CODES FOR ALL SEASONS Emina Soljanin, Bell Labs
IN THE CLOUD? CODES Emina @ Bell Labs
Codes at all Levels and for all Layers Files have chunks. . . . Chunks have packets. . . . CODING Packets have symbols. . . . Codes we can’t refuse. 3 / 52
Theory vs. Systems (Practice, Networking) The moon is more useful than the sun! Why? We need the light more during the night than during the day. 4 / 52
Codes for . . . Q n i o J k r . . . o F . . . all . . . CLOUDS, FAILURES, CROWDS Q r o t p a . . . R . . . all users. MULTICAST Q R A H . . . . . . all SNRs. FADING Great codes are not born great, they grow great? 5 / 52
An IR-HARQ Example CLAUDE SHANNON THE FATHER OF INFORMATION THEORY at the transmitter transmission # 1 transmission # 2 transmission # 3 transmission # 4 at the receiver CLAUDE SHANNON THE FATHER OF INFORMATION THEORY 6 / 52
IR-HARQ in Practice ◮ Considered one of the most important features of cdma2000 ◮ on the forward link in Release 0 and ◮ on both the reverse and forward links in Revision A ◮ The mother code is the rate 1 / 5 turbo code of IS-95. ◮ Decoding is ◮ attempted if the total received power is deemed sufficient, ◮ done by performing only a few iterations of message passing. ◮ There are ◮ prescribed puncturing patterns and error prediction methods, ◮ either predetermined or no transparent transmission rules. 7 / 52
Garden State (Video) Content Delivery Multicast is often the most cost effective, e.g., on stadiums. Verizon demoed LTE Multicast over the 2014 Super Bowl week. 8 / 52
Fountain (Rateless) Codes Claim to Fame Enable reliable communications over multiple, unknown channels: 1. They are rateless ⇒ their redundancy can be flexibly adapted to changing channel/network conditions of e.g. as in wireless. 2. In the case of multiple erasure channels as in Multimedia Broadcast/Multicast Service (MBMS), they can be made to universally achieve the channel capacity for all erasure rates. Fountain codes are simple to encode and decode . Isn’t Hybrid ARQ Enough? 9 / 52
Rateless 10 / 52
What is Ratelessness and is it Overrated? Ratlessness matters and means different things to different people: ◮ encoder can produce potentially infinite stream of symbols ◮ each code symbol is statistically identical Can fixed rate codes be made rateless? throughput 1 . BEC capacity R/ ( 1 − ǫ ∗ ) R/ ( 1 − ǫ ∗ ) . LDPC (rate R, BP threshold ǫ ∗ ) R R R R/ 2 . LT (length 10 4 , BP) ǫ √ ǫ ∗ 1 ǫ ∗ ǫ ∗ ǫ ∗ 11 / 52
The (Un)Bearable Ratelessness of Coding Two eMBMS Phases: Multicast Delivery & Unicast Repair: 12 / 52
What are Raptor Codes? for the conception, development, and analysis of practical rateless codes 13 / 52
Content Download Model and Objective Goal: Download a content chunk of k p packets with k s symbols per packet. Inner and Outer Codes & Channels: ◮ At the inner level, there is a (possibly rateless) ( n s , k s ) code. Symbols are sent through a channel with erasure probability ǫ s . ◮ At the outer level, there is a (possibly rateless) ( n p , k p ) code. What channel do the packets go through? To guarantee the file download, one code has to be rateless. If the inner code is rateless, there is no need for any outer coding. Feedback is instantaneous, noiseless, and possible after each symbol. 14 / 52
Infinite Incremental Redundancy (IIR) ◮ The inner (physical layer) code is rateless ⇔ coded symbols are transmitted until k s of them are received. ⇒ #transmissions/packet is ( k s , 1 − ǫ s ) negative Binomial ( NB ). ◮ The outer (packet level) code is not needed (no dropped packets). ◮ #transmissions/chunk T IIR is the sum of k p i.i.d. NB ( k s , 1 − ǫ s ) ⇒ also a negative Binomial with parameters ( k p · k s , 1 − ǫ s ) ⇒ = k s · k p T IIR � � E . 1 − ǫ s 15 / 52
Finite Redundancy (FR) ◮ A fixed-rate code ( n s , k s ) is used at the physical layer (per packet). ⇒ Each pcket is erased independently with probability n s � n s � � ǫ j s ( 1 − ǫ s ) n s − j . ǫ p = j j = n s − k s + 1 ◮ The outer (packet level) code is rateless ⇔ coded packets are transmitted until k p of them are received. ⇒ #packet-transmissions/chunk is NB ( k p , 1 − ǫ p ) . ◮ #symbol-transmissions/chunk T FR is not NB , but still, k p � T FR � = · n s . E 1 − ǫ p How do we compare = k s · k p = n s · k p T IIR � T FR � � � E vs. E ( n s � k s & ǫ s � ǫ p ) 1 − ǫ s 1 − ǫ p 16 / 52
IIR vs. FR in Point-to-Point Transmission � T IIR � � T FR � E � E Proof: Let S be the number of received symbols after n s transmission. ⇒ S is Binomial with expectation is n s · ( 1 − ǫ s ) . ⇒ 1 − ǫ p = P ( S � k s ) � n s · ( 1 − ǫ s ) , k s by the definition of ǫ p and Markov’s inequality. ♠ Ratless coding at the inner rather than at the outer layer results in fewer channel uses on average for chunk download. 17 / 52
Multicast Scenarios Point-to-Point Like: Infinite Incremental Redundancy (IIR): ◮ transmit coded symbols until k s are received by all users . ⇒ no coding or re-transmission at the packet level is needed. Fixed Redundancy (FR): ◮ a fixed a rate code ( n s , k s ) is used at the physical layer. ⇒ coding or retransmission at the packet level is required. ◮ transmit coded packets until k p are received by all users . Rateless coding may sometimes be used at both layers, e.g., & transmit coded symbols until k s are received by some ℓ users , & & and transmit coded packets until k p are received by all users . & 18 / 52
IIR vs. FR With u Users (normalized average download time) k p = 100 , k s = 100 , ǫ s ∈ { 0.1, 0.2, 0.3.0.4, 0.5 } . 2.8 Fixed Rate (FR) Infinite Incremental Redundancy (IIR) 2.6 2.4 2.2 normalized E[T] 2 1.8 1.6 1.4 1.2 1 0 1 2 3 4 10 10 10 10 10 number of users 19 / 52
IIR vs. FR in Multicast (normalized average download time) k p = 100 , k s = 1000 , ǫ s ∈ { 0.1, 0.2, 0.3.0.4, 0.5 } . 2.4 Fixed Rate (FR) Infinite Incremental Redundancy (IIR) 2.2 2 normalized E[T] 1.8 1.6 1.4 1.2 1 0 1 2 3 4 10 10 10 10 10 number of users 20 / 52
Codes for the Cloudy Season We want our content to be reliably stored & quickly accessible , and that requires energy for storage transmission computation Coding minimizes resources needed to maintain the guaranteed reliability. But what does it do to accessibility? 21 / 52
Getting Content from the Cloud(s) 22 / 52
( n , k ) Multiple Broadcasts Content Access Model ◮ Users request the same content (file F ), stored in the cloud. ◮ Server s , s = 1, . . . , n , 1. acquires F from the cloud at the time W s ( W s are iid RVs, exponentially distributed with the mean W ) 2. delivers F by broadcast to the users in time D s ( D s are iid RVs, highly concentrated around the mean D ) ⇒ File F download time from server s is W s + D s . ◮ F is split into k blocks and encoded into n blocks s.t. any k out of n blocks are sufficient for content reconstruction. ◮ Each user’s request is sent to all n servers. When do k out of n servers delver their F/k -size blocks? 23 / 52
Order Statistics ◮ For iid RVs { X 1 , X 2 , · · · , X n } , the k th smallest among them is an RV, known as the k th order statistics X k , n . ◮ When X i ’s are exp ( W ) , the mean and variance of X k , n are E [ X k , n ] = W ( H n − H n − k ) and V [ X k , n ] = W 2 ( H n 2 − H ( n − k ) 2 ) , where H n and H n 2 are (generalized) harmonic numbers n n 1 1 � � H n = and H n 2 = j 2 . j j = 1 j = 1 When k decreases, for fixed n E [ X k , n ] decreases; good for diversity but what about parallelism? 24 / 52
( n , k ) Multiple Broadcasts Response Time Theorem: 1. The mean download completion time is given by T n , k = W ( H n − H n − k ) + D k , where H n = 1 + 1 2 + · · · + 1 n . 2. T n , k depends on W vs. D , and is minimized when √ D 2 + 4 nWD k ∗ ≈ − D + 2 W 25 / 52
What is Really the Optimal k ? 4.0 ● 3.5 W=2, D=1 W=1, D=2 expected response time 3.0 ● 2.5 ● ● ● 2.0 ● ● 1.5 ● ● ● ● ● ● 1.0 ● ● ● ● ● 2 4 6 8 10 k VS. 26 / 52
Who’d Know? You know a lot of geniuses, y’know. You should meet some stupid people once in a while, y’know. You could learn something! 27 / 52
Queueing for Content 28 / 52
Two Queueing Models Single M/M/1 Queue: ◮ Requests arrive at rate λ according to a Poisson process. ◮ Job service times have an exponential distribution with rate µ . ◮ Many metrics of interest are well understood for this model, e.g. the response time is exponential with rate µ − λ . Fork-Join FHW Queue: ◮ Jobs are split on arrival and must be joined before departure. ◮ In the FHW (Flatto, Hahn, Wright) model, at each queue, ◮ Requests arrive at rate λ according to a Poisson process. ◮ Job service times have an exponential distribution with rate µ . ◮ It is seen as a key model for parallel/distributed systems, e.g., RAID. ◮ There is a renewed interest in the problem (e.g., map-reduce). ◮ Few analytical results exist, but various approximations are known. 29 / 52
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