SLIDE 1
Age of Information in Random Access Channels
Xingran Chen Konstantinos Gatsis Hamed Hassani Shirin Saeedi Bidokhti University of Pennsylvania
2020 IEEE ISIT, Los Angeles, California, USA
SLIDE 2
- Communication networks have witnessed rapid growth in the past few decades
cyber-physical systems, the Internet of Things, smart cities, healthcare systems
Time-sensitive
remote sensing, estimation, control
- Transmission policies keeping freshest information — Age of Information
markovity of the underlying physical processes
[Kadota-Sinha-Biyikoglu-Singh-Modiano-18], [Kadota-Sinha-Modiano-19], [Hsu-Modiano-Duan-19], [Kadota-Modiano-20], [Kaul-Yates-17], [Talak-Karanman-Modiano-18], [Kosta-Pappas-Ephremides-Angelakis-19], [Jiang-Krishnamachari-Zheng-Zhou-Niu-18] [Jiang-Krishnamachari-Zhou-Niu-18], [Yates-Kaul-20]
- We design for the first time decentralized age-based transmission policies
- Provide analytical results on the age of information
Background & Motivation
SLIDE 3
- Communication networks have witnessed rapid growth in the past few decades
cyber-physical systems, the Internet of Things, smart cities, healthcare systems
Time-sensitive
remote sensing, estimation, control
- Transmission policies keeping freshest information — Age of Information
markovity of the underlying physical processes
[Kadota-Sinha-Biyikoglu-Singh-Modiano-18], [Kadota-Sinha-Modiano-19], [Hsu-Modiano-Duan-19], [Kadota-Modiano-20], [Kaul-Yates-17], [Talak-Karanman-Modiano-18], [Kosta-Pappas-Ephremides-Angelakis-19], [Jiang-Krishnamachari-Zheng-Zhou-Niu-18] [Jiang-Krishnamachari-Zhou-Niu-18], [Yates-Kaul-20]
- We design for the first time decentralized age-based transmission policies
- Provide analytical results on the age of information
Background & Motivation
SLIDE 4
- Communication networks have witnessed rapid growth in the past few decades
cyber-physical systems, the Internet of Things, smart cities, healthcare systems
Time-sensitive
remote sensing, estimation, control
- Transmission policies keeping freshest information — Age of Information
markovity of the underlying physical processes
[Kadota-Sinha-Biyikoglu-Singh-Modiano-18], [Kadota-Sinha-Modiano-19], [Hsu-Modiano-Duan-19], [Kadota-Modiano-20], [Kaul-Yates-17], [Talak-Karanman-Modiano-18], [Kosta-Pappas-Ephremides-Angelakis-19], [Jiang-Krishnamachari-Zheng-Zhou-Niu-18] [Jiang-Krishnamachari-Zhou-Niu-18], [Yates-Kaul-20]
- We design for the first time decentralized age-based transmission policies
- Provide analytical results on the age of information
Background & Motivation
SLIDE 5
- Communication networks have witnessed rapid growth in the past few decades
cyber-physical systems, the Internet of Things, smart cities, healthcare systems
Time-sensitive
remote sensing, estimation, control
- Transmission policies keeping freshest information — Age of Information
markovity of the underlying physical processes
[Kadota-Sinha-Biyikoglu-Singh-Modiano-18], [Kadota-Sinha-Modiano-19], [Hsu-Modiano-Duan-19], [Kadota-Modiano-20], [Kaul-Yates-17], [Talak-Karanman-Modiano-18], [Kosta-Pappas-Ephremides-Angelakis-19], [Jiang-Krishnamachari-Zheng-Zhou-Niu-18] [Jiang-Krishnamachari-Zhou-Niu-18], [Yates-Kaul-20]
- We design for the first time decentralized age-based transmission policies
- Provide analytical results on the age of information
Background & Motivation
SLIDE 6
- Communication networks have witnessed rapid growth in the past few decades
cyber-physical systems, the Internet of Things, smart cities, healthcare systems
Time-sensitive
remote sensing, estimation, control
- Transmission policies keeping freshest information — Age of Information
markovity of the underlying physical processes
[Kadota-Sinha-Biyikoglu-Singh-Modiano-18], [Kadota-Sinha-Modiano-19], [Hsu-Modiano-Duan-19], [Kadota-Modiano-20], [Kaul-Yates-17], [Talak-Karanman-Modiano-18], [Kosta-Pappas-Ephremides-Angelakis-19], [Jiang-Krishnamachari-Zheng-Zhou-Niu-18] [Jiang-Krishnamachari-Zhou-Niu-18], [Yates-Kaul-20]
- We design for the first time decentralized age-based transmission policies
- Provide analytical results on the age of information
Background & Motivation
SLIDE 7
- A new metric to quantify the freshness of information (2011)
[Kaul-Yates-Gruteser 11]
- : timestamp of the most recently received update;
.
status update
status update
u(t) h(t) = t − u(t)
t′
k
kth tk kth
lim
T→∞
1 T ∫
T
h(t)
Age of Information (AoI)
SLIDE 8
- A new metric to quantify the freshness of information (2011)
[Kaul-Yates-Gruteser 11]
- : timestamp of the most recently received update;
.
status update
status update
u(t) h(t) = t − u(t)
t′
k
kth tk kth
lim
T→∞
1 T ∫
T
h(t)
Age of Information (AoI)
SLIDE 9
- A new metric to quantify the freshness of information (2011)
[Kaul-Yates-Gruteser 11]
- : timestamp of the most recently received update;
.
status update
status update
u(t) h(t) = t − u(t)
t′
k
kth tk kth
lim
T→∞
1 T ∫
T
h(t)
Age of Information (AoI)
SLIDE 10
- A new metric to quantify the freshness of information (2011)
[Kaul-Yates-Gruteser 11]
- : timestamp of the most recently received update;
.
status update
status update
u(t) h(t) = t − u(t)
t′
k
kth tk kth
lim
T→∞
1 T ∫
T
h(t)
Age of Information (AoI)
SLIDE 11
- A new metric to quantify the freshness of information (2011)
[Kaul-Yates-Gruteser 11]
- : timestamp of the most recently received update;
.
status update
status update
u(t) h(t) = t − u(t)
t′
k
kth tk kth
lim
T→∞
1 T ∫
T
h(t)
Age of Information (AoI)
SLIDE 12 System Model
- statistically identical source nodes
- Slotted time
- Stochastic arrival/generation process
- Collision channel, collision feedback
- One unit transmission delay
- Find transmission policy that minimizes Normalized Expected Weighted Sum
AoI (NEWSAoI)
M θ π lim
K→∞
1 KM2
M
∑
i=1 K
∑
k=1
hπ
i (k)
sensor arrival rate
i θ
SLIDE 13 System Model
- statistically identical source nodes
- Slotted time
- Stochastic arrival/generation process
- Collision channel, collision feedback
- One unit transmission delay
- Find transmission policy that minimizes Normalized Expected Weighted Sum
AoI (NEWSAoI)
M θ π lim
K→∞
1 KM2
M
∑
i=1 K
∑
k=1
hπ
i (k)
sensor arrival rate
i θ
SLIDE 14 System Model
- statistically identical source nodes
- Slotted time
- Stochastic arrival/generation process
- Collision channel, collision feedback
- One unit transmission delay
- Find transmission policy that minimizes Normalized Expected Weighted Sum
AoI (NEWSAoI)
M θ π lim
K→∞
1 KM2
M
∑
i=1 K
∑
k=1
hπ
i (k)
sensor arrival rate
i θ
SLIDE 15 System Model
- statistically identical source nodes
- Slotted time
- Stochastic arrival/generation process
- Collision channel, collision feedback
- One unit transmission delay
- Find transmission policy that minimizes Normalized Expected Weighted Sum
AoI (NEWSAoI)
M θ π lim
K→∞
1 KM2
M
∑
i=1 K
∑
k=1
hπ
i (k)
sensor arrival rate
i θ
SLIDE 16 System Model
- statistically identical source nodes
- Slotted time
- Stochastic arrival/generation process
- Collision channel, collision feedback
- One unit transmission delay
- Find transmission policy that minimizes Normalized Expected Weighted Sum
AoI (NEWSAoI)
M θ π lim
K→∞
1 KM2
M
∑
i=1 K
∑
k=1
hπ
i (k)
sensor arrival rate
i θ
SLIDE 17 System Model
- statistically identical source nodes
- Slotted time
- Stochastic arrival/generation process
- Collision channel, collision feedback
- One unit transmission delay
- Find transmission policy that minimizes Normalized Expected Weighted Sum
AoI (NEWSAoI)
M θ π lim
K→∞
1 KM2
M
∑
i=1 K
∑
k=1
hπ
i (k)
sensor arrival rate
i θ
SLIDE 18
Evolution of Age
Source AoI Destination AoI
wi(k + 1) = { wi(k) + 1
no new packet arrives a new packet arrives
hi(k + 1) = { wi(k) + 1
a packet is delivered
hi(k) + 1
no packet is delivered
SLIDE 19 Lower Bound
Theorem: For any transmission policy, NEWSAoI is lowered bounded by 1) 2)
where denote the sum-capacity of the underlying random access channel
[Tasybakov-Likhanov] Probl. Peredachi Inf, vol. 23
- RA with CSMA
- RA without feedback
NEWSAoI ≥ 1
Mθ
small arrival rates NEWSAoI ≥
1 2CRA + 1 2M
large arrival rates
CRA
CRA ≤ 0.568 (M → ∞) CRA ≤ 1 CRA ≤ 1 e (M → ∞)
SLIDE 20 Lower Bound
Theorem: For any transmission policy, NEWSAoI is lowered bounded by 1) 2)
where denote the sum-capacity of the underlying random access channel
[Tasybakov-Likhanov] Probl. Peredachi Inf, vol. 23
- RA with CSMA
- RA without feedback
NEWSAoI ≥ 1
Mθ
small arrival rates NEWSAoI ≥
1 2CRA + 1 2M
large arrival rates
CRA
CRA ≤ 0.568 (M → ∞) CRA ≤ 1 CRA ≤ 1 e (M → ∞)
SLIDE 21 Decentralized Age-based Policies
Slotted ALOHA: transmitters send packets immediately upon arrival they are “backlogged” after a collision a backoff probability
Small arrival rate: slotted ALOHA
In this work, we focus on Rivest’s stabilized slotted ALOHA, denote by an estimate
- f the number of backlogged nodes.
n(k)
pb(k) = min (1, 1 n(k) ) n(k) = min (n(k − 1) + Mθ + (e − 2)−1, M) c(k) = 1 min ( max (Mθ, n(k − 1) + Mθ − 1), M) c(k) = 0
Theorem: Suppose and define . Any stabilized slotted ALOHA scheme achieves Moreover, (stabilized) slotted ALOHA are asymptotically optimal in terms of NEWSAoI.
θ ≤ 1 eM η = lim
M→∞ Mθ
lim
M→∞ JSA(M) = 1
η .
Theorem: Suppose and define . Any stabilized slotted ALOHA scheme achieves Moreover, (stabilized) slotted ALOHA are asymptotically optimal in terms of NEWSAoI.
θ ≤ 1 eM η = lim
M→∞ Mθ
lim
M→∞ JSA(M) = 1
η .
SLIDE 22 Decentralized Age-based Policies
Slotted ALOHA: transmitters send packets immediately upon arrival they are “backlogged” after a collision a backoff probability
Small arrival rate: slotted ALOHA
In this work, we focus on Rivest’s stabilized slotted ALOHA, denote by an estimate
- f the number of backlogged nodes.
n(k)
pb(k) = min (1, 1 n(k) ) n(k) = min (n(k − 1) + Mθ + (e − 2)−1, M) c(k) = 1 min ( max (Mθ, n(k − 1) + Mθ − 1), M) c(k) = 0
Theorem: Suppose and define . Any stabilized slotted ALOHA scheme achieves Moreover, (stabilized) slotted ALOHA are asymptotically optimal in terms of NEWSAoI.
θ ≤ 1 eM η = lim
M→∞ Mθ
lim
M→∞ JSA(M) = 1
η .
Theorem: Suppose and define . Any stabilized slotted ALOHA scheme achieves Moreover, (stabilized) slotted ALOHA are asymptotically optimal in terms of NEWSAoI.
θ ≤ 1 eM η = lim
M→∞ Mθ
lim
M→∞
NEWSAoI(M) = 1
η .
SLIDE 23 Decentralized Age-based Policies Decentralized Age-based Policies
, then NEWSAoI of slotted ALOHA is around .
- Is the slotted ALOHA unstabilized (large AoI) when
?
- Can we get benefits (small AoI) by increasing arrival rate?
- What should the transmitters do in order to ensure a small age of information
when ?
θ = 1 eM e θ > 1 eM θ > 1 eM
Large arrival rate: Age-based Thinning
SLIDE 24 Decentralized Age-based Policies Decentralized Age-based Policies
, then NEWSAoI of slotted ALOHA is around .
- Is the slotted ALOHA unstabilized (large AoI) when
?
- Can we get benefits (small AoI) by increasing arrival rate?
- What should the transmitters do in order to ensure a small age of information
when ?
θ = 1 eM e θ > 1 eM θ > 1 eM
Large arrival rate: Age-based Thinning
SLIDE 25 Decentralized Age-based Policies Decentralized Age-based Policies
, then NEWSAoI of slotted ALOHA is around .
- Is the slotted ALOHA unstabilized (large AoI) when
?
- Can we get benefits (small AoI) by increasing arrival rate?
- What should the transmitters do in order to ensure a small age of information
when ?
θ = 1 eM e θ > 1 eM θ > 1 eM
Large arrival rate: Age-based Thinning
SLIDE 26 Decentralized Age-based Policies Decentralized Age-based Policies
, then NEWSAoI of slotted ALOHA is around .
- Is the slotted ALOHA unstabilized (large AoI) when
?
- Can we get benefits (small AoI) by increasing arrival rate?
- What should the transmitters do in order to ensure a small age of information
when ?
θ = 1 eM e θ > 1 eM θ > 1 eM
Large arrival rate: Age-based Thinning
SLIDE 27 Decentralized Age-based Policies Decentralized Age-based Policies
- We defined age-gain as:
- Decentralized age-based policies: transmitter send packets when it has large
.
- Adaptive threshold policy: node :
- Stationary threshold policy: node :
- Active nodes follow slotted ALOHA protocol and inactive nodes remain silent
δi(k) = hi(k) − wi(k) i δi(k)
i {
active
δi(k) ≥ 𝚄(k)
inactive
0 ≤ δi(k) < 𝚄(k) i {
active
δi(k) ≥ 𝚄*
inactive
0 ≤ δi(k) < 𝚄*
Large arrival rate: Age-based Thinning
SLIDE 28 Decentralized Age-based Policies Decentralized Age-based Policies
- We defined age-gain as:
- Decentralized age-based policies: transmitter send packets when it has large
.
- Adaptive threshold policy: node :
- Stationary threshold policy: node :
- Active nodes follow slotted ALOHA protocol and inactive nodes remain silent
δi(k) = hi(k) − wi(k) i δi(k)
i {
active
δi(k) ≥ 𝚄(k)
inactive
0 ≤ δi(k) < 𝚄(k) i {
active
δi(k) ≥ 𝚄*
inactive
0 ≤ δi(k) < 𝚄*
Large arrival rate: Age-based Thinning
SLIDE 29 Decentralized Age-based Policies Decentralized Age-based Policies
- We defined age-gain as:
- Decentralized age-based policies: transmitter send packets when it has large
.
- Adaptive threshold policy: node :
- Stationary threshold policy: node :
- Active nodes follow slotted ALOHA protocol and inactive nodes remain silent
δi(k) = hi(k) − wi(k) i δi(k)
i {
active
δi(k) ≥ 𝚄(k)
inactive
0 ≤ δi(k) < 𝚄(k) i {
active
δi(k) ≥ 𝚄*
inactive
0 ≤ δi(k) < 𝚄*
Large arrival rate: Age-based Thinning
SLIDE 30 Decentralized Age-based Policies Decentralized Age-based Policies
- We defined age-gain as:
- Decentralized age-based policies: transmitter send packets when it has large
.
- Adaptive threshold policy: node :
- Stationary threshold policy: node :
- Active nodes follow slotted ALOHA protocol and inactive nodes remain silent
δi(k) = hi(k) − wi(k) i δi(k)
i {
active
δi(k) ≥ 𝚄(k)
inactive
0 ≤ δi(k) < 𝚄(k) i {
active
δi(k) ≥ 𝚄*
inactive
0 ≤ δi(k) < 𝚄*
Large arrival rate: Age-based Thinning
SLIDE 31 Decentralized Age-based Policies Decentralized Age-based Policies
Adaptive Age-based Thinning
node :
i {
active
δi(k) ≥ 𝚄(k)
inactive
0 ≤ δi(k) < 𝚄(k)
Node is m-order at time if
i k δi(k) = m
Fraction of nodes with order : ; estimate:
m ℓm(k) ̂ ℓm(k)
Estimate
Expected age-gain distribution after packet arrival Expected fraction of nodes that have just become m-order
Estimate
am(k)
Find Threshold
T(k) {ˆ `m(k − 1)}∞
m=0
c(k)
Collision feedback
{ˆ `m(k)}∞
m=0
{ˆ `m(k−)}∞
m=0
{ ̂ ℓm(k − 1)}∞
m=0
{ ̂ ℓm(k+)}∞
m=0
{ ̂ ℓm(k)}∞
m=0
{ ̂ am(k)}∞
m=0
𝚄(k) = max {t|∑
m≥t
̂ am(k) ≥ 1 eM}
SLIDE 32 Decentralized Age-based Policies Decentralized Age-based Policies
Adaptive Age-based Thinning
node :
i {
active
δi(k) ≥ 𝚄(k)
inactive
0 ≤ δi(k) < 𝚄(k)
node is m-order at time if
i k δi(k) = m
Fraction of nodes with order : ; estimate:
m ℓm(k) ̂ ℓm(k)
Estimate
Expected age-gain distribution after packet arrival Expected fraction of nodes that have just become m-order
Estimate
am(k)
Find Threshold
T(k) {ˆ `m(k − 1)}∞
m=0
c(k)
Collision feedback
{ˆ `m(k)}∞
m=0
{ˆ `m(k−)}∞
m=0
{ ̂ ℓm(k − 1)}∞
m=0
{ ̂ ℓm(k+)}∞
m=0
{ ̂ ℓm(k)}∞
m=0
{ ̂ am(k)}∞
m=0
𝚄(k) = max {t|∑
m≥t
̂ am(k) ≥ 1 eM}
SLIDE 33 Decentralized Age-based Policies Decentralized Age-based Policies
Adaptive Age-based Thinning
node :
i {
active
δi(k) ≥ 𝚄(k)
inactive
0 ≤ δi(k) < 𝚄(k)
node is m-order at time if
i k δi(k) = m
expected fraction of nodes with order : ; estimate:
m ℓm(k) ̂ ℓm(k)
Estimate
Expected age-gain distribution after packet arrival Expected fraction of nodes that have just become m-order
Estimate
am(k)
Find Threshold
T(k) {ˆ `m(k − 1)}∞
m=0
c(k)
Collision feedback
{ˆ `m(k)}∞
m=0
{ˆ `m(k−)}∞
m=0
{ ̂ ℓm(k − 1)}∞
m=0
{ ̂ ℓm(k+)}∞
m=0
{ ̂ ℓm(k)}∞
m=0
{ ̂ am(k)}∞
m=0
𝚄(k) = max {t|∑
m≥t
̂ am(k) ≥ 1 eM}
SLIDE 34 Decentralized Age-based Policies Decentralized Age-based Policies
Adaptive Age-based Thinning
node :
i {
active
δi(k) ≥ 𝚄(k)
inactive
0 ≤ δi(k) < 𝚄(k)
node is m-order at time if
i k δi(k) = m
expected fraction of nodes with order : ; estimate:
m ℓm(k) ̂ ℓm(k)
Estimate
Expected age-gain distribution after packet arrival Expected fraction of nodes that have just become m-order
Estimate
am(k)
Find Threshold
T(k) {ˆ `m(k − 1)}∞
m=0
c(k)
Collision feedback
{ˆ `m(k)}∞
m=0
{ˆ `m(k−)}∞
m=0
{ ̂ ℓm(k − 1)}∞
m=0
{ ̂ ℓm(k+)}∞
m=0
{ ̂ ℓm(k)}∞
m=0
{ ̂ am(k)}∞
m=0
𝚄(k) = max {t|∑
m≥t
̂ am(k) ≥ 1 eM}
SLIDE 35 Decentralized Age-based Policies Decentralized Age-based Policies
Adaptive Age-based Thinning
node :
i {
active
δi(k) ≥ 𝚄(k)
inactive
0 ≤ δi(k) < 𝚄(k)
node is m-order at time if
i k δi(k) = m
expected fraction of nodes with order : ; estimate:
m ℓm(k) ̂ ℓm(k)
Estimate
Expected age-gain distribution after packet arrival Expected fraction of nodes that have just become m-order
Estimate
am(k)
Find Threshold
T(k) {ˆ `m(k − 1)}∞
m=0
c(k)
Collision feedback
{ˆ `m(k)}∞
m=0
{ˆ `m(k−)}∞
m=0
{ ̂ ℓm(k − 1)}∞
m=0
{ ̂ ℓm(k+)}∞
m=0
{ ̂ ℓm(k)}∞
m=0
{ ̂ am(k)}∞
m=0
𝚄(k) = max {t|∑
m≥t
̂ am(k) ≥ 1 eM}
SLIDE 36
By the stationarity of the scheme, the limit of and exist as . Denote by and .
{ℓm(k)}∞
m=0
{ℓm(k+)}∞
m=0
k → ∞ {ℓ*
m}∞ m=0
{ℓm(k+)}∞
m=0
𝚄* = max (1,⌊eM − 1/θ + 1⌋)
Stationary Age-based Thinning: For source , compute . If , then it does not transmit packets; if , then it transmits a packet by slotted ALOHA.
i δi(k) = hi(k) − wi(k) δi(k) < 𝚄* δi(k) ≥ 𝚄*
(Theorem 2) For any , .
θ = 1/o(M) lim
M→∞ JSAT(M) = e/2
Consider a stationary transmission policy that does not employ coding across packets.
π
Develop a variant of the transmission policy in which only the most recent packets of each transmitter are preserved and all packets are discarded. Denote this policy by .
π π(1)
When , denote the channel capacity by .
M → ∞ Cπ(1) 𝚄* = max (1,⌊M/Cπ(1) − 1/θ + 1⌋)
(Theorem 3) For any , .
θ = 1/o(M) lim
M→∞ JSAT(M) =
1 2Cπ(1)
Decentralized Age-based Policies Decentralized Age-based Policies Decentralized Age-based Policies
Stationary Age-based Thinning
node :
i {
active
δi(k) ≥ 𝚄*
inactive
0 ≤ δi(k) < 𝚄*
By the stationarity of the scheme, the limit of and exist as . Denote by and .
{ℓm(k)}∞
m=0
{ℓm(k+)}∞
m=0
k → ∞ {ℓ*
m}∞ m=0
{ℓ*+
m }∞ m=0
𝚄* = max (1,⌊eM − 1/θ + 1⌋)
Theorem: For any , .
θ = 1/o(M) lim
M→∞ JSAT(M) = e/2
Consider a stationary transmission policy that does not employ coding across packets.
π
Develop a variant of the transmission policy with buffer size 1. Channel capacity .
π C 𝚄* = max (1,⌊M/C − 1/θ + 1⌋)
(Theorem 3) For any , .
θ = 1/o(M) lim
M→∞ JSAT(M) = 1
2C
SLIDE 37 Decentralized Age-based Policies Decentralized Age-based Policies Decentralized Age-based Policies
Stationary Age-based Thinning
node :
i {
active
δi(k) ≥ 𝚄*
inactive
0 ≤ δi(k) < 𝚄*
- By the stationarity of the scheme, the limit of
and exists.
and .
{ℓm(k)}∞
m=0
{ℓm(k+)}∞
m=0
{ℓ*
m}∞ m=0
{ℓ*+
m }∞ m=0
Theorem: 𝚄* = max (1,⌊eM − 1/θ + 1⌋)
- A stationary transmission policy that does not employ coding across packets.
- Develop a variant of the transmission policy with buffer size 1. Channel capacity .
π π C
Theorem: 𝚄* = max (1,⌊M/C − 1/θ + 1⌋) Theorem: For any , .
θ = 1/o(M) lim
M→∞
NEWSAoI(M) = 1
2C
Theorem: For any , .
θ = 1/o(M) lim
M→∞
NEWSAoI(M) = e/2
SLIDE 38 Decentralized Age-based Policies Decentralized Age-based Policies Decentralized Age-based Policies
Stationary Age-based Thinning
node :
i {
active
δi(k) ≥ 𝚄*
inactive
0 ≤ δi(k) < 𝚄*
- By the stationarity of the scheme, the limit of
and exists.
and .
{ℓm(k)}∞
m=0
{ℓm(k+)}∞
m=0
{ℓ*
m}∞ m=0
{ℓ*+
m }∞ m=0
Theorem: 𝚄* = max (1,⌊eM − 1/θ + 1⌋)
- A stationary transmission policy that does not employ coding across packets.
- Develop a variant of the transmission policy with buffer size 1. Channel capacity .
π π C
Theorem: 𝚄* = max (1,⌊M/C − 1/θ + 1⌋) Theorem: For any , .
θ = 1/o(M) lim
M→∞
NEWSAoI(M) = 1
2C
Theorem: For any , .
θ = 1/o(M) lim
M→∞
NEWSAoI(M) = e/2
SLIDE 39 Decentralized Age-based Policies Decentralized Age-based Policies Decentralized Age-based Policies
Stationary Age-based Thinning
node :
i {
active
δi(k) ≥ 𝚄*
inactive
0 ≤ δi(k) < 𝚄*
- By the stationarity of the scheme, the limit of
and exists.
and .
{ℓm(k)}∞
m=0
{ℓm(k+)}∞
m=0
{ℓ*
m}∞ m=0
{ℓ*+
m }∞ m=0
Theorem: 𝚄* = max (1,⌊eM − 1/θ + 1⌋)
- A stationary transmission policy that does not employ coding across packets.
- Develop a variant of the transmission policy with buffer size 1. Channel capacity .
π π C
Theorem: 𝚄* = max (1,⌊M/C − 1/θ + 1⌋) Theorem: For any , .
θ = 1/o(M) lim
M→∞
NEWSAoI(M) = 1
2C
Theorem: For any , .
θ = 1/o(M) lim
M→∞
NEWSAoI(M) = e/2
SLIDE 40 Decentralized Age-based Policies Decentralized Age-based Policies Decentralized Age-based Policies
Stationary Age-based Thinning
node :
i {
active
δi(k) ≥ 𝚄*
inactive
0 ≤ δi(k) < 𝚄*
- By the stationarity of the scheme, the limit of
and exists.
and .
{ℓm(k)}∞
m=0
{ℓm(k+)}∞
m=0
{ℓ*
m}∞ m=0
{ℓ*+
m }∞ m=0
Theorem: 𝚄* = max (1,⌊eM − 1/θ + 1⌋)
- A stationary transmission policy that does not employ coding across packets.
- Develop a variant of the transmission policy with buffer size 1. Channel capacity .
π π C
Theorem: 𝚄* = max (1,⌊M/C − 1/θ + 1⌋) Theorem: For any , .
θ = 1/o(M) lim
M→∞
NEWSAoI(M) = 1
2C
Theorem: For any , .
θ = 1/o(M) lim
M→∞
NEWSAoI(M) = e/2
SLIDE 41 Decentralized Age-based Policies Decentralized Age-based Policies Decentralized Age-based Policies
Stationary Age-based Thinning
node :
i {
active
δi(k) ≥ 𝚄*
inactive
0 ≤ δi(k) < 𝚄*
- By the stationarity of the scheme, the limit of
and exists.
and .
{ℓm(k)}∞
m=0
{ℓm(k+)}∞
m=0
{ℓ*
m}∞ m=0
{ℓ*+
m }∞ m=0
Theorem: 𝚄* = max (1,⌊eM − 1/θ + 1⌋)
- A stationary transmission policy that does not employ coding across packets.
- Develop a variant of the transmission policy with buffer size 1. Channel capacity .
π π C
Theorem: 𝚄* = max (1,⌊M/C − 1/θ + 1⌋) Theorem: For any , .
θ = 1/o(M) lim
M→∞
NEWSAoI(M) = 1
2C
Theorem: For any , .
θ = 1/o(M) lim
M→∞
NEWSAoI(M) = e/2
SLIDE 42
Numerical Results
SLIDE 43
Numerical Results
Throughput/rate Age of Information
SLIDE 44
Numerical Results
SLIDE 45
Numerical Results
SLIDE 46
Numerical Results
SLIDE 47
Numerical Results
SLIDE 48
Numerical Results
SLIDE 49
Numerical Results
SLIDE 50
Numerical Results
SLIDE 51
Numerical Results
SLIDE 52
Numerical Results
SLIDE 53
Numerical Results
SLIDE 54
Thank you!
