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Differential Encoding for Real-Time Status Updates Sanidhay Bhambay - - PowerPoint PPT Presentation
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
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Potential Scenarios
Cloud Server
◮ Cyber-physical systems: Environmental/health monitoring ◮ Internet of Things: Real-time actuation/control
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Link Model
Source Encoder X(t) Channel X n Feedback Channel Decoder Y n Monitor ˆ X(t − n)
Context
◮ Point-to-point communication with limited to no feedback ◮ Reliability through finite block-length coding
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Source Model
1 11 21 31 41 51 5 6 7 time, t Source state, X(t)
◮ Source state X(t) can be represented by m bits ◮ State difference between n realizations can be represented by
k < m bits
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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
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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
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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)
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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: p1 = EP(n, n − m, E) ◮ Incremental updates: p2 = EP(n, n − k, E) ◮ Monotonicity: 0 < p2 < p1 < 1
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Performance Metric
◮ Last successfully decoded source state at time t was
generated at U(t)
◮ Information age1A(t) at time t as
A(t) = t − U(t).
◮ Limiting value of average age
lim
t→∞
1 t
t
- s=1
A(s).
1Kaul, S., Yates, R., & Gruteser M., “Real-time status: How often should one update?”. IEEE INFOCOM,
2012, pp. 2731–2735..
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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.
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Renewal Reward Theorem
◮ Time instant Si of the ith successful reception of the true
update
◮ For all three schemes, the ith inter-renewal time
Ti = Si − Si−1 is iid
◮ Accumulated age in ith renewal period
S(Ti) =
Si−1
- t=Si−1
A(t) is also iid
◮ By renewal reward theorem, the limiting average age is
EA lim
t→∞
1 t
t
- s=1
A(s) = ES(Ti)/ETi.
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Age Sample Path: True Updates
1 11 21 31 41 51 10 20 30 nZ1 nZ2 n n t Age, A(t) ◮ Inter-renewal time Ti = nZi ◮ Number of true update in ith renewal interval Zi ◮ {Zi : i ∈ N} is iid geometric with success parameter (1 − p1)
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Mean Age
Theorem
Limiting average age for the true update scheme is a.s. EA lim
t→∞
1 t
t
- s=1
A(s) = (n − 1)/2 + n/(1 − p1).
Proof.
Cumulative age for ith renewal interval is S(nZi) =
nZi−1
- j=0
(n + j) = n2Zi + nZi(nZi − 1)/2.
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Age Sample Path: Incremental Updates Without Feedback
1 11 21 31 41 51 61 10 20 30 40 50 n( ¯ W1 − 1) nq T1 = nqZ1 n t Age, ¯ A(t) ◮ Inter-renewal time Ti = nqZi ◮ Number of successfully decoded contiguous incremental
updates ¯ Wi − 1 in the ith renewal interval
◮
¯ Wi is the number of successfully decoded updates in ith renewal interval
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Mean Age
Theorem
Limiting average age for the incremental updates without feedback is E ¯ A lim
t→∞
1 t
t
- s=1
¯ A(s) = ET 2
i
2ETi + n2E ¯ Wi( ¯ Wi − 1) 2ETi −
- nE( ¯
Wi − 2) + 1 2
- .
Proof.
Cumulative age S(Ti) in the ith renewal interval is S(Ti) =
¯ Wi−1
- j=1
n−1
- k=0
(n + k) +
Ti−1
- j=n( ¯
Wi−1)
(n + j − n( ¯ Wi − 1)), = n2 ¯ Wi( ¯ Wi − 1) 2 + T 2
i
2 −
- n( ¯
Wi − 2) + 1 2
- Ti.
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Age Sample Path : Incremental Updates With Feedback
1 11 21 31 41 51 61 10 20 30 40 nW1 nZ1 T1 = nZ1 + nW1 n t Age, ˆ A(t) ◮ Inter-renewal time Ti = nZi + nWi ◮ Number of incremental updates Wi in ith renewal interval ◮ {Wi : i ∈ N} are iid geometric with success parameter p2
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Mean Age
Theorem
Limiting average age for the incremental updates with feedback is E ˆ A lim
t→∞
1 t
t
- s=1
ˆ A(s) = (3n − 1) 2 + n(EZ 2
i + EZi)
2(EWi + EZi).
Proof.
Cumulative age S(Ti) over the ith renewal period Ti is S(Ti) =
Wi−1
- j=1
n−1
- k=0
(n + k) +
Ti−n(Wi−1)−1
- k=0
(n + k) = (3n − 1)Ti/2 + n2(Zi + 1)Zi/2.
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Analytical Comparison
Theorem
The mean age for the three schemes satisfy, E ˆ A ≤ EA ≤ E ¯ A.
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Numerical Comparision
System Parameters
◮ Random coding scheme ◮ Code length n = 120 ◮ Number of information bits m = 105 ◮ q ∈ {2, 6} 110 112 114 116 118 120 180 200 220 240 Number of information bits, m Limiting average age True update Incremental w/o feedback q = 2 Incremental w/o feedback q = 6 Incremental with feedback
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20 40 60 80 100 200 300 400 500 600 Differential information bits, k Limiting average age True update Incremental w/o feedback q = 2 Incremental w/o feedback q = 6 Incremental with feedback
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2 · 10−2 4 · 10−2 6 · 10−2 8 · 10−2 0.1 200 250 300 350 Erasure probability, ǫ Limiting average age True update Incremental w/o feedback q = 2 Incremental w/o feedback q = 6 Incremental with feedback
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