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

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

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 More Packets decoded, but higher delay vs. 1 1 Linear Combination of Packets 2 and 3 Linear Combination of Packets 2 and 3

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

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

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

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

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

System Model Packets transmitted Packets delivered Time Inter-delivery time T Feedback Delay d= Packet Delivery Model

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

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

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

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

Immediate Feedback (d = 1) Packets decoded Packets delivered Time T d= 1 A simple ARQ scheme is optimal

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

No Feedback (d = ∞) Packets tx Packets delivered T Proposed Scheme: Transmit a combination of the first packets in slot n Linear Combination of packets 1 and 2

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,

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

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

Example: Block-wise feedback Packets decoded Packets delivered Time T Proposed Scheme for d = 3, a = 2 a = 3 is repetition coding a = 1 is greedy network coding

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

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

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

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

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

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

System Model Packet Delivery Model Packets Received Packets Delivered 4 4 Delivery in bursts of in-order packets Out-of-order packets are buffered 27

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

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

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

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 U1U1 U2U2 U3U

Structure of Coding Schemes Required Packet The earliest undecoded packet of U i. Denoted by U1U1 U2U2 U3U Claim 2 Add to a combination with packets only if U j has decoded all packets. Priority order 32

Structure of Coding Schemes Required Packet The earliest undecoded packet of U i. Denoted by U1U1 U2U2 U3U Claim 2 Add to a combination with packets only if U j has decoded all packets. Priority order 33

Structure of Coding Schemes Required Packet The earliest undecoded packet of U i. Denoted by U1U1 U2U2 U3U Claim 2 Add to a combination with packets only if U j has decoded all packets. Priority order 34

Structure of Coding Schemes Required Packet The earliest undecoded packet of U i. Denoted by U1U1 U2U2 U3U Claim 2 Add to a combination with packets only if U j has decoded all packets. Priority order 35

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

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

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 ) ’ 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

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’ 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

Throughput-Smoothness Trade-off 40 Throughput Smoothness Exponent

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’ 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

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

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

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

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

Markov Model of Packet Decoding ’ 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

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

Smoothness Exponent λ 2 where is the second largest eigen-value of the state transition matrix of this chain ’ -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

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

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

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

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

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

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

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

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