Real-Time Status Updates for Correlated Source Sudheer Poojary Sanidhay Bhambay Parimal Parag Electrical and Communication Engineering Indian Institute of Science, Bangalore IEEE Information Theory Workshop November 08, 2017 1/ 15
Why timely update? Real-Time Update Cloud Server ◮ Critical to know the status update before decision making ◮ Cyber-physical systems: Environmental/health monitoring ◮ Internet of Things: Real-time actuation/control 2/ 15
How to measure timeliness? Real-Time Update 30 Age, A ( t ) 20 10 nZ 1 nZ 2 n n t 1 11 21 31 41 51 ◮ Last correctly decoded message generated at U ( t ) ◮ Smaller the age A ( t ) = t − U ( t ), more timely the message ◮ Goal: Minimize limiting average age lim t →∞ 1 � t s =1 A ( s ) t 3/ 15
Link Model Control Channel X n Y n ˆ M ( t ) M ( t − n ) j j − 1 Source Encoder Channel Decoder Monitor Feedback Channel Context ◮ Point-to-point communication with limited feedback ◮ Reliability through finite block-length coding ◮ Control channel with information about coding scheme 4/ 15
Source Model 7 Source state, M ( t ) 6 5 time, t 1 11 21 31 41 51 ◮ Sampled source M j ∈ ∆ m Markov with transition matrix P ◮ Probability of the state difference M j +1 − M j ∈ ∆ k independent of the initial state 5/ 15
Update Protocol True Update Encode current state M j ∈ ∆ m to n bit codeword X n Incremental Update ◮ State difference M j − M j − 1 ∈ ∆ k almost surely ◮ If last update successfully received, then encode state difference to n bit codeword X n Generalized Incremental Update If state difference M j − M j − 1 ∈ ∆ k and last update successfully received, then send incremental update, else send true update 6/ 15
Problem Statement Question Find the differential encoding threshold k for timely update of a Markov source that minimizes limiting average age Answer ◮ Higher threshold: more differential encoding opportunities ◮ Lower threshold: more error protection 7/ 15
Coding and Channel Model Finite-length Code ◮ Finite length code of n bits with permutation invariant code Bit-wise Erasure Channel 1 − ǫ 1 1 ǫ e ǫ 0 0 1 − ǫ ◮ Each transmitted bit of the codeword X n erased iid with probability ǫ ◮ Number of erasures per codeword E Binomial ( n , ǫ ) 8/ 15
Decoding and Reception Receiver Timing Reception at time t + n of n bits sent at time t after n channel uses Probability of Decoding Failure ◮ True updates: p t = E P ( n , n − m , E ) ◮ Incremental updates: p d = E P ( n , n − k , E ) ◮ Monotonicity: 0 < p d < p t < 1 Differential Encoding Probability Probability of source state difference being represented by k bits, p e = P i (∆ k ) for all states i 9/ 15
Renewal Reward Theorem ◮ Time instant S i of the i th successful reception of the true update ◮ For all three schemes, the i th inter-renewal time T i = S i − S i − 1 is iid ◮ Accumulated age in i th renewal period also iid S i − 1 � S ( T i ) = A ( t ) t = S i − 1 ◮ By renewal reward theorem, the limiting average age is t 1 A ( s ) = E S ( T i ) � E A � lim . t E T i t →∞ s =1 10/ 15
Age Sample Path: True Updates 40 30 A ( t ) nW s nW f Age, ˜ 1 1 20 nZ 1 10 T 1 = nZ 1 + nW s 1 + nW f 1 n t 1 11 21 31 41 51 61 ◮ True updates Z i , iid geometric with success prob (1 − p t ) ◮ Successful incremental updates W s i , iid geometric with success probability p e (1 − p d ) ◮ Failed incremental updates W f i , iid Bernoulli with success p e p d probability 1 − p e (1 − p d ) 11/ 15
Mean Age 40 30 A ( t ) nW s nW f Age, ˜ 1 1 20 nZ 1 10 T 1 = nZ 1 + nW s 1 + nW f 1 n t 1 11 21 31 41 51 61 Theorem Limiting average age for the true update scheme is a.s. i ) 2 + n E ( W f t i + Z i ) 2 2 + n E ( W s 1 A ( s ) = n − 1 � E A � lim . 2( E W s i + E W f t i + E Z i ) t →∞ s =1 12/ 15
Uniform IID Source 38 True update Generalized incremental ( iid case) Incremental update 36 Limiting average age 34 32 30 28 2 4 6 8 10 12 14 Differential information bits (k) System Parameters ◮ Differential encoding prob p e = 2 k 2 m ◮ Random coding, erasure probability ǫ = 0 . 1 ◮ Code length n = 20, information bits m = 15 13/ 15
State Homogeneous Markov Source True update 36 Generalized incremental with α = 0 . 7 Generalized incremental with α = 0 . 1 Limiting average age 34 Incremental update 32 30 28 2 4 6 8 10 12 14 Differential information bits (k) System Parameters ◮ Transition probability P i , i ± 1 = α 2 , P i , i = 1 − α ◮ Random coding, erasure probability ǫ = 0 . 1 ◮ Code length n = 20, information bits m = 15 14/ 15
Discussion and Concluding Remarks Main Contributions ◮ Integration of coding and renewal techniques to study timely communication for delay-sensitive traffic ◮ We model channel unreliability by the erasure channel ◮ Model source correlation by Markov process ◮ True and incremental updates are special cases Avenues of Future Research ◮ Extend results to correlated finite-state erasure and error channels ◮ Impact of other coding schemes on timeliness 15/ 15
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