NCC 2009, Jnury 16-18, IIT Guwhti 170 Throughput of Wireless Rely Networks with Interference Processing M. Bm, rikrishn Bhshym nd Andrew Thngrj, eprtment of Electricl Engineering, Indin Institute of Technology, Mdrs, Chenni, Indi 600036. m, skrishn, ndrew@ iitm.c.in Astrct Consider wireless rely network, where ll nodes except the source nd destintion ct s relys. The prolem of evluting the cpcity for source-destintion pir in this network nd determining the corresponding optiml trnsmission strtegy hve ttrcted considerle reserch ttention recently. However, even for single rely, complete solutions re still unknown. A populr method to otin prtil results is to fix the rely strtegy. In this pper, we consider sitution where relys dopt the decode-nd-forwrd pproch with the possiility of network coding. Also, the nodes cn receive informtion from multiple trnsmitters simultneously. This is ccomplished y llowing physicl lyer interference processing t nodes. This processing yields incresed sum rte when compred to interference voidnce or treting interference s noise. A liner optimiztion model is proposed to determine the mximum chievle throughput with interference processing. Numericl evlution done on some exmple networks shows significnt throughput gins. I. INTROUCTION Consider multihop wireless network represented y grph G = (V, E).Inthispper,wewishtodeterminethe rtetwhichsinglesourcenode V cncommunicte tosinknode V.Othernodesin V \{, }ctsrelys inforwrdingthemessgefrom to. The cpcity of simple three-node rely network is still n open prolem. everl relying strtegies were proposed to simplify this prolem. These include strtegies such s mplify- nd-forwrd relys or compress-nd-forwrd relys. Most of the strtegies use Multiple Input Multiple Output (MIMO) system coopertion nd the results re symptotic in ignl-to-noise-rtio(nr)[1],[2],[3]. In this work, we consider hlf-duplex wireless nodes with the relys performing decode nd forwrd communiction long with network coding. However, there is no coopertion ssumed either t the trnsmitter nodes or t the receiver nodes. Further, we operte on the finite NR regime. Inournetworkmodel,ech nodetrnsmits t the sme rte using the wireless multicst dvntge[4] to rech its neighours in single hop. This dvntge of wireless nodes is modeled nd exploited in[5] using hyperrcs. While hyperrcs chrcterize one-to-mny trnsmission, they do not llow mny-to-one or mny-to-mny trnsmission. With dvnced physicl lyer processing, node cn get informtion from multiple trnsmitters simultneously, i.e. nodes cn operte with interference s well[6]. In this pper, we model mny-tomny trnsmissions using hyperedges. A hyperedge is denoted y (I, J), where I is theset oftrnsmittersnd J is the set of receivers. Brodcst Chnnels(BC), Multiple-Access Chnnels(MAC) nd Interference Chnnels(IC) re exmples of chnnels tht cn e modeled s hyperedges. Our hyperedge model is different from the physicl model of [7] where interference is treted s noise. Using the cpcity region of hyperedge (I, J) determined in[8], we formulte n optimiztion model for determining the network coding unicst throughput from the source to the sink in the network. Numericl evlution of throughput using the optimiztion model on comintion network nd dimond network shows significnt gins in the network throughput. The throughput gin for comintion network is round 30 %wheresitisround16%forthedimondnetwork.we ound the throughput y trnsforming the given network into lyered network. The pper is orgnised s follows: ection II descries vrious schemes to hndle interference t the physicl lyer nd the link lyer. The wireless network model is descried in ection III. The liner progrmming optimiztion model for unicst cpcity is detiled in ection IV. Numericl evlutions of the optimiztion model is explined in ection V. Bound on network throughput is derived in ection VI. Advntges of interference processing is delt in ection VII nd ection VIII concludes the pper. II. INTERFERENCE IN WIRELE NETWORK Consider the network shown in Fig. 1(). Let Nodes nd wnt to communictenode c. Let C c nd C c e thecpcitiesofthelink (, c)nd (, c)nd C c = C c. If oth nodes nd trnsmit simultneously, it cuses interference t the sink node c. The interference t the Node ccnehndledinmnywys.inwirelessnetworks,slotted trnsmission nd link scheduling re mostly commonly done to void interference. This time shring rte region is the region OABinFig.1(). Alterntively, the interfering signl cn e treted s noise while decoding the desired user signl t the physicl lyer. ThisrteregionistheregionOCEsshowninFig.1().Region OBFGA is otined y processing the interference using dvnced physicl lyer processing. From the rte regions, we oserve tht processing the interference hs etter sum rte thn the other methods. This promises sustntil gins in the throughput. Hence, in this pper, we study the wireless
NCC 2009, Jnury 16-18, IIT Guwhti 171 c () An exmple network R c B (C c, 0) C O E F G A (C c, 0) () Rte region Fig. 1. An exmple network nd rte regions. R c network throughput when the nodes re llowed to perform interference processing. III. WIRELE NETWORK MOEL Consider n-node wireless pcket networks. All nodes re hlf-duplex nd hve uniform trnsmission rnge c n. The networkismodeledsgrph, G = (V, E),where V = {1, 2,, n}isthesetofverticesnd E = {(i, j) : d ij c n, i, j V }isthesetofedges.here, d ij istheeucliden distnceetweennode indnode j.alink (i, j) E is lossless ndhs cpcity C ij pckets perunit-time.let N i = {j V : d ij c n }ethesetofneighorsofnode i.insuchnetwork G,weconsidersingleunicstsession fromsource V todestintion V. Considerhyperedge H = (I, J)where I = {,..., t M } isthesetoftrnsmittersnd J = {r 1, r 2,..., r N }istheset ofreceivers.let V H = I Jethesetofverticesin Hnd E H {(t i, r j ) : t i I, r j J}ethesetofedgesin H. Forech v V H,definetheneighouringsets Γ + (v) = {t i V H : (t i, v) E H }nd Γ (v) = {r j V H : (v, r j ) E H }. Let d ± v = Γ ± (v) etheincoming/outgoingdegrees.let A denotethesetofllhyperedgesin G. t M R 1 R 2 R M r 1 r 2 r N 1 r N Fig. 2. An exmple of hyperedge long with trnsmission rtes AnexmpleofsuchhyperedgeisshowninFig.2.The trnsmitter t k Isendscommonmessgetrte R k tothe receiversin Γ (t k )ndnottollreceiversin J.Thisissingle hop communiction using the inherent wireless rodcst nture.thepowerndndwidthreminthesmesthtof thepoint-to-pointcommuniction.receiver r j J needsto decodethecodewordsfromthetrnsmittersin Γ + (r j ).These receivers employ Interference-Awre Physicl Lyer(I-APL) processing techniques like successive interference cncelltion for successful decoding. IV. OPTIMIZATION MOEL Forthesingleunicstsessionfromthesource V tothe destintion V,wenowdescriethelinerprogrmming model to determine the network throughput. Before tht, we quickly explin the selection of non-interfering grphs for voiding the interference etween hyperedges nd the slotted trnsmission scheme. A. Non-interfering sugrphs To void interference etween hyperedges, slotted trnsmission is employed. A set of non-interfering hyperedges, clled non-interferingsugrphof G,isctiveinechslot.Two hyperedges (I 1, J 1 )nd (I 2, J 2 ) Arenon-interfering,if (i) ( i1 I 1 N i1 ) J 2 =,nd (ii) ( i2 I 2 N i2 ) J 1 =. Otherwise, the hyperedges re sid to e interfering. Theconflictgrphpprochof[9]isusedtoformthenoninterfering sugrphs. Ech node in the conflict grph correspondstohyperedgefrom A.Twonodesintheconflictgrph re connected if the corresponding hyperedges interfere in G. Itisesytoseethteveryindependentsetinsuchconflict grph will correspond to non-interfering sugrph of G. In our numericl evlution, for simplicity, we use n lgorithm tht genertes one rndom independent set of the conflict grph ineveryrun[9].notethtwellowinterferencewithin hyperedge, ut void interference etween hyperedges. B. Trnsmission slots nd pcket injection rte To proceed with the model, we suppose tht M mximl non-interfering sugrphs of G hve een generted(using M runs of the conflict grph independent set lgorithm) nd denote them A 1, A 2,, A M. The multicst session is modeledtoespredover Mtrnsmissionslotsinonetime unit.thedurtionofslot kcorrespondsto λ k frctionoftime ndthenon-interferingsugrph A k isctiveduringslot k (λ k couldezero). If (I, J) A k,nodes i I trnsmittonodesin Γ (i) duringslot k.let IJ indictethetotlfrctionoftimefor which the hyperedge (I, J) is ctive over M trnsmission slots.let z iij (pcketsperunittime)ethevergertet whichpcketsreinjectedynode i Iintothehyperrc (I, J)vergedoverllslots.Thetrnsmissionrtes z iij re proportionltotheon-time IJ.Amulticstthroughputof f ischievedif f pcketsresentfromthesource snd receivedyllsinks t Tinonetimeunit. C. Liner progrmming for unicst throughput Wessumethtlllinksin Ghveequlcpcityi.e. C ij = Lforll (i, j) E.Theconstrintsinthelinerprogrmre descried elow. cheduling constrints: The scheduling constrints on IJ usetheindictorfunction g k (I, J)defineds g k (I, J) = 1 if (I, J) A k nd0otherwise.intuitively,thescheduling constrintsimplytht IJ isupper-oundedythetotltime forwhichthehyperedge (I, J)isctive.
NCC 2009, Jnury 16-18, IIT Guwhti 172 Flowconstrints:Theflowvrile x iijj denotestheverge t i t M informtionflowrtefromnode itonode j Γ (i)long hyperedge (I, J)towrdsthesink.Thevergeflowsto ech sink stisfy the flow constrints t ech node. Cpcity constrints: For ech hyperedge (I, J), the cpcity r 1 r i r N r 1 regionissttedinlemm1of[8].inthtcpcityregion, wefixthetimeshringrndomvrile Qsconstntfor simplicity. Therefore, for ech hyperedge (I, J), the verge flowstoechsinkfromnode i Ilongdifferent j Γ (i) lies within the fesile rte region for the rodcst chnnel r 1 r 2 r 3 r 1 r 2 t 3 r 3 (BC)fromNode ito Γ (i).alsoforechhyperedge (I, J), r 1 t 3 r 1 the trnsmission rtes z iij re within the multiple ccess chnnel(mac)from Γ + (j)tonode j J. The liner progrm is s follows: Mximize the throughput fsujectto r 2 r 1 r 2 r 2 Fig. 3. Hyperedge chnnels used for numericl evlution. cheduling Constrints: λ k g k (I, J) IJ 0, (I, J) A, k {(I,J) A: i I} j Γ (i) λ k 1, λ k 0, k Flow Conservtion Constrints: x iijj {(J,I) A : i I} j Γ +(i) x jjii = f if i = fif i =, i V, 0otherwise Cpcity Conservtion Constrints: BC: z iij x iijj 0, i I, (I, J) A, MAC: j Γ (i) z iij 0, x iijj 0, z iij IJ I(X(B); Y j X(B c )), i B j J, (I, J) A. B Γ + (j), Every fesile solution to the liner progrmming prolem corresponds to vlid network code of throughput of f pckets per unit time over M slots with ech slot ctive for λ k frctionl time units[5],[10]. V. NUMERICAL EVALUATION We evlute the throughput using the proposed optimiztion modelon (i)dimondnetworknd (ii)(5, 4)comintion network.wessumellthelinksreawgnchnnelswith cpcity one. The numer of hyperedges in wireless networks is typiclly very lrge. Hence, we consider only the hyperedges shown in Fig. 3 long with their sugrphs for ese of numericl evlution. We run the conflict grph scheduling lgorithm 3,000 times to generte sufficient numer(sy M) of non-interfering sugrphs which otin good estimte of network throughput. A. imond network Consider the dimond network shown in Fig. 4(). The optimiztion model is evluted with the seven hyperedge chnnels shown in Fig. 3 to determine the throughput from the source to the destintion. The non-interfering sugrphs chosen re shown in Fig. 4(). The throughput is ounded y the rodcst cut t the source nd the multiple-ccess cutttheink.therefore, f min(1, 1.4037).Theunicst throughputwiththesesevenhyperedgesis f = 7 12 wheresit is f = 1 2 withinterferencevoidnce.noticetht,themac chnnelin A 2 isveryusefulinotininghigherthroughput. () imond network. A 1, λ 1 = 7 12 A 2, λ 2 = 5 12 () Non-interfering sugrphs. Fig. 4. Unicst throughput of dimond network. Trnsmission scheme: Consider twelve time slots nd seven informtion its. In the first seven slots source rodcsts lltheseveninformtionitstonodes nd.themac chnnelsin A 2 opertetthepoint (R, R ) = (1, 2 5 )to enle interference processing. The nodes nd select the codeooksccordingly.inslots 8 12,Nodes nd use their MAC codeooks to trnsmit ll seven informtion its to sink i.e., 5(R + R ) = 7. B. (5, 4) comintion network Consider the (5, 4) comintion network shown in Fig. 5(). The optimiztion model is evluted with the seven hyperedge chnnels shown in Fig. 3 to determine the throughput from the source to the destintion. The non-interfering sugrphs
NCC 2009, Jnury 16-18, IIT Guwhti 173 2 3 4 5 6 3 4 5 6 3 4 5 6 7 8 9 10 () (5, 4) comintion network. A 1, λ 1 = 0.6491 A 2, λ 2 = 0.3509 () Non-interfering sugrphs. Fig. 5. Unicst throughput of (5, 4) comintion network. chosen re shown in Fig. 5(). The unicst throughput is oundedytherodcstcuttthesourcendthemultipleccess cut t the sink. Therefore, f min(1, 1.8502). The unicst throughput with these seven hyperedges is f = 0.6491 9 14 wheresitis f = 1 2 withinterferencevoidnce. Noticetht,theMACchnnelin A 2 isveryusefulinotining higher throughput. This implies tht interference processing t the receivers helps in otining higher throughput Trnsmission scheme: Consider fourteen time slots nd nine informtion its. In the first nine slots source rodcsts ll thenineinformtionitstonodes 3, 4, 5nd 6.TheMAC chnnelsin A 2 opertetthepoint (R 3, R 4, R 5, R 6 ) = (1, 0.40, 0.25, 0.15) to enle interference processing. The Nodes 3, 4, 5nd 6selectthecodeooksccordingly.Inslots 10 14,Nodes 3, 4, 5nd 6usetheirMACcodeooksto trnsmitllnineinformtionitstosink i.e., 5(R 3 + R 4 + R 5 + R 6 ) = 9. VI. BOUN ON NETWORK THROUGHPUT In this section, we suggest pproches tht simplify the computtions nd provide pproximte solutions for lrge networks. For networks with lrge numer of nodes, the liner progrmming prolem ecomes unsolvle ecuse of the exponentil numer of vriles nd constrints. To do this, thegivennetwork Gistrnsformedintolyerednetwork G L inmnnerexplinedelow.echlyerin G L ishyperedge. Let methenumeroflyersin G L.Theconstructionof G L is s follows: (i) trt with source node s the root node on G L. Therefore,thehyperedgessocitedwithlyer 1is (I 1, J 1 ) = (, N ). (ii)any k th intermeditelyerin G L isconstructeds: I k = J k 1, J k = N(I k ) \ I k, for k = 2, 3,....Here N(I k ) = i Ik N i.thisprocedurecontinuesuntilthesinknodeiscontinedinthe m th hyperedge i.e., J m.thelyerednetworkislsousefultoderive ound on the network throughput. A. Upper ound on the lyered network throughput The informtionflow f in G L fromsource node to sinknode isthroughthelyersinit.therefore,theflow f isrestrictedytheminimumofthemximummountof informtion flow in ech lyer i.e., f min 1 k m Rk s, where R k sisthemximummountofinformtionflowinthe k th lyer(hyperedge).noticetht,thisounddoesnotccount for interference etween hyperedges nd hlf duplex nture of wireless nodes. Now, we investigtethe sum rte of the k th hyperedge (lyer) (I k, J k ).Thisisthemximummountofinformtion flow from the trnsmitters in t i I k to the receivers in r j J k.let X i ethecodewordsentytrnsmitter t i nd Y j ethecodewordreceivedyreceiver r j. Echtrnsmitter t i Isendscommoninformtiontoll itsreceiverstrte R i.therefore,thisrte R i isoundedy: R i C i, (1) where C i = min j Γ (t i) I(X i ; Y j ).incemutulinformtion is non-negtive quntity, the simplest upper ound on the sumrteofthehyperedgewithinterferenceprocessing R k s is the following: R k s = t i I k R i t i I k C i. (2) However, tighter ounds on the sum rte of the hyperedge (I k, J k ) with interference processing cn e otined y considering dditionl ounds on the MAC reception t the receivers from[8]. VII. AVANTAGE OF INTERFERENCE PROCEING We consider some exmple hyperedge chnnels to illustrte the enefits of physicl lyer interference processing.
NCC 2009, Jnury 16-18, IIT Guwhti 174 A. Orthogonl chnnels REFERENCE Inthehyperedge H,let M = Nnd t i econnectedonly tothereceiver r i,forll i = 1, 2,..., M.Thisformsset of M prllel independent chnnels in H. Therefore, the sum rte in H with interference processing is: R s = M C i. i=1 This is lso the sum-rte chievle with interference voidnce. It implies tht interference processing is not useful in orthogonl chnnels when compred to interference voidnce. B. Complete iprtite grph uppose the hyperedge chnnel H is complete i.e., ech trnsmitter t i Iisconnectedtollreceivers r j J.The sum rte possile with interference processing is ounded s R s = M R i, i=1 min r I(X(Γ+(rj)) ; Y j ). (3) j J In interference voidnce, receiver is llowed to receive from only one trnsmitter.therefore, the sum-rte with interference voidnce is the mximum of the trnsmission rtes mx ti I R i. In(3),ifthelinksreAWGNchnnelswithequltrnsmissionpower P,theoundecomes 1 R s min r j J 2 log 2 ( 1 + MP σ 2 j where σj 2 is the noise vrince t the receiver r j. With interference voidnce, the corresponding ound is ( ) 1 R s mx r j J 2 log 2 1 + P σj 2. For the complete iprtite grph exmple,we see tht the sum rte cn potentilly improve ecuse of interference processing. In section V, we notice significnt gins in the network throughput ecuse of the incresed sum-rte in ech hyperedge. In generl, when the receivers in J receives signls from multiple trnsmitters, it is very useful to use interference processing. VIII. CONCLUION We considered wireless rely networks with the relys dopting decode-nd-forwrd strtegy nd performing network coding. We llowed the trnsmitters to do common rodcst trnsmission nd the receivers to do interference processing ut did not require either MIMO coopertion or symptoticlly lrge NR. An optimiztion model ws formulted to determine the network throughput. Numericl evlution on some exmple networks promises sustntil gin in network throughput ecuse of the interference processing t the receivers. ), [1] A.. Avestimher,. N. iggvi, nd. N. C. Tse, Approximte cpcity of gussin rely networks, in Proc. of IEEE symposium on Informtion theory, July 2008, pp. 474 478. [2] K.reerm,.Brienjith,ndP.V.Kumr, MTofmulti-hopcoopertive networks-prt ii: Lyered nd multi-ntenn networks, in Proc. of IEEE symposium on Informtion theory, July 2008, pp. 2076 2080. [3], MT of multi-hop coopertive networks-prt i: K-prllel-pth networks, in Proc. of IEEE symposium on Informtion theory, July 2008, pp. 2081 2085. [4] J. Wieselthier, G. Nguyen, nd A. Ephermides, On the construction of energy- efficient rodcst nd multicst trees in wireless networks, in Proc. of INFOCOM, 2000, pp. 585 594. [5] J.-.Prk,..Lun,F.oldo,M.Gerl,ndM.Medrd, Performnce ofnetworkcodingindhocnetworks, inproc.ofieeemilcom2006, Octoer 2006. [6] T. M. Cover nd J. A. Thoms, Elements of informtion theory. John Wiley& ons(asi), 2004. [7] P.GuptndP.R.Kumr, Thecpcityofwirelessnetworks, IEEE Trns. Inform Theory, vol. 46, no. 2, pp. 388 404, Mrch 2000. [8] M. Bm,. Bhshym, nd A. Thngrj, Enhncing network coding throughput using interference-wre physicl lyers, sumitted to IEEE Trns. on Wireless Communictions, July 2008. [Online]. Aville: www.ee.iitm.c.in/ skrishn [9] K. Jin, J. Pdhye, V. Pdmnhn, nd L. Qiu, Impct of interference on multi-hop wireless network performnce, in Proc. of Moicom, 2003. [10] R. Ahlswede, N. Ci,. Y. R. Li, nd R. W. Yeung, Network informtion flow, IEEE Trns. Inform. Theory, vol. 46, no. 4, pp. 1204 1216, July 2000.