Cluster-based Cooperative Communication with Network Coding in Wireless Networks

Transcription

1 Cluster-based Cooperative Communiation with Network Coding in Wireless Networks Zygmunt J. Haas Shool of Eletrial and Computer Engineering Cornell University Ithaa, NY 4850, U.S.A. Abstrat Cooperative ommuniation is a promising way to redue probability of paket loss. The massive deployment of nodes in wireless sensor network renders suh networks espeially attrative for exploiting the advantage of ooperative diversity. Similarly, when used appropriately, network oding ould also improve the probability of orret reeption. In this paper, we introdue the luster-based Cooperative Coding (CC) protool, whih is based on the integration of ooperative ommuniation and network oding. In partiular, in the CC protool, network nodes are grouped into multiple lusters and nodes within the same luster ooperate in transmitting and reeiving pakets. Suh an integration redues the amount of redundant information being forwarded to ensure high probability of orret end-to-end reeption, when link-level retransmission of erroneous pakets is not allowed (i.e., no linklevel feedbak). In partiular, our analysis shows how to optimize the performane of the network by properly sizing the lusters. Compared to shemes without ooperation (whether with or without network oding), our simulation results demonstrate the signifiant performane improvement of the proposed sheme. Tuan-Che Chen Shool of Eletrial and Computer Engineering Cornell University Ithaa, NY 4850, U.S.A. t397@ornell.edu due to redued time- and spae-orrelation of the transmission fading, the overall reliability of the reeived signal is inreased. Depending on the type of diversity used at the physial layer, radios may need to be able to synhronize reeption of multiple signals. In ooperative ommuniation, lustering ould be used to group nodes whih are loated lose to eah other. The massive deployment of the nodes in wireless sensor network provides an effetive senario for node lustering. All nodes in a luster ooperate to transmit and reeive pakets to/from other ooperative lusters. Compared with other shemes, the luster-based approah redues the omplexity of resoure management of the ooperation among the luster s nodes. Reent researh in network oding has revealed its potential in inreasing the apaity of wired and wireless networks. The apaity gain is ahieved through oding of information reeived from multiple soures [4]. I. INTRODUCTION In multi-hop wireless sensor networks, the information from the soure to the destination is relayed by intermediate nodes. Traditionally, the routing protools hoose a path - a sequene of nodes between the soure and the destination - and then forward pakets along the path. To ombat the linklevel paket loss and to avoid signifiant end-to-end throughput degradation, networks use link-level retransmissions. However, due to orrelation of errors in retransmitted pakets espeially in wireless networks, retransmission is often quite ineffetive. It ould also be quite ineffiient, leading to signifiant waste of network apaity and energy, and onsiderably inreasing the end-to-end delay. Thus, in numerous instanes, suh as real-time traffi for example, link-level retransmission may not be the right approah for inreasing the end-to-end transmission reliability. In ontrat, ooperative ommuniation [] has reently reeived signifiant attention as a way to improve the reliability of wireless links. For instane, the ooperative sheme in [2] suggests that the traditional routing may not be the best approah. Cooperative ommuniation exploits the broadast nature of wireless ommuniations, where with a single transmission, a number of ooperating nodes reeive and relay the data. Due to spatial diversity a reeiver an then ombine multiple relayed signals (diversity ombining) or hoose the best signal (seletion diversity) at the physial layer to improve the overall hannel quality [3]. Moreover, Figure : Example of ooperative lusters in a wireless network In this paper, we integrate network oding to introdue the luster-based Cooperative Coding (CC) protool. Unlike ooperative ommuniation shemes at the physial layer, CC integrates oding and ooperative diversity at the link layer, whih ould be implemented with standard radio hardware. Indeed, it is natural to explore the ombining of network oding with ooperative diversity at the link layer, beause in ooperative ommuniation the broadasting property of the wireless medium allows a node to overhear pakets from multiple relays. The relay nodes an enode overheard information from different soures and forward the oded paket to other nodes. When enough information is reeived at the destination nodes, the destination nodes an then reover the original pakets. In our model of CC, there are multiple nodes in the reeiver and in the sender lusters of eah hop. Fig. illustrates an example of ooperative transmissions from the soure to the destination through multiple lusters, where pakets are relayed

2 from a luster to a luster. We assume that the intra-luster distanes are muh smaller ompared with the inter-luster distanes. As opposed to the traditional ase in whih eah hop is omposed of a point-to-point link, in ooperative transmission, eah hop is replaed with many-to-many links. A node that hears paket transmission from the nodes in the previous luster will relay the paket to the next luster towards the destination. Therefore, the routing path an be represented as having a width, whih is determined by the number of nodes in a luster. This ooperation in relaying the pakets inreases the probability that the paket will reah the destination. To ahieve this goal, in this paper, a simple lustering and medium aess ontrol are introdued based on the work in [9]. In addition, to redue the number of paket transmissions, CC randomly mixes the reeived pakets and relays the oded paket by ooperating with the nodes within the same luster. Thus a more reliable ommuniation an be ahieved. We show how to ompute the number of ooperating nodes in a luster as to optimize the end-to-end performane. We also ompare our proposed CC protool with shemes that do not employ ooperation, whether with or without network oding. Our analytial results, validated by simulations, show that CC leads to signifiant performane improvement of throughput and of suessful paket reeption probability. Thus, we demonstrate that the CC protool an exploit the integration of luster ooperation and network oding in improving the network performane. The paper is organized as follows. In Setion II, we review the related work. Setion III presents an example of how paket forwarding ould benefit from ooperation and network oding. The proposed CC protool is then introdued in Setion IV. Setion V presents a mathematial analysis to alulate the proper number of nodes in a luster for optimizing the end-toend performane. Setion VII disusses performane evaluation, followed by onlusion in Setion VIII. II. RELATED WORK A. Cooperative Communiation In the wireless network environment, where transmission is subjet to loss, exploiting multi-user diversity is one of the tehniques to ombat the transmission impairments. At the physial layer, nodes overhearing a transmission simultaneously relay the signal. Suh ooperative ommuniation was investigated in numerous works (e.g., [3], [5], [6]), and either amplifying-and-forwarding or deodingenoding-and-forwarding is performed by the relay nodes. Several works have proposed luster-based ooperative ommuniation. In [7] [9], the luster design and orresponding energy onservation are investigated. Given energy onstraint for eah link, [0] proposes a sheme to minimize end-to-end outage probability. The CC sheme builds on the idea of these luster ooperation shemes, but adopts a fundamentally different approah it uses flow-based network oding on the link layer. Also, in ontrast with most ooperative diversity approahes, the CC sheme does not require synhronized signal transmissions. Opportunisti routing is a tehnique that realizes some gains of ooperative diversity at the link layer. The testbed of the ExOR protool [] was first shown to improve performane over the traditional deterministi forwarding. However, in ExOR, to prevent medium ollisions, a strit transmission sheduling is imposed at the ost of redued spatial reuse. The MORE protool [2] addresses this issue and further improves the throughput by using network oding. In MORE, there is no partiular next-hop for opportunisti routing. One a node overhears a oded paket, it is involved in forwarding the paket. In ontrast, CC builds a strutural way to transmit oded pakets by grouping the nodes into the luster. The next-hop is limited to a finite group of nodes. Therefore, unneessary transmissions are avoided, and the omplexity for resoure management of nodes is redued. The CC sheme takes the advantages of luster-based forwarding, spatial diversity, and network oding. B. Network Coding As opposed to traditional networks with single relay hop, in network oding intermediate nodes intelligently mix the reeived pakets, so that the resulting transmitted pakets ontain information of multiple messages. One of the first works that studied network oding was the paper by Ahlswede et al. [4], whih analyzed apaity bound of networks. Several papers (e.g., [4], [3], [4]) show that network oding an ahieve the maximum multiast apaity in wired network. In the ontext of wireless networks, network oding was proposed to improve the performane of multiast [5] and broadast [6]. Testbed experiments were onduted to demonstrate the throughput gain for uniast appliations [7]. Through theoretial analysis, [8] quantified the potential throughput gain of oding-aware routing. III. MOTIVATION To justify the integration of ooperative ommuniation with network oding, this setion presents an example that demonstrates the improvement in performane of suh an integrated sheme. Consider Fig. 2, where the soure node (sr) attempts to deliver two pakets, a and b, to a destination. Assume that all the nodes in the ooperative luster reeived two pakets and that the probability of a suessful transmission over any link to a target node in the next luster is 0.5. Note that the target node an be one of the nodes in the next luster or the destination node. As disussed in Setion I, we assume that link-layer retransmissions are not feasible. In Fig. 2(a), pakets a and b have been both reeived (overheard) by all the nodes in the luster. To maximize the probability of reeption of both pakets, the nodes in the luster transmit eah paket the same number of times. Thus, for example, nodes, 2, 3, and 4 transmit pakets a, b, a, and b, respetively. In this ase, the probability that both pakets a and b are reeived by the target is (-(-0.5) 2 ) (-(-0.5) 2 ) = With network oding, as in Fig. 2(b), eah node in the ooperative luster having overheard the oded pakets, (a+b) and (3a+2b), from the previous luster (the sr node in this ase), reates and transmits a linear ombinations of the reeived oded pakets. In this way, as long as the target reeives two independent oded pakets, the two original pakets a and b an be reovered. Thus, the probability that both pakets a and b an be aquired by the target is (-(-0.5) 4-4 (-0.5) 4 ) = , whih is higher than the orresponding probability of the non-oding sheme in Fig.

3 2(a). Network oding offers an elegant solution to improve reliability in luster-based forwarding. Without network oding eah relayed transmission ontains only information of one original paket. With network oding loss of some of the transmissions ould be ompensated for, beause eah relayed transmission ontains linear ombination of some of the original pakets. Hene, the original pakets ould be reovered from the orretly reeived transmissions. For example, as in Fig. 2(b), assume that node, node 2, node 3, and node 4 reate and transmit the linear ombinations (4a+3b), (2a+b), (5a+4b), and (7a+5b), respetively. Despite the fat that one or even two of the transmitted linear ombinations are lost in transmission, the target node an still reover the original pakets a and b from the orretly reeived linear ombinations. On the other hand, in Fig. 2(a), when the transmission of nodes and 3 (or transmissions of nodes 2 and 4) are lost, paket a (or b) annot be reovered by the target node. higher performane gains, more nodes are needed to help relay the pakets. Furthermore, not all the ombinations transmitted by nodes in a luster need to be suessfully retrieved by any single relay in the next luster. The multiple relays with different oded pakets in the same luster an further ooperate to forward different oded pakets as long as suffiient information is relayed to the next luster. In Setion V, we address the seletion of the luster size, so as to ensure high deoding probability at the destination. IV. THE CLUSTER-BASED COOPERATIVE CODING PROTOCOL In this setion, we present the proposed CC protool. We first introdue how CC inorporates routing and medium aess ontrol in a ooperative manner, and how to apply network oding as part of the paket forwarding operation. Then, we desribe how the ooperative lusters are formed. Figure 3: One-node-width path A. Routing and Medium Aess Control with Network Coding In CC, a one-node-width path is first disovered between the soure and the destination, as illustrated in Fig. 3. Suh an initial path an be found by traditional routing protools; e.g., AODV (Ad ho On Demand Distane Vetor) [9] or DSR (Dynami Soure Routing) [20]. The energy transmission for eah link an be used as the link s ost to disover minimal ost path. Next, the nodes on the initial path beome luster heads and reruit other nearby nodes to form lusters. CC is designed to transmit data in a blok of oded pakets from a luster to a luster. Reruiting of luster nodes is done per blok of forwarded pakets. Reruiting of nodes of the next luster ommenes one all the oded pakets of the blok are reeived by the previous luster. Then, the newly generated oded pakets of the blok are forwarded, where the nodes within the sending and the reeiving lusters ooperate in transmitting and reeiving. ) The Soure: The soure node partitions the data into bloks of m pakets. The m unoded pakets in a blok are alled native pakets. A native paket is denoted by x i, {, 2...,m}. The soure transmits oded pakets to the nodes in the next luster. A oded paket x' j is a linear ombination of the native pakets, generated as: Figure 2: Example of the benefits of network oding and ooperation Of ourse, the performane of a protool that build upon the above ooperative and oded ommuniated depends on a number of parameters. One suh key parameter is the size of the ooperative luster. Consider again the example of Fig. 2(b), where the suessful reeipt probability is To ahieve x' j = m i= where the ji 's are the oeffiients piked randomly, and the addition and multipliation are operations over a Galois Field, GF(2 q ). We embed a ode vetor, j = ( j, j2,, jm ), and the blok id, into the x' j paket's header. The soure maintains a ounter with some initial value m', where m' > m. Eah time the ij x i

4 soure transmits a oded paket, the ounter is dereased by. The soure keeps transmitting randomly oded pakets until the ounter reahes zero. 2) Reruiting and Forwarding: The reruiting and forwarding operations run per hop from the soure to the destination. The luster head of a reeiving luster initiates the onstrution of the next luster after it has reeived the oded pakets of the same blok from the sending luster. The reeiving luster now beomes the urrent sending luster onsisting of the same nodes whih reeived the pakets from the previous hop. For eah node of the sending luster, the sending luster head first shedules the speifi time when it an transmit the oded paket to the reeiving luster. The reeiving luster head then reruits adjaent nodes to form the reeiving luster, and selets the nodes with higher ost C j. We define C j for node j, whih may possibly be reruited for the reeiving C = I p, where p ij is the loss luster, as: ( ) j ij ij node i sending luster probability of sending a paket from node i to node j, and I ij is the indiation funtion whether node j is available for reeiving pakets when node i in the sending luster is transmitting. Node j may not be available beause it has been sheduled to transmit or reeive pakets for other luster of a different route. The p ij -s are periodially evaluated by a node for all of its neighbors via ping probes. I ij depends on how the sending luster head shedules the nodes to transmit pakets. Therefore, the nodes whih are likely to reeive more pakets from the sending luster are hosen to form the reeiving luster. Assuming that a node has reeived the oded pakets x's, the new oded paket an be generated as '' m j = x =, x' j j where j '-s are random numbers hosen from GF(2 q ). In this way, the x'' is also a random linear ombination of the native pakets, sine x m m m m '' = = = j j ( i= jix i ) j ji ) xi = ( g x. i= j= i= Like the soure node, when forwarding the oded paket, a node will embed the new ode vetor, within the x'' paket's header. g j = (g, g 2,, g m ) 3) At the destination: When the destination obtains a oded paket, it will first hek if the paket is innovative. A paket is onsidered innovative if it is linearly independent from the previous pakets of the same blok that the destination has reeived. If the oded paket is not innovative, it will be disarded. Eah oded paket represents a linear equation of the m native pakets and the oding oeffiients are known to the destination via the embedded ode vetor. Thus, as long as m innovative pakets have been olleted, the destination is able to reover the native pakets. The deoding proess in the destination involves solving the following set of linear equations, for example by Gaussian elimination algorithm. When the rank of the matrix is m, (i.e., there is no linear dependene between the m oded pakets) the linear equations an be uniquely solved. m i i M m M m2 L L O L m 2m M mm x x' x2 x' 2 = M M xm x' m In these equations, an x' i is a oded paket reeived by the destination, while its orresponding ode vetor is i = ( i, i2,, im ), and x i -s are native pakets. The nodes in the sending luster may have reeived more than one paket from the previous luster, so that they an reate its own ombination. To generate a random linear ombination, a node ombines all the reeived pakets of the same blok with randomly seleted oeffiients. Figure 4: The luster-to-luster transmission model B. System Model Fig. 4 demonstrates our system model when a soure node transmits pakets to a destination. There are k lusters between the soure and the destination nodes. Cluster i onsists of n i ooperating nodes whih are relatively lose to eah other. Furthermore, the nodes in the same luster are assumed to be within the transmission range of eah other. The j th node in luster i is represented as node (i, j), where i {, 2,..., k} and j {, 2,..., n i }. Two nodes from neighboring lusters may not neessarily be onneted at all times. We define onnetivity as the ondition evaluated at a speifi time that a node in the reeiving luster an reeive a paket from a node in the sending luster when the sending node has been sheduled to transmit the paket at the speifi time. We define r ij, assoiated with the node (i, j), as the number of nodes in the luster i + that are onneted (i.e., an reeive the transmission) from the node (i, j). For example, in Fig. 4, r 2 = 3. Three nodes, the nodes (2, ), (2, 3), and (2, 5), an reeive transmission from the node (, 2). Moreover, r kj, j {, 2..., n k }, is an indiation whether the node (k, j) is onneted to the destination (the k th luster is the last luster). r kj an be either zero or one. Also, we denote r s as the number of nodes in luster whih are onneted to the soure node. In general, the parameters r ij and r s an hange with time based on the CC sheduler.

5 As disussed in Setion IV-A2, for any two nodes from adjaent lusters that are onneted, a transmission loss probability is defined. This probability depends on the quality of the wireless link [2], and we denote, p (i,j)(i+,q), i {, 2..., k - }, j {, 2..., n i }, q {, 2..., n i +}, as the loss probability of a transmission on the link between the node (i, j) and the node (i +, q). These probabilities for the links between the soure and the nodes in luster are defined as p s(,q), q {, 2..., n }. The orresponding probabilities of the links between the nodes in the last luster and the destination are denoted by p (k,q)d, q {, 2..., n k }. Table I lists the parameters used in the paper. n i k r ij r s p (x)(y) m TABLE I. PARAMATERS Number of nodes in a luster i Number of lusters between the soure and destination Number of nodes in luster i+ that are onneted to node (i,j) Number of nodes in luster that are onneted to the soure node The loss probability of a transmission over a link between node x and node y Number of native pakets in a blok V. ANALYSIS OF THE NUMBER OF NODES IN A CLUSTER Although a larger luster results in better performane (i.e., larger end-to-end suessful reeipt probability), it also leads to more transmissions and, thus, larger overhead. Thus, there is a tradeoff between the performane and the number of luster nodes. The goal is to maximize the deoding probability at the destination node, while maintaining the traffi below some level. To this end, we ompute the probability that all the m native pakets in the same blok an be deoded when the size of luster i is n i. In our analysis, we make the following Fundamental Assumption: a paket reeived by the destination and transmitted by the node in luster k is innovative with high probability; i.e., the first m pakets reeived by the destination are with high probability linearly independent. Therefore, we assume that as long as the destination has gathered m pakets, the m native pakets an be deoded. We will show in Setion VII that this assumption is, indeed, justified. The assumption is based on the observation in [22], where it has been shown that the probability that a oded paket is useful to another node is -(/2 q ), when eah relay node has abundant buffer to store oded pakets. Reall that 2 q is the size of the Galois Field. Typially, -(/2 q ) is very lose to for pratial values of q. The seletion of the value of q depends on the tradeoff between suffiient linear independene and other parameters suh as the overhead of the paket header, the buffer size, and the ease of implementation. Usually, q is hosen to be 8 (i.e., one byte is required to enode a oeffiient). Thus, the size of the Galois Field is 2 q = 2 8 = 256. We define V ij, where < i k, j n, as the probability that node (i,j) hears at least one oded paket transmitted from the nodes in the luster i-. Also, V j, j n i, is the probability that at least one oded paket sent from the soure node is reeived by a node (,j). We assume that the network onnetivity is uniformly distributed, meaning that the probability that a node in the luster i is onneted to the node ((i+),j) is equal for any j. Moreover, a node (,j) in luster an suessfully hear the paket from the soure with rs probability ( ps(, j) ). Also, the probability that node n (i,j) an suessfully reeive a paket from the node (i-,t) is r( i ) t given by: ( p( i, t )( i, j) ). Then, we obtain that n i m' rs ( ps( i, j) ) =, i ni V = ij n i V < ( i ) t r( i ) t ( p( i, t)( i, j) ), i k t = ni where we reall that m' stands for the number of oded pakets the soure atually transmits to luster. Therefore, V ij an be iteratively omputed from the above equation. Next, let P s denote the probability that at least m pakets are reeived by the destination. Using the above formula for V ij, we alulate the value of P s. Based on our Fundamental Assumption, P s also represents the probability that all m native pakets an be deoded at the destination. With the above analysis, we an ompute P s for a given number of nodes in eah luster. In addition, the expeted number of native pakets deoded in a blok of size m is given by mp s. VI. LINEAR DEPENDENCE OF CODED PACKETS The destination ould obtain a oded paket whih is transmitted from the node in luster k, and then try to ondut the deoding proess to retrieve the m native pakets as long as the m oded pakets have been gathered. In Setion V, we make the assumption in the analysis, that the first m oded pakets reeived by the destination an be deoded to retrieve the m native pakets. That is the first m oded pakets are linearly independent. Suh an assumption is based on the observation that with very high probability eah oded paket one reeived by the destination is innovative with respet to the pakets in the same blok. However, there may still exist linear dependene among the first m oded pakets, and the destination needs to keep reeiving more oded pakets until the rank of the deoding matrix reahes m. This depends on how the oded pakets are generated from the random linear ombinations by eah node when they are forwarded. When a node has more reeived pakets stored in its buffer, with large probability it generates a oded paket, whih is innovative to the node in the next luster. Therefore, if there are more nodes with more oded pakets, there is a higher probability that the destination obtains an innovative oded paket. Whether the destination ould obtain m linearly independent oded pakets by reeiving the least number of pakets, depends on several parameters, suh as n, r, and p. Generally, for larger n and r and for smaller p, the probability inreases that the destination an retrieve the m native pakets from the first m oded pakets. This is beause more pakets are suessfully transmitted to the next luster and hene ould be used to generate the linearly independent pakets. In Setion

6 VII, we will show how these parameters affet the linear dependene. In order to make the destination ollet m linearly independent pakets by reeiving fewest pakets, we modify CC into Enhaned CC (ECC) with an operation that heks the linear independene of the oded pakets. In ECC, one a node generates the random linear ombination from its stored pakets, it first heks whether the ombination is linearly independent from the oded pakets whih have been already transmitted by any other node in the same luster. If not, it would attempt to regenerate a new random linear ombination until an innovative oded paket is obtained. However, sine the node has limited number of pakets in its buffer or the previously transmitted oded pakets from the nodes in the same luster an span the whole linear spae, it may not be able to generate an innovative paket. Therefore, the regenerating proess would be onduted up to a ertain number of trials. In ECC, heking the linear independene ould be implemented in our luster model, beause all the nodes in the same luster are lose to eah other. Hene, eah node knows whih oded pakets have been already transmitted by the other nodes in the same luster. VII. PERFORMANCE EVALUATION In this setion, we report the results of our performane evaluation of the proposed CC protool and of the omparison of the CC protool with other shemes. These results were obtained by an extensive simulation. Furthermore, the simulation results validate our analysis in Setion V. In our simulations, we used 2 8 as the size of the Galois Field over whih network oding operations are performed. We onsider a network with k lusters and n nodes in eah luster. A homogeneous ase, where x, y : p (x)(y) = p, is onsidered. All the simulation results were obtained by averaging 0, 000 runs. In eah run, m = 0 native pakets in one blok are routed from the soure node to the destination node. We ompare the performane of the CC protool with two other ases, CNC and NCNC, as defined below. CNC (Cooperative with Non-Coding) is the sheme whih implements ooperative ommuniation, but without network oding. Therefore, in CNC, native pakets are diretly transmitted, and ooperation is ahieved by a node in a luster transmitting a paket, whih has not been already transmitted by any other node in the same luster. Sine all the nodes in the same luster are within the transmission range of one another, eah node knows whih native pakets have been already transmitted by the other nodes in the same luster. As in CC, eah node in a luster is allowed to forward one paket only. After all the native pakets in the same blok have already been transmitted by the other nodes, the next transmitting node randomly hooses from the buffered native pakets that have not been transmitted more than one. The NCNC (Non-Cooperative with Non-Coding) sheme forwards pakets without ooperation and without oding. Thus NCNC is the traditional routing protool, where pakets are transmitted only to the predetermined next hop relay node, without ooperation. For eah native paket in a blok, the soure hooses a relay node in eah luster and forwards the native paket along the hosen relay nodes. Therefore, as in CC and CNC, there are at most n transmissions of pakets of the same blok from eah luster. A. The effet of n Fig. 5 shows how throughput varies with different number of nodes in a luster. Throughput is defined as the total number of native pakets whih have been suessful reovered at the destination from among the original m = 0 native pakets. Fig. 5(b) shows results for p = 0., while Fig. 5(a) presents results for p = In both ases, the throughput inreases with n, sine as there are more nodes in the luster, it is more probably that a paket will reah the destination. Additionally, sine in CC at least 0 oded pakets are required for deoding, when the destination reeives less than 0 pakets, none of the pakets ould be reovered. Therefore, for n lose to m, suh as n = 0 the performane of the CC protool is worse than the other shemes. However, for n >, the throughput in CC is higher than both of the other shemes and quikly reahes values lose to 0, as in Fig. 5(a). This happens beause transmitting more native pakets in CNC only benefits the throughput if the same native paket is lost in a prior transmission. Otherwise, the transmission is a waste. On the other hand, with oding and with more transmissions, the CC sheme an reover the loss with very high probability. In ontrast, NCNC whih does not exploit ooperation and oding obtains the lowest throughput. Similar behavior is also observed in Fig. 5(b). Fig. 6 presents P s, versus n. P s defines the probability that all the 0 native pakets an be deoded at the destination. The figure demonstrates that the smaller is the value of p or the larger is the value of n, the larger is P s. The CC sheme signifiantly outperforms the CNC and the NCNC shemes in terms of P s. Espeially for p = 0.05 and n 3, P s of the CC sheme relatively quikly and losely approahes the value of.0. These results suggest that the proposed CC protool an be appliable for error-sensitive appliations. One of the major objetives of the CC sheme is to redue the number of transmissions. Thus, in Fig. 7 we investigate the total number of transmissions in the network. The figure shows the number of transmissions whih are required to ahieve P s = 0.8 for the different shemes. In this figure, p is set to It is apparent from the results that the CC sheme needs the least number of transmissions. In other words, to ahieve the same P s, the CC protool uses the smallest bandwidth among the three shemes. B. The effet of k We now further ompare the performane of the shemes for different length (number of hops) of the route; i.e., different number of lusters between the soure and the destination nodes. Fig. 8 and Fig. 9 illustrate how the throughput and P s vary as k inreases, respetively. For the results in these figures, eah luster onsists of n = 3 nodes. The figures show that the CC sheme outperforms the CNC sheme due to the network oding operation, as it has been previously disussed. Besides, the performane of the CC and the CNC shemes is relatively onstant as a funtion of k. The reason is that the nodes in the same luster help eah other to relay the pakets by ooperation, making the luster-to-luster transmission more reliable. Therefore, despite longer route and more hops to the destination, information an still be

7 forwarded reliably between lusters. However, due to lak of ooperation, the NCNC sheme is more prone to paket loses in eah hop. Thus for longer routes, eah hop has more signifiant effet on the overall end-to-end performane. when r (inluding r ij and r s ) is set 8. The analytial urve represents the numerial values omputed from Setion V. The simulation urve is obtained from the average of 0,000 independent runs. We observe that for different values of p, the analytial and the simulation results math nearly perfetly. Furthermore, the math of the simulation and the analytial results onfirms the validity of our Fundamental Assumption. Finally, as already pointed out previously, lower p and larger n leads to larger P s. Figure 7: Comparison of total number of transmissions of the different shemes Figure 5: Comparison of throughput as a funtion of n Figure 8: Comparison of throughput as a funtion of k Figure 6: Comparison of P s as a funtion of n C. Validation of the analysis Next, we validate the auray of our analysis of P s. In Fig. 0, we plot P s of the CC protool as a funtion of n and p Figure 9: Comparison of P s as a funtion of k

8 One of the main motivations for reduing the number of nodes in a luster is to ontrol the traffi. In Fig. 0, we an see that when n reahes a partiular threshold, the improvement in P s is only marginal. For example, for p = 0. and n = 4, the value of P s is very lose to.0. Thus, there is only a limited benefit in inreasing n above the value of 4. In other words, the benefit of inreasing the size of the luster above this threshold is not worth the inreased number of transmissions that would result from the larger lusters. Figure 0: P s vs. n for different p in CC From the system design view, sine more nodes in a luster auses more traffi in the network, the lusters should be made as small as possible, given the level of required network performane, suh as reliability or throughput. A good rule-of-thumb for p of 0. would be to keep the luster size at about 5 nodes, whih would result in the value of P s lose to.0. VIII. CONCLUSTION In this paper, we introdued the luster-based Cooperative Communiation sheme, whih integrates ooperative ommuniation with network oding. The basi idea behind the sheme is to exploit the ooperation to improve ommuniation reliability and to leverage network oding to redue the number of paket transmissions. We analyzed the probability of suessful reeption of transmitted pakets, and we showed how to optimize the number of nodes in the luster, as to trade off between performane and overhead. We also derived a general rule-ofthumb that the size of the luster should be kept at around 5 nodes. We ompared the performane of the proposed sheme with shemes that do not inorporate ooperation or whih do not inorporate network oding and we onlude that our sheme exhibit superior performane relative to the other simulated shemes. IX. ACKNOWLEDGEMENTS This work was supported by the NSF grant CNS and by the AFOSR ontrat number FA /Z X. REFERENCES [] A. Nosratinia, T.E. Hunter, and A. Hedayat, Cooperative ommuniation in wireless Networks, IEEE Commun. Mag., vol. 42, pp , Ot [2] R. U. Nabar, H. Bolskei, and F. W. Kneubuhler, Fading relay hannels: Performane limits and spae-time signal design, IEEE J. Sel. Areas Commun., vol. 22, no. 6, pp , June [3] J. N. Laneman, D. N. C. Tse, and G. W. Wornell, Cooperative diversity in wireless networks: effiient protools and outage behavior, IEEE Trans. Inf. Theory, vol. 50, no. 2, pp , De [4] R. Ahlswede, N. Cai, S.-Y. R. Li, and R. W. Yeung, Network information flow, IEEE Trans. Inf. Theory, vol. 46, no. 4, pp , July [5] A. Sendonaris, E. Erkip, and B. Aazhang, User ooperation diversity - part I: system desription, IEEE Trans. Commun., vol. 5, no., pp , Nov [6] M. Gastpar and M. Vetterli, On the apaity of large Gaussian relay networks, IEEE Trans. Inf. Theory, vol. 5, no. 3, pp , Mar [7] S. Cui, A. J. Goldsmith, and A. Bahai, Energy-effiieny of MIMO and ooperative MIMO tehniques in sensor networks, IEEE J. Sel. Areas Commun., vol. 22, no. 6, pp , Aug [8] L. S. Pillutla and V. Krishnamurthy, Joint rate and luster optimization in ooperative MIMO sensor networks, in IEEE 6th Workshop on Signal Proessing Advanes in Wireless Communiations, Philadelphia, PA, USA, Mar. 2005, pp [9] M. Elhawary and Z. J. Haas, Energy effiient protool for ooperative transmission, submit to publiation. [0] A. del Coso, U. Spagnolini, and C. Ibars, Cooperative distributed MIMO hannels in wireless sensor networks, IEEE J. Sel. Areas Commun., vol. 25, no. 2, pp , Feb [] S. Biswas and R. Morris, ExOR: opportunisti multi-hop routing for wireless networks, in Pro. of ACM SIGCOMM, Philadelphia, PA, USA, Aug. 2005, pp [2] S. Chahulski, M. Jennings, S. Katti, and D. Katabi, Trading struture for randomness in wireless opportunisti routing, in Pro. of ACM SIGCOMM, Kyoto, Japan, Aug. 2007, pp [3] R. Koetter and M. Medard, An algebrai approah to network oding, IEEE/ACM Trans. Netw., vol., no. 5, pp , Ot [4] S.-Y. R. Li, R. W. Yeung, and N. Cai, Linear network oding, IEEE Trans. Inf. Theory, vol. 49, no. 2, pp , Feb [5] D. S. Lun, N. Ratnakar, R. Koetter, M. Medard, E. Ahmed, and H. Lee, Ahieving minimum-ost multiast: a deentralized approah based on network oding, in Pro. of IEEE INFOCOM, Miami, FL, USA, Mar. 2005, pp [6] C. Fragouli, J. Widmer, and J.-Y. L. Boude, A network oding approah to energy effiient broadasting: from theory to pratie, in Pro. of IEEE INFOCOM, Barelona, Spain, Apr. 2006, pp.. [7] S. Katti, H. Rahul, W. Hu, D. Katabi, M. Medard, and J. Crowroft, XORs in the air: pratial wireless network oding, in Pro. of ACM SIGCOMM, Pisa, Italy, Sept. 2006, pp [8] S. Sengupta, S. Rayanhu, and S. Banerjee, An analysis of wireless network oding for uniast sessions: the ase for oding-aware routing, in Pro. of IEEE INFOCOM, Anhorage, AL, USA, May 2007, pp [9] C. E. Perkins and E. M. Royer, Ad ho On-Demand distane vetor routing, in the 2nd IEEE Workshop on Mobile Computing Systems and Appliations, New Orleans, LA, USA, Feb. 999, pp [20] D. B. Johnson and D. A. Maltz, Dynami soure routing in Ad Ho wireless networks, in Mobile Computing, edited by Tomasz Imielinski and Hank Korth, hapter 5, Kluwer Aademi Publishers, 996, pp [2] D. Aguayo, J. Biket, S. Biswas, G. Judd, and R. Morris, Link-level measurements from an 802.b mesh networkg, in Pro. of ACM SIGCOMM, New York, NY, USA, 2004, pp [22] S. Deb, M. Medard, and C. Choute, Algebrai gossip: a networkoding approah to optimal multiple rumor mongering, IEEE Trans. Inf. Theory, vol. 52, no. 6, pp , June 2006.