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Optimizing Age of Information in WirelessNetworks with Throughput Constraints
Igor Kadota, Abhishek Sinha and Eytan Modiano
IEEE INFOCOM, April 19, 2018
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Outlineโข Age of Information and Motivation
โข Network Model
โข Scheduling Policies and Performance Guaranteesโข Stationary Randomized Policyโข Max-Weight Policyโข Whittleโs Index Policy
โข Numerical Results
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Example:
Age of Information (AoI)
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Delivery of Packets to BSPacket Generation at i
i
Time
๐ผ๐ผ[1] Interdelivery Time
Packet Delay Single Source
Single Destination
L[1] L[2]
How fresh is the information at the destination?
Example:
Age of Information (AoI)
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i
Time
Time
๐ผ๐ผ[1] Interdelivery Time
Packet Delay
L[1]
Single Source
Single Destination
AoIL[1] L[2]
Delivery of Packets to BSPacket Generation at i
Example:
Age of Information (AoI)
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i
Time
Time
๐ผ๐ผ[1] Interdelivery Time
Packet Delay
L[1]
Single Source
Single Destination
AoIL[1] L[2]
Delivery of Packets to BSPacket Generation at i
Example:
Age of Information (AoI)
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i
Time
Time
๐ผ๐ผ[1] Interdelivery Time
Packet Delay
L[1] L[2]
Single Source
Single Destination
AoIL[1] L[2]
Delivery of Packets to BSPacket Generation at i
AoI: time elapsed since the most recently delivered packet was generated.
Relation between AoI, delay and interdelivery time?
Age of Information (AoI)
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Time
Time
๐ผ๐ผ[1] Interdelivery Time
Packet Delay
L[1] L[2]
AoIL[1] L[2]
โข Example: (M/M/1): (โ/FIFO) systemControllable arrival rate ๐๐ and fixed service rate ๐๐ = 1 packet per second.
Minimum throughput requirement DOES NOT guarantee regular deliveries.
AoI, Delay and Interdelivery time
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Server (๐๐)
๐๐
Source
Destination
[1] S. Kaul, R. Yates, and M. Gruteser, โReal-time status: How often should one update?โ, 2012.
๐๐ ๐ผ๐ผ[๐๐๐๐๐๐๐๐๐๐] ๐ผ๐ผ[๐๐๐๐๐๐๐๐๐๐๐๐๐๐๐๐. ] Average AoI0.01 1.01 100.000.53 2.13 1.890.99 100.00 1.01
โข Example: (M/M/1): (โ/FIFO) systemControllable arrival rate ๐๐ and fixed service rate ๐๐ = 1 packet per second.
Low time-average AoI when packets with low delay are delivered regularly.
AoI, Delay and Interdelivery time
9[1] S. Kaul, R. Yates, and M. Gruteser, โReal-time status: How often should one update?โ, 2012.
Server (๐๐)
๐๐
๐๐ ๐ผ๐ผ[๐๐๐๐๐๐๐๐๐๐] ๐ผ๐ผ[๐๐๐๐๐๐๐๐๐๐๐๐๐๐๐๐. ] Average AoI0.01 1.01 100.00 101.000.53 2.13 1.89 3.480.99 100.00 1.01 100.02
Source
Destination
Network - Example
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Wireless Parking Sensor
Wireless Tire-Pressure Monitoring System
Wireless Rearview Camera
Network - Description
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1
M๐ถ๐ถ๐๐
๐ถ๐ถ๐ด๐ดSensors / Nodes
๐๐๐๐
๐๐๐ด๐ด
๐๐๐๐๐๐๐ด๐ด
Central Monitor
1) Low network-wide AoI
2) Weights ๐ถ๐ถ๐๐ represent priority
3) Minimum throughput requirement, ๐๐๐๐
4) Channel is shared and unreliable, ๐๐๐๐
Values of M,๐ถ๐ถ๐๐,๐๐๐๐,๐๐๐๐ are fixed and known
Network - Scheduling Policy ๐๐
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During slot k:
1) BS selects a single node i [๐ข๐ข๐๐ ๐๐ = 1]
2) Selected node samples new data and then transmits
3) Packet is successfully delivered to the BS with probability ๐๐๐๐ [๐๐๐๐ ๐๐ = 1]
4) Packet Delay = 1 slot
Class of non-anticipative policies ฮ . Arbitrary policy ๐ ๐ โ ๐ท๐ท.
Central MonitorRunning policy ๐ ๐
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M๐ถ๐ถ๐๐
๐ถ๐ถ๐ด๐ดSensors / Nodes
๐๐๐๐
๐๐๐ด๐ด
๐๐๐๐๐๐๐ด๐ด
โข Age of Information associated with node iat the beginning of slot k is given by ๐๐๐๐(๐๐).
โข Recall: Packet Delay = 1 slot
โข Evolution of AoI:
Network - Age of Information
13Slots
Slots
Delivery of packets from sensor i to the BS
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๐ก๐ก๐ข๐ข(๐๐ + ๐๐) = ๏ฟฝ๐๐,
๐๐๐๐ ๐๐ + 1,
if ๐๐๐๐ ๐๐ = 1
otherwise
๐๐๐๐(๐๐)
Network - Objective Function
โข Expected Weighted Sum AoI when policy ๐๐ is employed:
โข Minimum Throughput Requirements:
โข Channel Interference:14
๐ผ๐ผ ๐ฝ๐ฝ๐พ๐พ๐๐ =1๐พ๐พ๐พ๐พ
๐ผ๐ผ ๏ฟฝ๐๐=1
๐พ๐พ
๏ฟฝ๐๐=1
๐๐
๐ถ๐ถ๐๐๐๐๐๐๐ ๐ (๐๐) , where ๐๐๐๐๐ ๐ ๐๐ is the AoI of node i and ๐ถ๐ถ๐๐ is the positive weight
๏ฟฝ๐๐๐๐๐๐: = lim๐พ๐พโโ
1๐พ๐พ๏ฟฝ๐๐=1
๐พ๐พ
๐ผ๐ผ[๐ ๐ ๐๐(๐๐)] โฅ ๐๐๐๐ ,โ๐๐ โ {1,2, โฆ ,๐พ๐พ}, where set ๐๐๐๐ ๐๐=1๐๐ is feasible.
๏ฟฝ๐๐=1
๐๐
๐๐๐๐(๐๐) โค 1 ,โ๐๐ โ {1,2, โฆ ,๐พ๐พ}
โข Policy ๐๐โ that solves (8) is AoI-optimal and achieves ๐๐๐๐๐๐โ
โข Policy ๐๐ โ ฮ that attains ๐๐๐๐T๐๐ is ๐๐-optimal when16
(Age of Information)
(Minimum Throughput)
(Channel Interference)
Scheduling Policies
๐๐๐๐๐๐โ โค ๐๐๐๐T๐๐ โค ๐๐๐๐๐๐๐๐โ
Summary of Results:
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Scheduling Policies
Scheduling Policy Technique Optimality Ratio Simulation Result
Optimal Stationary Randomized Policy Renewal Theory 2-optimal ~ 2-optimal
Max-Weight Policy LyapunovOptimization 4-optimal close to optimal
Whittleโs Index Policy RMAB Framework 8-optimal close to optimal
Summary of Results:
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Scheduling Policies
Scheduling Policy Technique Optimality Ratio Simulation Result
Optimal Stationary Randomized Policy Renewal Theory 2-optimal ~ 2-optimal
Max-Weight Policy LyapunovOptimization 4-optimal close to optimal
Whittleโs Index Policy RMAB Framework 8-optimal close to optimal
Stationary Randomized Policies
โข Policy R: in each slot k, select node i with probability ๐๐๐๐ โ 0,1 .
โข Packet deliveries from node i is a renewal process with ๐ฐ๐ฐ๐๐~ ๐บ๐บ๐๐๐บ๐บ(๐๐๐๐๐๐๐๐)โข Sum of ๐๐๐๐ ๐๐ over time is a renewal-reward process.
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๐๐๐๐(๐๐)
4321
๐ฐ๐ฐ๐๐
Stationary Randomized Policies
โข Policy R: in each slot k, select node i with probability ๐๐๐๐ โ 0,1 .
โข Packet deliveries from node i is a renewal process with ๐ฐ๐ฐ๐๐~ ๐บ๐บ๐๐๐บ๐บ(๐๐๐๐๐๐๐๐)โข Sum of ๐๐๐๐ ๐๐ over time is a renewal-reward process.
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๐๐๐๐(๐๐)
4321
lim๐พ๐พโโ
1๐พ๐พ๏ฟฝ๐๐=1
๐พ๐พ
๐ผ๐ผ ๐๐๐๐(๐๐) =๐ผ๐ผ ๐ฐ๐ฐ๐๐๐๐/2 + ๐ฐ๐ฐ๐๐/2
๐ผ๐ผ ๐ฐ๐ฐ๐๐=
1๐๐๐๐๐๐๐๐
๐ฐ๐ฐ๐๐
Stationary Randomized Policies
โข Policy R: in each slot k, select node i with probability ๐๐๐๐ โ 0,1 .
โข Packet deliveries from node i is a renewal process with ๐ฐ๐ฐ๐๐~ ๐บ๐บ๐๐๐บ๐บ(๐๐๐๐๐๐๐๐)โข Sum of ๐๐๐๐ ๐๐ over time is a renewal-reward process.
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(Age of Information)
(Minimum Throughput)
(Probability)
Stationary Randomized Policies
โข Policy R: in each slot k, select node i with probability ๐๐๐๐ โ 0,1 .
โข Packet deliveries from node i is a renewal process with ๐ฐ๐ฐ๐๐~ ๐บ๐บ๐๐๐บ๐บ(๐๐๐๐๐๐๐๐)โข Sum of ๐๐๐๐ ๐๐ over time is a renewal-reward process.
โข Optimal Stationary Randomized Policy R* uses probabilities ๐๐๐๐โ i=1M that can
be obtained offline with Algorithm 1 (omitted in this presentation).
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Stationary Randomized Policies
โข Policy R: in each slot k, select node i with probability ๐๐๐๐ โ 0,1 .
โข Packet deliveries from node i is a renewal process with ๐ฐ๐ฐ๐๐~ ๐บ๐บ๐๐๐บ๐บ(๐๐๐๐๐๐๐๐)โข Sum of ๐๐๐๐ ๐๐ over time is a renewal-reward process.
โข Optimal Stationary Randomized Policy R* uses probabilities ๐๐๐๐โ i=1M that can
be obtained offline with Algorithm 1 (omitted in this presentation).
Theorem: for any network configuration, policy R* is 2-optimal.
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Summary of Results:
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Scheduling Policies
Scheduling Policy Technique Optimality Ratio Simulation Result
Optimal Stationary Randomized Policy Renewal Theory 2-optimal ~ 2-optimal
Max-Weight Policy LyapunovOptimization 4-optimal close to optimal
Whittleโs Index Policy RMAB Framework 8-optimal close to optimal
Max-Weight Policy
โข Minimum Throughput Requirements:
โข Throughput debt:
โข Lyapunov Function:
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๐๐๐๐(๐๐ + ๐๐) = ๐๐๐๐๐๐ โ๏ฟฝ๐ก๐ก=1
๐๐
๐ ๐ ๐๐(๐๐)
๏ฟฝ๐๐๐๐๐๐: = lim๐พ๐พโโ
1๐พ๐พ๏ฟฝ๐๐=1
๐พ๐พ
๐ผ๐ผ[๐ ๐ ๐๐(๐๐)] โฅ ๐๐๐๐ ,โ๐๐ โ {1,2, โฆ ,๐พ๐พ}, where set ๐๐๐๐ ๐๐=1๐๐ is feasible.
๐ฟ๐ฟ ๐๐ โ12๏ฟฝ๐๐=1
๐๐
๐ถ๐ถ๐๐๐๐๐๐๐๐ ๐๐ + ๐๐ ๐ฅ๐ฅ๐๐+(๐๐) 2 , where V is a constant
and ๐ฅ๐ฅ๐๐+ ๐๐ = max 0, ๐ฅ๐ฅ๐๐(๐๐)
Max-Weight Policy
โข Max-Weight is designed to reduce the Lyapunov Drift.
โข Lyapunov Function:
โข Lyapunov Drift:
โข Policy MW: in each slot k, select the node with highest value of ๐๐๐๐(๐๐), where:
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๐๐๐๐ ๐๐ =๐ถ๐ถ๐๐๐๐๐๐
2๐๐๐๐ ๐๐ ๐๐๐๐(๐๐) + 2 + ๐๐๐๐๐๐๐ฅ๐ฅ๐๐+(๐๐)
๐ฟ๐ฟ ๐๐ โ12๏ฟฝ๐๐=1
๐๐๐ถ๐ถ๐๐๐๐๐๐๐๐ ๐๐ + ๐๐ ๐ฅ๐ฅ๐๐+(๐๐) 2
โ ๐๐ โค โ๏ฟฝ๐๐=1
๐๐๐ผ๐ผ ๐๐๐๐ ๐๐ ๐๐๐๐ ๐๐ , ๐ฅ๐ฅ๐๐ ๐๐ ๐๐=1
๐๐ ๐๐๐๐ ๐๐ + ๐ต๐ต๐๐(๐๐)
โ ๐๐ โ ๐ผ๐ผ ๐ฟ๐ฟ ๐๐ + 1 โ ๐ฟ๐ฟ ๐๐ ๐๐๐๐ ๐๐ , ๐ฅ๐ฅ๐๐(๐๐) ๐๐=1๐๐
Max-Weight Policy
โข Policy MW: in each slot k, select the node with highest value of ๐๐๐๐(๐๐), where:
Theorem: MW satisfies ANY feasible set of throughput requirements ๐๐๐๐ ๐๐=1๐๐
Theorem: for every network with ๐๐ โค 2โ๐๐=1๐๐ ๐ถ๐ถ๐๐ /๐พ๐พ, the MW is 4-optimal.
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๐๐๐๐ ๐๐ =๐ถ๐ถ๐๐๐๐๐๐
2๐๐๐๐ ๐๐ ๐๐๐๐(๐๐) + 2 + ๐๐๐๐๐๐๐ฅ๐ฅ๐๐+(๐๐)
Max-Weight Policy vs Whittleโs Index Policy
โข Policy MW: in each slot k, select the node with highest value of ๐๐๐๐(๐๐), where:
Theorem: MW satisfies ANY feasible set of throughput requirements ๐๐๐๐ ๐๐=1๐๐
Theorem: for every network with ๐๐ โค 2โ๐๐=1๐๐ ๐ถ๐ถ๐๐ /๐พ๐พ, the MW is 4-optimal.
โข Policy Whittle: in each slot k, select the node with highest value of ๐ถ๐ถ๐๐(๐๐), where:
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๐๐๐๐ ๐๐ =๐ถ๐ถ๐๐๐๐๐๐
2๐๐๐๐ ๐๐ ๐๐๐๐(๐๐) + 2 + ๐๐๐๐๐๐๐ฅ๐ฅ๐๐+(๐๐)
๐ถ๐ถ๐๐ ๐๐ =๐ถ๐ถ๐๐๐๐๐๐
2๐๐๐๐ ๐๐ ๐๐๐๐ ๐๐ +
2๐๐๐๐โ 1 + ๐ฝ๐ฝ๐๐
Summary of Results:
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Scheduling Policies
Scheduling Policy Technique Optimality Ratio Simulation Result
Optimal Stationary Randomized Policy Renewal Theory 2-optimal ~ 2-optimal
Max-Weight Policy LyapunovOptimization 4-optimal close to optimal
Whittleโs Index Policy RMAB Framework 8-optimal close to optimal
Numerical Results
โข Metric:โข Expected Weighted Sum AoI : ๐ผ๐ผ ๐ฝ๐ฝ๐พ๐พ๐๐
โข Network Setup with M nodes. Node i has:โข channel reliability ๐๐๐๐ = ๐๐/๐พ๐พ [increasing]โข weight ๐ถ๐ถ๐๐ = (๐พ๐พ + 1 โ ๐๐)/๐พ๐พ [decreasing]โข throughput requirement ๐๐๐๐ = ๐๐๐๐๐๐/M, where ๐๐ in 0; 1
โข Each simulation runs for ๐พ๐พ = ๐พ๐พ ร 106 slotsโข Each data point is an average over 10 simulations
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๐๐ = ๐๐.๐๐
2x
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๐ด๐ด = ๐๐๐๐
2x
Final Remarksโข In this presentation:
โข Age of Information and Network Modelโข Three low-complexity scheduling policiesโข Performance guaranteesโข Numerical Results: Max-Weight has superior performance
โข In the paper: โข Derive Universal Lower Bound on Age of Informationโข Discuss Indexability and Whittleโs Index Policyโข Additional simulation results
โข Recent result not in the paper: Drift-Plus Penalty Policy is 2-optimal33
1
M๐ถ๐ถ๐๐
๐ถ๐ถ๐ด๐ด
๐๐๐๐
๐๐๐ด๐ด
๐๐๐๐๐๐๐ด๐ด
๐ ๐