On network coding and routing in dynamic wireless multicast networks

Transcription

1 On network coding nd routing in dynmic wireless multicst networks Trcey Ho, Ji-Qi Jin Cliforni Institute of Technology Psden, CA Emil: {tho, Hrish Viswnthn Lucent Technologies, Bell Ls Murry Hill, NJ Emil: Astrct We compre multicst network coding nd routing for time-vrying wireless network model with interferencedetermined link cpcities We use dynmic ck pressure lgorithms tht re optiml for intr-session network coding nd routing respectively Our results suggest tht under such conditions, the gp in multicst cpcity etween network coding nd routing cn decrese reltive to collision-sed wireless model with fixed link cpcities I INTRODUCTION In this pper we consider dynmic multicsting in timevrying wireless networks We investigte the cpcity enefit of network coding reltive to multicst routing, ie forwrding nd repliction over one or more multicst trees Network coding hs een shown to e necessry for chieving multicst cpcity in some wired networks such s the multicst network of Figure 1 [1]; mny known exmples of such networks re extensions or generliztions of this sic exmple Reference [16] gives n nlogous exmple, shown in Figure 2, of sttic wireless network with fixed link rtes nd hlf-duplex constrint, ie ech node cn either send or receive single trnsmission t ny one time In this exmple, the optiml network coding solution, shown in Figure 3, chieves 4/3 times the multicst throughput of the optiml routing solution, shown in Figure 3 As we consider more relistic wireless network models, we move progressively further from the sttic wireless model tht is closest to the originl wired model For instnce, if we consider the effect of interference on link cpcities, it is not cler to wht extent the network coding nd routing solutions of Figure 3 re ffected, since their trnsmit scenrios involve different sets of interfering links Also, chnnel conditions often vry nd trffic is usully ursty ecuse either the sources generte trffic in ursts or the network nodes employ queuing nd scheduling cross multiple sessions In such scenrios, optiml scheduling, routing nd coding should vry dynmiclly depending on the current stte of the network chnnel conditions nd uffer occupncy Routing, scheduling, rte, nd power control in networks with ursty trffic hs recently received significnt ttention in the context of wireless networks [2], [10], [12], [13], [14], [15], [19], [21], [22] Much of the recent work in this re uilds on the ides of [3], [18] tht descrie lgorithms for routing nd scheduling flows using queue sizes, or differences Fig 1 Wired multicst network exmple, with single node A multicsting informtion to two sink nodes E nd F in queue size etween the queues t the source nd the destintion of link, s the metric to select etween different flows Such n pproch is usully sid to e ck-pressure sed since hevily loded nodes downstrem slow down the flow coming down from nodes upstrem Such pproches re optiml in the sense of llowing trnsmission t the mximum possile rrivl rtes into the network for which the queues t the vrious network nodes cn e stilized While the ck-pressure pproch hs mostly een pplied in the context of unicst trnsmissions, it hs lso een extended to the cse of multicst trnsmissions [2], [17] However, in the multicst cse without network coding the lgorithms re significntly more complex, even for wired networks We hve recently extended the ove ck-pressure sed dynmic routing nd scheduling lgorithms to include network coding nd correlted sources [8] Rndom network coding [6], introduced for the flow model, extends nturlly to timevrying network with ursty trffic nd provides distriuted implementtion For networks with one or more multicst sessions, ech consisting of set of sources nd sinks such tht dt from ll the sources is intended for ll the sinks,

2 Fig 2 Wireless multicst network exmple, with single node A multicsting informtion to two sink nodes E nd F The coordintes represent physicl distnces which re vried in our simultion experiments Fig 3 Illustrtion of optiml schedules for coding nd routing An lterntive schedule for routing is given in c Ech ox shows set of wireless links tht cn e simultneously ctivted, termed trnsmit scenrio TS our lgorithm for routing, network coding, scheduling nd rte control chieves stility for ll input rtes tht cn e stilized with intr-session network coding Aprt from the potentil cpcity gin of llowing network coding within multicst sessions, we lso otin much reduced complexity compred to the existing lgorithms of [17], which involves enumertion of ll multicst trees used, nd [2], which involves mintining virtul queue for every suset of sinks for every session In our pproch, ech node hs just one virtul queue for ech sink of ech session for independent sources or for ech source-sink pir of ech session for correlted sources Routing, network coding nd scheduling decisions re mde loclly y compring, for ech link, the difference in length of corresponding virtul queues, summed over ech session s queues For correlted sources, the sinks loclly determine nd control trnsmission rtes cross the sources This gives completely distriuted lgorithm for wired networks; in the wireless cse, scheduling nd power control mong interfering trnsmitters is done centrlly This pper is orgnized s follows We present the system nd network coding model in Section II Optiml ck pressure lgorithms for multicst with nd without network coding re descried in Section III In Section IV, we compre the two lgorithms on the exmple of Figure 2 We conclude with summry in Section V A Wireless network model II MODEL We use model similr to tht in [15] We consider network composed of set N of N = N nodes with communiction links etween them tht re fixed or timevrying ccording to some specified ergodic processes, nd trnsmission of set of multicst sessions C through the network Ech session c C is ssocited with set S c N of sources, nd n exogenous process of dt rrivls t ech of these sources which must e trnsmitted over the network to ech of set T c N of sinks Trnsmissions re ssumed to occur in slotted time, with time slots of length T Decisions on routing, scheduling, etc re mde t most once slot For simplicity, we ssume fixed length pckets nd link trnsmission rtes tht re restricted to integer multiples of the pcket-length/time-slot quotient Tht is, n integer numer of pckets cn e trnsmitted in ech slot The link trnsmission rte µ ij from node i to node j, with other nodes n N trnsmitting independent informtion simultneously, is given y the Shnnon formul [5] P i S ij µ ij P, S = log 1 + N 0 + n N P ns nj where P l is the power trnsmitted y node l, S lj is the chnnel gin from node l to node j nd N 0 is dditive white Gussin noise power over the signling ndwidth We ssume tht the chnnel conditions re fixed over the durtion of slot, nd known t the eginning of the slot For simplicity of exposition we ssume tht the chnnel nd rrivl processes re independent nd identiclly distriuted cross slots; strightforwrd generliztion to ergodic processes is possile using similr pproch s tht in [15] We denote y, Z wireless rodcst link where is the originting node nd Z is the set of receiving nodes Link rtes µp, S = µ Z P, S re determined y the vector of trnsmit powers P t = P Z t nd chnnel stte vector St St is ssumed to e constnt over ech time slot, ie, stte trnsitions occur only on slot oundries t = kt, k integer We lso ssume tht St tkes vlues from finite set nd is ergodic; we denote y π S the time verge proility of stte S P t is lso held constnt over ech time slot, nd is chosen from compct set Π of power lloctions representing limits on trnsmit power per node nd/or cross nodes

3 B Network coding We use the pproch of distriuted rndom liner network coding [4], [6], [7], in which network nodes form output dt y tking rndom liner comintions of input dt The contents of ech pcket, s liner comintion of the input pckets, re specified y coefficient vector in the pcket heder, updted y pplying to the coefficient vectors the sme liner trnsformtions s to the dt The coefficient vector is thus function of the rndom code coefficients specifying the liner comintions t intermedite nodes A sink is le to decode when it receives full set of pckets with linerly independent coefficient vectors For simplicity, we consider the cse where no restrictions re plced on coding mong pckets from the sme multicst session This symptoticlly chieves optiml cpcity, ut in the worst-cse decoding my not e possile until the end of the entire session III DYNAMIC BACK-PRESSURE ALGORITHMS A Multicst with intr-session network coding In [8] we give dynmic lgorithm tht uses queue stte informtion to mke network coding nd trnsmit scenrio decisions, without requiring ny knowledge of the input or chnnel sttistics Bck-pressure policy In ech time slot [t, t + T, the following re crried out: Scheduling: For ech link, Z, one session c Z = rg mx c is chosen Let wz = β T c Z mx β T c mx mx Z mx Z U c Z β U cβ U cβ, 0 U c Z β, 0 1 Power control: The stte St is oserved, nd power lloction P t = rg mx µ Z P, StwZ 2 P Π,Z is chosen Network coding: For ech link, Z, rndom liner comintion of dt corresponding to ech session, sink pir c Z, β T c for which mx Z Z U c Z β U c Z β > 0 is sent t the rte offered y the power lloction Ech destintion node d Z ssocites the received informtion with the virtul uffers corresponding to sinks β T c Z for which d = rg mx Z U c Z β U c Z β If the originting queues re empty within the time slot, no dt is sent In network where simultneous trnsmissions interfere, optimizing 2 requires centrlized solution In prctice, the optimiztion 2 cn e done heuristiclly using greedy pproch similr to tht in [11], [20] ut with the dded guidnce of weights wz for prioritiztion mong cndidte links, Z If there re enough chnnels for independent trnsmissions, the optimiztion cn e done independently for ech trnsmitter This lgorithm stilizes ny set of input rtes stilizle with intr-session network coding [8]: Theorem 1: If input rtes λ c i stisfy λc i + ɛ Λ, the ck-pressure policy stilizes the system nd gurntees n verge totl uffer occupncy upper ounded y T BN ɛ, where { B = τ mx 1 A } c 2 E i + µ out mx + µ in 2 N T mx 2 i,c B Bck pressure lgorithm for multicst routing We compre our ck pressure multicst network coding lgorithm with ck pressure lgorithm for optiml multicst routing, for the cse of single multicst session with two sinks The routing lgorithm is similr to tht in [2] in tht ech node mintins queue for every suset of sinks The lgorithms differ in their policies for updting queues, s the lgorithm of [2] is for dversril wired networks, wheres our lgorithm is for non-dversril wireless networks In generl, the numer of queues is exponentil in the numer of sinks We descrie the lgorithm for the cse of two sinks, where ech node mintins three queues: two individul queues contining dt tht is to e trnsmitted to ech of the sinks nd common queue contining dt tht is to e trnsmitted to oth sinks We denote y U k i, k = 1, 2, 3, respectively the lengths of these three queues t node i, nd refer to dt in the respective queues s commodity k = 1, 2, 3 dt Bck pressure is used to control the rnch points of the multicst distriution trees used In ech time slot [t, t + T, the following re crried out: Blncing: At the strt of ech timeslot, for ech nonsink node i, if U 3 i U 1 i U 2 i > 0, then n mount U 3 i U 1 i U 2 i /3 of dt is removed from the common queue nd the sme mount of dt is dded to ech of the individul queues t i Scheduling: For ech link, Z, If then we set k Z = rg mx k mx Z U k mx U k Z U k Z > U 3 Z U k 2 j=1 wz = mx U k Z U k Z Z Otherwise, we set k Z = 0 nd w Z = U 3 2 j=1 min U j Z min U j, Z

4 Power control: The stte St is oserved, nd power lloction P t = rg mx µ Z P, StwZ P Π,Z is chosen Routing: For ech link, Z, if kz 0 nd mx Z U k Z U k Z > 0, commodity kz dt is sent from to rg mx Z U k Z U k Z t the rte offered y the power lloction Otherwise, if kz = 0 nd U 3 2 j=1 min Z U j > 0, dt from the common queue t is rodcst on, Z t the offered rte, nd corresponding mount is dded to min Z U j, j = 1, 2 If the originting queues re empty within the time slot, no dt is sent This lgorithm symptoticlly chieves the optiml multicst throughput for routing The proof is omitted for revity A Experimentl setup IV SIMULATION EXPERIMENTS We run the two multicst lgorithms of the previous section on the network of Fig 2, with single node A multicsting informtion to two sink nodes E nd F We ssume common trnsmission power nd hlf-duplex constrint t ech network node The chnnel gin for from node i to node j is modeled s: trnsmission rte of R, defined s the mximum vlue for which the network is stle t rte R nd unstle t rte R + 01 The network ws considered stle for given trnsmission rte nd policy if the mximum cklog in ll queues of ll nodes ws ounded y 200 when run for 20,000 timeslots B Results The simultion results re shown in Tle 1 nd Figures 4-6 From Tle 1, we see tht under this time-vrying wireless network model with interference, the cpcity gin of the network coding strtegy of [16] over routing is lower thn the 4/3 fctor otined in the fixed-rte collision model Figures 4-6 show the proportion of different trnsmit scenrios used y the two policies, which vries depending on the experimentl prmeters In the network coding cse, the proportion of trnsmit scenrios gives rough indiction of the proportion of time network coding is used TABLE I MAXIMUM TRANSMISSION RATES R FOR CODING AND ROUTING snr 2 R for coding R for routing / / / / S ij = f ij 2 d k ij where f ij is the Ryleigh fding stte nd d ij is the distnce etween i nd j, nd k is the propgtion power loss exponent We ssume µ = E[ f 2 ] is the sme for ll i, j The two lgorithms symptoticlly chieve, for liner coding nd routing respectively, the optiml multicst throughput chievle with the following seven trnsmit scenrios, chosen so s to cover the network coding nd routing schedules of Figure 3: trnsmit scenrio 1, where A trnsmits to B, nd C rodcsts to D nd F trnsmit scenrio 2, where A trnsmits to C, nd B rodcsts to D nd E trnsmit scenrio 3, where D rodcsts to E nd F trnsmit scenrio 4, where A rodcsts to B nd C trnsmit scenrio 5, where B trnsmits to E, nd C trnsmits to F, respectively trnsmit scenrio 6, where B trnsmits to E trnsmit scenrio 7, where C trnsmits to F To investigte the effects of network geometry nd SNR, the two lgorithms were run for numer of different vlues for prmeter 1, defined in Figure 2, nd SNR For ech choice of prmeter vlues, series of simultion runs ws crried out to find the mximum stle multicst rte For ech run, the throughput ws incresed y 01, until the source queue ecme unstle, giving n pproximte mximum 3 Fig 4 The proportion of different trnsmit scenrios used y the network coding lgorithm for snr = 10 10dB, 1 = 2 = = 2/2, k = 2, µ = 15, trnmssion rte = 14 V CONCLUSION We hve compred multicst network coding nd routing for time-vrying wireless network model with interference Our results suggest tht when link cpcities re ffected y interference, nd power control, scheduling, network coding nd routing re dynmiclly controlled in response to network conditions, the gp in multicst cpcity etween network

5 Fig 5 The proportion of different trnsmit scenrios used y the network coding lgorithm for snr = dB, 1 = 2 = = 2/2, k = 2, µ = 15, trnsmission rte = 23 Fig 6 The proportion of different trnsmit scenrios used y the routing lgorithm for snr = 10 10dB, = 1 = 2 = 2/2, k = 2, µ = 15, trnmssion rte = 13 coding nd routing cn decrese reltive to collision-sed wireless model with fixed link cpcities In such cses, the min dvntge of network coding my e the reduction in complexity of optimiztion nd opertion The coding dvntge my lso increse in situtions where routing, power control nd scheduling re not done optimlly, s hs een shown for the multiple unicsts cse [9] [3] Bruch Aweruch nd Tom Leighton A simple locl control pproximtion lgorithm for multicommodity flow In Proceedings of 34th IEEE Conference on Foundtions of Computer Science, 1993 [4] P A Chou, Y Wu, nd K Jin Prcticl network coding In Proceedings of 41st Annul Allerton Conference on Communiction, Control, nd Computing, Octoer 2003 [5] Thoms Cover nd Joy Thoms Elements of Informtion Theory Wiley, 1991 [6] T Ho, R Koetter, M Médrd, D R Krger, nd M Effros The enefits of coding over routing in rndomized setting In Proceedings of 2003 IEEE Interntionl Symposium on Informtion Theory, June 2003 [7] T Ho, M Médrd, J Shi, M Effros, nd D R Krger On rndomized network coding In Proceedings of 41st Annul Allerton Conference on Communiction, Control, nd Computing, Octoer 2003 [8] T Ho nd H Viswnthn Dynmic lgorithms for multicst with intrsession network coding In Proc 43rd Annul Allerton Conference on Communiction, Control, nd Computing, 2005 [9] S Ktti, D Kti, nd M Médrd The importnce of eing opportunistic In Proc 43rd Annul Allerton Conference on Communiction, Control, nd Computing, 2005 [10] Thierry Klein nd Hrish Viswnthn Centrlized power control in multihop wireless networks In Proceedings of IEEE Interntionl Symposium on Informtion Theory, 2003 [11] Syndev Mukherjee nd Hrish Viswnthn Throughput rnge trdeoff of wireless mesh ckhul networks IEEE Journl on Selected Ares in Communictions, to pper, 2006 [12] Michel Neely Energy optiml control for time vrying networks In Proceedings of the IEEE INFOCOM, 2005 [13] Michel Neely, Eytn Modino, nd Chih-Ping Li Firness nd optiml stochstic control for heterogeneous networks In Proceedings of the IEEE INFOCOM, 2005 [14] Michel Neely, Eytn Modino, nd C Rohrs Pcket routing over prllel time-vrying queues with ppliction to stellite nd wireless networks In Proceedings of the Thirty-Ninth Annul Allerton Conference on Communictions nd Control, 2001 [15] Michel Neely, Eytn Modino, nd Chrles E Rohrs Dynmic power lloction nd routing for time-vrying wireless networks In Proceedings of IEEE Infocom, 2003 [16] Y E Sgduyu nd A Ephremides Joint scheduling nd wireless network coding In Proc 1st Workshop on Network Coding, Theory nd Applictions, 2005 [17] Sswti Srkr nd Lendros Tssiuls A frmework for routing nd congestion control for multicst informtion flows IEEE Trnsctions on Informtion Theory, 2002 [18] Lendros Tssiuls nd Anthony F Ephremides Stility properties of constrined queuing systems nd scheduling policies for mximum throughput in multihop networks IEEE Trnsctions on Informtion Theory, 1992 [19] Hrish Viswnthn nd Krishnn Kumrn Rte scheduling for multiple ntenn downlink wireless systems In Proceedings of the Thirty- Ninth Annul Allerton Conference on Communictions nd Control, 2001 [20] Yunnn Wu, Philip A Chou, Qin Zhng, Kml Jin, Wenwu Zhu, nd Sun-Yun Kung Network plnning in wireless d hoc networks: A cross-lyer pproch IEEE Journl on Selected Ares in Communictions, Specil Issue on Wireless Ad Hoc Networks, 2005 [21] Edmund Yeh nd Aron Cohen Throughput nd dely optiml resource lloction in multiccess fding chnnels In Proceedings of IEEE Interntionl Symposium on Informtion Theory, 2003 [22] Edmund Yeh nd Aron Cohen Throughput optiml power nd rte control in queued multiccess nd fding chnnels In Proceedings of IEEE Interntionl Symposium on Informtion Theory, 2004 REFERENCES [1] R Ahlswede, N Ci, S-YR Li, nd RW Yeung Network informtion flow IEEE Trnsctions on Informtion Theory, 46: , 2000 [2] Bruch Aweruch, André Brinkmnn, nd Christin Scheideler Anycsting nd multicsting in dversril systems: routing nd dmission control, preliminry technicl report, 2002