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On Coding for Real-Time Streaming under Packet Erasures

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On Coding for Real-Time Streamingunder Packet Erasures

- Derek Leong*#, AsmaQureshi*, and Tracey Ho*
- *California Institute of Technology, Pasadena, California, USA
- #Institute for Infocomm Research, Singapore
- ISIT 2013
- 2013-07-09

Introduction: Real-Time Streaming System

- Real-time streaming system where
- messages created at regular time intervals at a source are encoded for transmission to a receiver over a packet erasure link, and
- the receiver must subsequently decode each message within a given delay from its creation time

Introduction: Erasure Models and Objectives

- Types of erasure models:
- Deterministic:
- Window-based: Limited number of erasures in each window [1]
- Bursty: Erasure bursts of a limited length with minimum guard interval lengths
- IID: Each packet is erased independently with the same probability
- Objective for deterministic erasure models:

Find an erasure correction code that achieves the maximum message size, among all codes that allow all messages to be decoded by their respective deadlines under all admissible erasure patterns

- Objective for i.i.d. erasure model:

For a given message size, find an erasure correction code that achieves the maximum probability of decoding a message by its deadline

[1] D. Leong and T. Ho, “Erasure coding for real-time streaming,” in Proc. IEEE ISIT, Jul. 2012.

Introduction: Related Work

- Related work:
- Martinianet al. [2,3] and Badret al. [4] provided constructions of streaming codes that minimize the decoding delay for certain types of bursty erasure models[2] E. Martinian and C.-E. W. Sundberg, “Low delay burst erasure correction codes,” in Proc. IEEE ICC, May 2002.[3] E. Martinian and M. Trott, “Delay-optimal burst erasure code construction,”in Proc. IEEE ISIT, Jun. 2007.[4] A. Badr, A. Khisti, W.-T. Tan, and J. Apostolopoulos, “Streaming codes for channelswith burst and isolated erasures,” in Proc. IEEE INFOCOM, Apr. 2013.
- Tree codes or anytime codes, for which the decoding failure probability decays exponentially with delay, are examined in [5,6,7][5] L. J. Schulman, “Coding for interactive communication,” IEEE Trans. Inf. Theory, Nov. 1996. [6] A. Sahai, “Anytime information theory,” Ph.D. dissertation, MIT, 2001.[7] R. T. Sukhavasi, “Distributed control and computing: Optimal estimation, error correcting codes, and interactive protocols,” Ph.D. dissertation, Caltech, 2012.

Problem Definition: Real-Time Streaming System

- Discrete-time data streaming system
- Independent messages of uniform size sare created at regular intervals of c time steps at the source; at each time step, the source transmits a single data packet of normalized unit size over the packet erasure link
- Receiver must decode each message within a delay of dtime steps from its creation time

messages are created everyc = 3 time steps …

… and have to be decoded withind = 8 time steps

Problem Definition: Erasure Patterns and Models

- An erasure pattern specifies a set of erased packet transmissions
- An erasure model describes a distribution of erasure patterns

Erasure pattern #2 = {2, 3, 13, 14, 15, 16}

Erasure pattern #1 = {7, 12, 20}

Bursty Erasures: Problem Definition

- Define the set of admissible erasure patterns to be those in which
- each erasure burst consists of at most z erased time steps
- consecutive bursts are separated by a guard interval or gapof at least d−z unerased time steps (each message sees at most z erasures)
- Interested in erasure correction codes that allow all messagesto be decoded by their respective deadlines under any admissible erasure pattern
- Specifically, we want to find a code that achieves the maximum message size s, for a given choice of (c, d, z)

Other bursty models possible…

Bursty Erasures: Results

- Previously in [1], we constructed a time-invariant intrasession code that is asymptotically optimal (as the number of messages n goes to infinity) in the following cases:
- d is a multiple of c
- d is not a multiple of c, and the maximum erasure burst length zis sufficiently short, i.e., z ≤ c − r(d,c)
- d is not a multiple of c, and the maximum erasure burst length zis sufficiently long, i.e., z ≥ d − r(d,c)
- Here, we construct time-invariant diagonally interleaved codes that are asymptotically optimal in several other cases

[1] D. Leong and T. Ho, “Erasure coding for real-time streaming,” in Proc. IEEE ISIT, Jul. 2012.

Bursty Erasures: Diagonally Interleaved Codes

2. Apply a systematic block code for d −z information symbols and z parity symbolson each diagonal.

3. Diagonally interleaved codes derived from systematic block codes with certain properties are asymptotically optimal for the bursty erasure model.

4. These sufficient code properties describe decoding deadlines for individual symbols.

1. About symbols, packets, and messages.

Here (c,d,z) = (3,11,4)

Bursty Erasures: Diagonally Interleaved Code Example

Consider the d symbols of one codeword

Bursty Erasures: Diagonally Interleaved Code Example

Rearrange these d symbols as follows

Here (c,d,z) = (5,57,42)

Bursty Erasures: Diagonally Interleaved Code Example

Actual information symbolsNondegenerate parity symbolsDegenerate parity symbols

Here (c,d,z) = (5,57,42)

Symbol decoding deadlines

Bursty Erasures: Diagonally Interleaved Code Example

Example of an erasure burst…

Here (c,d,z) = (5,57,42)

IID Erasures: Problem Definition

- Each packet transmitted over the link is erased independently withthe same probability pe
- Want to find an erasure correction code that achieves themaximum message decoding probability, for a given message size s
- Primary performance metric:Decoding probability, i.e., the probability that a given message is decodable by its decoding deadline
- Secondary performance metric:Burstiness of undecodable messages, i.e., the conditional probability that the next message is undecodable by its decoding deadline given that the current message is undecodable by its decoding deadline
- In the interest of practicality, we restrict our attention to time-invariant codes

for applications sensitive to bursts of decoding failures

IID Erasures: Decoding Probability

- Assume decoder memory size is unbounded
- The probability of decoding a given message k can be expressed in terms of conditional entropies as follows
- Here, finding an optimal code is a combinatorial problem involving probabilistic erasure patterns, in contrast to the deterministic“worst-case” problem formulation for the other erasure models

unerased packets received up to the decoding deadline for message k

message k

criteria for successful decoding

condition over all erasure patterns

probability of the corresponding erasure pattern

IID Erasures: Decoding Probability Upper Bound

- We derive the following upper bound on the decoding probability for any time-invariant code:
- Proof combines the bounding technique from our work on thesliding window erasure model [1] with a combinatorial analysis on the maximum number of erasure patterns that can be supported

for any message k ≥ encoder memory size mE

IID Erasures: Symmetric Codes

- For intrasession codes
- coding is allowed within the same message but not across different messages
- the unit-size packet at each time step is divided into blocks allocated to different messages; a suitable code (e.g., MDS) is subsequently applied
- For symmetriccodes, we define a spreading parameter m , and divide the packet at each time step evenly among all active messages

Spreading parameter m is a multiple of c: Same-size blocks

m = 9

c= 3

m = 9

IID Erasures: Symmetric Codes

- For intrasession codes
- coding is allowed within the same message but not across different messages
- the unit-size packet at each time step is divided into blocks allocated to different messages; a suitable code (e.g., MDS) is subsequently applied
- For symmetriccodes, we define a spreading parameter m , and divide the packet at each time step evenly among all active messages

Spreading parameter m is not a multiple of c: Big and small blocks

m = 8

c= 3

m = 8

IID Erasures: Symmetric Codes Performance

- Decoding ProbabilityPlots of the decoding failure probability against... message size s packet erasure probability pe(for pe = 0.05)(for s= 1)

maximal spreading performs well whens and pe are small

minimal spreading performs well whens and pe are large

IID Erasures: Symmetric Codes Performance

- Burstiness of Undecodable MessagesPlots of the conditional probability against... message size s packet erasure probability pe(for pe = 0.05)(for s= 1)

results agree with our intuition about overlapping effective coding windows

minimal spreading performs well over a wide range ofs and pe

IID Erasures: Symmetric Codes Performance

- Trade-off between performance metrics
- When the message size and packet erasure probability are small (a regime of interest), maximal spreading achieves a high decoding probability, but it also exhibits a higher burstiness of undecodable messages
- For applications sensitive to bursty undecodable messages,a symmetric code with a suboptimal decoding probability butlower burstiness may be preferred

Conclusion

- For the bursty erasure model
- We constructed diagonally interleaved codes that are asymptotically optimal over all codes in several specific cases
- For the i.i.d. erasure model
- We derived an upper bound on the decoding probabilityfor any time-invariant code, and
- We analyzed the performance of symmetric codes(observed phase transitions, good codes, trade-offs)

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