal.Then,capacityγ Λ canbeachievedby a ),y [j K+b ,(j+1) K+b ) , anda sub-square(i,j) t t t · ∈ ∗ ∗ } switching links to the new independent set when its weight is insideitisthesetofcells: cell(x,y)x [i K+a +M,(i+ t { | ∈ ∗ larger than the previous one(otherwise, previous set of links 1) K + a M),y [j K + b + M,(j + 1) K + t t ∗ − ∈ ∗ ∗ will be kept for scheduling). The algorithm should generate b M) . The corresponding link set Y (or L ) consists t ij ij − } the new scheduling S(t) from the old scheduling S(t 1) of links with both ends inside super-subSquare(i,j) (or sub- − and current queue length Q(t). square(i,j)). Let constant K = 2M +J, 0 a ,b < K. t t ≤ 4 TABLE I: Summary of notations J sidelengthofsub-square Lij linksetofsub-square(i,j) Yij linksetofsuper-subSquare(i,j) K sidelengthofsuper-subSquare Xij(t) newscheduling forLij att Sij(t) scheduling forYij att Zi locallinkset OPTi∗j(t) localoptimalMWISLforLij S∗(t) globaloptimalMWISLatt R longestlinklength Si∗j(t) intersection ofLij andS∗(t) S(t) globalscheduling attimeslott d sidelength ofcell kuvk linklength rS(l) relative interference lgetfromlinksetS aS(l) affectness lgetfromlinksetS Q(t) queuelengthvector W(S) weightoflinksetS ISl interference link lsuffered fromlinkset S Iml ax themaximuminterference lcanbear Imax theminimumofIml ax nodethen participatesin computingthe new localscheduling, (cid:54)(cid:88)(cid:83)(cid:72)(cid:85)(cid:16)(cid:86)(cid:88)(cid:69)(cid:54)(cid:84)(cid:88)(cid:68)(cid:85)(cid:72) (cid:54)(cid:88)(cid:69)(cid:16)(cid:86)(cid:84)(cid:88)(cid:68)(cid:85)(cid:72) (cid:54)(cid:88)(cid:83)(cid:72)(cid:85)(cid:16)(cid:86)(cid:88)(cid:69)(cid:54)(cid:84)(cid:88)(cid:68)(cid:85)(cid:72)(cid:54)(cid:88)(cid:69)(cid:16)(cid:86)(cid:84)(cid:88)(cid:68)(cid:85)(cid:72) denoted as (t), for its sub-square(i,j) using the weight ij X Q(t).LetS (t 1)bethesetoflinksfromS(t 1)(theglobal ij − − solutionattime slott 1)fallingin the super-subSquare(i,j) − instead of sub-square(i,j). If S (t 1) Q(t) > (t) ij ij − · X · Q(t), let S (t)=S (t 1), else S (t)= (t), the global ij ij ij ij − X solution is the union of S (t) from all super-subSquares. ij The pseudo-codes are given in Algorithm 1. Since every nodeknowsthelocalityfromwhichitwillcollectinformation, (cid:71) (cid:45)(cid:71) t (t+1) (cid:71) (cid:45)(cid:71) it then participates the corresponding local computation, and Shifting at last it sends (if it is a coordinator) or receives (if not) (a) Partition(K,at,bt). Here the (b) Partition(K,a(t+1),bt). the results. It may require multihop propagation to collect gray area is the local area the Change the partition through andbroadcasttheneededinformationinsidesuper-subSquares. link set of which participate in shifting to the right by one cell, computingthenewschedule;the ensuring that links in the white The details are omitted here. Please refer to [27] for de- links in the white area keep area in previous partitions have tailed implementation. The coordinatorcomputes a new local silent. opportunity to be scheduled. scheduling link set by enumeration in constant time 2|Lij|, becausethesizeofinterference-freelinksforasub-square(i,j) Fig. 1: The partition and shifting process. The partition is bounded by a constant (we claim it in Lemma 1). changes at different time slots by shifting vertically or hor- izontally by a step of one cell-length. Algorithm 1 Distributed Scheduling by node v under the a , b are adjustable variables, referred to as the horizontal t t linear power setting and vertical shifting respectively. M is a constant, relating to the distance for independent scheduling in local areas. We 1: state = White; active = No; Coordinator = No; will formallydefine it later in Lemma 2. We define the above 2: Calculate which cell node v resides in regarding to the current partition(K,a ,b ); process as Partition(K,a ,b ). By separately adjusting a , b t t t t t t by a step of single cell, we can getK2 differentpartitionsfor 3: if v is the closest node to the center of super-subSquare then a plane totally. Note that herein nodes do not shift, it looks like that 4: Coordinator = Yes; a virtually partitioned cover having the same size with the 5: end if originaltopologyshifts verticallyorhorizontally.Initially,the 6: if Coordinator = Yes then virtualcoverandtheoriginaltopologyofthenetworkcoincide 7: Collect Q(t) and Sij(t−1). completely.Whenthevirtualcovershifts,nodeslocatedin the 8: Compute Xij(t) in sub-square(i,j); edge area willcertainly getuncoveredif the distance between 9: if Sij(t−1)·Q(t)>Xij(t)·Q(t) then the edge of topology and the nearest nodes is less than Kd. 10: Sij(t)=Sij(t−1); To tackle this, we just need to assume a larger virtual cover, 11: else e.g.,thedistancebetweentheedgeoftopologyandthenearest 12: Sij(t)=Xij(t); nodes is equal or greater than Kd. 13: end if Then at each time slot nodes cooperate to compute a 14: Broadcast RESULT(Sij(t)) in super-subSquare(i,j); 15: end if globally feasible scheduling as follows. 16: if state = White then Time slot t = 0: Every node first decides in which cell it resides by a partition using (a ,b ) = (0,0); then it 17: if receive message RESULT(Sij(t)) then participates in the process of comput0ing0a local scheduling 18: if v ∈Sij(t) then 19: state = Red; active = Yes; S (0) for the sub-square(i,j) it belongs to. Let the solution ij 20: else S(0) of time slot 0 be the union of the local solutions S (0) ij 21: state = Black; active = No; for all sub-squares. 22: end if Timeslott 1:Everynodedecidesinwhichcellitresides ≥ 23: end if by a partition starting from (a ,b ). The shifting strategy for t t 24: end if a and b works as follows. We let a =t mod K; and b = t t t t (b + 1) mod K if a = 0, or it keeps unchanged. Each t t 5 C. Theoretical analysis and proof Lemma 3: ( [21]) There is a polynomial-timeprotocolthat takesa p-signalset andrefinesintoa p-signalset, for p >p, We will then prove that our algorithm is correct and has a ′ ′ increasing the numberof slots by a factor of at most 4(p′)2. constant approximation ratio to the optimal capacity region. p Given a network (V,E), supposing Zi is a set of disjoint That is, if a p′-signal set of links can be scheduled simul- local link sets inside for scheduling, w∪here Z E and Z taneously in a single slot, then we can use some algorithm to i i Zj =φ if i=j, for any link l Zi, if l is acti∈vated, then ∩ partitiona p-signalintoatmost4(p′)2 slotssuchthateachset 6 ∈ p Il =Iiln+Iolut oafpl-isnikgsnianlesvetercyanslobteisreafipn′e-dsiginntaolsaettm.Tohsits4l(emp′m)2apim-spiglineaslthseatt where Il denotes cumulative interference from all other p ′ throughapolynomial-timealgorithm,e.g.,afirst-fitalgorithm. activatedlinksinthenetwork,Il denotesthetotalinterference in Next in Lemma 4 we present a constant approximation from simultaneously transmitting links inside Z and Il i out relationship between each locally computed scheduling link denotes the total interference from transmissions outside. set and its counterpart of the global optimal scheduling set. Therefore, we can do independent scheduling inside Z owuitthsioduetZco,nisfidIelratigoentsobfoIuolnudtefdrobmy acocnocnusrtraenntt, it.rea.n,smissionsi tioLnermatmioato4:thTehweewigehigthotfothfeXiinjt(etr)sehcatsioancsoentsbtayntthaeplporcoaxlilminak- i out set L and the global optimal MWISL S (t). Il (1 ε) Il , Il ε I , 0<ε<1, l Z . ij ∗ in ≤ − · max out≤ · max ∈ i Proof:Thoughthe total interferencethateverylink l can lWinkesleitnIEmaxcadnentoolteerathnetmduarxinimguamsuinctceersfsefruelnctreatnhsamtitshseiolonn,gaensdt twoelersahtoewatthaatsiutcdcoeesssfulilttlterainnsflmuiesnsicoenobnelcoocmalesco(m1p−utaεt)iIomlnaoxf, Il representthemaximuminterferencethatanactivatedlink optimal link scheduling set for L . max ij l can tolerant during a successful transmission. NormallyanyISLisa1-signalset.Nevertheless,inorderto We will give a formal statement that Il of each activated keep independence of sub-squares, the locally computed ISL out link l is indeed bounded by εImax in Lemma 2. However, to shouldbe a p-signalset, where p is biggerthan 1. Thatis, for prove Lemma 2, we shall claim the following Lemma 1 in the affectness of a normal ISL it holds that: advance. Lemma 1: In the linear power setting, the number of σ Il interference-free links for a local link set Zi inside a square aS(l)=cl rl∗(l) max 1, (2) with a size length JR is bounded by a constant. Let OPTi ·l∗ S ≤ 1−σξ/Pl · Pl ≤ refer to the optimal MWISL for Zi. That is, X∈ whereas the affectness of a locally computed ISL for sub- (√2JR)κ 1 ξ rβ−κ square must satisfy that, |OPTi|≤ (1 ε) (cid:20)σ − ·cη (cid:21)+1, − σ (1 ε) Il locWalePorlpeottoimf|:OaTPlhMTisiW|puIbrSodoLefnfioosrteaZvaain,ilatuhbpelpeneiwrnbeAohupanpvdeen,odfixthAe.size of the aSij(t)(l)=cl·l∗∈XSij(t)rl∗(l)≤ 1−· σξ−/Pl · mPalx ≤1−(ε3.) Lemma 2: In a given network (V,E) under the physical Therefore, by Lemma 3, any p-signal set can be refined interferencemodelinthelinearpowersetting,iftheEuclidean into at most 4(p′)2 p-signal sets, where p >p. So a normal distance between any two disjoint local link sets is at least p ′ ′ M R, then activated links in each local link set suffer a MWISL can be refined into 4 1 -Signal link sets at bou×nded cumulative interference from all other activated link most.Sincethe 1 -Signall(i1n−kε)s2et(r1e−tuεr)nedbyenumeration sets, i.e., for each activated link l in local link set Zi, ismostweighted(,1s−oε)thelocallycomputedlinkschedulingsets Iolut≤ε·Imax, 0<ε<1, Xij(t) has a weight 1 (1 ε)2 where M is a constant, satisfying M 2πcηRβ−κ·|OPTi|ub κ W( ij(t)) − W(OPTij(t)). (4) ≥(cid:20) (κ−2)εImax (cid:21). X ≥ 4 Proof: This proof is available in Appendix B. LetS (t)=S (t) L denotethe intersectionby L and i∗j ∗ ∩ ij ij To analyze the theoretical performance of our method, we S (t), where S (t) is the globaloptimal MWISL at time slot ∗ ∗ first review the following definitions. t. Let W(S) denote the summed weight of links in a link set Definition 2: (affectness [21]) The relative interference of S. It is obvious that link l on l is the increase caused by l in the inverse 4 ∗ ∗ coofntvheeniSenINceR, daetfiln,enram(le)ly=rl∗0(.l)Le=t cIll∗=/(Plηkuvkσ−κ). For W(Si∗j(t))≤W(OPTij(t))≤ (1−ε)2W(Xij(t)). (5) be a constant that indlicates the extentl to w1h−icσhξ/t(hPelηakumvbki−eκn)t Thus it proves the lemma. Theorem 1: S(t) = S (t) computed by our algorithm is ij noise approaches the required signal at receiver t . The ∪ v an independentlink set underthe physicalinterferencemodel affectness of link l caused by a set S of links, is the sum inthelinearpowersetting.Theweightof S(t), i.e.,W(S(t)), of relative interference of the links in S on l, scaled by c , or l isaconstantapproximationoftheweightoftheglobaloptimal aS(l)=cl·l∗P∈Srl∗(l). MWISL with probability of at least 1/K2. Definition 3: (p-signal set [21]) We define a p-signal set to Proof: The proof consists of two phases. We first prove beonewheretheaffectnessofanylinkisatmost1/p.Clearly, that S(t)= S (t) is an independentset. We nextderivethe ij ∪ any ISL is a 1-signal set. approximation bound S(t) achieves. 6 Thedistancebetweenanydistinct (t )isnolessthan2M mn ∗ X l3 cells. Thenwe considerthe distancebetweena disjointsubset of Φpq (t ) and a (t ). Since (t ) locates in l4 Partition pqi′j′ i′j′ ∗ Xmn ∗ Xmn ∗ at time slot t* subS-sqSuare(m,n), which is M cells away from the border of l1 l2 super-subSquare(m,n),thedistance betweena disjointsubset of Φpq (t ) and a (t ) is still no less thanM cells. Partition pqi′j′ i′j′ ∗ Xmn ∗ at time slot t*-1 CoSmpSrehensively,S(t∗) consistsof disjointsubsets which are separated by at least M cells. Note that a disjoint subset of S(t ) does not equalize to a ∗ Sij(t∗) since a Si′j′(t∗ 1) may be reserved completely in − Fig. 2: Partitions at time slot t∗−1 and time slot t∗ differentsuper-subSquaresattime slot t∗. Here we denotethe disjoint subset by ψ (t ), and S(t )= ψ (t ). Phase I: We rely on induction to infer that at every time i ∗ ∗ i ∗ By Lemma2, foreachlinkl ψ (t ), where ψ (t ) comes slot S(t) is a union of disjoint activated local link sets that ∈ i ∗S i ∗ are separated by at least M cells from each other. Then we from the former part of equation (9), we have ISl(t∗) ≤ Imlax. Meanwhile,foreachl ψ (t ),wherel ψ (t )comesfrom have that S(t) is an independent link set under the physical ∈ i ∗ ∈ i ∗ interferencemodelbyLemma2.Thefollowingarethedetails. thelaterpartof(9),itholdsthat ISl(t∗) ≤Imlax. Thenwehave For any link l S(t), assuming l S (t), the total inter- Il Il , l S(t ), (10) ∈ ∈ ij S(t∗) ≤ max ∀ ∈ ∗ ferencelsuffersfromalltheothersimultaneouslytransmitting indicating that S(t ) is an independent set. links in S(t) is denoted by Il . ∗ S(t) Next we consider situations at time slot t +1. Similarly, Attimeslot0,everylocalactivatedlinksetS (0)= (0) ∗ is kept 2M cells away from each other, sioj S(0)Xiisj an S(t∗+1) composes of disjoint subsets separated by no less than M cells. Using the same techniqueas at time slot t , we independent set by Lemma 2. ∗ can get that S(t +1) is still an independent set. At time slot 1, either S (0) or (1) is chosen to be a ∗ ij Xij By induction we can infer that S(t) is a union of disjoint part of S(1). For those super-subSquares whose S (1) = ij activated subsets separated by M cells at least, thus it is an Si′j′(0) ∩ Yij(0), their distance is kept at least 2M cells independentset. Hereinwefinish thefirstphraseoftheproof. away. For those super-subSquares whose S (1) = (1), ij Xij Phase II: We derivetheapproximationratiobythepigeon- their distance is also kept at least 2M cells away. And the hole principle, Lemma 4, and Proposition 1. distance between the two kinds of link set is at least 2M 1 − Let D(t) denote the link set of the removed strips at cells away. So S(1) is an independent set. Partition(K,a ,b ). D (t) represents a subset of S (t), links At some time slot t , t > 1, for some super-subSquares, t t ∗ ∗ ∗ ∗ ofwhichfallinsideD(t),i.e.,D (t)=D(t) S (t).Recalling S (t ) consists of disjoint subsets from several different ∗ ∗ ij ∗ ∩ Si′j′(t∗ 1) which fall into Yij(t∗), i.e., that there are K2 different partitions for a plane totally. If − we tried all these partitions in a single slot, each cell(i,j) Sij(t∗)=S(t∗−1)∩Yij(t∗)= {Si′j′(t∗−1)∩Yij(t∗)}. (6) would appear in the “removed” strips at most 2KM times. i[′j′ Thus the weight of all D (t) in the K2 partitions should be ∗ For instance, as illustrated in Fig. 2, link sets {l1,l2} 2KMW(S∗(t)), i.e., and l ,l respectively get scheduled in two differ- { 3 4} K2 1 ent super-subSquares (i.e., super-subSquare(1,2) and super- − subSquare(2,2)) at time slot t 1, the links l ,l ,l ,l W(Di∗(t))≤2KMW(S∗(t)). (11) ∗ − { 1 2 3 4} i=0 are then get scheduled in the same super-subSquare(1,2) at X Since actually we only experience one partition at time time slot t . ∗ slot t, by the pigeonhole principle the corresponding We then let Partition(K,a ,b ) has a probability at least 1/K2 to be an t t Φiij′j′(t∗)=Si′j′(t∗−1)∩Yij(t∗) (7) optimalpartition,theweightofwhoseremovedlinksinD∗(t) is no greater than that of other partitions. Instantly we have for brevity. Each Si′j′(t∗ 1) is kept at least M cells away from each other, so is eac−h Φij . the following with probability of 1/K2 at least, i′j′ Clearly, S(t ) can be divided into two separated subsets, ∗ boynenfeowrSlmy(etc∗do)=mbypustSoemd(eXt∗si)ju(bts∗e)t,sio.ef.,S(t∗ −1), the other form(e8d) W(Di∗(t))≤ K12 ·KXi2=−01W(Di∗(t))≤ 2KMW(S∗(t)), (12) ij [ij and then, 2M W( S∗(t))=W(S∗(t) D∗(t)) (1 )W(S∗(t)). (13) =[pq i[′j′Φpi′qj′(t∗)[ m[n, Xmn(t∗) (9) ∪ ij \ ≥ − K mn6=pq . FollowingLemma 4,the followingholdswith a probability     of 1/K2, Since Φpq (t )isasubsetofS(t 1),itiscomposed pqi′j′ i′j′ ∗ ∗− (1 2M)W(S∗(t)) 4 W( (t)). (14) by disjoiSnt sSubsets with a mutualdistance of M cells at least. − K ≤ (1 ε)2 ∪Xij − 7 So by Proposition 1 we get that the approximation ratio for constant. Given a set of links and weights associated with the the optimal is 4 links, the algorithm works as follows: (1−2·MK)(1−ε)2. Phase I: Remove every link whose associated weight is at most wmax where w denotes the maximum weight among D. Time and communication complexity n max all links and n is the number of all given links. Let w min We then give a brief analysis on time complexity and denote the minimum weight from the remaining links. communication complexity. PhaseII:Partitiontheremaininglinksintologwmax groups We define the time complexity as the time unitsrequiredby wmin such that the links of the i th group G have weights within the runningofAlgorithm1 insideeachlocalareain the worst − i [2iw ,2i+1w ]. For each group G of links, it finds an case[28].Itincludestimeunitsforlocalinformationcollection min min i independent set of links among it by adopting the method in and local computation time at the center node. At every time [32]. Totally it will get logwmax ISLs, one for each group. slot, the coordinators shall collect queue information and last wmin Thenitchoosesthe onewith themaximumweightamongthe schedulingstatusofalllinksinsidethesuper-subSquare.After logwmax ISLs as the final solution. computation,thecoordinatorsbroadcasttheschedulingresults wmin TheremoveoperationofPhaseIsimplyexcludesthoselinks throughout the super-subSquares. It may require multihop with smaller weight so that we can get a smaller number of propagation to collect and broadcast the needed information groups when partition is done according to weight. In that inside super-subSquares. We implement it as follows. way, it is more likely to get a better approximation ratio to To avoid collision, the coordinator can first compute a tree the optimal in practical, though the theoretical guarantee is that determinesthe sequenceof transmissions at each node in not explicitly improved. We also notice that the resulted link the super-subSquare based on topology information. As the schedulingset is an ISL by the methodin [32]. With the help root node it broadcasts the results to its children (or child). ofthe followinglemma presentedin [32], we will getthatthe The children acquire the orderof transmissions and relay this the resulted link set is indeed with limited number of links. message as the specified order. The message is relayed until Lemma 5: ([32])Consideralink l=(u,v)andasetN of it reaches every node. Then the required information (or the nodes other than u whose distance from u is at most ρ uv . schedulingresults)canbecollected(orbroadcast)accordingto k k If link l succeeds in the presence of the interference from N, the sequence.We supposethat the the parentnode can packet then the children’s messages into its own packet for collection. In (ρ+1)κ uv theworstcaseeachnodehastotransmitonebyoneatdifferent N 1 (k k)κ mini slot, then it causes O(n ) time complexitywhere n is | |≤ σ − R ij ij (cid:20) (cid:21) the number of nodes inside super-subSquare(i,j). Herein we The corollary below asserts a upper bound of the number just give a basic scheme, the time complexity may be further of successfully transmitting links in a local link set with size reduced with the help of other techniques. JR JR when utilizing Algorithm 1 in [24]. After all required information gathered, the local compu- × Corollary 1: Thecardinalityoftheresultedlinkscheduling tation complexity at the center node is O(2|Lij|), which can set by Algorithm 1 in [24] is upper bounded, i.e., be ignored comparing to the time for information collection. ThusthetotaltimecomplexityisO(n ).The communication ij (√2JR +1)κ r complexity is the total number of basic messages transmitted (t) r 1 ( )κ ij |X |≤ σ − R during each scheduling in the worst case [28]. The commu- (cid:20) (cid:21) nication complexity is O(V ), and the number of messages | | Proof:In the methodpresentedin [32], it addsfirstly the transmitted in each local area is O(n ). ij shortest link among all the candidates to the scheduling set. Andthedistancebetweenanypairofnodeswillbenogreater IV. THE ALGORITHM IN THEUNIFORM POWERSETTING than √2JR. Therefore, let r be the shortest link length and We now extend the framework to the uniform power set- then ρ= √2JR, we derive the upper bound. r ting. Instead of enumeration, we compute a MWISL of the We let (t) denote a upper bound of the size of the ij ub candidate links for each sub-square by adopting the method |X | local computed ISL for Z by Algorithm 1 in [24]. Then we i proposed in [24]. The reader may wonder why we do not present the lemma below to show that in the uniform power use enumerationdirectly as we do in the linearpowersetting. setting, if any two disjoint local link sets of a given network The difficulty lies behind the fact that the cardinality of the areseparatedbyacertainconstantdistanceaway,independent optimal MWISL in each sub-square is no longer bounded by scheduling inside each local link set without consideration of a constant in the uniform power setting, which drives M no interference outside is possible. longer bounded by a constant and simple enumeration time Lemma 6: In a given network (V,E) under the physical consuming.Except for the difference,the generalstructure of interference model in the uniform power setting, if the Eu- thealgorithmisthesamewiththatofthelinearpowersetting. clidean distance between any two disjoint local link sets is The following we will give a detailed analysis, and derive a at least M R , then activated links in each local set suffer logarithmic bound on throughput capacity. × negligiblecumulativeinterferencefromallotheractivatedlink We first describe the main idea of Algorithm 1 in [24] for sets, i.e., for each activated link l in local link set Z , i computationofMWISLinsideeachsub-square,andshowthat the cardinality of the resulted link set is upper bounded by a Il ε I , 0<ε<1, out ≤ · max 8 whereI isthemaximuminterferencethatthelongestlinks Since µ=log V , it is clear that Algorithm 1 can achieve max | | in E can tolerant during a successful transmission, M is a O( 1 )oftheoptimalcapacityregionintheuniformpower log V constant, satisfying M 2πηP·|Xij(t)|ub κ1 setting| b|y Proposition 1. ≥(cid:20)(κ−2)εImaxRκ (cid:21) . The time complexity and communicationcomplexity is the Proof: The proof is available in Appendix C. same as the corresponding of the linear power assignment. In light of Lemma 6, the partition strategy to enable distributedimplementationwillremaineffectiveintheuniform V. PERFORMANCE EVALUATION power setting. Wedosimulationexperimentstoverifythecorrectness,and Next, we reveal the relationship between the newly com- evaluate the throughput performance of our proposed algo- putedschedulingsetforsub-square(i,j)andthecorresponding rithms. The general setting of our experiments is as follows. S (t). It will help us to derive an approximation ratio of the i∗j We consider a network with 500 nodes, half of which as global solution generated by our algorithm later. sendersrandomlylocatedonaplanewithsize200 200units, Lemma 7: Theweightofthenewlycomputedlinkschedul- × the other half as receivers positioned uniformly at random ing set inside each sub-square(i,j), i.e., W( (t)), has a approximationratioof 4µ totheweightoftXheijintersection inside disks of radius R around each of the senders. Packets (1 ε)2 arrive at each link independently in a Poisson process 1 with set by the corresponding−local link set L and the global ij the same average arrival rate λ. Initially, we assign each link optimal MWISL S (t). ∗ k packets where k is randomly chosen from [100,300]. The Proof:Thenewlycomputedlocalschedulingsetforeach path loss exponent is set to be 3. sub-square should be a (1 ε)-signal set, so its weight is We conduct three series of experiments to fully evaluate at least (1−ε)2 times the corr−esponding1-signalset. Since the our algorithms. In the first series, we verify the correctness 4 solutionforMWISLwithuniformpowerassignmentproposed ofouralgorithms,thatis, whethereachactivatedlinkreceives in [24] finds an ISL with affectness no greater than 1, it is boundedinterferencefromallotherlinksoutsidethelocal link obvious that set it belongs to. The other two series of experiments focus 4µ on the evaluationof average throughputperformancein terms W(S (t)) W(OPT (t)) W( (t)), (15) i∗j ≤ ij ≤ (1 ε)2 Xij of total backlog (the total number of unscheduled packets). − The total backlog fluctuates slightly in a region if the arrival where µ = log V is the approximation bound of the algo- | | rate vector lies in the achievable capacity region of a link rithm in [24]. schedulingalgorithm.Oritincreasesdramaticallyifthearrival Then, we will state an exact bound on the throughput rate vector exceeds the achievable capacity region. If the performance of our algorithm in Theorem 2. total backlog increases unboundedly to infinity, the network Theorem 2: The union of the local computed scheduling will become unstable. Thus we can roughly approximate link sets, i.e., S(t) = S (t), is feasible under the uniform ij an average supportable arrival rate through the changes of power setting. The weight of S(t) achieves a fraction of S backlog. O(log V ) times the optimal solution. | | In the second series, we study how some related variables Proof: We first show that the resulted scheduling link affect the performance of the algorithms. In the third series, set by our algorithm is an independent link set with uniform we further study the performance efficiency of the proposed powerateverytimeslot.Usingthesametechniqueintheproof algorithms by comparisons with two centralized algorithms: of Theorem 1, we can inductively conclude that each global 1) Greedy Maximal Schedule (GMS), which is observed scheduling set S(t) is composed by disjoint link sets which to perform throughput optimally in simulations [4]. It are keptat least M cells away from each other. Therefore,by greedily adds to the scheduling set the most weighted Lemma 6, we can get that for each link l S(t), the total ∈ linkthatisindependentwith thepreviouslyaddedlinks. interference it receives satisfies that, 2) RandomizedAlgorithms, where each time slot links are Il (1 ε) Il +ε I Il (16) randomlypicked, and addedto currentschedulingset if S(t) ≤ − · max · max ≤ max. they do not conflicts with previous selected links. The It then follows that S(t) is an independentlink set. process is repeatedly performed until no new links can Wenextprovetheapproximationboundontheperformance be added. of our algorithm. Remember that a Partition(K,a ,b ) has a t t probability of 1/K2 to be a optimal partition. In that case, The reasons why we do not compare with distributed im- with the help of Lemma 6 it follows that plementationof the two centralizedalgorithmsare as follows. Undoubtedly, these two are distributed in nature according (1 ε)2(K 2M) W(∪Xij(t)) ≥ − 4Kµ− W(∪Si∗j(t)) (17) to some locality-sensitive approaches [28], if considering binary interference models. And previous works on binary (1 ε)2(K 2M) − − W(S∗(t)) (18) interference models indeed implement them in a distributed ≥ 4Kµ manner, i.e., [34] proposing a local greedy link scheduling Hence, we get algorithmand[8]makingschedulingdecisionsbyprobability. (1 ε)2(K 2M) 1 Thekeypointisthattheeffectofthiscategoryofinterference P W(S(t)) − − S∗(t) . (19) ≥ 4Kµ ≥ K2 (cid:0) (cid:1) 1The results hold under any arrival process satisfying the strong law of largenumbers,usingthefluidmodelapproach [33][12]. 9 0.07 Inference outside 1 Affectness inside εImax. From Fig. 3(b), we can furtherconfirm that the trans- 0.06 Minfaexriemnucme otoultesridaeble 0.8 TMoatxailm auffmec atnffeescstness inside mittinglinkssuffernegligibleinterferenceoutside.For Fig. 4, 0.05 Interference000...000234 Affectness00..46 orweuectessiihvdoeewsaavttheerdaiagfvfeeerraeafngfteectoitmnuetessissdlepoetisrntitenrrafnFesrimgen.ict4tei(napg)e,rlaitnnradkntsihnmeiFtttioignt.agl4la(ibnn)kd. 0.2 0.01 Theseresultscooperatetoshowthattheoutsideinterferencea 00 1 2 3 L4ink ID5 6 7 8 9 00 1 2 3 T4ime sl5ot 6 7 8 9 transmittingreceivesisindeedboundedbyεImax,andindicate the correctness of Algorithm 1. (a) Interference vs.links (b) Affectness vs.links Next we set anotherset of experimentsto study the perfor- Fig. 3: The interference/affectness each randomly chosen transmit- manceofAlgorithm1aloneunderdifferentvaluesofvariables ting link receives with the linear power assignment. (a) shows the thatmay impactthe averagethroughputperformanceactually. outsideinterferenceeachlinkreceives.(b)showstheinsideaffectness The variables K and ε dominate the theoretical bound of M for each link, and the total affectness each receives from all other Algorithm 1. When ε is fixed, a larger K implies a bigger M simultaneously transmitting links. fraction of the optimal capacity region, but with a smaller probability of 1 to achieve it. ε denotes a weighting factor 0.07 Inference outside 1 Affectness inside K2 0.06 Maximum tolerable Total affectness between the interference a activated link suffers inside and inference ouside 0.8 Maximum affectness inside 0.05 outsidethesub-square.Abiggervalueof εindicatesasmaller Inteference000...000234 Affectness00..46 vvaalluuee ooff MMK athsemoraeltliecralεlyl.eaTdhserteofoareb,igthgoeurgfhracutniodnearlthcaepfiacxietdy 0.2 region, the probability to achieve this region gets smaller 0.01 01 0 20 30 40 50 60 70 80 90 100 01 0 20 30 40 50 60 70 80 90 100 because of the resulted bigger M. Since the running time of Time slot Time slot the simulationis muchshorterthanthe time forthe algorithm (a) Averageinterferencevs.timeslot (b) Averageaffectness vs.timeslot to achieve the theoretical value, the probability will impact the actual average throughput in our experiments. Typically, Fig.4:Averageinterferencepertransmittinglinkreceivesatdifferent a small probability of 1 may cause poorer throughput time slots with the linear power assignment. (a) shows the average K2 performance. An experiment study of the two variables are outsideinterferencepertransmittinglinkreceives.(b)showstheaver- illustrated in Fig. 5 and Fig. 6. ageinsideaffectness,andtheaveragetotalaffectnesspertransmitting In Fig. 5 we compare the total backlog by increasing the link receives from all other simultaneously transmitting links. feasible valuesof K where ε servesas an constant. Fig. 5(a), M is localized and binary. It is easy to keep feasibility of the 5(b),5(c)respectivelyshowsthecomparisonsoftotalbacklog final global scheduling set by inactivating all other links in at time slot 1000 with increasing arrival rate when ε = 0.2, the interference range of an activated link surely or w.h.p.. ε = 0.4, ε = 0.8. Note that the value of K must be greater M However, it is not so obvious and easy to guarantee under than 2, or the size of the sub-squares will be 0. It shall also the physical interference model. It is hard to localize these be noticed that the feasible values of K are different when ε M approachesbecause the interference effect is globaland addi- varies,sinceεaffectsthevalueofM.Itthenexplainswhywe tive. Simplyinactivatingall otherlinks in some neighborhood set different values for K for varying ε. All the three figures M of an activated link will not help much to keep the final show that the throughput performance generally improves as schedulingsetfeasible.Consideringthesedifficulties,wefocus K increases, which coincides with the theoretical results we M on comparisons with the centralized visions. derive. Though the relevant probability shall become smaller as K increases, there is no obvioussign shown in Fig. 5. (a), M A. Under Linear Power Setting 5.(b).WecanseeanobviousimpactinFig.5(c)where K can M Inthissubsectionweconductthethreeseriesofexperiments besetbiggervalues.Itshowstheaveragethroughputbecomes for the linear power assignment. a little worse at K =7 than K =6. M M We first plot the numerical results for the first series of experiments in Fig. 3, 4. For convenience, we call the x 105 4x 105 interference/affectnesseachlinkreceivesfromallother simul- 3.5 eeepppsssiiilllooonnn===000...248 3.5 eeepppsssiiilllooonnn===000...248 3 3 tsaaufnfpeeecort-unsseulsybsS.tqrSauinnasrcmeeiittthtienbgemlolainxngikmssutoomutastosidleeor/uaintbssliieddeeinittnehtreefercfreoernrercenescpeoo/finnedsaiincdhge Total backlog12..255 Total backlog12..255 1 linkforsuccessfultransmissionisdifferent,thecorresponding 1 0.5 0.5 inside and total interference each link receives is shown as 0 0.2 0.4 0.6 0.8 1 00 0.2 0.4 0.6 0.8 1 Average arrival rate Average arrival rate the value of affectness for better illustration. Recall that, for (a) K =3 (b) K =4 a successful transmission, the inside and total affectness shall M M be less than 1 ε and 1 respectively. Fig. 6: Total backlog vs. average arrival rate vs. different values of − Inthissetofexperiments,wesetεasmallvalueof0.3.For ε at time slot 1000 in the linear power setting Fig. 3, we randomly choose a time slot and 10 transmitting links. As shown in Fig. 3(a), the outside interference each We briefly illustrate the impact of ε at fixed K in Fig. 6. M chosen link receives is much smaller than the constant bound Thoughthetheoreticallyachievablecapacityregionshall have 10 3.5x 105 3.5x 105 3.5x 105 K/M=3 K/M=3 K/M=3 3 KK//MM==34.5 3 KK//MM==34.5 3 KK//MM==45 K/M=6 Total backlog12..1255 Total backlog12..1255 Total backlog12..1255 K/M=7 0.5 0.5 0.5 00 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Average arrival rate Average arrival rate Average arrival rate (a) ε=0.2 (b) ε=0.4 (c) ε=0.8 Fig. 5: Total backlog vs. average arrival rate vs. different values of K at time slot 1000 in the linear power setting M been greater with a smaller ε, it seems to show contradicted The setting of the first series of experimentsis the same as resultsin Fig. 6(a).The apparentcontradictionlies behindthe the linear power setting. The correspondingnumerical results probability of 1 which becomes quite small because of a are plotted in Fig. 9, 10. From the two figures it concludes K2 much bigger M caused by a smaller ε. Fig. 6(b) shows a that the transmitting links receive much smaller interference similar consequence caused by the crucial impact of ε both outside than the theoreticalbound.Therefore,it indicates that on the achievablecapacityregionandthe relevantprobability. our algorithm is correctly applicable to the uniform power We then focus on comparisons with the two centralized setting as well. algorithms in Fig. 7, 8. We set ε = 0.8, K = 6 to conduct Then we then conduct another set of experiments to study M the following simulations. These simulation results indicate how K and ε affect the performance. M that our distributed scheduling algorithm(denoted by DS in We showtheeffectof K atdifferentvaluesof εwhereε= M figures)achievescomparableperformancewiththecentralized 0.2,0.4,0.8respectivelyinFig.11.Fig.11(a),11(b)and11(c) GMS and the heuristic RA. plotthetotalbacklogchangesasincreasingaveragearrivalrate under different values of K. The trends are a little different M 3x 105DS 3x 104DS from those in the linear power setting. From the three figures 2.5 GRAMS 2.5 GRAMS wecanseethattheaveragethroughputperformancegetsbetter Total backlog1.125 Total backlog1.125 aopsfaritKnl1yc2rcesaahusosinwedgsbMnKyothuoenbdvaeilogruoasrifitihnxmfleudfeoεnr.cceTohomenpduthetiecnrgereanssueinwltgss.pcIhrtoembdauablyiinlibgtyes 0.5 0.5 inside sub-squares since it selects candidate links based on 00 0.2 0.4 0.6 0.8 1 00.1 6 0.165 0.17 0.175 0.18 their distance with previous selected links. Thus a larger area Average arrival rate Average arrival rate implies more links get scheduled. This improvement remits (a)Totalbacklogvs.averagearrivalrate (b) Zoominof(a) the influence of the decreasing probability of 1 K2. Fig.7:Totalbacklogvs.averagearrivalratevs.differentalgorithms at time slot 1000 in the linear power setting 0.05 1 Inference outside Affectness inside Maximum tolerable Total affectness 0.04 inference outside 0.8 Maximum affectness inside tthharIetneoFauilrgg.oarl7gitohwrmiteshmcaosmmapavyaerreaagptephreaorxtroiimvtaaallterblaaytceksulionpgcproecrahtsaeansg.meIsatxosihmfoutwhmes Interference00..0023 Affectness00..46 averagerate nogreaterthan 0.2,and the othertwo supportno 0.01 0.2 greater than 0.3. We zoom in a subgraph of the Fig. 7(a) in 00 1 2 3 4 5 6 7 8 9 00 1 2 3 4 5 6 7 8 9 Link ID Time slot the Fig. 7(b), where the average arrival rate is in [0.16,0.18]. (a) Interference vs.links (b) Affectness vs.links It shows that our algorithm can keep the total backlog stable in the region. Fig. 8 then illustrates detailed comparisons of Fig. 9: The interference/affectness each randomly chosen transmit- achievable capacity region for the three algorithms. It shows tinglink receives with theuniform power assignment. (a) shows the thatouralgorithmcansupportacomparabletrafficarrivalrate outsideinterferenceeachlinkreceives.(b)showstheinsideaffectness vector with the two centralized algorithms. From the three for each link, and the total affectness each receives from all other subfigures we can see that our proposed algorithm can keep simultaneously transmitting links. the total backlogstable at an arrivalrate no greaterthan 0.18, while the counterpart of the greedy algorithm and random Thesimilartrendoccurswhenεincreasesatdifferentvalues algorithm is 0.28 and 0.26. of K We give a brief illustration in Fig. 12. Both the figures M. show that a larger ε generatesbetterperformanceat fixed K M. Since the affectness of local ISLs computed by the algorithm B. Under Uniform Power Setting of[24]insideeachsub-squaremaybemuchsmallerthan1 ε, − Now we conduct the same three series of experiments for it explains why ε has little influence on the theoretical bound the uniform power assignment. actually. But ε has much more influence on the value of M

Throughput Optimizing Localized Link Scheduling for Multihop Wireless Networks Under Physical Interference Model

Similar Throughput Optimizing Localized Link Scheduling for Multihop Wireless Networks Under Physical Interference Model