Network Coding as a Dynamical System

Transcription

1 Network Codng as a Dynamcal System Narayan B. Mandayam IEEE Dstngushed Lecture (jont work wth Dan Zhang and a Su) Department of Electrcal and Computer Engneerng Rutgers Unversty

2 Outlne. Introducton 2. Dfferental Equaton (DE) framework for RNC 3. Use Cases of DE framework 4. Concentraton Property 5. Applcatons n Resource Allocaton 6. Concluson 2

3 What s Network Codng? Butterfly Example How to acheve multcast capacty? Each lnk has unt capacty Node, 2 want to delver b, b 2 to 5 and 6 Take 5 seconds and center lnk used twce 3

4 What s Network Codng? Butterfly Example Contd. Can we do better? XOR at node 3 Center lnk used once Fnsh transmsson n 3 seconds 4

5 Wreless Network Codng & Multcast Advantage How to explore multcast advantage? Each wreless lnk s broadcastng wth unt capacty Node and 3 want to exchange a bt Node and 3 can reach each other ONLY through node 2 Takes 4 Transmssons 5

6 Multcast Advantage Contd. How to explot multcast advantage? XOR at node 2 3 transmssons prevously 4 6

7 Random Lnear Network Codng Every outgong packet c s a coded packet c s a lnear combnaton of source packets c s s s, s2,, sm over GF( q) Every wreless node performs the same codng operaton 2 coded packet source packets coeffcent vector s m b 7

8 Random Lnear Network Codng Every outgong packet c s a coded packet c s a lnear combnaton of source packets c s s s, s2,, sm over GF( q) Every wreless node performs the same codng operaton 2 coded packet source packets coeffcent vector s m b 8

9 Random Lnear Network Codng Every outgong packet c s a coded packet c s a lnear combnaton of source packets c s s s, s2,, sm over GF( q) Every wreless node performs the same codng operaton 2 coded packet source packets coeffcent vector s m b 9

10 Decodng: Random Lnear Network Codng Codng coeffcent vector b sent along wth coded packet ey quantty to track Number of lnearly ndependent coeffcent vectors Call t rank Decodablty m lnearly ndependent coeffcent vectors Full rank s s s c c c b b b 2 m 2 m 2 m 0

11 Modelng: Hypergraph for Wreless Networks G ( N, E), E {(, ) N, N} Hyperarc (,{2,3}) Hyperarc (, ): sender multple recevers Broadcast nature of wreless Transmsson range Drectonal antenna (beamformng)

12 Defnton of Capactated Hypergraph A packet goes through the hyperarc (, ) wth probablty P, P, : receved by at least one node n set Recepton can be correlated e.g., channel correlaton, jont detecton For ndependent recepton Capacty of (, ) P z,, P j P,, j z, 2 z,{2,3} z,3 Tx rate of node (MAC) 2 3 2

13 Rank Evoluton of RNC Decodng reles on the rank of node from whch also defne rank of an arbtrary set S S span{ b, b,, b S, N n }, rank at N rank at dm( S ) dm( S ) Defne V =E[N ] and V =E[N ] Flud approxmaton V N and V N How wll V and V evolve over tme? When wll V reach m? 3

14 Dfferental Equatons for RNC In Δt a packet s sent from node n c wth probablty λ Δt It s receved by at least one node n wth probablty P, Ths ncomng packet ncrements V wth probablty To see ths V V V V V V V S S S S q q q q q q q S S S S } { } { dm dm dm 4

15 Dfferental Equatons for RNC On average, the packet ncrements V by A Dfferental Equaton (DE) for rank evoluton: c V V q P t t V t t V } {, } {, c V V q z t t V t t V V 5

16 Dfferental Equatons for RNC V c z, q V V { } Innovatve packet probablty A system of N 2 DE s, one for every nonempty Can be solved numercally wth ntal condtons Good for performance evaluaton of varous RNC schemes Can be analyzed Good for theoretcal nsghts too Enables Crosslayer Desgn 6

17 Illustraton: Boundary Condtons Specfc networkng scenaros yeld boundary condton multcast source m packets 2 5 V 0 m, 0,, o.w. 3 4

18 Illustraton 2: Boundary Condtons Specfc networkng scenaros yeld boundary condton 2 multcasts 2 mage sources m packets sngle source m 2 packets 2 5 m, {,3},4, m2, 4,{,3}, V 0 m m2, {,3},4, 0, o.w. 3 4

19 A Toy Example A wreless network wth ndependent receptons Transmsson of m packets start at t = 0, compute V 4 (t) P,2 2 3 P 2,4 P 3,4 4 P,2 V V V 2,4, 2,4 3,4, 3,4 2,3,4, 2,3,4 z P,, z B.C. V 4 2,4 V4 V2,4 V4 V3, 4 2,4 q z3,4 q V2,4 m V2,4 V2,3,4 q z3, 2,4 q V3,4 m V3,4 V2,3,4 q z2, 3,4 q P, q V m 2,4, 2,4 3,4, 3,4,3, 2,3,4, 2,3,4, 2,3 2,3 P,2 P,3 4 2 P V z z z z 2,4, 0 V 0 V 0 V 0 0 2,4 z 3,4 P 2,3, 4 P 3 3,4 3,4 z, z, P 2,3,4 P P P 9,2

20 Analytcal Result: Multcast Throughput Usng RNC Solve the DEs (wth B.C.) for V (t) Throughput s the dervatve of V (t) Throughput gven by mn cut! 2 multcast source m packets V = z, - q V -V {}È V t V å Ï B.C. V ( 0) = : c m, : ( ) mn ìï í îï m, Î, 0, o.w. t m, t 0,. {}, t, t 0, m cmn {},, t m c {},, mn. 20

21 What s Mn Cut of a Hypergraph? A cut for (S, ) s a set T s.t. Cut sze T c S 2 5 T c z, T c T Mn cut: smallest cut sze 3 4 c mn S, mn ct T s a ( S, ) cut Example: Cut T for ({},{3}), T={3,4} c T z,{3,4} z2,{3,4} z5,{3,4 } 2

22 Example - Wreless P2P,,2,3,4 Peers download from server m 400 Peers transmt to each other to enhance throughput c mn {},{4}

23 Example Multple Sources,,2,3 m 200, m å ( ) V = z, - q V -V {}È Ï B.C. V ( 0) = ì ï ï í ï ï î 200, = {},{, 2}, 300, = {3},{2,3}, 500, = {,2,3}, 0, = {2}. 23

24 Example Complex Networks,,2,,0, m 00 24

25 Example Correlated Recepton C mn (, 4) P,2 0.49, P,3 0.49, P,{2,3 } 0.5 Channel 2 and channel 3 hghly correlated 25

26 Effect of Number of Source Packets m? m=00 m=200 m=400 m=800 26

27 Is there a Concentraton Result? Numercal examples suggest concentraton property As no. of source packets ncrease, DE soluton becomes ncreasngly accurate Rank processes concentrate to DE soluton Prevous studes showed throughput gven by c mn src, dst s acheved asymptotcally wth no. of source packets Can we prove ths result wth DE framework? 27

28 Yes We Can! Let node be source, D the destnaton set, m packets to delver and T the total Tx tme DE for the varance d var[ N ] de[ N ] { } 2, cov[, N z N N q dt dt Use t to bound varance Use Chebyshev nequalty to show throughput converges to mn cut n probablty lm P( m/ T cmn (, D)), 0 m var[ N ] lm sup d for some, 0 2 d t t ] 28

29 Can we use DE Framework for Cross-layer Resource Allocaton n RNC? Model RNC as a dynamcal system: Hyperarc capacty of Tx rate at node Packet recepton rate MAC Innovatve packet probablty Recepton probablty (A pkt from can be recvd by at least node n ) PHY DE framework closely models rank evoluton of RNC n terms of PHY and MAC parameters 29

30 How does Power Control Impact RNC? In general, power control to acheve certan objectve, e.g.: Mantan certan SINR value Mnmze total power Wth RNC, Tx powers nfluence network codng performance Larger transmt power: Good for ts own transmsson Increases nterference for other transmssons n the same network 30

31 Motvaton - Necessty of Power Control n RNC Consder 6-node topology, RNC Node delvers pkts to {4,5,6} Each node Tx at pkt/ms t=0, every node s Tx power set to 3dBm Tx powers,, are ncreased sequentally Observatons Incrementng power sn t always good for throughput Is there an optmal strategy? Set P Tx, =4dBm Set P Tx,3 =4dBm Set P Tx,4 =4dBm 3

32 Network Codng Aware Power Control Fnd an optmal allocaton of Tx powers such that multcast throughput can be maxmzed RNC Improved Throughput Challenge: Need a good model of RNC performance n terms of Tx powers DE framework! Power Control 32

33 Problem Setup - Max-mn-throughput Power Control Formulate max-mn throughput problem usng DE: Mn. throughput among the dests. set DE framework for RNC Sze-N Tx power vector Objectve: adjustng powers so that mnmum throughput s maxmzed 33

34 Gradent-based Algorthm Idea: Suppose dest k has the mnmum nstantaneous throughput Fnd the gradent of the throughput of k Adapt powers towards the drecton of the gradent Postve constant servng as gan parameter Power control feedback loop: Gradent of mnmum throughput 34

35 Gradent-based Algorthm: Approxmaton of the Gradent Challenge: hard to drectly evaluate Recall Why hard? Underlyng PHY schemes may change over tme Result n changed The explct expresson for may even be unknown n practce Depends on PHY specfcs lke modulaton, FEC, dversty... 35

36 Gradent-based Algorthm: Approxmaton of the Gradent Approach: Estmate n a centralzed manner For node to adjust power Increment s power, others reman the same. Construct a power vector Power ncrement for node Estmate the gradent: Vector wth -th component beng and 0 elsewhere Adjust the power n the drecton of gradent 36

37 Numercal Results Smulaton setup: Use numercal DE solver to evaluate the algorthm 6-node topology, nodes perform RNC Source: node, and dest. nodes {4,5,6} src has 2000 packets to multcast to dests. Assumpton: Certan MAC: each node transmts at pkt/ms Certan PHY: BPS sgnalng and Gaussan nterference model Tx power vares between 0dBm and 5dBm

38 Throughput Performance Wthout Power Control (PC), throughput s a constant Wth PC: nstantaneous throughput s mproved compared to no PC Wth PC: throughput Converges around t=60ms Each node Tx at pkt/ms, mn cut between. src. and dst. s pkt/ms RNC wth PC acheves optmal throughput! 38

39 Transmt Powers Tx power ntally set to 3dBm Powers are adjusted wth an upper bound of 5dBm Tends to converge around t=40ms 39

40 Motvaton - Necessty of CSMA backoff control n RNC Consder Node delvers pkts to {4,5,6} CSMA as the MAC protocol Exp. backoff f channel busy At t=000, 2000, 3000ms, mean backoff tme of node, 4, 6 reduced, by 30%, 40% and 50%, respectvely Transmsson aggressveness (TA) ncreased Observatons Increased TA: May mprove neghbor throughput May also be bad: reduced channel avalablty Throughput (packets/ms) Reduce node s backoff tme node 4 node 5 node 6 Reduce node 4 s backoff tme Tme (mllseconds) Reduce node 6 s backoff tme 40

41 CSMA backoff control - Throughput Performance Wthout control: 5 Mean backoff tme of nodes s fxed Wthout control: 3 4 throughputs reman at 0.06pkt/ms. Wth control: nstantaneous throughput s mproved compared wth no control case Wth control: throughput converges to 0.22 around t=4000ms Throughput gan: >200%. 4

42 Concludng Remarks Impact of PHY algorthms on network codng performance Rate of Rank Evoluton Impact of MAC on network codng performance Interference effects Resource Allocaton Power Control, Schedulng V V V } z q, { Crosslayer Desgn problems Solvng systems dfferental of equatons Approprate boundary condtons Emulaton/Software Utlty under Development Network Topologes Rado Channels PHY, MAC and Resource Allocaton Algorthms 42

43 References on DE Framework for Network Codng D. Zhang, N. B. Mandayam, and S. Parekh, DEDI: A framework for analyzng rank evoluton n random network codng, n IEEE Internatonal Symposum on Informaton Theory, Austn, TX, Jun. 200 D. Zhang and N. B. Mandayam, Analyzng Multple Flows n a Wreless Network wth Dfferental Equatons and Dfferental Inclusons, IEEE Wreless Network Codng Workshop (WNC) 200, Boston, MA, Jun. 200 D. Zhang and N. B. Mandayam, Resource Allocaton for Multcast n an OFDMA Network wth Random Network Codng, IEEE Internatonal Conference on Computer Communcatons (INFOCOM), Shangha, Jun. 20 D. Zhang and N. B. Mandayam, Analyzng Random Network Codng wth Dfferental Equatons and Dfferental Inclusons, IEEE Trans. Inform. Theory, vol. 57, no. 2, pp , December 20. Su, D. Zhang and N. B. Mandayam, "Network Codng Aware Power Control," 46th Conference on Informaton Scences and Systems (CISS), Prnceton, NJ, March 202 D. Zhang,. Su and N. B. Mandayam, "Network Codng Aware Resource Allocaton to Improve Throughput," IEEE Internatonal Symposum on Informaton Theory (ISIT), Boston, MA, Jul Su, D. Zhang and N. B. Mandayam, Dynamc Rado Resource Management for Random Network Codng: Power Control and CSMA Backoff Control" submtted to Transactons on Wreless, March

44 Acknowledgments U.S. Natonal Scence Foundaton Offce of Naval Research IEEE COMSOC Collaborators: Dan Zhang, a Su () Deb Serng 44