Cross-Layer Optimizations for Intersession Network Coding on Practical 2-Hop Relay Networks, Lecture notes of Network Design

Download Cross-Layer Optimizations for Intersession Network Coding on Practical 2-Hop Relay Networks and more Lecture notes Network Design in PDF only on Docsity! Cross-Layer Optimizations for Intersession Network Coding on Practical 2-Hop Relay Networks Chih-Chun Wang, Abdallah Khreishah Center of Wireless Systems and Applications (CWSA) School of ECE, Purdue University Ness B. Shroff Departments of ECE and CSE The Ohio State University Abstract—Full characterization of Intersession Network Cod- ing (INC), i.e., coding across multiple unicast sessions, is no- toriously challenging. Nonetheless, the problem can be made tractable when considering practical constraints that restrict the types of INC schemes of interest. This paper characterizes the INC capacity of 2-session wireless 2-hop relay networks with a packet erasure channel model and a round-based feedback sched- ule motivated by the usage of “reception reports” in practical protocols such as COPE. The capacity regions are formulated as linear programming problems, which admit simple concatenation with other competing techniques such as opportunistic routing (OpR), and cross-layer (CL) optimization. Extensive numerical evaluation is conducted on 1000 random topologies, which compares and quantifies the throughput benefits of INC, OpR, and CL, and their arbitrary combinations. The results show that by combining all three techniques of INC, OpR, and CL, the throughput of a wireless 2-hop relay network can be improved by 100–500% over the benchmark single-path routing solution depending on the number of sessions to be coded together. I. INTRODUCTION Network coding (NC) has promised strict throughput im- provement over non-coding solutions [1], [2], [3]. Although the benefits of NC within a single multicast session is elegantly characterized by the min-cut/max-flow theorem, to realize the throughput benefits for multiple unicast sessions, NC must be performed across different sessions, termed intersession network coding (INC). Full characterization of INC is no- toriously challenging and existing investigation shows that the existence of a closed-form solution is highly unlikely for general network settings [4]. Nonetheless, the problem can be made tractable when considering practical constraints that restrict the types of INC schemes of interest. In this work, we are interested in a special wireless 2-hop relay network that is motivated by empirical INC protocols such as COPE [3]. Wireless mesh networks (WMN) have been considered as a cost-effective solution to the last mile problem. Among many techniques that improve the throughput of a WMN, the following three have demonstrated significant throughput en- hancement: 1-hop INC (the COPE protocol [3]); Intrasession- coding-based opportunistic routing (OpR) (the MORE pro- tocol [1], [5]) that takes advantage of the spatial diversity of wireless channels by opportunistically deciding the packet route; Cross-layer (CL) optimization [6] that allows distributed resource allocation that jointly takes into account the traffic demands and the interference constraints of wireless channels. This work was supported by the NSF Grant CNS-0905331. (a) Wireless broadcast (b) PrR-based averag- ing (c) Opportunistic-scheduling- based averaging Fig. 1. Two different wireless-to-wireline conversion methods. Most existing results on joint INC and CL are based on converting the wireless networks to the corresponding wireline counterpart (see [7] and the reference therein). Two conversion methods are commonly used: the predetermined- routing-based (PrR-based) conversion and the opportunistic- scheduling-based (OpS-based) conversion. Take Fig. 1(a) for example, source s broadcasts a packet wirelessly and nodes a and b can hear the packet with probabilities 0.3 and 0.4, respectively and independently. The PrR-based conversion predetermines the packet route and uses error control cod- ing or Automatically Repeat reQuest (ARQ) to average the random overhearing events into deterministic average rates as in Fig. 1(b). The OpS-based conversion assumes that s knows before transmission which of the receivers a and b will successfully overhear the to-be-transmitted packet. The deterministic rates to a, to b, and to (a, b) simultaneously, are thus expressed as a percentage of ts, the amount of time for which s is broadcasting, as described in Fig. 1(c). Neither PrR nor OpS model the practical schemes closely, as random overhearing is not considered in the PrR conversion while the assumption of perfectly knowing the channel condition is too optimistic for WMNs.1 In this work we model the wireless channel by a more realistic packet erasure channel (PEC) and characterize the capacity of a 2-session 2-hop relay network under round-based feedback. II. THE SETTING We use X (and sometimes Y ) to denote a symbol in GF(q). A 1-to-K packet erasure channel (PEC) takes an input X ∈ GF(q) and outputs a K-dimensional vector in ({X}∪ {∗})K , 1For a single multicast session, random linear network coding achieves the capacity of OpS even without the knowledge about the channel condition [8]. Nonetheless, the capacity of INC when not knowing the channel condition is strictly smaller than the capacity of OpS. (a) With OpR (b) Without OpR Fig. 2. Illustration of 2-session relay networks. 0 1 2 0 0.5 1 Distance S uc ce ss P ro b. p Fig. 3. The overhear- ing probability versus the distance. where the k-th coordinate being “∗” denotes that the input symbol X is erased for the k-th receiver. We consider only i.i.d. PECs of which the erasure pattern is independent for each channel usage. For example, a 1-to-2 PEC can be described by four parameters ps;12, ps;12c , ps;1c2, and ps;1c2c , which denote the probabilities that X is received by both receivers (rx) 1 and 2, by only rx 1, by only rx 2, and by neither rx 1 nor 2. This notation can be easily extended to the marginal success probability ps;1 ∆= ps;12 + ps;12c and to the union ps;1∪2 ∆= ps;12 + ps;12c + ps;1c2. A 2-session relay network is described in Fig. 2(a), in which source s1 would like to send nR1 packets X1, · · · , XnR1 to destination d1, and s2 would like to send nR2 packets Y1, · · · , YnR2 to d2. r is a relay node. Each of s1, s2, and r can use the corresponding PEC n times, respectively, and we are interested in the largest achievable rate pair (R1, R2) that guarantees decodability at d1 and d2 with close-to-1 probability for sufficiently large n and the finite field size q. To model the “reception report” suggested in COPE, we enforce the following sequential, round-based feedback sched- ule: Each of s1 and s2 transmits n symbols, respectively. After the transmission of 2n symbols, two reception reports are sent from d1 and d2, respectively, back to the relay r so that r knows which packets have successfully arrived which destinations. After the reception reports, no further feedback is allowed and the relay r has to make its own decision how to use the available n PEC usages to guarantee decodability at d1 and d2. In our setting, we also assume that the success probability parameters of all PECs and all the coding operations are known to all nodes. The only unknown part is the values of the X and Y symbols. For the purpose of illustration, a simplified network setting is also depicted in Fig. 2(b), in which the packets sent by si will not be overheard by the 2-hop-away destination di. In this simplified setting, the question thus becomes given the following parameters: ps1;rd2 , ps1;rdc 2 , ps1;rcd2 , and ps1;rcdc 2 ; ps2;rd1 , ps2;rdc 1 , ps2;rcd1 , and ps2;rcdc 1 ; pr1;d1d2 , pr;d1dc 2 , pr;dc 1d2 , and pr;dc 1dc 2 , what is the maximum achievable rate under the round-based feedback model. For future reference, we say Fig. 2(a) admits OpR as the packets can be overheard by the two-hop destinations while Fig. 2(b) does not admit OpR. Since the settings are symmetric, we sometimes assume that pr;d1 ≥ pr;d2 which can be achieved by relabelling the sessions. Fig. 4. A broadcast PEC problem with side informa- tion. (a) (b) Fig. 5. (a) An example of random node placement for M = 2 sessions. (b) The con- straint on the topological relationship between si and di. III. THE CAPACITY RESULTS Consider Fig. 2(b) and the scenario in which s1 and s2 have finished transmission and the reception reports have been sent to r. The question now becomes a broadcast PEC problem with side information (SI) as depicted in Fig. 4. That is, the packets X [2] 1 , · · · , X [2] nR1;2 have been overheard by d2 and the packets X [2c] 1 , · · · , X [2c] nR1;2c have not. Similarly, Y [1] 1 , · · · , Y [1] nR2;1 have been overheard by d1 and Y [1c] 1 , · · · , Y [1c] nR2;1c have not. X [2] and Y [1] packets can later serve as SI when decoding at d2 and d1, respectively. The total rates for each session are R1 = R1;2 + R1;2c and R2 = R2;1 + R2;1c . We then have the following theorem: Theorem 1: For any rate vectors (R1;2, R1;2c , R2;1, R2;1c) for the broadcast PEC with SI in Fig. 4 and assuming that pr;d1 ≥ pr;d2 , one can communicate the values of all X and Y packets to d1 and d2, respectively, within n channel usage if and only if (R1;2, R1;2c , R2;1, R2;1c) satisfies R1;2 + R1;2c + R2;1c ≤ pr;d1 (1) R2;1 + R2;1c + pr;d2 pr;d1 R1;2c ≤ pr;d2 . (2) Remark: The broadcast capacity of Gaussian channel with SI has been considered in many papers (see [9] as a representative work). In addition to the difference of the settings of PECs and Gaussian channels, this paper will also consider the best coding strategy at sources s1 and s2 and characterize the capacity as a linear programming problem. It is worth noting that for 3-session Gaussian broadcast channels with general SI, the capacity remains an open problem. On the other hand, in [10] we show that the capacity of 3-session broadcast PEC can be computed in a similar way as outlined in this paper. Recently, the capacity of 2-session broadcast PECs with instant packet-by-packet feedback (but without SI) is studied in [11]. Sketch of the proof of Theorem 1: Achievability: For any vector (R1;2, R1;2c , R2;1, R2;1c) sat- isfying (1) and (2), perform the following 2-staged coding scheme sequentially. Stage 1: Whenever we can use the broadcast PEC, we mix the packets of the three groups: X [2] 1 to X [2] nR1;2 , Y [1] 1 to Y [1] nR2;1 , and Y [1c] 1 to Y [1c] nR2;1c by random linear network coding (RLNC) [8] and generate one outgoing symbol. Repeat this random packet generation until we have sent out the following amount