Differential Encoding for Real-Time Status Updates Sanidhay Bhambay - PowerPoint PPT Presentation

Differential Encoding for Real-Time Status Updates Sanidhay Bhambay Sudheer Poojary Parimal Parag Electrical and Communication Engineering Indian Institute of Science, Bangalore The IEEE Wireless Communication and Networking Conference March 22, 2017 1/ 23

Why timely update? Cloud Server ◮ Critical to know the status update before decision making 2/ 23

Potential Scenarios Cloud Server ◮ Cyber-physical systems: Environmental/health monitoring ◮ Internet of Things: Real-time actuation/control 3/ 23

Link Model ˆ X ( t ) X ( t − n ) X n Y n Source Encoder Channel Decoder Monitor Feedback Channel Context ◮ Point-to-point communication with limited to no feedback ◮ Reliability through finite block-length coding 4/ 23

Source Model 7 Source state, X ( t ) 6 5 time, t 1 11 21 31 41 51 ◮ Source state X ( t ) can be represented by m bits ◮ State difference between n realizations can be represented by k < m bits 5/ 23

Problem Statement Question How to encode message at the temporally correlated source for timely update? Should one send the current state or the difference between the current and the past state? Answer It depends on the feedback 6/ 23

Coding Model ◮ Finite length code of n bits with permutation invariant code Updates ◮ True Update: current state X ( t ) of m bits is encoded to n bit codeword X n ◮ Incremental Update: the state difference X ( t ) − X ( t − n ) of k bits encoded to n bit codeword X n 7/ 23

Channel Model ◮ Each transmitted bit of the codeword X n erased iid with probability ǫ Erasure Distribution Number of erasures is Binomial with parameter ( n , ǫ ) 8/ 23

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 1 = E P ( n , n − m , E ) ◮ Incremental updates: p 2 = E P ( n , n − k , E ) ◮ Monotonicity: 0 < p 2 < p 1 < 1 9/ 23

Performance Metric ◮ Last successfully decoded source state at time t was generated at U ( t ) ◮ Information age 1 A ( t ) at time t as A ( t ) = t − U ( t ) . ◮ Limiting value of average age t 1 � lim A ( s ) . t t →∞ s =1 1 Kaul , S. , Yates , R. , & Gruteser M. , “Real-time status: How often should one update?”. IEEE INFOCOM , 2012 , pp. 2731–2735. . 10/ 23

Update Transmission Schemes True Updates ◮ Each opportunity send true update Incremental Updates without Feedback ◮ Periodically send the true update after q updates ◮ In between true updates, send incremental updates . Incremental Updates with Feedback ◮ Send the true update after each decoding failure ◮ In between true updates, send incremental updates . 11/ 23

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 S i − 1 � S ( T i ) = A ( t ) t = S i − 1 is also iid ◮ By renewal reward theorem, the limiting average age is t 1 � E A � lim A ( s ) = E S ( T i ) / E T i . t t →∞ s =1 12/ 23

Age Sample Path: True Updates 30 Age, A ( t ) 20 10 nZ 1 nZ 2 n n t 1 11 21 31 41 51 ◮ Inter-renewal time T i = nZ i ◮ Number of true update in i th renewal interval Z i ◮ { Z i : i ∈ N } is iid geometric with success parameter (1 − p 1 ) 13/ 23

Mean Age Theorem Limiting average age for the true update scheme is a.s. t 1 � E A � lim A ( s ) = ( n − 1) / 2 + n / (1 − p 1 ) . t t →∞ s =1 Proof. Cumulative age for i th renewal interval is nZ i − 1 � ( n + j ) = n 2 Z i + nZ i ( nZ i − 1) / 2 . S ( nZ i ) = j =0 14/ 23

Age Sample Path: Incremental Updates Without Feedback 50 40 A ( t ) 30 nq Age, ¯ n ( ¯ W 1 − 1) 20 10 T 1 = nqZ 1 n t 1 11 21 31 41 51 61 ◮ Inter-renewal time T i = nqZ i ◮ Number of successfully decoded contiguous incremental updates ¯ W i − 1 in the i th renewal interval ¯ ◮ W i is the number of successfully decoded updates in i th renewal interval 15/ 23

Mean Age Theorem Limiting average age for the incremental updates without feedback is t + n 2 E ¯ W i ( ¯ A ( s ) = E T 2 1 W i − 1) E ¯ � ¯ i A � lim 2 E T i 2 E T i t t →∞ s =1 � W i − 2) + 1 � n E ( ¯ − . 2 Proof. Cumulative age S ( T i ) in the i th renewal interval is ¯ W i − 1 n − 1 T i − 1 � � � ( n + j − n ( ¯ S ( T i ) = ( n + k ) + W i − 1)) , j = n ( ¯ j =1 k =0 W i − 1) = n 2 ¯ W i ( ¯ + T 2 � � W i − 1) W i − 2) + 1 n ( ¯ i 2 − T i . 2 2 16/ 23

Age Sample Path : Incremental Updates With Feedback 40 30 A ( t ) nW 1 Age, ˆ 20 nZ 1 10 T 1 = nZ 1 + nW 1 n t 1 11 21 31 41 51 61 ◮ Inter-renewal time T i = nZ i + nW i ◮ Number of incremental updates W i in i th renewal interval ◮ { W i : i ∈ N } are iid geometric with success parameter p 2 17/ 23

Mean Age Theorem Limiting average age for the incremental updates with feedback is t + n ( E Z 2 1 A ( s ) = (3 n − 1) i + E Z i ) E ˆ ˆ � A � lim 2( E W i + E Z i ) . t 2 t →∞ s =1 Proof. Cumulative age S ( T i ) over the i th renewal period T i is T i − n ( W i − 1) − 1 W i − 1 n − 1 � � � S ( T i ) = ( n + k ) + ( n + k ) j =1 k =0 k =0 = (3 n − 1) T i / 2 + n 2 ( Z i + 1) Z i / 2 . 18/ 23

Analytical Comparison Theorem The mean age for the three schemes satisfy, E ˆ A ≤ E A ≤ E ¯ A . 19/ 23

Numerical Comparision System Parameters ◮ Random coding scheme ◮ Code length n = 120 ◮ Number of information bits m = 105 ◮ q ∈ { 2 , 6 } 240 True update Incremental w/o feedback q = 2 Limiting average age Incremental w/o feedback q = 6 220 Incremental with feedback 200 180 110 112 114 116 118 120 Number of information bits, m 20/ 23

600 True update Incremental w/o feedback q = 2 Limiting average age 500 Incremental w/o feedback q = 6 Incremental with feedback 400 300 200 20 40 60 80 100 Differential information bits, k 21/ 23

350 True update Incremental w/o feedback q = 2 Limiting average age Incremental w/o feedback q = 6 300 Incremental with feedback 250 200 2 · 10 − 2 4 · 10 − 2 6 · 10 − 2 8 · 10 − 2 0 . 1 Erasure probability, ǫ 22/ 23

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 ◮ Incremental updates only when there is feedback availability Avenues of Future Research ◮ Extend results to structured sources ◮ Extend results to correlated finite-state erasure and error channels ◮ Impact of other coding schemes on timeliness 23/ 23