OFDMA Cross Layer

Resource Control

Gwanmo Ku

Adaptive Signal Processing and Information Theory Research Group

Jan. 25, 2013

Outline

OFDMA Cross Layer Resource Control

Objective Functions

- System Throughput (L1), Total Transmit Power (L1)

Constraints

- Transmit Power Constraint (L1)

- Quality of Service (User Demand, Fairness), Buffer Status (L2-3)

- Stability (L2-3)

Generalized Cross Layer Control (GCLC)

Stochastic Network Optimization (SNO)

Network Utility Maximization (NUM)

2/20

OFDMA Cross Layer Resource Control

3/20

System Model (𝑴 users, 𝑲 subcarriers)

Base Station (eNB)

Mobile (UE)

𝒖𝟏

𝒖𝑴

…

Higher Layer

Buffer

PHY

Higher Layer

Buffer

PHY

OFDMA

Ian Wong & Brian Evans

GCLC

𝒓𝒎

OFDMA Resource Control

4/20

Objective Functions

System Throughput Maximization

Transmit Power Minimization

Constraints

Transmit Power Constraint

Quality of Service

User Demands : Each User Required Data Rate

Fairness : Minimum User Data Rate

Stability based on Buffer Status

OFDMA Resource Allocation

5/20

Notations

𝒎 ∈ {𝟏,… ,𝑴} User Index

𝒌 ∈ {𝟏,… ,𝑲} Subcarrier Index

𝒑𝒎,𝒌 : Power Control Coefficient

𝜸𝒎,𝒌 : SINR for user index 𝒎 and subcarrier index 𝒌

𝑷𝑻 Total Transmit Power Constraint

𝒓𝒎 Required Each User Data Rate

𝒓𝟎 Required Minimum User Data Rate

𝒃𝒎 Buffer Service Rate

𝑹𝒎 Overall Coding Rate for User 𝒎

OFDMA Resource Allocation

6/20

System Throughput Maximization

Power Control

𝒑𝒎,𝒌 = argmaxE 𝑤𝑚

𝑀

𝑚=1

log(1 + 𝑝𝑚,𝑘𝛾𝑚,𝑘)

𝐾

𝑘=1

𝐒𝐲𝐬𝐭𝐞𝐦 𝐓𝐡𝐫𝐨𝐮𝐠𝐡𝐩𝐮𝐭

𝐸 𝑝𝑚,𝑘

𝐾

𝑘=1

𝑀

𝑚=1

≤ 𝑃𝑇

s.t

max (𝛽𝑚𝑅𝑚𝑟0, 𝛽𝑚𝑅𝑚𝑟𝑚) ≤ 𝑤𝑚 log(1 + 𝑝𝑚,𝑘𝛾𝑚,𝑘)

𝐾

𝑘=1

Stability

OFDMA Resource Allocation

7/20

Work by Ian Wong and Brian Evans

System Throughput Maximization with Tx. Power Constraint

𝑝𝑚,𝑘 = argmax 𝐄 𝑤𝑚

𝑀

𝑚=1

log(1 + 𝑝𝑚,𝑘𝛾𝑚,𝑘)

𝐾

𝑘=1

𝐄 𝑝𝑚,𝑘

𝐾

𝑘=1

𝑀

𝑚=1

≤ 𝑃𝑇

𝑤𝑚

𝑀

𝑚=1

= 1

s.t

OFDMA Resource Allocation

8/20

Optimization Framework

Dual Optimization

𝐿 𝑝 ⋅ , 𝜆 = 𝐄 𝑤𝑚

𝑀

𝑚=1

log(1 + 𝑝𝑚,𝑘𝛾𝑚,𝑘)

𝐾

𝑘=1

+𝜆 𝑃𝑇 − 𝐄( 𝑝𝑚,𝑘

𝐾

𝑘=1

)

𝑀

𝑚=1

𝑔∗ = min𝜆≥0

Θ(𝜆)

Θ 𝜆 = max𝑝 ⋅ ∈𝑃𝑇

𝐿(𝑝 ⋅ , 𝜆)

OFDMA Resource Allocation

9/20

Dual

Θ 𝜆 = max𝑝 ⋅ ∈𝑃𝑇

𝐿(𝑝 ⋅ , 𝜆)

= 𝜆𝑃𝑇 + max𝑝 ⋅ ∈𝑃𝑇

𝐄 𝑤𝑚 log 1 + 𝑝𝑚,𝑘𝛾𝑚,𝑘 − 𝜆𝑝𝑚,𝑘

𝑀

𝑚=1

𝐾

𝑘=1

= 𝜆𝑃𝑇 + max𝑝𝑘 ⋅ ∈𝑃𝑘

𝐄 𝑤𝑚 log 1 + 𝑝𝑚,𝑘𝛾𝑚,𝑘 − 𝜆𝑝𝑚,𝑘

𝑀

𝑚=1

𝐾

𝑘=1

= 𝜆𝑃𝑇 + 𝐸 max𝑝𝑘 ⋅ ∈𝑃𝑘

𝑤𝑚 log 1 + 𝑝𝑚,𝑘𝛾𝑚,𝑘 − 𝜆𝑝𝑚,𝑘

𝑀

𝑚=1

𝐾

𝑘=1

= 𝜆𝑃𝑇 + 𝐾𝐸𝛾𝑘max

𝑚∈{1,…,𝑀}max

𝑝𝑚,𝑘≥0(𝑤𝑚 log 1 + 𝑝𝑚,𝑘𝛾𝑚,𝑘 − 𝜆𝑝𝑚,𝑘)

multilevel water filling

𝑚𝑎𝑥 𝑑𝑢𝑎𝑙 𝑢𝑠𝑒𝑟 𝑠𝑒𝑙𝑒𝑐𝑡𝑖𝑜𝑛

OFDMA Resource Allocation

10/20

A Simple Closed Form

𝑝 𝑚,𝑘(𝜆) =1

𝛾0,𝑚 𝜆−

1

𝛾𝑚,𝑘

+

𝑥 + = max(0, 𝑥)

𝛾0,𝑚 𝜆 =𝜆 ln 2

𝑤𝑚 Cutoff value

𝑔∗ = min𝜆≥0

[𝜆𝑃𝑇 + 𝐾 𝐄𝛾𝑘𝑔𝑘 𝛾𝑘 , 𝜆 ]

𝑔𝑘 𝛾𝑘 , 𝜆 = max𝑚∈{1,…,𝑀}

{𝑔𝑚,𝑘 𝛾𝑚,𝑘 , 𝜆 }

𝑔𝑚,𝑘 𝛾𝑚,𝑘 , 𝜆 = 𝑤𝑚 log 1 + 𝑝 𝑚,𝑘 𝜆 − 𝜆 𝑝 𝑚,𝑘 𝜆

OFDMA Resource Allocation

11/20

Optimal Solution

𝑔𝑚,𝑘 𝛾𝑚,𝑘 , 𝜆 = 𝑤𝑚 log 1 + 𝑝 𝑚,𝑘 𝜆 − 𝜆 𝑝 𝑚,𝑘 𝜆

=𝑤𝑚

ln 2ln

𝛾𝑚,𝑘

𝛾0,𝑚 𝜆−

𝑤𝑚

ln 2+

𝜆

𝛾𝑚,𝑘𝑢(𝛾𝑚,𝑘 − 𝛾0,𝑚 𝜆 )

𝑢 𝑥 = 0 𝑥 < 01 𝑥 ≥ 0

𝜆∗ = 𝑎𝑟𝑔min𝜆≥0

[𝜆𝑃 + 𝐾 𝑔𝑘𝑓𝑔𝑘𝑔𝑘 𝑑𝑔𝑘

∞

0

]

𝑝 𝑚,𝑘(𝜆∗) =

1

𝛾0,𝑚 𝜆∗−

1

𝛾𝑚,𝑘

+

𝑝𝑚,𝑘∗ = 𝑝 𝑚,𝑘 𝜆∗ 1(𝑚 = 𝑚𝑘

∗ )

𝑚𝑘∗ = argmax

𝑚∈{1,…,𝑀}𝑤𝑚 log 1 + 𝑝 𝑚,𝑘 𝜆∗ 𝛾𝑚,𝑘 − 𝜆∗𝑝 𝑚,𝑘 (𝜆∗)

Cross Layer Control

12/20

Generalized Cross Layer Control (GCLC)

Proposed by Georgiadis, Neely, and Tassiulas

Focus on Stability based on Queuing Statistics

• Stochastic Network Optimization

• Network Utility Maximization

Network Stability

• Differential Equation of Queuing Statistics

• Lyapunov Stability

Cross Layer Control

13/20

Stochastic Network Optimization

Buffer for user 𝑚

Arrival Rate 𝝀𝒎 Service Rate 𝝁𝒎

Backlog Queue 𝑸𝒎 (𝒕)

Network State Variable 𝑺(𝒕)

Control Action 𝑰 𝒕 ∈ 𝑰𝑺(𝒕) feasible control region under 𝑺(𝒕)

𝐐 𝑡𝑆 𝑡 ,𝐼(𝑡)

𝐐(𝑡 + 1)

Cross Layer Control

14/20

Stochastic Network Optimization

Stability Issue

𝑄𝑚 𝑡 + 1 ≤ max 𝑄𝑚 𝑡 − 𝑅𝑚𝑜𝑢𝑡 𝐼 𝑡 , 𝑆 𝑡 , 0 + 𝑅𝑚

𝑖𝑛(𝐼 𝑡 , 𝑆 𝑡 )

𝑶𝒖𝒕𝒈𝒐𝒊𝒏𝒈 𝑸𝒖𝒆𝒖𝒆 𝑬𝒏𝒕𝒆𝒓𝒊𝒏𝒈 𝑸𝒖𝒆𝒖𝒆

lim𝑡→ ∞

sup1

𝑡 𝐄{𝑄𝑚 𝜏 }

𝑡−1

𝜏=0

< ∞

Lyapunov Stability

If there exist 𝑩 > 𝟎 and 𝝐 > 𝟎, such that for all

times slot 𝒕 we have :

Then network is strongly stable, and

𝐄 ∆ 𝑡 Q 𝑡 ≤ 𝐵 − 𝜖 𝑄𝑚(𝑡)

𝑀

𝑚=1

lim𝑡→ ∞

sup1

𝑡 𝐄{Q 𝜏 }

𝑡−1

𝜏=0

<𝐵

𝜖

15/20

Cross Layer Control

16/20

Find 𝑰(𝒕)

Find 𝚲 by Lyapunov Drift

Drift Definition

𝐼∗ 𝑡 = argmax𝐼 𝑡 ∈𝐈𝑆(𝑡)

𝑊𝑎𝑏∗ 𝑡 𝜇𝑎𝑏

∗ (𝑡)

𝑎𝑏

𝑊𝑎𝑏∗ 𝑡 = max

𝜇𝑎𝑏

𝑄𝑎 𝑡 − 𝑄𝑏 𝑡 +

maximum queue backlog differential

∆ 𝑡 = 𝐿 𝑡 + 1 − 𝐿(𝑡)

𝐿 𝑡 =1

2 𝑄𝑚

2 (𝑡)

𝑀

𝑚=1

Lyapunov Drift

17/20

∆ 𝑡 = 𝐿 𝑡 + 1 − 𝐿(𝑡)

=1

2 [𝑄𝑚

2 𝑡 + 1 −

𝑀

𝑚=1

𝑄𝑚2 (𝑡)]

After applying 𝑄𝑚2 (𝑡 + 1)

Find Lyapunov Bound with Conditional Expectation

∆ 𝑡 𝐐 𝑡 ≤ …

lim𝑡→ ∞

sup1

𝑡 𝐄{𝑄𝑚 𝜏 }

𝑡−1

𝜏=0

< ∞

Network Utility Maximization (NUM)

18/20

Rate 𝐫 ∈ 𝚲 with Maximum Utility

𝐫∗ = argmax𝐫≤𝛌

𝑔 𝐫 |𝐫 ∈ 𝚲

𝑔(𝐫) : Utility Function

Minimize Cost

Generalized Cross Layer Control

19/20

General Form of GCLC

Cost Variable Vector 𝐱 : Maximum cost constraints 𝐐

Utility Variable Vector 𝐲 :Minimum utility constraints 𝐇

Stable Region 𝐫 ∈ 𝚲

• Arrival Rate Vector 𝛌

Minimize net cost

• Natural cost function 𝒇(𝒙) and Concave Utility function 𝒈(𝒚)

min𝐫≤𝛌

𝑓 𝐱 − 𝑔(𝐲)|𝐪 𝐱 ≤ 𝐐, 𝐡 𝐲 ≥ 𝐇, 𝐫 ∈ 𝚲

OFDMA Resource Control via GCLC

20/20

OFDMA via GCLC

Cost Variable Vector 𝐱 : power coefficients

• 𝐐 = 𝐏𝐓 : Total Power Constraint

Utility Variable Vector 𝐲 : user data rate 𝐲 = 𝐫

• 𝐇 : Quality of Service (User demands, Fairness)

Stable Region based on Queuing Statistics

Minimize net cost : Maximize System Throughput

min𝐫≤𝛌

𝑓 𝐱 − 𝑔(𝐲)|𝐪 𝐱 ≤ 𝐐, 𝐡 𝐲 ≥ 𝐇, 𝐫 ∈ 𝚲