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On Real-Time Status Updates over Symbol Erasure Channels Parimal - - PowerPoint PPT Presentation
On Real-Time Status Updates over Symbol Erasure Channels Parimal - - PowerPoint PPT Presentation
On Real-Time Status Updates over Symbol Erasure Channels Parimal Parag Austin Taghavi Jean-Francois Chamberland Electrical Communication Engineering Indian Institute of Science Electrical and Computer Engineering Texas A&M University
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Updates over Erasure Channel
Remote Sensor Destination Unreliable Channel Phenomenon
1 1 ε e
System Model
◮ Source has K bit message to send at all times ◮ One bit per channel use can be sent ◮ Bit-wise erasures iid Bernoulli with probability ǫ ◮ Number of i erasures in N channel usage is Binomial (N, ǫ)
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Problem Statement
Compare the optimal coding for finite block length and hybrid ARQ schemes for timely update.
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Timeliness Metric
U(t) generation time of last successfully received message, then information age A(t) = t − U(t)
Empirical information age
Limiting empirical information age is ¯ A lim
T→∞
1 T T A(t)dt
Renewal Reward Theorem
For a renewal interval T and accumulated reward T
0 A(t)dt
EA = ¯ A = E T
0 A(t)dt
ET
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Finite Block Length Coding
◮ Maps message m ∈ FK 2 to codeword x ∈ FN 2 ◮ Parity check matrix H such that Hx = 0 for any codeword x ◮ Number of erasures |E| in a block N are Binomial (N, ǫ) ◮ Decoding failure event is iid Bernoulli with probability
ρ = E1{ˆ x(y) = x} EPf (N − K, |E|)
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Single Transmission Scheme
Remote Sensor Destination Unreliable Channel Phenomenon
1 1 ε e
◮ Encode K length message to N length codeword ◮ Transmit N length codeword over N channel usage ◮ Transmit new codeword at next transmission opportunity ◮ Number of transmission attempts M before a success is
Geometric with success probability 1 − ρ
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Single Transmission Scheme
5 10 15 20 25 30 35 40 45 5 10 15 20 25 30
NM1 NM2 N N
t Timeliness, T(t)
◮ Mean age is N + E[Mi(NMi+1)] 2EMi
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Hybrid Automatic Repeat Requests
◮ Maps message m ∈ FK 2 to codeword x ∈ FaN 2
- f depth a
◮ Decoding failure event is iid Bernoulli with probability
¯ fa EPf (aN − K, |E|)
◮ Number of transmission attempts M before a success is
Geometric with success probability 1 − ¯ fa
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Hybrid Automatic Repeat Requests
◮ Successive decoding failures with effective number of erasures
|E|i + (a − i)N
◮ Decoding failure event after transmission of depth i of
aN-length codeword is Bernoulli with probability ¯ fi = EPf (aN − K, |E|i + (a − i)N)
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Hybrid Automatic Repeat Requests
◮ Number of erasures in first iN bits is |E|i ≤ |E|i−1 + N ◮ Number of effective erasure decreasing with each transmission
|E|1 + (a − 1)N ≥ |E|2 + (a − 2)N ≥ · · · ≥ |E|a
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Hybrid Automatic Repeat Requests
5 10 15 20 25 30 35 40 45 5 10 15 20 25 30
(aM1 + R1)N (aM2 + R2)N NR1 NR2
t Timeliness, T(t)
◮ Mean age is NERi−1 + E[(aMi+Ri)((aMi+Ri)N+1)] 2E(aMi+Ri)
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Mean Timeliness Result
40 60 80 100 120 140 160 180 100 200 300 400 Block Length, N Average Timeliness Performance for number of information bits K = 80 per message ε = 0.05 ε = 0.1 ε = 0.2 ε = 0.3
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