1. Throughput-Smoothness Trade-offs in Streaming Communication Gauri Joshi (MIT) Yuval Kochman (HUJI) Gregory Wornell (MIT) 1 13 th Oct 2015 Banff Workshop on Coding Theory for Streaming

2. Streaming dominates Internet traffic o Unlike traditional file transfer, packets are required in-order o Fast in-order delivery comes at the cost of throughput 2

3. Using Coding to Correct Erasures o First developed to provide reliability against o Channel noise (Communications) o Component failures (Storage) o Coding gives higher throughput, but increases delay o Eg. Rateless Fountain Codes: Delay prop. to size of stream 3 1 1 1 2 2 2 2 1 1 321321 321321 2121 2121 321321 321321 1 More Packets decoded, but higher delay vs. 1 1 Linear Combination of Packets 2 and 3 Linear Combination of Packets 2 and 3

4. Broad Research Objective o Ensure fast in-order packet delivery with efficient use of available bandwidth o Tools: Coding, Markov chains, Renewals, Large Deviations 4 1 1 1 2 2 2 2 1 1 321321 321321 2121 2121 321321 321321 1 More Packets decoded, but higher delay vs. 1 1 Linear Combination of Packets 2 and 3 Linear Combination of Packets 2 and 3

5. Outline 5 Point-to-point Streaming o Immediate Feedback o No Feedback o Block-wise delayed Feedback Multicast Streaming with Immediate Feedback o Structure of Optimal Coding Schemes o Two Users: Piggybacking on a Primary User o Two Users: Tuning the user priorities

6. System Model S S R R Source Model Source has an infinite packet stream Transmit coded combination per slot 6 Assume linear independence of transmitted combinations

7. System Model p Channel Model i.i.d packet erasure channels with success probabilities 7 S S R R

8. System Model Feedback Model Block-wise error-free feedback after every d slots 8 S S R R 2 2 3 3 1 d=3 bits of feedback

9. System Model 9 1 1 3 3 2 2 4 4 Packets transmitted Packets delivered 2 2 3 3 4 4 1 1 Time 1 2 3 4 5 Inter-delivery time T 6 5 5 2323 2323 Feedback Delay d= 4 5 5 5 5 7 8 2323 2323 Packet Delivery Model

10. o Captures the smoothness of in-order packet delivery o Exponent of in-order decoding delay can be shown to be λ Throughput and Delay Metrics Smoothness Exponent T is the time between successive in-order delivery instants or 10

11. Throughput and Delay Metrics Throughput Rate of packet delivery to the user o High throughput  Efficient use of bandwidth o Maximum possible = Erasure Channel Capacity = p 11

12. Related Previous Work Decoding delay with constraints o Streaming without feedback for adversarial and burst erasure channels [Martinian 2004], [Badr-KTA 2013] Playback delay o Throughput-delay trade-offs for uncoded tx. [Yao-KW 2004], [Mahadaviani-KJW 2015] o Coded tx, real-time packet generation [Joshi-KW 2012] This work o Delay metric: Smoothness exponent o How block-wise feedback affects the throughput-delay trade-off 12

13. Outline 13 Point-to-point Streaming o Immediate Feedback o No Feedback o Block-wise delayed Feedback Multicast Streaming with Immediate Feedback o Structure of Optimal Coding Schemes o Two Users: Piggybacking on a Primary User o Two Users: Tuning the user priorities

14. Immediate Feedback (d = 1) 14 1 1 2 2 2 2 3 3 2 2 Packets decoded Packets delivered 2 2 3 3 4 4 1 1 Time 1 2 3 4 5 T 4 4 6 4 4 7 d= 1 A simple ARQ scheme is optimal

15. Immediate Feedback (d = 1) One in-order packet decoded in every successful slot Throughput Smoothness Exponent and hence, 15

16. No Feedback (d = ∞) 16 1 1 2 2 2121 2121 321321 321321 Packets tx Packets delivered 4 4 3 3 1 1 T 43214321 43214321 5432154321 5432154321 2121 2121 321321 321321 654321654321 654321654321 654321654321 654321654321 5 5 6 6 Proposed Scheme: Transmit a combination of the first packets in slot n Linear Combination of packets 1 and 2

17. CONVERSE: Varying the rate r of adding packets to the combinations does not improve the trade-off No Feedback (d = ∞): Trade-off Achieved 17 o For, decoding occurs w.p. 1 and thus o Probability that there is no decoding in n slots o Hence,

18. Throughput-Delay Trade-offs (d=1 and d=∞) 18 Boundaries of the achievable trade-off with finite feedback delay d Channel success probability p= 0.6 Smoothness Exponent

19. Block-wise feedback every d slots o We can plan the transmission block-by-block o Combinations of the first un-decoded packets Proposed scheme: Given, in each block of slots, o Repeat the first un-decoded packet times o Send combinations of the first un-decoded packets. 19 Combination of repetition and greedy network coding

20. Example: Block-wise feedback 20 1 1 2 2 1 1 2 2 Packets decoded Packets delivered 4 4 3 3 1 1 Time 1234 5 T 3232 3232 6 4 4 7 2121 2121 2 2 4 4 5454 5454 89 Proposed Scheme for d = 3, a = 2 a = 3 is repetition coding a = 1 is greedy network coding

21. Block-wise feedback Throughput Smoothness Exponent 21 Repetition coding Independent coded combinations Independent coded combinations In-order decoding does not occur if the repetition part is erased

22. Effect of Block-wise Feedback on the trade-off 22 For d=2 and d=3, can prove optimality over all time-invariant codes Channel success probability p= 0.6 a=3a=2 a=1 Regime of interest Smoothness Exponent MAIN INSIGHTS o Worthwhile to invest resources in a frequent feedback mechanism o A simple combination of repetition and greedy coding gives close to optimal performance

23. Outline 23 Point-to-point Streaming o Immediate Feedback o No Feedback o Block-wise Delayed Feedback Multicast Streaming with Immediate Feedback o Structure of Optimal Coding Schemes o Two Users: Piggybacking on a Primary User o Two Users: Tuning the user priorities

24. System Model S S U1U1 U1U1 U2U2 U2U2 UKUK UKUK Source Model Source has an infinite packet stream Multicast a coded combination per slot 24

25. System Model S S U1U1 U1U1 U2U2 U2U2 UKUK UKUK p1p1 p2p2 pKpK Channel Model i.i.d packet erasure channels with success probabilities p i Independent across the users 25

26. System Model S S U1U1 U1U1 U2U2 U2U2 UKUK UKUK Feedback Model Immediate error-free 1-bit feedback after every slot 1 1 1 1 1 1 p1p1 p2p2 pKpK 26

27. System Model Packet Delivery Model 1 1 2 2 3 3 3232 3232 4 4 1 1 2 2 3 3 Packets Received Packets Delivered 4 4 Delivery in bursts of in-order packets Out-of-order packets are buffered 27

28. Related Previous Work Instantaneous Decodability Delay Decoding Delay In-order Delivery Delay [Keller-DF 2008] Index Coding Poisson Packet Generation [Sundararajan-SM 2009] [Fu-SM 2012] [Sundararajan-SM 2008] [Barros-CMW 2009] This work [El Rouyaheb-S 2010], [Maleki-CJ 2012], Shanmugam-D 2013] Same Erasure Probabilities Smoothness of In-order Delivery Smoothness of In-order Delivery [Joshi-KW Netcod 2014] 28

29. Performance Metrics o Performance metrics: where T is the time between successive in-order decoding instants o Optimal trade-off impossible to achieve simultaneously at all users 29

30. Outline 30 Point-to-point Streaming o Immediate Feedback o No Feedback o Block-wise Delayed Feedback Multicast Streaming with Immediate Feedback o Structure of Optimal Coding Schemes o Two Users: Piggybacking on a Primary User o Two Users: Tuning the user priorities

31. Structure of Coding Schemes Required Packet The earliest undecoded packet of U i. Denoted by Claim 1 Only transmit combinations of required packets of the users 1 1 2 2 U1U1 U2U2 U3U3 3 3 5 5 1 1 3 3 4 4 5 5 1 1 2 2 2+3+4 2+3 2+4 2 2 3 3 4 4 3+4 2 3 4 31

32. Structure of Coding Schemes Required Packet The earliest undecoded packet of U i. Denoted by 1 1 2 2 U1U1 U2U2 U3U3 3 3 5 5 1 1 3 3 4 4 5 5 1 1 2 2 4 4 2 3 4 Claim 2 Add to a combination with packets only if U j has decoded all packets. Priority order 32

33. Structure of Coding Schemes Required Packet The earliest undecoded packet of U i. Denoted by 1 1 2 2 U1U1 U2U2 U3U3 3 3 5 5 1 1 3 3 4 4 5 5 1 1 2 2 4 4 2 3 4 Claim 2 Add to a combination with packets only if U j has decoded all packets. Priority order 33

34. Structure of Coding Schemes Required Packet The earliest undecoded packet of U i. Denoted by 1 1 2 2 U1U1 U2U2 U3U3 3 3 5 5 1 1 3 3 4 4 5 5 1 1 2 2 2+4 2 3 4 Claim 2 Add to a combination with packets only if U j has decoded all packets. Priority order 34

35. Structure of Coding Schemes Required Packet The earliest undecoded packet of U i. Denoted by 1 1 2 2 U1U1 U2U2 U3U3 3 3 5 5 1 1 3 3 4 4 5 5 1 1 2 2 2+4 2 3 4 Claim 2 Add to a combination with packets only if U j has decoded all packets. Priority order 35

36. Outline 36 Point-to-point Streaming o Immediate Feedback o No Feedback o Block-wise Delayed Feedback Multicast Streaming with Immediate Feedback o Structure of Optimal Coding Schemes o Two Users: Piggybacking on a Primary User o Two Users: Tuning the user priorities

37. Two Users: Piggybacking on a Primary User o Source always priority to U 1 o What is the best trade-off for user U 2 ? Optimal Coding Scheme Transmit when and U 2 has decoded but U 1 has not. Otherwise transmit. 37

38. Markov Model of Packet Decoding Probabilities a = p 1 p 2 b = p 1 (1-p 2 ) c = (1-p 1 )p 2 d = (1-p 1 )(1-p 2 ) 0 1 2 2’ 1’ c b a+b b c a b c a a+d c+d d a c d a+d Gaps in U 2 ’s decoding Advantage states: When U 2 and not U 1 has decoded s r1 38

39. Markov Model of Packet Decoding 39 Probabilities a = p 1 p 2 b = p 1 (1-p 2 ) c = (1-p 1 )p 2 d = (1-p 1 )(1-p 2 ) Gaps in U 2 ’s decoding The Markov chain is positive-recurrent if p 2 > p 1 0 1 2 2’ 1’ c b a+b b c a b c a a+d c+d d a c d a+d Advantage states: When U 2 and not U 1 has decoded s r1

40. Throughput-Smoothness Trade-off 40 Throughput Smoothness Exponent

41. Markov Model of Packet Decoding 41 Probabilities a = p 1 p 2 b = p 1 (1-p 2 ) c = (1-p 1 )p 2 d = (1-p 1 )(1-p 2 ) Gaps in U 2 ’s decoding The Markov chain is positive-recurrent if p 2 > p 1 0 1 2 2’ 1’ c b a+b b c a b c a a+d c+d d a c d a+d Advantage states: When U 2 and not U 1 has decoded s r1

42. Smoothness Exponent λ 2 42 o Absorbing state F reached when U 2 decodes an in-order packet o where is the second largest eigen value of the state transition matrix of this chain 0 I I’ a+b b a+c b c d a+b+d c+d d a+c F 1

43. Throughput-Smoothness Trade-off 43 Throughput Smoothness Exponent Increases with p 2

44. Outline 44 Point-to-point Streaming o Immediate Feedback o No Feedback o Block-wise Delayed Feedback Multicast Streaming with Immediate Feedback o Structure of Optimal Coding Schemes o Two Users: Piggybacking on a Primary User o Two Users: Tuning the user priorities

45. Tuning the priority for each user 45 Want the flexibility to tune user priorities Proposed Scheme o When U 1 is ahead (r 1 > r 2 ) favor U 2 with prob. q 2 o When U 2 is ahead (r 2 > r 1 ), favor U 1 with prob. q 1 Special Cases o (q 1, q 2 ) = (1,0): U 2 Piggybacking on primary user U 1 o (q 1, q 2 ) = (0,0): Greedy code, throughput optimal for both users

46. Markov Model of Packet Decoding 46 0 1 2 2’ 1’ -1’ -2 -2’ (1-q 1 )c c q 1 (a+b) q 2 (a+c) b b q 1 (a+b) q 2 (a+c) (1-q 2 )b c a b c(1-q 2 ) a a+d (1-q 2 )a+q 2 b+d (1-q 1 )a+q 1 c+d d d a a c b o q 1 = 1, q 2 = 0: User 2 piggybacking on User 1 o Increasing q i tilts that side of the chain towards state 0

47. Throughput-Delay Trade-offs 47 Throughput Smoothness Exponent o Add an absorbing state which is reached when U 1 (or U 2 ) decodes an in-order packet o Find the rate of convergence to of the system to that state

48. Smoothness Exponent λ 2 where is the second largest eigen-value of the state transition matrix of this chain 48 0 1 1’ -2 q 2 (a+c) b q 1 (a+b) b c(1-q 2 ) (1-q 2 )a+b+d (1-q 1 )b+q 1 c+d d F a+c q 1 (a+b) (1-q 1 )(a+c) d a+ca+c 1

49. Varying q 1 = q 2 when p 1 = p 2 49 q 1 = q 2 varying from 0 to 1 Increasing q prevents gaps in decoding from growing on either sides of the Markov model Increasing q i Smoothness Exponent

50. Varying q 2 while keeping q 1 = 1 50 p 1 = 0.5, p 2 = 0.6, q 1 = 1, and q 2 varying from 0 to 1 Increasing q 2 Increasing q 2 increases λ 2 at the cost of throughputs of both users

51. Summary Point-to-point Streaming o Frequent feedback improves smoothness a lot o Combination of repetition and greedy coding gives good trade-off Multicasting to two users o Inter-dependence between the two trade-offs o Tune priorities to stabilize the trade-offs 51

52. Open Questions o Dynamic Bandwidth schemes o Multicast with delayed feedback, and for more users o Considering Packet dropping o Streaming from Distributed Sources 52

53. No Feedback Case o Can extend the analysis for the point-to-point case o In slot n transmit a combination of packets, where Gauri Joshi (MIT)Throughput-Delay Trade-offs in Streaming 53 U1U1 U1U1 U2U2 U2U2 UNUN UNUN S S p1p1 p2p2 pNpN

54. How does feedback delay d affects the throughput-smoothness trade-off? 54 Channel success probability p= 0.6 a=3a=2 a=1 Smoothness Exponent

55. Varying q 2 while keeping q 1 = 1 55 p 1 = 0.5, p 2 = 0.4, q 1 = 1, and q 2 varying from 0 to 1 Gaps in decoding of U 2 go to infinity Gaps in Decoding stabilized by increasing q 2 Increasing q 2 Beyond some q 2 the chain becomes stable Case: p 2 < p 1

56. Challenges in Multicast Streaming o A packet that is the next in-order required packet of one user, may be redundant for another user o Can give priority to a user at the cost of the user How to strike a balance such that they all users good throughput-delay trade-offs? U1U1 U1U1 UKUK UKUK S S p1p1 pKpK U2U2 U2U2 p2p2 56