Multi Packet Reception and Network Coding

Transcription

1 The 2010 Military Counications Conference - Unclassified Progra - etworking Protocols and Perforance Track Multi Packet Reception and etwork Coding Aran Rezaee Research Laboratory of Electronics Massachusetts Institute of Technology Cabridge, MA. Eail: aranr@it.edu Linda Zeger MIT Lincoln Laboratory Lexington, MA. Eail: zeger@ll.it.edu Muriel Médard Research Laboratory of Electronics Massachusetts Institute of Technology Cabridge, MA. Eail: edard@it.edu Abstract We consider throughput and delay gains resulting fro network coding used to copleent ultipacket reception in a fully connected network with broadcast traffic. The network is coprised of nodes, J of which have data packets to be distributed to all other nodes. Owing to half-duplex constraints, a transitting node is not able to receive data fro other transitting nodes in the sae tie slot. This requires a node to be back-filled with the inforation that it is issing. We consider singlepacket reception, (for which network coding alone yields no gain), ulti-packet reception without network coding, and cobined ulti-packet reception and network coding. We show that the initial transissions and the back-filling process can be greatly sped up through a cobination of network coding and ulti-packet reception. In particular, we show that MPR capability of 2 will not reduce the total delivery tie for packets within a data network unless network coding is used. We also deonstrate that a cobination of network coding and ulti-packet reception can reduce this tie by a factor of, where is the MPR capability of the syste. We deonstrate that network coding can substitute for soe degree of MPR and achieve the sae, or alost the sae, perforance as the higher degree of MPR without network coding. In effect, network coding allows alost double the traffic for a given level of MPR. I. ITRODUCTIO In networks, ulti-user interference is an iportant liitation. In order to alleviate it, the use of ulti-packet reception (MPR) has been proposed [1], [2]. MPR ay be ipleented in a variety of ways, ranging fro orthogonal signaling schees such as OFDM to spread spectru techniques such as frequency hopping or DS-CDMA. Such systes allow a liited nuber of packets to be siultaneously received at a node with MPR capability. In this paper we analyze the effects on network perforance of a range of MPR capabilities, without restricting ourselves This work is sponsored by the United States Air Force under Air Force Contract FA C Opinions, interpretations, recoendations, and conclusions are those of the authors and are not necessarily endorsed by the United States Governent. Specifically, this work was supported by: 640th Electronics Systes Squadron of the Electronic Systes Center, Air Force Materiel Coand, and Inforation Systes of D,DR&E. Generous contributions of Irwin Mark Jacobs and Joan Klein Jacobs Presidential Fellowship has been critical in the success of this project. to specific physical layer ipleentations. Multi-packet reception can help when used with wireless MAC protocols (e.g. ALOHA), in which conventional odels consider the transission of ultiple packets to be a collision and the throughput can be optiized through different backoff echaniss [3]. There has been a renewed interest in capture schees as explained in [4], [5], and [6], the siplest for of which can be used to capture the packet fro the signal that has higher energy at the receiver. It has been shown, in [7] and [8], that network coding can be used to iprove throughput by allowing ixing of data at interediate nodes in a network. If network coding and MPR are utilized together, we expect a higher throughput because MPR at a node enables cobination of diverse packets which can be coded together and consequently, each transission can on average disseinate ore inforation to the network. The ain focus of our paper is to show when and where we should cobine network coding with ulti-packet reception to iprove perforance. The paper is organized as follows. In Section II, the network odel and paraeters are introduced. In Section III, we characterize single-packet reception. In Section IV, ulti-packet reception without network coding is described. In Section V, we present the gains associated with the cobination of network coding and ulti-packet reception. Finally, in Section VI, we conclude the paper with a brief coparison of the results. II. ETWORK MODEL AD PARAMETERS Consider a wireless network represented by a set of nodes. We assue that all nodes are within transission range of each other. That is to say, given that node i is transitting, any node j i in the network can receive the transitted packets if j itself is not transitting. ote that our odel preserves the half-duplex constraint. Within this network, there is a set of J nodes (J ) thathave packets for transission. Each of these J nodes has k packets and transission occurs in tie slots. Let us define the MPR coefficient,, to denote the axiu nuber of siultaneous receptions that is pos /10/$ IEEE 1393

2 sible per node per tie slot. For each node i, we define T (i) to be the first tie slot in which it has received every packet transitted by the J transitting nodes. Two tie variables are defined to easure the perforance of the syste. First is the total tie T tot required for disseination of inforation to all nodes; this tie is a easure of delay perforance of the last node that acquires all the packets. Second is the average tie T avg for disseination of inforation; this tie easures the average delay perforance of the syste. In Sections IV, V, and VI we introduce two auxiliary tie variables: T 1 which is the nuber of tie slots until each of the J transitting nodes has transitted its packets and T 2 to denote the nuber of tie slots to coplete the backfilling. To avoid trivialities, two assuptions are ade regarding the size of the network. First, J which states that the nuber of non-transitting nodes is lower bounded by the MPR coefficient. Second, 2 which ensures that the nuber of nodes in the network is at least twice the MPR capability of the syste. In suary: : uber of nodes in the network. J: uber of transitting nodes within the network. k: uber of packets to be transitted per node. : uber of possible receptions per node per tie slot. T (i): The first tie slot in which node i has received every packet. T 1 : uber of tie slots until each of the J transitting nodes has transitted its packets. T 2 : uber of tie slots to coplete the back-filling process. T tot : Total tie required for disseination of inforation (in tie slots). T avg : Average tie for a node to receive the last packet of the transitted inforation (in tie slots). C, C : Denote network coding, or lack of it, respectively. MPR, MPR : Denote ulti-packet reception, or lack of it, respectively. J 2 III. SIGLE PACKET RECEPTIO ( =1) Since = 1, all transitting nodes will take turns transitting their packet. Thus for k =1: T MPR tot = J (1) More generally, if each of the J transitting nodes has k>1 packets to transit, the total tie will be increased by a factor of k and: T MPR tot = Jk (2) Consider the case where each of the J transitting nodes successively transits all of its k packets. It can be seen that the last transitting node will have every packet after (J 1)k transissions and every other node will have it after Jk transissions. Thus: T MPR avg = 1 T (i) i=1 = Jk k (3) Table I deonstrates how a typical transission takes place when J = 5, and = 8 without MPR. Each node is denoted by a letter, A through H, and tie slots are enuerated by t 1 to t 5. During each tie slot, the transitting node is explicitly arked with an arrow. For exaple X A, shows that node A transitted during t 1 and its packet was received by every other node during the sae tie slot as denoted by X A in the other rows. ode (i) t 1 t 2 t 3 t 4 t 5 T (i) A X A X B X C X D X E 5 B X A X B X C X D X E 5 C X A X B X C X D X E 5 D X A X B X C X D X E 5 E X A X B X C X D X E 4 F X A X B X C X D X E 5 G X A X B X C X D X E 5 H X A X B X C X D X E 5 TABLE I TRASMISSIO SCHEDULE OF A ETWORK WITH =8, J =5, AD =1 ote that since MPR is not present, coding cannot help. In other words, if each node codes its packets before transission, there will be no reduction in the total transission tie or the average transission tie. Hence: avg tot = avg = tot IV. MULTI-PACKET RECEPTIO WITHOUT ETWORK CODIG Consider J transitting nodes, each with one packet to be distributed to every other node. Since receivers are liited by receptions per tie slot, the set of transitting nodes is partitioned into groups of nodes such that all nodes in a given group can transit siultaneously. If J is not a ultiple of, the last group will have J od nodes. Let T 1 be the nuber of tie slots it takes until 1394

3 each of the J transitting nodes has transitted its packet. There will be J such groups, thus: J T 1 = (4) Because of half-duplex constraints, the J transitting nodes need to be back-filled. ote that transissions occur in distinct groups of nodes, and each transitting node has issed a axiu of 1 packets during its transission slot. Since J in our odel, the network will back-fill the previously defined groups consecutively by utilizing of the non-transitting nodes. Each group can be back-filled in one tie slot and backfilling will take the sae nuber of slots as T 1.LetT 2 denote the nuber of slots to coplete the back-filling: J T 2 = u(j 1) (5) where u(j) is the unit step function defined as: { 0 : J 0 u(j) = 1 : J>0 Table II shows a saple transission schedule for =8, J =5,and =2. We note that this saple strategy does not fully utilize the MPR since node E transits alone. In contrast, our scheduling discussed in Part B below ore fully utilizes MPR. ode (i) t 1 t 2 t 3 t 4 t 5 T (i) A X A X C,X D X E X B 4 B X B X C,X D X E X A 4 C X A,X B X C X E X D 5 D X A,X B X D X E X C 5 E X A,X B X C,X D X E 2 F X A,X B X C,X D X E X A X C 3 G X A,X B X C,X D X E X B X D 3 H X A,X B X C,X D X E 3 TABLE II TRASMISSIO SCHEDULE OF A ETWORK WITH =8, J =5, AD =2WITHOUT ETWORK CODIG ow consider the case where each transitting node has k packets. We will present the results for two separate cases, naely for J and J>. A. J When the nuber of transitting nodes is less than the MPR coefficient, we are not able to fully utilize the MPR capability of the syste and the initial transissions will occur in groups of J nodes per tie slot and: T 1 = k (6) As discussed previously, J and we will backfill the J transitting nodes by of the non-transitting nodes. Thus: Jk T 2 = u(j 1) (7) Thus, the total copletion tie is: tot = T 1 + T 2 = k + Jk u(j 1) (8) The average copletion tie can be upper bounded by noting that the J non-transitting nodes will have all the data by T 1 and the J transitting nodes will have every packet in at ost T 1 + T 2 tie slots. Thus: avg ( J)T 1 + J(T 1 + T 2 ) Jk u(j 1) (9) k + J B. J> In this case, the optial strategy is to transit new packets in each tie slot. This is achieved by using all possible cobinations of transitting nodes. In other words, there are ( J ) distinct groups of out of J nodes. Any given node is present in ( J 1 1) groups and is excluded fro ( ) J 1 groups. Assue that k is an integral ultiple of ( J 1 1).Then each node transits k/ ( J 1 1) packets during each transission round, and T 1 can be obtained as: ( ) J k T 1 = ( J 1 ) = Jk (10) 1 Recall that the J transitting nodes need to be backfilled because of half-duplex constraints. Since the transitting groups during tie T 1 are distinct and known to other nodes, we can back-fill the groups consecutively. Let T 2 denote the back-filling tie: T 2 = Jk (11) Thus, the total copletion tie is: ( ) Jk tot = T 1 + T 2 =2 (12) 1395

4 The average copletion tie can be upper bounded followingthesaearguentusedinparta. Thus: avg ( J)T 1 + J(T 1 + T 2 ) Jk ( 1+ J ) (13) V. MULTI-PACKET RECEPTIO WITH ETWORK CODIG Following the odel presented in Section IV, we will introduce network coding as an instruent to reduce the total transission tie. We present a strategy that utilizes network coding only in back-filling. Table III illustrates an exaple in detail; this exaple, like that in Table II, is suboptial since node E transits alone and hence the MPR capability is not fully leveraged. However, our scheduling in Part B ore fully utilizes the MPR. Coefficients α 1 through α 5 are used as coding coefficients and are chosen according to the size of the required finite field. ode (i) t 1 t 2 t 3 t 4 T (i) A X A X C,X D X E X B 4 B X B X C,X D X E X A 4 C X A,X B X C X E X D 4 D X A,X B X D X E X C 4 E X A,X B X C,X D X E 2 F X A,X B X C,X D X E α 3 X C + α 4 X D + α 1 X A + α 2 X B + α 5 X E G X A,X B X C,X D X E 3 H X A,X B X C,X D X E 3 TABLE III TRASMISSIO SCHEDULE OF A ETWORK WITH =8, J =5, AD =2WITH ETWORK CODIG As before, let us analyze the proble separately for J and J>. A. J Since we only use network coding in backfilling, T 1 is the sae as the case of MPR with no network coding: T 1 = k (14) Recall that J, so we can find at least nodes that did not participate in any of the previous transissions and have received each of the Jk transitted packets. The back-filling is accoplished by having each of these nodes transit a linear cobination of all the packets that it has received so far. Every tie these nodes transit, independent coded packets are sent out and, as a result, the original J transitting nodes will get 3 new degrees of freedo in each tie slot and back-filling will be copleted in (J 1)k/ tie slots. Thus: (J 1)k T 2 = (15) The total transission tie for this strategy is: tot = T 1 + T 2 (J 1)k = k + (16) To calculate the average copletion tie, recall that all non-transitting nodes will have every packet by T 1 and the J transitting nodes will have the data after T 1 + T 2 tie slots. Thus: avg = ( J)T 1 + J(T 1 + T 2 ) = k + J (J 1)k (17) B. J> As in Section IV, let us assue that k is a ultiple of ( ) ( J 1 1 and the transitting nodes will send k/ J 1 ) 1 uncoded packets during the first ( J ) tie slots, therefore T 1 will have the sae value as in Section IV: ( ) J k T 1 = ( J 1 ) = Jk (18) 1 During each transission, exactly nodes transitted siultaneously and any given node within a transitting group was unable to receive the packets fro the other 1 nodes. Since each node transitted k packets, it is issing ( 1)k packets in total. Following the sae reasoning used in part A of this section, the back-filling will be copleted in ( 1)k/ tie slots. Thus: ( 1)k T 2 = Thus, the total copletion tie is: tot = T 1 + T 2 (19) = k (J + 1) (20) The average copletion tie can be calculated as: avg = ( J)T 1 + J(T 1 + T 2 ) = Jk ( 1+ 1 ) (21) This strategy deonstrates how network coding can be used with MPR to reduce the total and average transission ties. 1396

5 VI. COMPARISO In practical networks the nuber of transitting nodes is usually uch greater than the MPR capability of the syste, hence we will copare the results of Sections III, IV, and V for the case where J>. Let us revisit the results when each transitting node has k packets. Without MPR: tot avg = tot = Jk (22) = avg = Jk k When MPR of is used without network coding: ( ) Jk tot = 2 avg Jk ( 1+ J When MPR is copleented with network coding: tot = k avg = Jk ) (23) (24) (25) (J + 1) (26) ( 1+ 1 ) (27) Coparing (22) and (24), we can see that MPR reduces T tot by a factor of /2. ote that when =2, the total transission tie reains unchanged. This is depicted in Fig. 1 where lack of network coding is represented by dashed lines. otice that the dashed lines do not change between =1and 2. We will later copare this result with the case that cobines MPR with network coding. To see the advantage of network coding, copare (24) to (26). The total transission tie is reduced by a factor of 2J/(J + 1) which becoes arbitrarily close to two with increasing J. LetusrevisitFig.1.An ellipse in the figure points out the close proxiity of two lines: one representing J =50with coding and the other J = 25 without coding. Coding yields the sae total disseination tie as a network with no coding and only half the traffic. ote that if network coding is used, we can reduce the total transission tie even when =2, which was not achievable without network coding. While Fig. 1 presents the value of T tot for k =1packet per transitting node, if k > 1 and k satisfies the integrality constraint discussed in Sections IV and V, the value of T tot shown in this figure would be ultiplied by k. We will follow the sae approach in calculations of T tot in Fig. 2 and Fig. 3. In Fig. 2, we show that when is fixed, T tot grows linearly with the nuber of nodes if the ratio J/ is kept constant. As shown in the figure, the total transission tie of a network with J = /2 transitting nodes that uses network coding is only slightly higher than that of a network with J = /4 transitting nodes that does not use coding. ( = 4) J = /2 with coding J = /3 with coding J = /4 with coding J = /2 without coding J = /3 without coding J = /4 without coding T tot MPR 10 T tot MPR J = 50 with coding J = 33 with coding J = 25 with coding J = 50 without coding J = 33 without coding J = 25 without coding 5 J = /2 with coding and J = /4 without coding uber of odes () Fig. 2. T tot as a function of for different ratios of J/ when =4 and k =1. Solid lines denote use of network coding and dashed lines are for the case without coding J = 50 with coding and J = 25 without coding Fig. 1. T tot as a function of for different values of J and k =1. Solid lines denote use of network coding and dashed lines are for the case without coding. Fig. 3 shows that for a given value of, the total tie T tot increases linearly with the nuber of transitting nodes J. It is interesting to note that the lines in the figure cross one another when J [1, 4]. This occurs because J in this range, and the behavior is governed by equations (8) and (16) where we have assued that k is divisible by to allow scalability for the results. If the traffic coes fro a fixed nuber of nodes J, the total transission tie T tot will not be affected by an increase in. 1397

6 T tot / k = 1 with coding = 1 without coding = 2 without coding = 3 with coding = 3 without coding = 2 with coding = 4 without coding = 4 with coding uber of transitting nodes (J) Fig. 3. T tot/k as a function of J for different values of. Finally, Fig. 4 illustrates how the nuber of packets k at each transitting node affects T tot for a network with 9 and J =5. otice the siilarity between Fig. 3 and Fig. 4. In essence T tot increases linearly with the nuber of packets k. Again, notice that total transission tie of =2with coding and =4without coding are close to each other. The aount of tie saved by adding network coding to a fixed level of MPR is seen to increase with the nuber of packets k. The sall discontinuities seen on each line are caused by the integrality constraint on k. T tot = 1 with coding = 1 without coding = 2 without coding = 4 with coding = 2 with coding = 3 with coding = 3 without coding = 4 without coding uber of packets (k) without network coding. It is iportant to note that a twofold MPR capability will not reduce the total disseination tie without network coding and is thus ineffective. We have also shown that no gain can be obtained, if network coding is used without MPR. We finally argued that the cobination of network coding and MPR can reduce the total transfer tie by as uch as a factor of. Wealso showed that a MPR equipped network that uses coding behaves siilarly to an equivalent network that has half of that traffic and does not use network coding. Our work deonstrates a nuber of significant gains that do not scale with, in contrast to previous works such as [9] and [10], in which the network is analyzed through scaling laws, and hence would not show such gains. Erasures in these networks are currently under investigation, and we expect an even greater gain fro the cobined usage of network coding and MPR when erasures are present and ust be handled. REFERECES [1] Z. Wang, H. Sadjadpour, and J. Garcia-Luna-Aceves, Capacitydelay tradeoff for inforation disseination odalities in wireless networks, in Inforation Theory, ISIT IEEE International Syposiu on, July 2008, pp [2] X. Wang and J. Garcia-Luna-Aceves, Ebracing interference in ad hoc networks using joint routing and scheduling with ultiple packet reception, in IFOCOM The 27th Conference on Coputer Counications. IEEE, April 2008, pp [3] D. Bertsekas and R. Gallager, Data etworks. Prentice Hall, Englewood Cliffs. J, [4] G. D. Celik, Distributed MAC protocols for networks with ultipacket reception capability and spatially distributed nodes, Master s thesis, MIT Departent of Electrical Engineering and Coputer Science, [5] Z. Hadzi-Velkov and B. Spasenovski, Capture effect in IEEE basic service area under influence of Rayleigh fading and near/far effect, vol. 1, Sept. 2002, pp [6] A. yandoro, L. Liban, and M. Hassan, Service differentiation in wireless LAs based on capture, vol. 6, Dec. 2005, pp [7] R. Ahlswede,. Cai, S.-Y. Li, and R. Yeung, etwork inforation flow, Inforation Theory, IEEE Transactions on, vol. 46, no. 4, pp , Jul [8] T. Ho, M. Médard, R. Koetter, D. Karger, M. Effros, J. Shi, and B. Leong, A rando linear network coding approach to ulticast, Inforation Theory, IEEE Transactions on, vol. 52, no. 10, pp , Oct [9] E. Ahed, A. Eryilaz, M. Médard, and A. E. Ozdaglar, On the scaling law of network coding gains in wireless networks, in Military Counications Conference, MILCOM IEEE, Oct. 2007, pp [10] J. Liu, D. Goeckel, and D. Towsley, The throughput order of ad hoc networks eploying network coding and broadcasting, in Military Counications Conference, MILCOM IEEE, Oct. 2006, pp Fig. 4. T tot as a function of k for different values of when 9 and J =5. VII. COCLUSIO In conclusion, we have shown that MPR can reduce the total tie for a file transfer by as uch as a factor of