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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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Outlineβ’ Age of Information and Motivation
β’ Network Model
β’ Scheduling Policies and Performance Guaranteesβ’ Stationary Randomized Policyβ’ Max-Weight Policyβ’ Whittleβs Index Policy
β’ 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
πΌπΌ[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
πΌπΌ[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
πΌπΌ[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
πΌπΌ[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
πΌπΌ[1] Interdelivery Time
Packet Delay
L[1] L[2]
AoIL[1] L[2]
β’ Example: (M/M/1): (β/FIFO) systemControllable arrival rate ππ and fixed service rate ππ = 1 packet per second.
Minimum throughput requirement DOES NOT guarantee regular deliveries.
AoI, Delay and Interdelivery time
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Server (ππ)
ππ
Source
Destination
[1] S. Kaul, R. Yates, and M. Gruteser, βReal-time status: How often should one update?β, 2012.
ππ πΌπΌ[ππππππππππ] πΌπΌ[ππππππππππππππππ. ] Average AoI0.01 1.01 100.000.53 2.13 1.890.99 100.00 1.01
β’ Example: (M/M/1): (β/FIFO) systemControllable arrival rate ππ and fixed service rate ππ = 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, βReal-time status: How often should one update?β, 2012.
Server (ππ)
ππ
ππ πΌπΌ[ππππππππππ] πΌπΌ[ππππππππππππππππ. ] 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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MπΆπΆππ
πΆπΆπ΄π΄Sensors / Nodes
ππππ
πππ΄π΄
πππππππ΄π΄
Central Monitor
1) Low network-wide AoI
2) Weights πΆπΆππ represent priority
3) Minimum throughput requirement, ππππ
4) Channel is shared and unreliable, ππππ
Values of M,πΆπΆππ,ππππ,ππππ are fixed and known
Network - Scheduling Policy ππ
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During slot k:
1) BS selects a single node i [π’π’ππ ππ = 1]
2) Selected node samples new data and then transmits
3) Packet is successfully delivered to the BS with probability ππππ [ππππ ππ = 1]
4) Packet Delay = 1 slot
Class of non-anticipative policies Ξ . Arbitrary policy π π β π·π·.
Central MonitorRunning policy π π
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MπΆπΆππ
πΆπΆπ΄π΄Sensors / Nodes
ππππ
πππ΄π΄
πππππππ΄π΄
β’ Age of Information associated with node iat the beginning of slot k is given by ππππ(ππ).
β’ Recall: Packet Delay = 1 slot
β’ Evolution of AoI:
Network - Age of Information
13Slots
Slots
Delivery of packets from sensor i to the BS
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π‘π‘π’π’(ππ + ππ) = οΏ½ππ,
ππππ ππ + 1,
if ππππ ππ = 1
otherwise
ππππ(ππ)
Network - Objective Function
β’ Expected Weighted Sum AoI when policy ππ is employed:
β’ Minimum Throughput Requirements:
β’ Channel Interference:14
πΌπΌ π½π½πΎπΎππ =1πΎπΎπΎπΎ
πΌπΌ οΏ½ππ=1
πΎπΎ
οΏ½ππ=1
ππ
πΆπΆπππππππ π (ππ) , where πππππ π ππ is the AoI of node i and πΆπΆππ is the positive weight
οΏ½ππππππ: = limπΎπΎββ
1πΎπΎοΏ½ππ=1
πΎπΎ
πΌπΌ[π π ππ(ππ)] β₯ ππππ ,βππ β {1,2, β¦ ,πΎπΎ}, where set ππππ ππ=1ππ is feasible.
οΏ½ππ=1
ππ
ππππ(ππ) β€ 1 ,βππ β {1,2, β¦ ,πΎπΎ}
β’ Policy ππβ that solves (8) is AoI-optimal and achieves ππππππβ
β’ Policy ππ β Ξ that attains ππππTππ is ππ-optimal when16
(Age of Information)
(Minimum Throughput)
(Channel Interference)
Scheduling Policies
ππππππβ β€ ππππTππ β€ ππππππππβ
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
Whittleβs 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
Whittleβs Index Policy RMAB Framework 8-optimal close to optimal
Stationary Randomized Policies
β’ Policy R: in each slot k, select node i with probability ππππ β 0,1 .
β’ Packet deliveries from node i is a renewal process with π°π°ππ~ πΊπΊπππΊπΊ(ππππππππ)β’ Sum of ππππ ππ over time is a renewal-reward process.
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ππππ(ππ)
4321
π°π°ππ
Stationary Randomized Policies
β’ Policy R: in each slot k, select node i with probability ππππ β 0,1 .
β’ Packet deliveries from node i is a renewal process with π°π°ππ~ πΊπΊπππΊπΊ(ππππππππ)β’ Sum of ππππ ππ over time is a renewal-reward process.
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ππππ(ππ)
4321
limπΎπΎββ
1πΎπΎοΏ½ππ=1
πΎπΎ
πΌπΌ ππππ(ππ) =πΌπΌ π°π°ππππ/2 + π°π°ππ/2
πΌπΌ π°π°ππ=
1ππππππππ
π°π°ππ
Stationary Randomized Policies
β’ Policy R: in each slot k, select node i with probability ππππ β 0,1 .
β’ Packet deliveries from node i is a renewal process with π°π°ππ~ πΊπΊπππΊπΊ(ππππππππ)β’ Sum of ππππ ππ over time is a renewal-reward process.
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(Age of Information)
(Minimum Throughput)
(Probability)
Stationary Randomized Policies
β’ Policy R: in each slot k, select node i with probability ππππ β 0,1 .
β’ Packet deliveries from node i is a renewal process with π°π°ππ~ πΊπΊπππΊπΊ(ππππππππ)β’ Sum of ππππ ππ over time is a renewal-reward process.
β’ Optimal Stationary Randomized Policy R* uses probabilities ππππβ i=1M that can
be obtained offline with Algorithm 1 (omitted in this presentation).
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Stationary Randomized Policies
β’ Policy R: in each slot k, select node i with probability ππππ β 0,1 .
β’ Packet deliveries from node i is a renewal process with π°π°ππ~ πΊπΊπππΊπΊ(ππππππππ)β’ Sum of ππππ ππ over time is a renewal-reward process.
β’ Optimal Stationary Randomized Policy R* uses probabilities ππππβ 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
Whittleβs Index Policy RMAB Framework 8-optimal close to optimal
Max-Weight Policy
β’ Minimum Throughput Requirements:
β’ Throughput debt:
β’ Lyapunov Function:
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ππππ(ππ + ππ) = ππππππ βοΏ½π‘π‘=1
ππ
π π ππ(ππ)
οΏ½ππππππ: = limπΎπΎββ
1πΎπΎοΏ½ππ=1
πΎπΎ
πΌπΌ[π π ππ(ππ)] β₯ ππππ ,βππ β {1,2, β¦ ,πΎπΎ}, where set ππππ ππ=1ππ is feasible.
πΏπΏ ππ β12οΏ½ππ=1
ππ
πΆπΆππππππππ ππ + ππ π₯π₯ππ+(ππ) 2 , where V is a constant
and π₯π₯ππ+ ππ = max 0, π₯π₯ππ(ππ)
Max-Weight Policy
β’ Max-Weight is designed to reduce the Lyapunov Drift.
β’ Lyapunov Function:
β’ Lyapunov Drift:
β’ Policy MW: in each slot k, select the node with highest value of ππππ(ππ), where:
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ππππ ππ =πΆπΆππππππ
2ππππ ππ ππππ(ππ) + 2 + πππππππ₯π₯ππ+(ππ)
πΏπΏ ππ β12οΏ½ππ=1
πππΆπΆππππππππ ππ + ππ π₯π₯ππ+(ππ) 2
β ππ β€ βοΏ½ππ=1
πππΌπΌ ππππ ππ ππππ ππ , π₯π₯ππ ππ ππ=1
ππ ππππ ππ + π΅π΅ππ(ππ)
β ππ β πΌπΌ πΏπΏ ππ + 1 β πΏπΏ ππ ππππ ππ , π₯π₯ππ(ππ) ππ=1ππ
Max-Weight Policy
β’ Policy MW: in each slot k, select the node with highest value of ππππ(ππ), where:
Theorem: MW satisfies ANY feasible set of throughput requirements ππππ ππ=1ππ
Theorem: for every network with ππ β€ 2βππ=1ππ πΆπΆππ /πΎπΎ, the MW is 4-optimal.
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ππππ ππ =πΆπΆππππππ
2ππππ ππ ππππ(ππ) + 2 + πππππππ₯π₯ππ+(ππ)
Max-Weight Policy vs Whittleβs Index Policy
β’ Policy MW: in each slot k, select the node with highest value of ππππ(ππ), where:
Theorem: MW satisfies ANY feasible set of throughput requirements ππππ ππ=1ππ
Theorem: for every network with ππ β€ 2βππ=1ππ πΆπΆππ /πΎπΎ, the MW is 4-optimal.
β’ Policy Whittle: in each slot k, select the node with highest value of πΆπΆππ(ππ), where:
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ππππ ππ =πΆπΆππππππ
2ππππ ππ ππππ(ππ) + 2 + πππππππ₯π₯ππ+(ππ)
πΆπΆππ ππ =πΆπΆππππππ
2ππππ ππ ππππ ππ +
2ππππβ 1 + π½π½ππ
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
Whittleβs Index Policy RMAB Framework 8-optimal close to optimal
Numerical Results
β’ Metric:β’ Expected Weighted Sum AoI : πΌπΌ π½π½πΎπΎππ
β’ Network Setup with M nodes. Node i has:β’ channel reliability ππππ = ππ/πΎπΎ [increasing]β’ weight πΆπΆππ = (πΎπΎ + 1 β ππ)/πΎπΎ [decreasing]β’ throughput requirement ππππ = ππππππ/M, where ππ in 0; 1
β’ Each simulation runs for πΎπΎ = πΎπΎ Γ 106 slotsβ’ Each data point is an average over 10 simulations
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ππ = ππ.ππ
2x
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π΄π΄ = ππππ
2x
Final Remarksβ’ In this presentation:
β’ Age of Information and Network Modelβ’ Three low-complexity scheduling policiesβ’ Performance guaranteesβ’ Numerical Results: Max-Weight has superior performance
β’ In the paper: β’ Derive Universal Lower Bound on Age of Informationβ’ Discuss Indexability and Whittleβs Index Policyβ’ Additional simulation results
β’ Recent result not in the paper: Drift-Plus Penalty Policy is 2-optimal33
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MπΆπΆππ
πΆπΆπ΄π΄
ππππ
πππ΄π΄
πππππππ΄π΄
π π