Mathematical Analysis of Peer to Peer Communication in Networks B. Hajek (with thank you to collaborators L. Massoulie, S. Sanghavi, and Z.

Transcription

1 Mathematical Analysis of Peer to Peer Communication in Networks B. Hajek (with thank you to collaborators L. Massoulie, S. Sanghavi, and Z. Zhu) 1

2 Outline of presentation I. Overview II. File exchange for fixed peer population III. File exchange with continuous arrival of peers A. Fluid analysis (after Massoulie and Vojnovic) B. Direct stochastic analysis (work with Ji Zhu) IV. Discussion 2

3 I. Overview 3

4 Traditional File Service 4

5 Peer-to-peer File Service Service capacity scales with demand! Scalable Robust 5

6 Rough composition of Internet traffic today* P2P file transfer of movies (50%), software distribution (20-30%), games, etc % Web pages (including social networking sites) 20-30% Web streaming (via flash, quicktime, etc.) 8% Voice over IP 1% s, gaming, newsgroups, other 5% *See Schulze and Mochalski, 09. 6

7 1 st Generation P2P Systems Entire file uploaded in one shot Users wait till entire file is received before relaying BUT if file size is large, 1. Long delay before users become useful. e.g. Gnutella, KaZaa 2. If users depart mid-way, a lot of upload time wasted. 7

8 2 nd Generation P2P Systems File split into pieces Peers can start serving pieces without waiting for entire file. More effective at serving large files. e.g. BitTorrent, edonkey 8

9 A Model for P2P Live Streaming [Zhao,Liu,Chiu 09] Play deadline

10 ISP Friendly P2P Protocols Trans-ISP traffic is costly to ISPs ISPs can throttle P2P based on deep packet inspection Would like small trans-isp traffic 10

11 Flash Crowds Number downloading 5K 4K 3K 2K 1K Rapid startup is common Much variation in size and duration Flash Crowd start 24 hours 48 hours 11

12 Broad questions To wrap up overview, here is an incomplete list of performance issues for P2P: Peer selection strategies Piece selection strategies Push vs. pull Effects of heterogeneous link speeds Effects of network topologies ISP friendly operation Applications: file transfer, live streaming, video on demand, online interactive gaming or instruction,... 12

13 II. File exchange for fixed peer population Pertains to early portion of flash crowd situation. 13

14 Speedup from File Splitting One file at one source to N peers, File consists of K pieces, Measure time in multiples of time to download one piece. If file is not split, best completion time is: If file is split, best completion time is: * *[Mundinger,Weber,Weiss 07] gives centralized algorithm. 14

15 Blind collection of pieces (no exchange of lists) If contacts result in random pieces, get coupon collection problem: C 1 C 4 C 5 C K-2 C K 15

16 Coding at source Source message: K pieces... C 1 C 2 C 3 C 4 C K Transformed to M coded pieces:... Ĉ 1 Ĉ 2 Ĉ 3 Ĉ 4 Ĉ M Peers only need to collect K out of M: Ĉ 1 Ĉ 4 Ĉ 5 Ĉ M-2 Ĉ M [Luby, Shakrollahi, Mitzenmacher, Urbanke,...] 16

17 Network coding [Ahlswede, Cai, Li, Yeung, ] Each peer collects and produces coded pieces: C 1 C 2 C 3 C 4 C K Ĉ 127 Ĉ 37 Ĉ Ĉ 28 Ĉ 187 Ĉ 187 Ĉ 157 Ĉ Ĉ 289 Ĉ 349 Ĉ 157 Ĉ 489 Ĉ 128 Ĉ 349 Can achieve one-to-all dissemination in C(K+logN) time without exchange of piece lists: [Deb,Medard,Choute 07]. Can achieve same result with same bounds by PUSH + PULL without network coding: [Sanghavi,H., Massoulie 07]. 17

18 Newton Institute, Cambridge N peers, N pieces Use exchange of piece lists Each peer is advocate for one piece Random contact Pull random useful piece, with priority for advocated piece Completion time N+ O(log(N)) with high prob. [Sanghavi,H.,Massoulie 07] (Believe completion time is at most N+2 with high probability.) Simple result for N peers, N pieces 22 Mar 10 18

19 III. File exchange with continuous arrival of peers A. Fluid analysis (after [Massoulié, Vojnovic 07]) 19

20 peers peers collecting pieces (coupons) pieces initial seeding seed 20

21 peers peers collecting pieces (coupons) pieces initial seeding seed 21

22 peers peers collecting pieces (coupons) pieces initial seeding seed 22

23 peers peers collecting pieces (coupons) pieces initial seeding seed 23

24 peers peers collecting pieces (coupons) pieces initial seeding 24

25 peers peers collecting pieces (coupons) pieces initial seeding seed 25

26 III. File exchange with continuous arrivals of peers A. Fluid analysis (after Massoulie and Vojnovic) B. Direct stochastic analysis (work with Ji Zhu) 26

27 peers collecting pieces (coupons) seed 22 Mar 10 Newton Institute, Cambridge 27

28 peers collecting pieces (coupons) seed 22 Mar 10 Newton Institute, Cambridge 28

29 22 Mar 10 29

30 peers collecting pieces (coupons) seed 22 Mar 10 Newton Institute, Cambridge 30

31 31

32 peers pieces one club (members have all pieces except piece one) normal young peers seed infected young peers 22 Mar 10 32

33 one club normal young peers seed infected young peers 33

34 one club normal young peers seed infected young peers 22 Mar 10 Newton Institute, Cambridge 34

35 one club normal young peers seed infected young peers 35

36 0 1 2 i K i j K-1 36

37 0 1 2 i K i j K-1 37

38 Discussion 38

39 Discussion (continued) (See Ji Zhu s poster Thursday) normal young peer one club (members have all pieces except piece one) infected young peer 39

40 Discussion (continued) 40

41 Discussion (continued) 41

42 42

43 43

44 Thanks! 44