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Fresh Samples through Queues: Age of Information and Remote Estimation

Fresh Samples through Queues: Age of Information and Remote Estimation. Yin Sun Dept. ECE, Auburn University Joint work with Tasmeen Zaman Ornee Benjamin Cyr. Age of Information: Definition. Age at receiver. Info. packets/ Data samples.

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Fresh Samples through Queues: Age of Information and Remote Estimation

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  1. Fresh Samples through Queues: Age of Information and Remote Estimation Yin Sun Dept. ECE, Auburn University Joint work with Tasmeen Zaman OrneeBenjamin Cyr

  2. Age of Information:Definition Age at receiver Info. packets/ Data samples Definition: At time , the Age of Information is time difference between the current time and the generation time of the latest received data sample • If sample is generated at and delivered at channel Receiver (Monitor/controller/ Database) Time difference between data generation and usage

  3. A Brief Historyon Age of Information • Defined in real-time databases [Song-Liu’90,…] • Queueing analysis [Kaul-Gruteser-Rai-Kenney’11, Yates-Kaul’12, Kam-Kompella-Ephremides’13,14, …] • Scheduling and age minimization [Altman et al.’10, Yates’15, He-Yuan-Ephremides’16, Sun et al.’16, Bedewy-Sun-Shroff’16&17,…] • Source coding [Zhong-Yates’16, Zhong-Yates-Soljanin’17,…] • Channel coding [Najm-Yates-Soljanin’17, Yates-Najm-Soljanin-Zhong’17,…] • Energy harvesting [Bacinoglu-Ceran-Uysal’15, Wu-Yang-Wu’17, Bacinoglu-Uysal’17, Arafa-Ulukus’17,…] • Freshness of Channel State Information [Costa-Valentin-Ephremides’15, Farazi-Klein-Brown’17,…] • Caching [Yates-Ciblat-Yener-Wigger’17, Kam et al.’17] • Game theory [Nguyen et al. 17, Xiao-Sun’18] • Age of information and InformationTheory[Baknina-Ozel-Yang-Ulukus-Yener’18] • Age of information and learning [Sert-Sonmez-Baghaee-Uysal-Biyikoglu’18, Ceran-Gunduz-Gyorgy’18] • Age of information and Control [Zhang-Wang’18, Soleymani-Baras-Johansson’18,…] • Human in the loop [Bastopcu-Ulukus’18, Bin-Liu’19]

  4. Number of papers on AoI 2015 2017 2018 2019 （by April 29） 2016 2014 2012 2013 2010 2011

  5. Is Age a Good Metric for Freshness? • Age: Time difference between data generation and usage • Issue: • Some information changes fast (e.g., car location) • Update frequently (sub-second), more bandwidth • Other information changes slowly (e.g., temperature) • Update infrequently (hourly), less bandwidth

  6. Tony’s Question • Tony asked a question in the Open Problem Session at ITA 2015 • …There is a correlation structure in the signal. Clearly, the actual age is not a good representation of freshness. • What is freshness, can we get an Information Theoretical measure of freshness? • There should be a concept of effective age…

  7. A Possible Answer: Nonlinear Age Functions • Penalty function : can be any non-decreasing function • Dissatisfaction for data staleness, eagerness for refreshing • Utility function : can be any non-increasing function • Utility value of fresh data • Examples: • Temporal Auto-Correlation Function [Kosta-Pappas-Ephremides-Angelakis’ISIT2017] • For example, auto-correlation function of CSI is a non-increasing age function for small age [Truong-Health’JCN2013]

  8. Nonlinear Age Functions (cont’d) • More Examples: • Estimation Error of Real-time Signal: Estimate signal value based on causally received samples. • If sampling times are independent of the signal, estimation error is an age penalty function. [Sun-Polyanskiy-Uysal’ISIT17] • Wiener process: [Yates-Kaul’ISIT12, Sun-Polyanskiy-Uysal’ISIT17] • Ornstein-Uhlenbeck process: continuous-time AR (1) [Ornee-Sun’19] • Note: If sampling times are determined based on causal knowledge of the signal, estimation error is not a function of age. Ornee and Sun, “Sampling for Remote Estimation through Queues: Age of Information and Beyond,” IEEE WiOpt, 2019.

  9. Nonlinear Age Functions (cont’d) • More Examples: • Mutual Information and Conditional Entropy: • Let be the set of samples received up to time • Mutual information: amount of information contained in [Sun-Cyr’SPAWC18] • Conditional Entropy: uncertainty after knowing [Soleymani-Hirche-Baras’CDC, ECC, WODES2016] • If sampling times are independent of the signal and signal is a stationary Markov chain, mutual information is an age utility function, conditional entropy is an age penalty function[Sun-Cyr’JCN2019] • Otherwise, they are not functions of age Sun and Cyr, “Information Aging through Queues: A Mutual Information Perspective,” IEEE SPAWC, 2018. Sun and Cyr, “Sampling for Data Freshness Optimization: Non-linear Age Functions,” JCNspecialissueonAgeofInfo., 2019.

  10. Nonlinear Age Functions (cont’d) • More Examples: • Estimation Error in Feedback Control System: • Samples of the system state are sent to the controller. • The controller determines based on causally received samples. • The estimation error is independent of control. • If sampling times are independent of the signal, is an age penalty function [Champati et. al’19, Klugel et. al’19];otherwise, it is not afunctionof age. • Nonlinear age functions date back to 2003. A short survey available in [Sun-Cyr’JCN2019] • Nonlinear age functions seem simple enough and general enough, but not always perfect Sun and Cyr, “Sampling for Data Freshness Optimization: Non-linear Age Functions,” JCNspecialissueonAgeofInfo., 2019.

  11. Our Researchon Age of Information • Optimal Sampling and Scheduling • From AoI to Sampling theory • Scheduling Only • Given arrival process • optimal scheduling in networks with multiple channels, hops, and/or sources channel channel Info./sample arrivals sensor [INFOCOM’16,TIT’17, ISIT’17, JCN special issue on AoI ’19, WiOpt’19, MobiHoc’19] receiver queue receiver sampling Scheduling Scheduling Scheduling [ISIT’16 & 17, AoI Workshop’18, TIT’19, TON’19]

  12. Sampling for Optimizing Nonlinear Age Functions Joint work with Benjamin Cyr

  13. Model:Fresh Samples through Queues • Two terminals: Sampler sends samples to Receiver • Channel: FIFO queue with i.i.d. transmission times • Feedback: ACK • Zero delay  sampler knows channel state (idle/busy) • Sample is generated at , with channel transmission time • has a generaldistribution • Can be discrete/continuous random variable (finite moments) • Robustness of CPS under occasionally long commun. delay ACK sampler receiver queue channel

  14. Problem: Continuous-time Sampling age penalty function Expected time-average age penalty Avg. sampling-rate constraint inter-sample time • is arbitrary non-decreasing function of the age • Age utility maximization handled by choosing • Sampling policy: sequence of sampling times • Space of causal policies: Sampling time decided by the history information of the system Question: Which sampling policy minimizes the age penalty?

  15. Problem: Continuous-time Sampling age penalty function Expected time-average age penalty Avg. sampling-rate constraint inter-sample time • Constrainedcontinuous-time MDP with continuous state space • Usually difficult to solve

  16. Solution: UnconstrainedContinuous-time Sampling Theorem 1. For unconstrained problem, optimalsampling times are given by where and is the root of Deterministic threshold policy Expected age penalty exceeds threshold After previous sample is delivered Bisection search (no “curse of dimensionality”) Threshold = optimal objective value General penalty function, general service time distribution, with finite moments MC simulation with Structure more insightful than [Sun-Uysal-Yates-Koksal-Shroff, TIT17] Sun and Cyr, “Sampling for Data Freshness Optimization: Non-linear Age Functions,” https://arxiv.org/abs/1812.07241, 2018.

  17. Bisection Search for • Goal: Solve • Method: Bisection search • where

  18. Solution: ConstrainedContinuous-time Sampling Theorem 2. The optimalsampling times are given by Theorem 1 if Otherwise, where is determined by solving Then, is given by Randomized threshold policy Bisection search. No curse of dimensionality Sun and Cyr, “Sampling for Data Freshness Optimization: Non-linear Age Functions,” https://arxiv.org/abs/1812.07241, 2018.

  19. Solution: ConstrainedContinuous-time Sampling Corollary. Suppose is strictly increasing. Then, the optimalsampling times are given by Theorem 1 if Otherwise, where is determined by solving two thresholds are the same Bisection search. No curse of dimensionality Sun and Cyr, “Sampling for Data Freshness Optimization: Non-linear Age Functions,” https://arxiv.org/abs/1812.07241, 2018.

  20. Solution: UnconstrainedDiscrete-time Sampling Theorem 3. The optimalsampling times are given by where and is the root of Deterministic threshold policy discrete-time Expected age penalty exceeds threshold After previous sample is delivered Bisection search No curse of dimensionality Threshold = optimal objective value are integers. Scalable with clock rate Sun and Cyr, “Information Aging through Queues: A Mutual Information Perspective,” IEEE SPAWC, 2018. Sun and Cyr, “Sampling for Data Freshness Optimization: Non-linear Age Functions,” https://arxiv.org/abs/1812.07241, 2018.

  21. Solution: ConstrainedDiscrete-time Sampling Theorem 4. The optimalsampling times are given by Theorem 3 if Otherwise, where is determined by solving Then, is given by Randomized threshold policy Bisection search. No curse of dimensionality Sun and Cyr, “Sampling for Data Freshness Optimization: Non-linear Age Functions,” https://arxiv.org/abs/1812.07241, 2018.

  22. Is the optimal sampling policy deterministic or randomized? • Why randomized? • If one policy cannot meet the sampling rate constraint with equality, a randomized mixture of two policies can.

  23. Sampling for Signal Updates Joint work with Tasmeen Zaman Ornee

  24. Motivation: Real-time Monitoring & Control • Real-time information usually in the form of a signal • Age-of-information: time-difference between source and destination • time-difference signal-difference controller observer/ reactor channel monitor

  25. Model: Timely Signal Updates • Sampler: Take samples of an Ornstein-Uhlenbeck (OU) process • continuous-time analogue of AR(1) model • Defined by • and is Wiener process • MMSE Estimator: Use causally received samples to estimate the real-time value of • Channel: FIFO queue with i.i.d. service times • ACK: Zero delay Goal: Find optimal sampling policy that minimizes estimation error between signal and estimate ACK estimator sampler Estimate Signal: Ornstein-Uhlenbeck process queue channel

  26. Problem: Optimal Sampling Estimation error • Sampling policy: • Causal policy space : Sampling time decided by the history of signal and channel processes (in math, stopping time) • Continuous-time MDP with continuous state space and a constraint • Quite challenging to solve Time-avg. estimation error (MSE) Avg. sampling-rate constraint inter-sample time Questions: Which sampling policy is optimal? What is the relationship with the age of information?

  27. Question 1: Which sampling policy is optimal? Theorem. For unconstrained problem, a threshold-based sampler is optimal: The optimal is determined by solving: Threshold function After previous sample is delivered Signal difference exceeds threshold Bisection search (no “curse of dimensionality”) = optimal objective value Ornee and Sun, “Sampling for Remote Estimation through Queues: Age of Information and Beyond,” IEEE WiOpt, 2019.

  28. EvaluatingExpectations By Dynkin's formula and optional stopping theorem, we get where

  29. Wiener Process vs. Ornstein-Uhlenbeck Process Theorem: (Wiener Process) The MSE-optimal sampling policy is optimal is determined by solving threshold function Theorem: (Ornstein-Uhlenbeck Process) The MSE-optimal sampling policy is optimal is determined by solving threshold function instantaneous estimation error instantaneous estimation error = optimal objective value = optimal objective value The same solution structure, different threshold functions Ornee and Sun, “Sampling for Remote Estimation through Queues: Age of Information and Beyond,” IEEE WiOpt, 2019.

  30. Question 2: Relation with Age of Information? • Recall that sampling time decided by the history of signal and channel processes • If the sampling times are independent of signal , then Our problem reduces to an age penalty minimizaition problem. (MSE = Age penalty function)

  31. Age-Optimal vs. MSE-Optimal Sampling Theorem: (Age-Optimal Sampling) The age-optimal sampling policy is optimal is determined by solving Theorem: (MSE-Optimal Sampling) The MSE-optimal sampling policy is optimal is determined by solving instantaneous estimation error expected estimation error Threshold function = optimal objective value = optimal objective value Solutions to constrained problems available in the papers. Sun and Cyr, “Sampling for Data Freshness Optimization: Non-linear Age Functions,” https://arxiv.org/abs/1812.07241, 2018. Ornee and Sun, “Sampling for Remote Estimation through Queues: Age of Information and Beyond,” IEEE WiOpt, 2019.

  32. Summary • Optimal sampler follows a deterministic or randomized threshold policy. • For signal-aware sampling, the threshold is on instantaneous estimation error, and it is a function of optimal objective value. • For signal-ignorant sampling, the threshold is on expected estimation error, and it is equal to the optimal objective value. • It is conjectured that this structure holds for other strong Markov signals. • Tool: Free-boundary method for solving optimal stopping problems of diffusion processes. • Sampler structure appears more insightful than [Sun-Uysal-Yates-Koksal-Shroff, TIT17] and[Sun-Polyanskiy-Uysal’ISIT17] • Insight improved as we investigate more  the charm of AoI • More general than the Queueing model • Results may hold for other channel models (e.g., erasure channel)

  33. The 3rd AgeofInformationWorkshop • April 26th 2020 in Beijing, with IEEE INFOCOM 2020 • General Chairs: • Ness B. Shroff (The Ohio State University) • ZhishengNiu (Tsinghua University) • TPC Chairs: • SennurUlukus (University of Maryland) • Sheng Zhou (Tsinghua University)

  34. Yin Sun, sun.745@osu.edu

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