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 Wireless Communications and Networking Conference Mar 22, 2017 1/ 14
Timely Status Updates Cloud Server Potential Scenarios ◮ Cyber-physical systems: Environmental/health monitoring ◮ Internet of Things: Real-time actuation/control 2/ 14
Updates over Erasure Channel Phenomenon Unreliable Channel 1 1 ε e 0 0 Remote Sensor Destination 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 ) 3/ 14
Problem Statement Compare the optimal coding for finite block length and hybrid ARQ schemes for timely update. 4/ 14
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 � T 1 ¯ A � lim A ( t ) dt T T →∞ 0 Renewal Reward Theorem � T For a renewal interval T and accumulated reward 0 A ( t ) dt � T A = E 0 A ( t ) dt E A = ¯ E T 5/ 14
Finite Block Length Coding ◮ Maps message m ∈ F K 2 to codeword x ∈ F N 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 x ( y ) � = x } � E P f ( N − K , | E | ) ρ = E 1 { ˆ 6/ 14
Single Transmission Scheme Phenomenon Unreliable Channel 1 1 ε e Remote Sensor 0 0 Destination ◮ 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 7/ 14
Single Transmission Scheme 30 25 Timeliness, T ( t ) 20 15 10 NM 1 NM 2 5 N N t 5 10 15 20 25 30 35 40 45 ◮ Mean age is N + E [ M i ( NM i +1)] 2 E M i 8/ 14
Hybrid Automatic Repeat Requests ◮ Maps message m ∈ F K 2 to codeword x ∈ F aN of depth a 2 ◮ Decoding failure event is iid Bernoulli with probability ¯ f a � E P f ( aN − K , | E | ) ◮ Number of transmission attempts M before a success is Geometric with success probability 1 − ¯ f a 9/ 14
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 ¯ f i = E P f ( aN − K , | E | i + ( a − i ) N ) 10/ 14
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 11/ 14
Hybrid Automatic Repeat Requests ( aM 1 + R 1 ) N 30 ( aM 2 + R 2 ) N 25 Timeliness, T ( t ) 20 15 10 NR 1 5 NR 2 t 5 10 15 20 25 30 35 40 45 ◮ Mean age is N E R i − 1 + E [( aM i + R i )(( aM i + R i ) N +1)] 2 E ( aM i + R i ) 12/ 14
Mean Timeliness Result Performance for number of information bits K = 80 per message 400 300 Average Timeliness 200 ε = 0 . 05 100 ε = 0 . 1 ε = 0 . 2 ε = 0 . 3 0 40 60 80 100 120 140 160 180 Block Length, N 13/ 14
Conclusions ◮ An analytical framework for parameter selection for timely updates over unreliable channels ◮ Demonstration of natural tradeoff between erasure resilience and timely delivery ◮ Optimal code rate very sensitive to channel characteristics ◮ Limited feedback does not improve timeliness update performance 14/ 14
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