Relablty Gan of Network Codng n Lossy Wreless Networks Majd Ghader Deartment of Comuter Scence Unversty of Calgary mghader@cs.ucalgary.ca Don Towsley and Jm Kurose Deartment of Comuter Scence Unversty of Massachusetts Amherst {towsley,kurose}@cs.umass.edu Abstract The caacty gan of network codng has been extensvely studed n wred and wreless networks. Recently, t has been shown that network codng mroves network relablty by reducng the number of acket retransmssons n lossy networks. However, the extent of the relablty beneft of network codng s not known. Ths aer quantfes the relablty gan of network codng for relable multcastng n wreless networks, where network codng s most romsng. We defne the exected number of transmssons er acket as the erformance metrc for relablty and derve analytcal exressons characterzng the erformance of network codng. We also analyze the erformance of relablty mechansms based on rateless codes and automatc reeat request ARQ, and comare them wth network codng. We frst study network codng erformance n an access ont model, where an access ont broadcasts ackets to a grou of K recevers over lossy wreless channels. We show that the exected number of transmssons usng ARQ, comared to network codng, scales as Θlog K as the number of recevers becomes large. We then use the access ont model as a buldng block to study relable multcast n a tree toology. In addton to scalng results, we derve exressons for the exected number of transmssons for fnte multcast grous as well. Our results show that network codng sgnfcantly reduces the number of retransmssons n lossy networks comared to an ARQ scheme. However, rateless codng acheves asymtotc erformance results smlar to that of network codng. Index Terms Relablty, network codng, ARQ, asymtotc analyss. I. INTRODUCTION In tradtonal networks, data ackets are transmtted by store-and-forward mechansms n whch the ntermedate nodes relays or routers only reeat data ackets that they have receved. Wth network codng NC, a network node s allowed to combne several ackets that t has generated or receved nto one or several outgong ackets. The orgnal aer of Ahlswede et al. ] showed the caacty gan of network codng for multcast n wrelne networks. Recently, network codng has been aled to wreless networks and receved sgnfcant attenton as a means of mrovng network caacty and cong wth unrelable wreless lnks 2]. In fact, the unrelablty and broadcast nature of wreless lnks make wreless networks a natural settng for network codng. In ste of sgnfcant research on the caacty gan of network codng, the relablty gan of network codng s largely unknown. In ths aer, we study the alcaton of network codng as an error control technque for relable multcastng n a wreless network. Our goal s to quantfy the beneft of usng network codng comared to tradtonal error control technques such as ARQ and rateless codng. Recently, there has been some work on characterzng the relablty beneft of network codng n lossy networks 3] 5]. However, our work dffers from exstng work n that we rovde tght asymtotc bounds on the erformance of relablty mechansms based on both ARQ and NC. Addtonally, the exstng work only rovdes erformance results n terms of the exected number of transmssons wthout rovdng any nsght about the scalng behavor of dfferent relablty mechansms. Moreover, n addton to the commonly studed access ont model, we analyze relable multcast n a tree toology as well. Tree-based multcast has been revously studed n the context of wred networks and ARQ mechansms 6], 7]. In ths aer, we resent analytcal and numercal results for the erformance of end-to-end and lnk-by-lnk relablty mechansms based on ARQ, FEC and NC n a tree toology. An nterestng model can be constructed by allowng the recevers of the tradtonal access ont model to communcate locally n order to recover lost ackets. Ths model rovdes an effcent structure for relable multcast when the access ont transmssons are costly or the communcaton qualty among the recevers s sueror to that of the access ont, a scenaro occurrng often n mltary and satellte communcatons. Due to sace lmtatons, analyss of the extended access ont model s not resented n ths aer. Interested readers are referred to 8] for detals. Our contrbutons n ths aer are as follows: We resent a detaled characterzaton of the erformance of dfferent relablty mechansms based on ARQ, FEC and NC for the access ont model and tree-based multcast model. We resent both analytcal and numercal results. We rovde asymtotc bounds on the erformance of dfferent relablty mechansms for the toologes consdered n the aer, and show how our results can be used to analyze more comlcated toologes. The rest of the aer s organzed as follows. In Secton II, we analyze the access ont model whch serves as a bass for the analyss n followng sectons. In Secton III, we study four dfferent relablty mechansms for multcast n a tree
Fg.. AP K nodes The access ont model wth K recevers. toology. These mechansms are based on the alcaton of ARQ and FEC n a lnk-by-lnk or end-to-end fashon. For the three models consdered here, we derve exressons for the exected number of transmssons, and rovde asymtotc results for the erformance of relablty mechansms based on ARQ and NC. Our conclusons as well as future work are dscussed n Secton IV. II. ACCESS POINT MODEL The access ont model conssts of a sngle source, called the access ont AP, broadcastng to a set of K recevers over a lossy wreless channel as dected n Fg.. Transmssons from the AP to recevers are lossy wth losses dstrbuted by ndeendent dentcal ernoull rocesses wth arameter. We assume the use of block codng for NC, where denotes the sze of the codng block. Wth NC, the AP transmts random lnear combnatons of the ackets belongng to the same codng block. Hence, recevers need to receve ndeendent lnear combnatons n order to decode the orgnal ackets lease see 9] for more nformaton on random lnear codng. Throughout ths aer, we assume that feedback s relable, and hence, do not consder the overhead and comlexty of the feedback mechansm. Interested readers are referred to 8] for an analyss of the overhead of random lnear network codng. A. Dstrbuton of the Number of Transmssons Let N denote the number of transmssons of a acket by the AP untl the acket s receved by all K recevers. It s straghtforward to comute P {N n} as follows for more detals, see 6], 7] for ARQ, and 4], 5] for NC. ARQ Performance: The robablty that a node does not receve any acket out of n ackets transmtted by the AP s n. Therefore, wth robablty n the node receves at least one of the ackets. All K nodes have ndeendent losses, therefore, the robablty that every node receves at least one acket s P {N n} = n K, n. 2 NC Performance: We assume the block sze for network codng s. The robablty that a node receves at least coded ackets out of n ackets transmtted by the AP s gven by a bnomal dstrbuton. Each node needs ackets n order to decode and extract the orgnal ackets. Therefore, we obtan n K n P {N n} = n, n. 2 = K. Asymtotc Analyss The exact exressons derved n the revous subsecton do not rovde nsght about the scalng behavor of the number of transmssons wth resect to K and. In ths subsecton, we derve asymtotc exressons for the exected number of transmssons for the access ont model. In artcular, we are nterested n the asymtotc erformance of the ARQ and NC mechansms as the number of recevers becomes very large,.e., K. For NC, we only consder nfntely large block szes and assume that = ΩK, namely block sze grows to nfnty faster than the number of recevers. Interested readers are referred to 0] for relmnary results on the erformance of NC wth fnte block szes,.e., = O. Let N k denote the number of transmssons at the AP untl node k receves all the ackets one acket n ARQ, and ackets n NC. We are nterested n characterzng the followng exectaton as the erformance metrc ] E N] = E N k. 3 ARQ Performance: In ths case, N k has a geometrc dstrbuton. In other words, the robablty that node k receves the n-th transmtted acket s gven by P {N k = n} = n. 4 Therefore, E N] s the exected value of the mum of K geometrc random varables. For the sake of analyss, we aroxmate each geometrc random varable N k by an exonentally dstrbuted random varable X k wth rate µ. In order to fnd µ, we solve the equaton P {N k n} = P {X k n}, whch yelds µ = ln. We then aroxmate E N k ] by E X k ]. To comute the aroxmaton error, denoted by ɛ, we note that Therefore, E N k ] = E X k ] = P {N k > n} 5 n+ n P {X k > x} dx 6 ɛ = E N k ] E X k ] = n + e n+µ e nµ µ = n + n+ n µ = + ln. Let {X k } k =,..., K denote a set of K ndeendent exonentally dstrbuted random varables wth arameter µ. Then, usng roertes of exonentally dstrbuted random 7
varables, t follows that see 0] for detals ] E X] = E X k = K k= kµ = µ HK, where, HK s the K-th harmonc number. It s well-known that for large K lm HK = ln K + γ, 9 K where, γ s Euler s constant. Hence, we have lm E N] = lm K K E X] + ɛ = ln K ln + 8 γ ln + + ln. 0 From ths we conclude that E N] = Θlog K, where log K = ln K ln /. 2 NC Performance: In ths case, N k has a negatve bnomal dstrbuton, that s n P {N k = n} = n. Ths means that ackets have been receved untl acket n, and acket n s receved too. We are nterested n characterzng the exected number of transmssons er acket: E N] = ] E. 2 N k However, t s dffcult to comute E N] usng the negatve bnomal formulaton n. Fortunately, we can reresent N k by a dfferent form, whch s then amenable to analyss. In an alternatve but equvalent form, N k can be consdered as the sum of IID geometrc random varables wth arameter. Each geometrc random varable reresents the number of transmssons at the AP untl one of the ackets s receved at node k. That s N k = G k + + G k, 3 where, G k n s the number of transmssons at the AP untl node k receves the n-th acket, gven that t has already receved n ackets. For geometrc random varables we have P { G k n = } =, 4 and, E Gn] k =. Now, we rewrte 2 as follows, E N] = ] ] E = E = E N k G k + + G k ]. N k 5 We are nterested n comutng E N] as. We have lm E N] = lm E G k + + G k ] = E lm G k + + G k ] 6. Note that the G k n s are IID and hence, by alyng the law of large numbers, we obtan G k + + G k lm y substtutng nto 6, we obtan lm E N] = E C. Relablty Gan of NC = ] = = Θ. 7 8 Let N ARQ and N NC denote the number of transmssons n the case of ARQ and NC, resectvely. Defne the relablty gan of network codng as follows: Relablty Gan = E N ARQ] E N NC ], 9 where, E N ARQ ] and E N NC ] are gven by 0 and 8, resectvely. Then, the relablty gan of network codng for the access ont model s of order Θlog K as K becomes large. In the followng secton, we show that the logarthmc gan of network codng s achevable n tree toology as well n 8] we have shown that the same scalng holds for an extended access ont toology n whch recevers are allowed to communcate locally n order to recover lost ackets wthout nvolvng the access ont. III. MULTICAST OVER A TREE TOPOLOGY In ths secton, we study the erformance of dfferent relablty mechansms for relable multcast over a tree toology as dected n Fg. 2. We base our analyss on the access ont model, and derve exact and asymtotc exressons for the relablty gan of network codng. The followng relablty mechansms are consdered: End-to-End ARQ: The root of the multcast tree retransmts each acket untl t s correctly receved by all the multcast recevers. All other nodes n the tree only forward ackets they receve from ther arents to ther chldren. 2 End-to-End FEC: Ths technque s commonly referred to as rateless codng ]. Smlar to end-to-end ARQ, only the root of the multcast tree s resonsble for retransmttng a acket untl t s receved by all recevers. All other nodes only forward the ackets they receve from ther arents to ther chldren. For FECbased schemes, we assume the use of a block codng technque to create coded ackets for transmsson. 3 Lnk-by-Lnk ARQ: Every node of the multcast tree s resonsble for the relable transmsson of ackets to ts chldren. That s, a node retransmts the acket t has receved from ts arent to ts chldren untl the acket s correctly receved by all of ts chldren. Note that some chldren may receve more than one coy of the acket because of the random nature of acket losses.
Fg. 2. S K nodes K h h Tree toology for relable multcast. 4 Lnk-by-Lnk FEC NC: We refer to ths technque as network codng NC because codng s erformed not only at the source but also wthn the network. That s, every node s resonsble for relable delvery of the block of ackets t has receved from ts arent to ts chldren. Wth NC, each node erforms rateless codng to delver a block of ackets to ts chldren. Note that lnk-by-lnk relablty mechansms are equvalent to the access ont model that we studed n the revous secton. Essentally, each node s resonsble for the relable delvery of the ackets t has receved from ts arent to ts chldren. Therefore, the exected number of transmssons at each node of the tree can be readly comuted from and 2, for ARQ and NC, resectvely. A. Dstrbuton of the Number of Transmssons In ths subsecton, we study the erformance of end-to-end relablty mechansms based on ARQ and FEC. Let N r denote the number of transmssons of a acket to the root of a subtree of heght r from ts arent before the acket s receved by all nodes of the subtree. For the source of a multcast tree of heght h, we nterret N = N h as the number of acket transmssons at the source untl the acket s receved by all of the multcast recevers. Defne F r n as follows: F r n = P {N r n}, 0 r h, n. 20 Smlar to 6], we develo recursve relatons to comute F r n. Frst, consder the case of r > 0, and denote the root of the subtree by s. The robablty that ackets out of n ackets that have been transmtted to node s are receved by node s s gven by a bnomal dstrbuton, P { n} = n n, 0 n. 2 Note that for the root of the multcast tree, the error robablty s zero,.e., = 0, and hence P { n} =, f = n, and P { n} = 0, otherwse. If node s receves ackets, t wll broadcast the receved ackets to ts chldren. For each chld, the robablty that all nodes of the subtree rooted at that chld receve a acket s gven by F r. Snce the chldren of a node have ndeendent acket losses, the robablty that all of the nodes of the subtrees rooted at chldren of node s receve a acket s gven by {F r } K, whch we denote by Fr K for notatonal smlcty. Therefore, by summng over all ossble values of, we obtan n n F r n = n F r, K 0 < r < h. =0 22 Hence, we have a recursve equaton for comutng F r n for r > 0. Interestngly, comutng F r n for r > 0 s ndeendent of the aled relablty mechansm. Next, we comute F 0 n for the leaves of the multcast tree as follows. End-to-End ARQ: The robablty that a leaf node does not receve any acket out of n transmtted ackets s gven by n. Therefore, wth robablty n the node receves at least one coy of the acket. Therefore, F 0 n = n, n. 23 2 End-to-End FEC: The robablty that a node receves at least coded ackets out of n transmtted ackets s gven by a bnomal dstrbuton. Therefore, n n F 0 n = n, n. 24 = So far, we have comuted F r n for all subtrees of heght r. As mentoned earler, for the root of the multcast tree, we have P {n n} =. Hence, the exresson for F h n can be smlfed as F h n = F K h n, where, F h n s gven by 22. That s P {N n} = F K h n.. Exected Number of Transmssons Usng the exressons for P {N n}, we can comute the exected number of transmssons at the root of the multcast tree,.e., E N]. Next, we comute the exected number of transmssons n the multcast tree not just at the root untl a acket s receved by all recevers. Let T denote the total number of transmssons n the multcast tree untl a acket s receved by all recevers. We are nterested n comutng E T ]. Lnk-by-Lnk Mechansms: Consder a multcast tree of heght h. At heght r of the tree, there are K h r nodes. For each of them, the exected number of transmssons E N] can be comuted from and 2, for ARQ and NC, resectvely. Ths results n E T ] = E N] h r= K h r = Kh E N]. 25 K Note that for K =, we have E T ] = he N]. 2 End-to-End Mechansms: Frst, we comute the exected number of transmssons n the tree for each transmsson at the root of the multcast tree. Let X r denote the number of transmssons n a subtree of heght r for each transmsson at the root of the subtree. Then, K E X r ] = + j=0 K j = + K E X r ], j K j je X r ] 26
where, X 0 = 0. The dea s that, for each acket receved at the root of a subtree, there s one transmsson at the root of the subtree. If j chldren out of the K chldren receve the acket, then n average, each of them wll have E X r ] transmssons n ther subtrees. Ths yelds E X r ] = Kqr Kq, 27 where, q =. Therefore, the exected number of transmssons er acket n the multcast tree s gven by E T ] = E X h ] E N] = Kqh E N]. 28 Kq Note that f Kq =, then t follows that E T ] = he N]. C. Asymtotc Analyss For the lnk-by-lnk relablty mechansms, the same bounds we derved for the access ont model aly to the tree tology as well. For end-to-end mechansms, we consder the case of havng K exonentally growng to nfnty,.e., log K. In ths case, we can aly the results from the access ont model to derve asymtotc exressons for the exected number of transmssons at the root of the multcast tree, and hence, wthn the tree. ARQ Performance: ased on the analyss of the access ont model, n order to delver a coy of a message to all leaf nodes,.e., nodes at heght 0, each node at heght needs to transmt the message Θlog K tmes. The key dea s that because log K, the law of large numbers can be aled n a fashon smlar to the access ont model. Hence, the nodes at heght 2 need to transmt only c = tmes more n order for each node at heght to receve log K coes of the acket. Therefore, we have E N] = Θ c h log K. 29 2 FEC Performance: We have the same argument for endto-end FEC excet that at each level of the tree, we need to retransmt a block of ackets only a constant number of tmes,.e., c = tmes. Therefore, we obtan E N] = Θ c h. 30 Table I has summarzed the relablty gan of network codng comared to other relablty mechansms for multcast n a tree toology. Interestngly, the asymtotc gan of network codng comared to end-to-end FEC,.e., rateless codng, s just a constant factor. However, deendng on the heght of the multcast tree h, and the loss robablty, the constant can be arbtrary large. IV. CONCLUSION In ths aer, we studed the beneft of network codng for relable multcast n lossy wreless networks. We analyzed the access ont model n whch an access ont broadcasts ackets to a set of K recevers. We showed that, for large codng blocks, the relablty gan of network codng comared Relablty Mechansm Relablty Gan Lnk-by-Lnk ARQ Θlog K End-to-End ARQ Θc h log K End-to-End FEC Θc h TALE I RELIAILITY GAIN OF NC IN A TREE TOPOLOGY OF HEIGHT h c =. to ARQ s of order Θlog K. We then used the access ont model to study the relablty gan of network codng n a tree-based relable multcast. For the tree toology, four dfferent relablty mechansms based on ARQ and NC were consdered. We showed that NC acheves the best erformance n terms of the requred number of transmssons n the tree. We also extended the access ont model by allowng nter recever communcaton to recover lost ackets, and showed that stll the log K relablty gan can be acheved whle mnmzng the number of transmssons at the access ont 8]. In the future, we would lke to extend our analyss to more comlcated network toologes such as a grd wth sgnfcant amount of ath dversty. V. ACKNOWLEDGEMENTS Ths research was suorted by DARPA CMANET and US/UK ITA rograms. REFERENCES ] R. Ahlswede, N. Ca, S.-Y. R. L, and R. W. Yeung, Network nformaton flow, IEEE Trans. Inform. Theory, vol. 46, no. 4,. 204 26, July 2000. 2] S. Katt, H. Rahul, W. Hu, D. Katab, M. M. Médard, and J. Crowcroft, XORs n the ar: Practcal wreless network codng, n Proc. ACM SIGCOMM, Psa, Italy, Setember 2006. 3] D. S. Lun, M. Medard, and M. Effros, On codng for relable communcaton over acket networks, n Proc. Allerton, Urbana Chamagn, USA, Setember 2004. 4] A. Erylmaz, A. Ozdaglar, and M. Medard, On delay erformance gans from network codng, n Proc. CISS, Prnceton, USA, March 2006,. 864 870. 5] D. Nguyen, T. Nguyen, and. ose, Wreless broadcastng usng network codng, n Proc. NetCod, san Dego, USA, January 2007. 6] P. hagwat, P. P. Mshra, and S. K. Trath, Effect of toology on erformance of relable multcast communcatons, n Proc. IEEE INFOCOM, Toronto, Canada, June 994,. 602 609. 7] J. Nonnenmacher and E. W. ersack, Performance modellng of relable multcast transmsson, n Proc. IEEE INFOCOM, Kobe, Jaan, Arl 997,. 47 479. 8] M. Ghader, D. Towsley, and J. Kurose, Relablty gan of network codng n lossy wreless networks, Deartment of Comuter Scence, Unversty of Calgary, Tech. Re. 2008-889-02, January 2008. Onlne]. Avalable: htt://www.cs.ucalgary.ca/ mghader/docs/relex.df 9] T. Ho, M. Medard, R. Koetter, D. R. Karger, M. Effros, J. Sh, and. Leong, A random lnear network codng aroach to multcast, IEEE Trans. Inform. Theory, vol. 52, no. 0,. 443 4430, October 2006. 0] M. Ghader, D. Towsley, and J. Kurose, Relablty beneft of network codng, Deartment of Comuter Scence, Unversty of Massachusetts Amherst, Tech. Re. TR-07-08, February 2007. Onlne]. Avalable: htt://www.cs.umass.edu/ mghader/docs/relablty.df ] J. yers, M. Luby, M. Mtzenmacher, and A. Rege, A dgtal fountan aroach to relable dstrbuton of bulk data, n Proc. ACM SIGCOMM, Vancouver, Canada, February 998.