Effects of MAC Approaches on Non-Monotonic Saturation with COPE - A Simple Case Study

Ef fects of MA C Approaches on Non-Monotonic Saturation with COPE - A Simple Case Study Jason Cloud* † , Linda Zeger † , Muriel Médard* *Research Laboratory of Electronics, Massachusetts Institute of T echnology , Cambridge, MA. † MIT Lincoln Laboratory , Lexington, MA. Email: {jcloud@, zeger@ll., medard@}mit.edu Abstract — W e construct a simple network model to provide insight into network design strategies. W e show that the model can be used to address various approaches to network coding, MA C, and multi-packet reception so that their effects on network throughput can be evaluated. W e consider sev eral topology components which exhibit the same non-monotonic saturation behavior found within the Katti et. al. COPE experiments. We further show that fairness allocation by the MA C can seriously impact perf ormance and cause this non-monotonic saturation. Using our model, we develop a MA C that provides monotonic saturation, higher saturation throughput gains and fairness among flows rather than nodes . The proposed model pro vides an estimate of the achievable gains for the cross-layer design of network coding, multi-packet reception, and MA C showing that super -additive throughput gains on the order of six times that of routing are possible. I . I N T RO D U C T I O N W ith the multitude of network technologies a v ailable that increase performance, it is dif ficult to efficiently integrate them into a coherent network. W e develop a simple model that provides insight into cross-layer network design, and illustrate its use through examples in volving the combination of the 802.11 medium access control (MAC), network coding (NC), and multi-packet reception (MPR). Our model not only provides strategies to integrate various technologies, but also predicts the achiev able gains possible. The dev elopment of the opportunistic inter-session NC scheme, COPE, by Katti et. al. [1] led to various models and analyses that attempted to explain COPE’ s experimental results. Le et. al. [3] and Sengupta et. al. [4] developed models to describe these results, but only considered coding a maximum of two packets together at a time or did not address the interaction between NC and MA C fairness. As a result, their models provide throughput gains that are considerably smaller than the experimental results and do not explain the non-monotonic beha vior sho wn in the upper half of Fig. 1. This work is sponsored by the Department of Defense under Air Force Contract F A8721-05-C-0002. Opinions, interpretations, recommendations, and conclusions are those of the authors and are not necessarily endorsed by the United States Government. Specifically , this work was supported by Information Systems of ASD(R&E). Contributions of the Irwin Mark Jacobs and Joan Klein Jacobs Presidential Fellowship have also been critical to the success of this project. Figure 1. Comparison of the empirical COPE performance data collected from a 20-node 802.11 wireless ad hoc network test bed (top), [1], and the resulting throughput using a model of the 802.11 MAC and 5-node cross topology component proposed by [2] (bottom). W e start with these elements to develop a model that can be used for cross-layer design of various network technologies. The NC+MPR+MAC, NC+MA C, and Routing+MAC curves are deriv ed from our model and provide an estimate of the achie vable performance gains. Zhao and Médard [2] sho wed that the fairness imposed by the 802.11 MA C helps to explain this non-monotonic behavior . In addition, they demonstrated that the majority of the throughput gain achieved from COPE is a result of coding three or more unencoded, or native, packets together at time. Their analysis showed that these gains are not reflected in three node network models and that at least fiv e nodes are required to accurately capture the ef fects of COPE. Fig. 1 shows that the 802.11 model and 5-node cross topology component from [2] is consistent with the results from [1]. Furthermore, Seferoglu et. al. [5] has used similar 5-node topology components, and variants of them, to analyze TCP performance over coded wireless networks. W ith this in mind, we dev elop our model using the basic 5-node cross components from [2] and [5] and other possible combinations of these 5-node topology components in order to help in our understanding of the effects of combining NC and various MAC implementations in larger networks. Finally , we demonstrate the capability of the model to predict the gains and highlight design challenges from incorpo- rating additional technologies, such as MPR, into the network. MPR is kno wn to enhance wireless network performance and has been extensi vely researched with unencoded traffic [6, 7, 8, 9]; but little on the joint use of MPR, NC, and MA C design exists. Garcia-Luna-Aceves et. al. [10] compar ed the use of NC to MPR, b ut did not consider their combined use; and Rezaee et. al. [11] provided an analysis of the combined use of NC and MPR in a fully connected network, but did not consider the effects of bottlenecks or multi-hop traffic. W e show that our model provides an intuitive method for determining an estimate of the achiev able gains from the combined use of MPR and NC in a congested, multi-hop network. The remainder of the paper is organized as follo ws: Section II provides a detailed description of the network model; Section III provides an example of NC and MPR for 5-node network topology components using the e xisting 802.11 MA C; Section IV demonstrates the flexibility of using the model when considering the design of various network elements; and we conclude in Section V. I I . N E T W O R K M O D E L S A N D PA R A M E TE R S Our main goal is to dev elop a simple model that gives insight into cross-layer design of wireless networks by using NC, v arious MAC approaches, and MPR as examples. T o this end, we identify the fundamental behavior of each aspect of the network and model each element using simple, intuiti ve methods so that we can e valuate the potential throughput gains. The performance of NC is modeled by considering the ability of a given node to combine multiple packets together as well as the primary implementation details of the particular NC scheme. W e use COPE [1] as a case study . COPE uses the broadcast nature of the wireless channel to opportunistically code packets from different nodes together using a simple XOR operation. Any node that receives an encoded message is able to decode it using the unencoded, or nati ve, packets captured from the wireless channel. W e model COPE such that packet transmissions are ne ver delayed. If a node does not hav e more than one packet to encode, it does not wait for another packet to arri ve. Rather, it sends the packet unencoded at the first opportunity . In addition, all packets headed towards the same next-hop will not be encoded together because the next- hop would not be able to decode these coded packets due to the lack of enough degrees of freedom. W e do not consider the complexity of the coding or decoding operations nor any other aspects of the NC implementation since their contributions to the overall network performance is small in relation to the specific implementation aspects mentioned above. The MA C is modeled by identifying its primary behavior in the network and simplifications are made by assuming optimal performance from its secondary and tertiary behaviors. This giv es us an intuiti ve approach in determining the potential throughput while ensuring that we understand the fundamental characteristics of the network. For instance, the 802.11 MAC employs a fairness mechanism that distributes channel re- sources equally among all competing nodes within a network. As we show in the follo wing sections, this is the primary cause of the non-monotonic saturation behavior in the experimental throughput sho wn in the upper half of Fig. 1. Since the fairness mechanism is the major contributor to ov erall network performance, we assume optimal performance from all other aspects of the MA C. For e xample, the non-monotonic behavior is a result of both collisions and fairness; but the total effects of collisions from either hidden nodes or identical back-of f times on throughput are small in relation to the effects of the 802.11 MA C fairness mechanisms. Furthermore, we do not consider the additional ef fects on ov erall throughput associated with the virtual 802.11 CS mechanisms (R TS/CTS) or other aspects of the MAC such as the potential of lost channel resources due to the MA C’ s random back-off. Instead, we assume optimal performance from each of these secondary and tertiary behaviors. These assumptions, as a result, provide upper bounds to the achiev able throughput in the various networks that employ the MAC. W e also sho w ho w additional techniques to increase netw ork performance, such as MPR, can be similarly modeled. MPR can be implemented in a variety of methods from Code Division Multiple Access (CDMA) or multiple-input-multiple- output (MIMO) to orthogonal frequency division multiple access (OFDMA). In subsequent sections, we model MPR by allowing each node to successfully receive m multiple packets at the same time. Finally , our model uses sev eral basic canonical topology components that contain only fiv e nodes where each node is both a source and a sink. T wo of the possible components are shown in Fig. 2. These components are of interest because they form the primary structures in larger networks that create bottlenecks and congestion. By looking at the traf fic that trav els through the center node, these components help us model the performance gains of multi-hop traf fic under both low and high loads. While all possible combinations of these basic canonical topology components should be ev aluated, we only focus on two of the components, shown in Fig. 2, since the analysis of the others are redundant and provide little to the clarification of our approach. Each component has specific constraints due to their struc- ture and will effect the performance of the MA C, NC, and MPR in different ways. The center node n 5 in each component is fully connected regardless of the topology , and traffic flows originating from the center require only a single hop to reach their destination. Within the cross topology component, each traffic flow originating from a given node is terminated at the node directly opposite the center; and in the “X” topology component, all flows originating from a node in a giv en set terminates at a node in the opposite set. Therefore, each flow must pass through the center regardless of topology . For (a) Cross Comp onen t (b) ”X” Comp onen t Overhear/Listen Primary Communication n 1 n 2 n 3 n 4 n 5 n 1 n 2 n 3 n 4 n 5 Figure 2. T wo of the basic network structures responsible for traffic bottlenecks and congestion in larger networks. All nodes are sources and all flows originating from n j , j ∈ [1 , 4] cross at n 5 . example, nodes n 1 , n 2 , and n 5 in the “X” topology component are fully connected and nodes n 3 , n 4 , and n 5 are also fully connected; b ut n 1 and n 2 are not connected to n 3 and n 4 . All traf fic between an y node { n 1 , n 2 } ∈ X 1 and any node { n 3 , n 4 } ∈ X 2 must trav el through the center . In an effort to simplify the following explanation, we make se veral additional simplifications which can be easily incorporated into the model. W e assume feedback is perfect, the load required for ackno wledgments are contained as part of the initial transmission’ s load, and the wireless channel is loss- less. W e also consider the additional constraint that each node is half-duplex, i.e. a node will ov erhear any transmission from its neighbors only if it is not transmitting. When considering the model for MPR, we allo w m packets to be sent from different sources in a single time slot with the constraint that we try to maximize the number of neighbors a node can ov erhear within any giv en time slot. In essence, we duplicate CSMA/CA for each m = 2 case in the sense that a node will transmit only if none of its neighbors are transmitting. For cases where m ≥ 3 , we pick the combination of transmitting nodes such that the average number of transmissions receiv ed by any given node within the network is maximized. Sections III and IV provide both an analysis of the max- imum achiev able throughput and simulations ov er various values of the total of fered load P to the network. The load P to the network from the set of source nodes i ∈ N , is defined as P = P i ∈N ρ i , where N is the set of nodes in the topology component and ρ i = k i / 100 is node i ’ s indi vidual load contribution, or the fraction of time required to send all of its k i packets to the next-hop. W e stochastically determine k i using a binomial distribution giv en P with parameters n i = K − P i − 1 j =1 k j and p i = 1 / ( N − i +1) , N = |N | , in each iteration of our simulation and average these results for each total offered load e v aluated. W e then use the model described in this section to determine the total network throughput S which is equiv alent to the total number of packets di vided by number of time slots needed to send e very packet to its intended destination. I I I . M U LT I - P AC K E T R E C E P T I O N A N D N E T W O R K C O D I N G P E R F O R M A N C E A N A L Y S I S W ith each of the network topology components shown in Fig. 2, we provide an example of the performance analysis with and without the use of NC and MPR when the 802.11 MA C is used. W e also consider both unicast and broadcast traffic where a unicast transmission is complete when all packets from each source successfully reaches their destination and a broadcast transmission is complete when all nodes hav e receiv ed every packet from all sources. A. Cr oss T opology Component Analysis Each node i ∈ [1 , 5] , requires ρ i of the time to send all of its packets one hop. The resulting total of fered load to the network is then P = P 5 i =1 ρ i . W e define the total network component load P T = ρ R + ρ M as the time required to send all packets through the topology component where ρ R is the required load to relay packets and ρ M is the required load to send each nati ve packet one hop. The load required to relay packets through the center is ρ R = 1 c P 4 j =1 ρ j where c is the number of packets that can be encoded together which is dependent on the number of neighbors a giv en node can ov erhear . In the case of the cross topology component and enough packets to code together , c = 4 for m = { 1 , 2 } and c = 2 for all m ≥ 3 . The load needed to send each node’ s unencoded packets one hop is defined as ρ M ≥ 1 / m P 4 j =1 ρ j + ρ 5 . Because we setup the model so that we maximize the number of neighbors any giv en node can overhear when using MPR, this expression is met with equality if each ρ j , j ∈ [1 , 4] , is equal. Otherwise, the load ρ M is lo wer bounded by this equation and is a function that is dependent on both the topology component’ s configuration and each node’ s load imbalance. Finally , let the fraction of allocated time slots a node receiv es as a result of the MA C be s i . The throughput S with unicast and broadcast traffic is shown as a function of P in Fig. 3. Each curve is obtained through simulation and is an average ov er the load distribution discussed in Section II. Each star is obtained by analysis and depicts the maximum achie vable throughput when the MPR and/or NC gain is maximized. When P T < 1 , each node is allocated enough time slots to send all of its packets, and the allocated load is s j = ρ j for j ∈ [1 , 4] and s 5 = ρ 5 + ρ R . The throughput S increases linearly as the network load increases, regardless of the use of MPR or NC, and reaches a maximum for each case when P T = 1 . The throughput then transitions into a saturated region for P T > 1 , where the allocated load for each node is s j ≤ ρ j and s 5 ≤ ρ 5 + ρ R . When NC is not used, the throughput is S = s 5 ; and when NC is used, the throughput will be a function of the number of packets that can be encoded together . 1) Routing (No Network Coding, m = 1 ): W e use routing as the performance baseline. Consistent with the results found in [1] and the analysis performed in [2], the throughput increases linearly within the non-saturated re gion, P ∈ [0 , 5 / 9 ) . At P = 5 / 9 , the throughput reaches a maximum of 5 / 9 (depicted by a star in Fig. 3). This occurs when each sources’ load reaches ρ i = 1 / 9 for i ∈ [1 , 5] . The total load of the center node, as a consequence, is ρ 5 + ρ R where ρ R = P 4 i =1 ρ j = 4 / 9 . Since P T = 1 , s j = ρ j and s 5 = ρ 5 + ρ R . The throughput saturates for P > 5 / 9 . Initially , the 802.11 MA C allocates time slots to nodes requiring more resources. 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Total Offered Load (P) Throughput (S) Routing NC MPR = 2 MPR = 4 NC + MPR = 2 NC + MPR = 4 MPR (m=2) Maximum NC+MPR (m=2) Maximum and NC+MPR (m=4) Maximum MPR (m=4) Maximum and NC Maximum Routing Maximum Figure 3. A verage unicast and broadcast throughput for a 5-node cross topology component. Each vertical double arrow shows the difference in the maximum and saturated throughput due to MAC fairness for each case. The throughput is therefore the amount of time n 5 is able to transmit, s 5 = 1 − P 4 i =1 s i , which decreases as P increases. The network component completely saturates when each node requires a large fraction of the av ailable time slots but the MA C restricts each node’ s access to the channel by ensuring fairness among each competing node, i.e. s i = 1 / 5 for i ∈ [1 , 5] . For large enough P , the throughput saturates to the total amount of information that n 5 can transmit, i.e., S = s 5 = 1 / 5 . 2) Network Coding Only ( m = 1 ): W e now allo w NC to be used by the center node. Each node transmits one at a time, allowing each node to recei ve four nati ve packets or four degrees of freedom (three degrees of freedom through the use of opportunistic listening plus one degree of freedom from the packet originating at the node). After each edge node has completed transmission, node n 5 transmits a single encoded packet, which is sufficient for each edge node to obtain the single degree of freedom it still requires to complete both the unicast and broadcast sessions. From Fig. 3, when P ∈ [0 , 5 / 9 ) , NC is seen to provide no additional gains ov er the use of routing alone since n 5 can forward each packet receiv ed without the MA C limiting its channel use. For P ∈ [ 5 / 9 , 5 / 6 ) , NC is instrumental in achieving the throughput shown. The MA C does not limit channel resources until the maximum throughput of S = 5 / 6 is reached when P T = P 5 i =1 ρ i + 1 4 P 4 j =1 ρ j = 1 where ρ i = 1 / 6 for i ∈ [1 , 5] . At this maximum, the MA C ensures fairness among all competing nodes and the throughput saturates. As P increases, the gain provided by NC diminishes. The number of packets reaching n 5 from each edge node is limited by the MA C while packets introduced into the network component by n 5 are not. The coding gain, therefore, approaches zero as P → ∞ . 3) Multi-P ac ket Reception of Or der 2 and 4 (No Network Coding and m = { 2 , 4 } ): MPR is similar to the routing case described earlier except we now allow a maximum of m edge nodes to transmit within a gi ven time slot. For m = 2 , the total time used by all of the edge nodes to transmit their packets to n 5 is 1 / 2 that needed by routing while the center node cannot transmit multiple packets simultaneously and must transmit each recei ved packet indi vidually . Using CSMA, which re- stricts nodes opposite each other to transmit at the same time, the point at which the protocol saturates for symmetric source loads occurs when P T = P 4 j =1 ρ j + 1 / 2 P 4 j =1 ρ j + ρ 5 = 1 where ρ i = 1 / 7 for i ∈ [1 , 5] . This yields the maximum throughput of S = 5 / 7 . The throughput saturates to the same throughput as routing for values of P T > 1 and the gain for m = 2 in the saturated regime is 1 due to the suboptimal saturation behavior of the protocol. The beha vior for m = 4 is the same as that for m = 2 except the maximum of S = 5 / 6 occurs when P T = P 4 j =1 ρ j + 1 / 4 P 4 j =1 ρ j + ρ 5 = 1 where ρ i = 1 / 6 . W e allow all edge nodes to transmit their packets to n 5 simultaneously , requiring a total of 1 / 6 of the time-slots. Node n 5 then sends each node’ s packet individually , including its own, to the intended recipient requiring the remainder of the time slots to finish each unicast/broadcast transmission. As P increases, the MA C limits each node’ s number of av ailable time slots and S saturates to 1 / 5 . Again, the gain in the saturated region for m = 4 is equal to the cases of m = 2 and routing. 4) Network Coding with Multi-P ack et Reception of Or der 2 and 4 ( m = 2 , 4 ): For m = 2 , the maximum throughput of S = 5 / 4 occurs when P T = 1 / 4 P 4 j =1 ρ j + 1 / 2 P 4 j =1 ρ j + ρ 5 = 1 where ρ i = 1 / 4 for i ∈ [1 , 5] . Each set of nodes, { n 1 , n 3 } and { n 2 , n 4 } , uses 1 / 4 of the total number of time slots to transmit to n 5 which then transmits a single encoded packet deriv ed from all four node’ s native packets in addition to its own nati ve packet. For P T > 1 , the throughput saturates to the saturated NC throughput due to the 802.11 MA C. While the maximum achiev able throughput is 25 / 16 times the NC without MPR throughput, the saturated gain for m = 2 is equal to the gain found when NC was used alone in this region. The throughput using NC and m = 4 for unicast traffic is equiv alent to NC and m = 2 . All four edge nodes transmit to n 5 which then transmits two encoded packets in addition to its own; or we limit the number of simultaneous transmissions to two thus allowing n 5 to code e verything together and send a single encoded packet to all of the edge nodes. Either strategy will achie ve the same gain although the dif ference occurs when considering either unicast (former option) or broadcast (later option). The maximum throughput for broadcast traffic using the first method is S = 1 , and S = 5 / 4 for the second which is consistent with the maximum unicast throughput. This dif ference indicates that increasing m when using NC may not be the optimal strategy . Although we do not show it here, the canonical topology components can be easily modified to include any number of nodes which would allow us to further look into the optimal strate gy for broadcast traf fic. B. “X” T opology Component The cross topology component gives insight into the performance of COPE and MPR in a dense network, and it represents the best case scenario when COPE is used since it maximizes the number of transmissions any giv en node receiv es. In order to understand the beha vior of COPE and MPR in sparser networks, we limit the number of each node’ s neighbors by analyzing the behavior of COPE and MPR in the “X” topology component shown Fig. 2(b). Fig. 4 sho ws both 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Total Offered Load (P) Throughput (S) Routing NC MPR = 2 MPR = 4 NC + MPR = 2 NC + MPR = 4 (Broadcast) NC + MPR = 4 (Unicast) MPR (m=2) Maximum and NC Maximum NC+MPR (m=4) Unicast Maximum Routing Maximum NC+MPR (m=2) Maximum and NC+MPR (m=4) Broadcast Maximum MPR (m=4) Maximum Figure 4. A verage broadcast and unicast throughput for a 5-node “X” topology component. Each vertical double arrow shows the difference in the maximum and saturated throughput due to MAC fairness for each case. the maximum and average throughput resulting from the use of the “X” topology component. W ithin this section, we focus only on the cases inv olving NC since it can be easily verified that the routing and m = { 2 , 4 } analysis for this topology component are the same as the cross topology component analysis. 1) Network Coding Only ( m = 1 ): Limiting the ability to overhear other edge nodes in the component results in the reduction in the number of possible packets that can be encoded together . Packets from different nodes within the same set, i.e. { n 1 , n 2 } ∈ X 1 and { n 3 , n 4 } ∈ X 2 , cannot be encoded together because all flo ws transitioning between X 1 and X 2 are ef fectiv ely headed towards the same ne xt-hop. This forces n 5 to code only a subset of packets together which increases the number of transmissions the center node must make. For example, the center node must make a minimum of two transmissions for e very four packets it receiv es from different edge nodes in order to ensure that each node has the necessary degrees of freedom to decode all of the packets. Like the cross component’ s throughput, the throughput of the “X” topology component increases linearly until it reaches its maximum at S = 5 / 7 . Assuming symmetric source loads, this maximum occurs when P T = 1 / 2 P 4 j =1 ρ j + P 5 j =1 ρ j = 1 where ρ i = 1 / 7 for i ∈ [1 , 5] . The throughput saturates for P T > 1 and the non-monotonic behavior in the saturated throughput regime is again due to the fairness aspect of the 802.11 MAC. By comparing the results obtained from both the cross and “X” topology components, it is evident that the performance of COPE is highly dependent on the network structure. As the network becomes sparser , the gain from COPE is diminished. 2) Network Coding with Multi-P ack et Reception of Or der 2 and 4 ( m = { 2 , 4 } ): For m = 2 , the throughput in- creases linearly until it reaches its maximum at S = 1 when P T = 1 / 2 P 4 j =1 ρ j + 1 / 2 P 4 j =1 ρ j + ρ 5 = 1 where ρ i = 1 / 5 for i ∈ [1 , 5] . The throughput then saturates to the NC throughput for P T > 1 . F or m = 4 , the maximum unicast throughput is S = 5 / 4 and is achiev ed when P T = 1 / 2 P 4 j =1 ρ j + 1 / 4 P 4 j =1 ρ j + ρ 5 = 1 where ρ i = 1 / 4 for i ∈ [1 , 5] . The center codes a maximum of two packets together from different edge node sets and transmits two encoded packets back to the set of 1 2 3 4 0.5 0.65 0.8 0.95 1.1 1.25 1.4 MPR Coefficient (m) Maximum Throughput NC, Broadcast Only NC, Unicast Only NC, Broadcast and Unicast NC Only + MPR Only No NC, Broadcast and Unicast Super Additive Gain From MPR and NC Figure 5. Maximum throughput of the 5-node “X” topology component as function of the MPR capability . Super-additiv e gains are achieved when using NC in conjunction with MPR. This is shown by comparing the throughput obtained using both NC and MPR with the throughput that would be obtained if the gain from using NC alone is added with the gain obtained from using MPR alone (NC Only + MPR Only). edge nodes in addition to its o wn nativ e pack et. This gi ves each edge node enough degrees of freedom to complete all unicast transmissions. When considering broadcast traffic, each node still requires a maximum of one additional degree of freedom. Allowing n 5 to code all of the edge node’ s native packets together and send one encoded transmission enables each node to extract the required degree of freedom and obtain the full set of messages from each source. The maximum throughput is therefore the same as the case for NC with m = 2 and is equal to S = 1 . Similar to the cross topology component, the av erage throughput for both cases discussed in this section does not reach the maxima found because of the stochastic load distribution, which results in asymmetric traffic flows across the center node that limits the ef fectiv eness of both COPE and the implementation of MPR that we chose. If each node has an equal amount of information to send, the maxima found in this section would be achie ved. The use of these components within our model allows us to determine the fundamental behavior of combining COPE and MPR in a larger network. For example, Fig. 5 shows a summary of our analysis by plotting the maximum unicast and broadcast throughput as a function of the MPR capability . It shows the super-additi ve throughput beha vior when MPR is used in conjunction with NC by comparing this throughput with the throughput that would be obtained by adding the individual gains obtained using MPR and NC separately . The use of the “X” topology component allows us to determine behavior that was otherwise masked in the cross component. It provided insight into the behavior of NC and MPR in sparser networks, methods in which to implement variants of COPE for broadcast traffic, and highlighted the super-additi ve gains from combining the two communication technologies. W e can also look into variants of the cross and “X” components; b ut the y are not discussed since they provide little clarification to the presentation of the model. In this example, howe ver , these variants provide insight into the robustness of MPR and COPE’ s throughput gains to topology changes. I V . I M P R OV I N G T H E M A C F A I R N E S S P RO TO C O L The use of our model highlighted a major drawback to the use of the 802.11 MA C in multi-hop networks. The non- monotonic throughput behavior that is e vident in Fig. 3 and 4 significantly reduces throughput as the offered load to the network increases. W e no w show how the model can be used to redesign the MA C so that we eliminate this sub-optimal behavior in the presence of NC, MPR, and their combination. Furthermore, we use the model to dev elop a MA C approach that provides fairness to flows rather than to nodes . It is obvious from Section III that if we want to eliminate the non-monotonic saturation throughput behavior , we want to allocate more channel resources to the center node than the edge nodes. W e choose to allocate resources proportional to the amount of non-self-generated traf fic flowing through each node when the network saturates. While allocating fewer re- sources to flo ws originating at the center and more resources to flows originated at edge nodes yields even higher throughput, this approach ensures that each flo w of information is gi ven the same priority . The center node will be allocated more resources than each edge node in order to relay information; but it must also limit the amount of self-generated traffic so that it equals the av erage per -node non-self-generated traffic being relayed. W e design the revised MA C using a slight modification of the components found in Fig. 2. For the cross topology component, we let there be N − 1 edge nodes and a single center , or relay , node. All edge nodes are connected with the center node and connected with all other edge nodes except the one directly opposite the center . For the “X” topology component, we also let there be N − 1 edge nodes and a single center node. The edge nodes are split into two sets X 1 and X 2 . All edge nodes within a given set are fully connected and also connected to the center node. Within the cross topology component, each node communicates with the node directly opposite the center . In the “X” topology component, each node communicates with a node in a different set. The allocated number of time slots each node receives so that the throughput S is maximized, subject to the flo w constraints and P N − 1 j =1 s j / m + s R = 1 , is di vided into three cases where s j is the fraction of time slots allocated to each edge node and s R is the fraction of time slots allocated to the center node. Similar to Section III, the throughput S = s R when NC is not used. When NC is used, S is a function of the number of packets that can be coded together , which is dependent on the density of the network, the MPR coefficient m , the use of CSMA, and the traffic type (unicast or broadcast). In order to simplify the explanation within this paper , we limit the examples we explore by considering only values of m = { 1 , 2 , 4 } and symmetric source loads. In the case of the “X” topology component, we also restrict our example to situations where the cardinality of each set X 1 and X 2 are equal. The cases include: Cross T opology Component with Unicast Traf fic or Broad- cast Traf fic: The cross topology component can be used to design part of the MAC for operation in dense networks. W ithout NC, the center node requires a number of time slots equal to the number of source nodes N . With NC, throughput is maximized by ensuring the center node codes the maximum number of nativ e packets together . Implementing MPR for m = 4 can potentially prevent each node from immediately decoding any coded message sent by the center . This is due to the potential of a giv en node transmitting at the same time as one of its neighbors. Generalizing for N and m as well as considering only integer numbers of time slots: s j = ( 1 d ( N − 1) /m e + N without NC 1 d ( N − 1) /m e + m with NC (1) and s R = ( N d ( N − 1) /m e + N without NC m d ( N − 1) /m e + m with NC (2) “X” T opology Component: Using the “X” topology com- ponent helps gain insight into the design of the MAC for operation in sparser networks. From the 802.11 e xample in Section III-B, we determined that the throughput dif fers for both unicast and broadcast traf fic. As a result, we define the fraction of time slots s U allocated to each node for unicast traffic as: s j = s U j = ( 1 d ( N − 1) /m e + N without NC 1 d ( N − 1) /m e +max( | X 1 | , | X 2 | )+1 with NC (3) and s R = s U R = ( N d ( N − 1) /m e + N without NC max( | X 1 | , | X 2 | )+1 d ( N − 1) /m e +max( | X 1 | , | X 2 | )+1 with NC (4) When considering broadcast traffic, additional degrees of freedom may need to be sent by the center to complete the session. For m = { 1 , 2 } , no additional degrees of freedom are required by any node. For the case inv olving NC and m = 4 , each edge node will require one additional degree of freedom in order to decode all of the encoded packets sent by the center . As a result, the denominator in the NC equations of (3) and (4) is replaced by d ( N − 1) / 4 e + max( X 1 , X 2 ) + 2 , as well as the numerator in the NC case of (4) with max( X 1 , X 2 ) + 2 . W e apply the revised fairness protocol to both the 5- node cross and “X” topology components using the same methods described in Section III. W e find that the through- put saturates at the maxima found in Section III for each topology component. Fig. 6 and 7 show both the unicast and broadcast throughput for the cross and “X” topology components, respecti vely , using our improved MA C approach. It is clear by comparing Figures 3 and 4 with Figures 6 and 7 respectiv ely that the new MAC eliminates the non-monotonic saturation beha vior . Furthermore, this comparison sho ws that the combination of NC, MPR with m = { 2 , 4 } , and the new MA C provides throughput gains on the order of six times that of routing alone with the 802.11 MAC in the saturated throughput regime. 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Total Offered Load (P) Throughput (S) Routing NC MPR = 2 MPR = 4 NC + MPR = 2 NC + MPR = 4 Routing Maximum NC Maximum and MPR = 4 Maximum MPR = 2 Maximum NC + MPR = 2 Maximum and NC + MPR = 4 Maximum Figure 6. 5-Node cross topology component unicast and broadcast throughput using the improv ed MA C. 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Total Offered Load (P) Throughput (S) Routing NC MPR = 2 MPR = 4 NC + MPR = 2 NC + MPR = 4 (Broadcast) NC + MPR = 4 (Unicast) NC + MPR = 2 Maximum and NC + MPR = 4 (Broadcast) Maximum Routing Maximum NC Maximum and MPR = 2 Maximum MPR = 4 Maximum NC + MPR = 4 (Unicast) Maximum Figure 7. 5-Node “X” component throughput using the improved MAC. V . C O N C L U S I O N W e dev eloped a simple, intuitive model by approximating key network elements with simple models of their underly- ing primary behavior . W e then used the model to ev aluate the performance of specific implementations of the 802.11 MA C with COPE and MPR in multi-hop networks. Gaining key insight into design strategies for combining the three technologies, each scenario presented ga ve a rough order of magnitude for the performance of implementing the MA C, NC, MPR, and their combination in larger networks. The model further shows that combining COPE with MPR results in super-additi ve throughput gains. W e then demonstrated that the non-monotonic saturation experienced in [1] is explained by the sub-optimal behavior of the 802.11 MA C, and used our model to de velop a MAC approach tailored tow ard the combined use of COPE and MPR that provides monotonic saturation behavior , as well as fairness to flows rather than nodes . R E F E R E N C E S [1] S. 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