Medium Access Control for Distributed Systems Faeze Heydaryan - - PowerPoint PPT Presentation

Medium Access Control for Distributed Systems

Faeze Heydaryan Joint work with Yanru Tang

Committee members:

• Prof. Rockey Luo
• Prof. Liuqing Yang
• Prof. Ali Pezeshki
• Prof. Haonan Wang

Coordinated vs Distributed Communication

Coordinated Communication

• Joint coding optimization
• long messages

Distributed Communication

• Opportunistic channel access
• Bursty short messages

1

Classical Information and Network Theory

2

Wireless network requires both efficiency and modularity.

Classical Information Theory Classical Network Theory

• Emphasizes on efficiency
• Joint coding optimization
• Long messages
• Emphasizes on modularity
• Layering architecture
• User coordination can be expensive.
• Binary transmission/ idlingde decisions at each

data link layer user

• Power/rate adaptation not supported.

Problems in Distributed Communication Networks

Enhanced Physical Link-layer Interface

3

Current Physical-Link Layer Interface Enhanced Physical-Link Layer Interface

• Single transmission option
• Multiple transmission options
• Possible support of power/rate adaptation

Distributed Channel Coding

Ensemble of channel codes at the physical layer Each code corresponding to a link layer transmission option

• Transmitters choose channel code individually.
• Receiver knows the code ensemble, but not the coding choices.
• Message decoding or collision report due to reliability requirement.

4

• If the transmitters happen to choose their options inside the region, the packet can be recovered with asymptotic

error probability of zero.

• If the transmitters happen to choose their options outside the region, collision can be reported with asymptotic

probability of one.

Achievable Region

Achievable region for multiple access system over a discrete time memoryless channel

Multiple Transmission Options at Link Layer

Back off by decreasing transmission probability in response to packet collison. Decrease communication rate in response to packet collision. With multiple transmission options: How system should respond to success transmission and packet collision? How to support such functions at the physical layer ? We propose a distributed MAC algorithm to optimize a general utility function with/without enhanced interface.

5

Current Protocol Efficient Approach Example Multiple access system with 𝐿 users + , 𝐿 users have messages (𝐿 is changing) If 𝑠

=

• log 1 +
• (bits/symbol) is fixed, maximum achievable sum rate is
• log 1 +
• .

If 𝑠

=

• log 1 +
• (bits/symbol) with rate adaptation, maximum achievable sum rate is
• log 1 +
• .

Existing MAC Protocols

6

ALOHA protocol Tree-splitting algorithm Back-off approach Existing MAC algorithms either consider throughput optimization and/or collision channel. We consider a general channel, a general utility, with/without enhanced physical link-layer interface.

Current Distributed MAC Algorithms

Set of colliding users Transmitting users Non-Transmitting users Users maintain a transmission probability Transmission is successful Probability increases Probability decreases No Yes

System Model

7

• Multiple access network with 𝐿 homogenous users.
• 𝐿 unknown to transmitters and receiver.
• Each user is backlogged with a saturated message queue.
• Time is slotted.
• Each user has 𝑁 transmission options + an idling option.

𝒒 𝑢 = 𝑞 𝑢 𝒆 𝑢 , 0 ≤ 𝑒 𝑢 ≤ 1, 𝑒 𝑢 = 1

• 𝒒 𝑢 : Transmission probability vector of user 𝑙

𝑞(𝑢): Transmission probability of user 𝑙 𝒆 𝑢 : Transmission direction vector of user 𝑙 Probabilities of choosing each option if user) 𝑙 transmits)

8

Stochastic Approximation Framework

Updating Rule: 𝒒 𝑢 + 1 = 1 − 𝛽 𝑢 𝒒 𝑢 + 𝛽 𝑢 𝒒 𝑢 = 𝒒 𝑢 + 𝛽(𝑢)(𝒒 𝑢 − 𝒒 𝑢 ) 𝑸 𝑢 = 𝒒 𝑢 𝒒 𝑢 … 𝒒 𝑢 𝑸 𝑢 = 𝒒 𝑢 𝒒 𝑢 … 𝒒 𝑢 𝑸 𝑢 + 1 = 𝑸 𝑢 + 𝛽(𝑢)(𝑸 𝑢 − 𝑸(𝑢)) Mean-Bias Condition Bias Term: 𝑯 𝑢 = 𝑯 𝑸 𝑢 = 𝐹 𝑸 𝑢 − 𝑸 (𝑢) 𝑯 𝑢 ≤ 𝐿𝛾(𝑢) Lipschitz Continuity Condition 𝑸 𝑸 − 𝑸 (𝑸) ≤ 𝐿 𝑸 − 𝑸 for all 𝑸 and 𝑸 Associated Ordinary Differential Equation (ODE):

𝑸()

• = − 𝑸 𝑢 − 𝑸

(𝑢) Equilibrium of ODE 𝑸∗ : 𝑸∗ = 𝑸 (𝑸∗) Target probability vector Noiseless version of 𝑸 𝑢

9

Theorem 1

Assumptions:

• Associated ODE has unique equilibrium 𝑸∗
• ∑

𝛽 𝑢 = ∞, ∑ 𝛽 𝑢 < ∞, ∑ 𝛽 𝑢 𝛾 𝑢 < ∞.

• Mean and Bias condition
• Lipschitz Continuity condition

Conclusion:

• 𝑸 𝑢 converges to 𝑸∗ with probability one.

Theorem 2

Assumptions:

• Associated ODE has unique equilibrium 𝑸∗
• ∃ 0 < 𝛽 < 𝛽 < 1, ∃ 𝑈 ≥ 0, 𝛽 ≤ 𝛽 𝑢 ≤ 𝛽, 𝛾 𝑢

≤ 𝛽, ∀𝑢 > 𝑈

• Mean and Bias condition
• Lipschitz Continuity condition

Conclusion:

• 𝑸 𝑢 converges weakly to 𝑸∗ in the sense that

∀𝜗 > 0, ∃𝐿: lim sup

→

𝑄𝑠 𝑸 𝑢 − 𝑸∗ ≥ 𝜗 < 𝐿𝛽.

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MAC Algorithm Design Challenges

Objective

Design a distributed MAC algorithm to satisfy Mean-Bias and Lipschitz Contunity conditions and to place unique equilibrium of associated ODE at a point that maximizes a chosen utility function.

• There is a virtual packet in each time slot.
• The receiver can estimate virtual packet success probability 𝑟(𝑢).

Examples

• Collision Channel:

Virtual packet success probability Idling probability 𝑟 𝑢 = 1 − 𝑞 𝑢 𝑟 𝑢

• General Channel + Random Block Coding:

Reception of virtual packet Detecting whether transmission vector of real users belong to a specific region

Assumptions

Design Choice

• Users should obtain the same target transmission probability vector

𝑸∗ = 𝟐⨂𝒒∗ = 𝟐⨂𝒒 (𝒒∗)

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Single Option: Channel Model

Single transmission option 𝑁 = 1 𝒒 = 𝑞, 𝑸 = 𝒒 = 𝑞 𝑞 … 𝑞 Channel Model

Real channel parameter set 𝐷 for 𝑘 ≥ 0 Virtual channel parameter set 𝐷 for 𝑘 ≥ 0 𝐷 ≥ 𝐷() 𝐷: Conditional success probability

• f a real packet should it be

transmitted in parallel with j other real packets. 𝐷: Success probability of a virtual packet should it be transmitted in parallel with j real packets. 𝐾 = 𝑏𝑠𝑕 min

• 𝐷 > 𝐷() + 𝜗

Definition

12

Single Option: Utility Optimization

Maximize a symmetric network utility

• Sum system throughput:

𝑉 𝐿, 𝑞, 𝐷 = 𝐿 𝐿 − 1 𝑘 𝑞 1 − 𝑞 𝐷

• Asymptotically optimal transmission probability satisfies lim

→ 𝐿𝑞 ∗ = 𝑦∗

𝑦∗ = 𝑏𝑠𝑕 max

• lim

→ 𝑉 𝐿, 𝑦

𝐿 , 𝐷 Set the system equilibrium at 𝑞∗ = 𝑛𝑗𝑜 𝑞,

∗ , 𝑞 = 𝑛𝑗𝑜 1, ∗ without knowing 𝐿.

Channel contention measure: 𝑟(𝑞, 𝐿) = ∑ 𝐿 𝑘 𝑞 1 − 𝑞 𝐷

• Weighed sum system throughput

𝑉 𝐿, 𝑞, 𝐷 = 𝐿 𝐿 − 1 𝑘 𝑞 1 − 𝑞 𝐷 − 𝐿𝑞𝐹

13

Single Option: Two Monotonicity Properties

Estimated users number: 𝐿

• Target transmission probability: 𝑞̂ = 𝑛𝑗𝑜 𝑞,

∗

• .

Contribution:Theorems

With 𝐷 ≥ 𝐷() for all 𝑘 ≥ 0, 𝑟 𝑞, 𝐿 is non-increasing in 𝑞. Furthermore

,

• < 0 for 𝐿 > 𝐾 and 𝑞𝜗(0,1) .

Theoretical channel contention measure 𝑟

∗

𝑂 = 𝐿 : Largest integer below 𝐿

• 𝑟

∗ 𝑞̂ = 𝑞̂ − 𝑞

𝑞 − 𝑞 𝑟 𝑞̂ + 𝑞 − 𝑞̂ 𝑞 − 𝑞 𝑟 𝑞̂ 𝑞 = 𝑛𝑗𝑜 𝑞,

∗ ,

𝑟 𝑞 = 𝑂 𝑘 𝑞 1 − 𝑞 𝐷

• If 𝑦∗ > 0 and 𝑐 ≥ 𝑛𝑏𝑦 1, 𝑦∗ − 𝛿 with 𝛿 being defined as

𝛿 = min

,,∗ ∑

• ()
• ∑
• ()
• then 𝑟

∗ 𝑞̂ is non-decreasing in 𝑞̂.

Furthermore if 𝑐 > 𝑛𝑏𝑦 1, 𝑦∗ − 𝛿 ,𝑟

∗ 𝑞̂ is strictly increasing in 𝑞̂

for 𝑞̂𝜗 (0, 𝑞).

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Single Option: Distributed MAC algorithm

Transmitters Receiver

Initialize transmission probability Measures the success probability

• f virtual packet 𝑟 and feeds it back to the

transmitters. Derive target transmission probability 𝑞̂ by solving 𝑟

∗ 𝑞̂ = 𝑟

Update transmission probability by 𝑞 = 1 − 𝛽 𝑞 + 𝛽𝑞̂ 𝑔𝑝𝑠 𝑙 = 1, … , 𝐿 Converge No Yes End

15

Single Option: Convergence and Optimality

• 𝑐 > 𝑛𝑏𝑦 1, 𝑦∗ − 𝛿
• Using proposed MAC algorithm
• ODE has unique equilibrium at 𝒒∗ = 𝑛𝑗𝑜 𝑞,

∗ 𝟐

• 𝑞̂(𝒒) satisfies Mean-Bias and Lipschitz Continuity conditions.

𝒒 converges to 𝒒∗

Theorem

Optimality concern Ideal transmission probability: 𝑞∗ = min {1,

∗ }

Transmission probability at the equilibrium: 𝑞∗ = 𝑛𝑗𝑜 𝑞,

∗

• 𝑐 should be small

𝛿 and 𝐾 not much smaller than 𝑦∗ 𝑞 close to one 𝐾 not much larger than 𝑦∗ Requires an appropriate virtual packet design

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Single Option: Simulation

Collision channel Sum system throughput Fading channel Sum system throughput weighted by transmission energy cost 𝐷 = 1, 𝐷 = 0 𝑔𝑝𝑠 𝑘 > 0 𝑦∗ = 1 𝐷 = 1, 𝐷 = 0 𝑔𝑝𝑠 𝑘 > 0 𝜗 = 0.01, 𝛿 = 𝐾 = 0 𝐷 = 1, 𝑔𝑝𝑠 𝑘 < 4, 𝐷 = 0.7 𝑔𝑝𝑠 4 ≤ 𝑘 < 6, 𝐷 = 0 𝑔𝑝𝑠 𝑘 ≥ 6 𝑦∗ = 3.29 𝐷 = 𝐷 = 0 𝑔𝑝𝑠 𝑏𝑚𝑚 𝑘 ≥ 0 𝜗 = 0.01, 𝛿 = 𝐾 = 3

17

Multiple Options :Channel Model

Multiple transmission options 𝑁 ≥ 1 𝒒 = 𝑞𝒆, 𝑸 = 𝒒

𝒒 … 𝒒

Channel Model

Real channel parameter function set 𝐷(𝒆) for 1 ≤ 𝑗 ≤ 𝑁, 𝑘 ≥ 0 Virtual channel parameter function set 𝐷(𝒆) for 𝑘 ≥ 0 𝐷(𝒆) ≥ 𝐷 (𝒆) 𝐷(𝒆): Conditional success probability of a real packet corresponding to 𝑗th transmission

• ption, should it be

transmitted in parallel with j other real packets. 𝐷(𝒆): Success probability

• f a virtual packet should it be

transmitted in parallel with j real packets. 𝐾 𝒆 = 𝑏𝑠𝑕 min

• 𝐷(𝒆) > 𝐷 (𝒆) + 𝜗

Definition

18

Multiple Options: Utility Optimization

Intend to design a distributed MAC algorithm to maximize a chosen utility 𝑉 𝐿, 𝒒, 𝐷(𝒆) by maintaining channel contention measure 𝑟 at the desired level. Channel contention measure: 𝑟(𝒒, 𝐿) = ∑ 𝐿 𝑘 𝑞 1 − 𝑞 𝐷 𝒆 , 𝒒 = 𝑞𝒆

• Example

𝒒∗ = 𝑞

∗, 𝑞 ∗ = 𝑏𝑠𝑕 max 𝒒

𝑉 𝐿, 𝒒, 𝐷(𝒆) 𝒒∗ = 𝑞∗𝒆∗, 𝒆∗ = 1,0 𝑔𝑝𝑠 𝐿 ≤ 4, 𝒆∗ = 0,1 𝑔𝑝𝑠𝐿 ≥ 10

19

Multiple Options: Conditions and Definition

Head and Tail Condition

There exist two integer-valued constants 0 < 𝐿 ≤ 𝐿 such that 𝐿 ≥ 𝐾(𝒆(𝐿)) and 𝒆 𝐿 = 𝒆(𝐿) for 𝐿 ≤ 𝐿, 𝐿 ≥ 𝐾(𝒆(𝐿)) and 𝒆 𝐿 = 𝒆(𝐿) for 𝐿 ≥ 𝐿.

Theoretical channel contention measure 𝒓𝒘

∗

𝐿 ≤ 𝐿, 𝐿 ≥ 𝐿 𝑟

∗ 𝐿

= 𝑞(𝐿 ) − 𝑞(𝑂 + 1) 𝑞(𝑂) − 𝑞(𝑂 + 1) 𝑟 𝑞 𝐿 , 𝑂 + 𝑞(𝑂) − 𝑞(𝐿 ) 𝑞(𝑂) − 𝑞(𝑂 + 1) 𝑟 𝑞 𝐿 , 𝑂 + 1 𝐿 < 𝐿 < 𝐿 𝑟

∗ 𝐿

= 𝑂 + 1 − 𝐿 𝑟 𝑞 𝐿 , 𝑂 + 𝐿 − 𝑂 𝑟 𝑞 𝐿 , 𝑂 + 1 With varying 𝒆 𝐿 , highly difficult to guarantee the two monotonicity properties.

Challenge

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Pinpoints Condition 𝑀 + 1 integers: 𝐿 = 𝐿 < 𝐿 < … < 𝐿 = 𝐿 For 𝑗 = 0, … , 𝑀 and 0 ≤ 𝜇 < 1, define 𝐿 = 1 − 𝜇 𝐿 + 𝜇𝐿

• 𝒆 = 1 − 𝜇 𝒆 𝐿

+ 𝜇𝒆 𝐿

• 𝑟

∗

= 1 − 𝜇 𝑟

∗ 𝐿

+ 𝜇𝑟

∗ 𝐿

• Interpolation Approach
• ∃ 𝜗 such that 𝑟

∗ 𝐿

− 𝑟

∗ 𝐿

≥ 𝜗.

• ∃ 𝜗 such that 𝐿

> 𝐾(𝒆).

• ∃ 0 < 𝑞 < 𝑞 < 1 such that 𝑞 ≤ 𝑞(𝐿

) ≤ 𝑞.

• Extension of definition of 𝑟(𝒒, 𝐿

) for non-integer 𝐿

• 𝑟 𝒒, 𝐿

= 𝐿 + 1 − 𝐿 𝑟 𝒒, 𝐿

• + 𝐿

− 𝐿

• 𝑟(𝒒, 𝐿

+ 1) This inequality should be satisfied 𝑟 𝑞𝒆, 𝐿 ≤ 𝑟

∗

≤ 𝑟 𝑞𝒆, 𝐿 . 𝑞 𝐿 is designed for 𝑀 + 1 integers 𝐿 = 𝐿 < 𝐿 < … < 𝐿 = 𝐿 to satisfy Pinpoints Condition. For every 𝐿 = 𝐿 , choose 𝑞 𝐿 as the solution of 𝑟 𝑞(𝐿 )𝒆, 𝐿 = 𝑟

∗ .

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Multiple Options: Distributed MAC Algorithm

Transmitters Receiver

Initialize transmission probability vector Measures the success probability of virtual packet 𝑟 and feeds it back to the transmitters. Derive a user number estimate 𝐿 by solving 𝑟

∗ 𝐿

= 𝑟 Update transmission probability vector by 𝒒 = 1 − 𝛽 𝒒 + 𝛽𝒒(𝐿 )𝑔𝑝𝑠 𝑙 = 1, … , 𝐿 Converge End Yes No At equilibrium: If 𝐿 = 𝐿 then 𝑟

∗ 𝐿

= 𝑟 If 𝐿 > 𝐿 then 𝑟

∗ 𝐿

< 𝑟 If 𝐿 < 𝐿 then 𝑟

∗ 𝐿

> 𝑟 Need to prove only monotonicity of 𝑟

∗(𝐿

)

22

Multiple Options: Convergence

Monotonicity and Gradient Condition

For 𝐿 ≤ 𝐿 ≤ 𝐿:

• 𝒒 𝐿

= 𝑞 𝐿 𝒆(𝐿 ) should be Lipschitz continuous in 𝐿

• 𝒒 𝐿

− 𝒒 𝐿

• ≤ 𝐿

𝐿

− 𝐿 .

• 𝑟

∗(𝐿

) should be continuous and strictly decreasing in 𝐿

• 𝑟

∗ 𝐿

− 𝑟

∗ 𝐿

• ≥ 𝜗 𝐿

− 𝐿 .

• ∃ 𝜗 > 0 ; 𝐿

> 𝐾(𝒆(𝐿 )).

• ∃ 0 < 𝑞 < 𝑞 < 1 such that 𝑞 ≤ 𝑞(𝐿

) ≤ 𝑞.

• Users update their transmission probability vectors by

adopting proposed MAC algorithm.

• Head and Tail Condition is satisfied.
• For 𝐿

≤ 𝐿 and 𝐿 ≥ 𝐿: 𝒒(𝐿 ) and 𝑟

∗(𝐿

) are designed like single transmission option

• For 𝐿 ≤ 𝐿

≤ 𝐿: 𝒒(𝐿 ) and 𝑟

∗(𝐿

) satisfy Monotonicity and Gradient Condition.

• Associated ODE has unique equilibrium at

𝑸∗ = 𝟐⨂𝒒(𝐿).

• Target transmission probability vector 𝒒

(𝑸) satisfies Mean-Bias and Lipschitz continuity Conditions

• Transmission probability vectors of all

users converge to 𝑸∗ = 𝟐⨂𝒒(𝐿).

Contribution: Theorem

Using Interpolation Approach, 𝑞 𝐿 and 𝑟

∗(𝐿

) satisfy Monotonicity and Gradient Condition. Theorem

23

Multiple Options: Simulation

Head Regime Tail Regime 𝒆 = 1 0 𝐷 = 1 ; 𝑘 ≤ 2, 𝐷 = 0 ; 𝑘 > 2 𝐷 = 𝐷; 𝑘 ≥ 0 𝑦

∗ = 2.27, 𝑐 = 1.01, 𝛿 = 2

𝒆 = 0 1 𝐷 = 1 ; 𝑘 ≤ 11, 𝐷 = 0 ; 𝑘 > 11 𝐷 = 1 ; 𝑘 ≤ 8, 𝐷 = 0 ; 𝑘 > 8 𝑦

∗ = 8.82, 𝑐 = 1.01, 𝛿 = 8

𝐿 = 𝐿 = 4, 𝐿 = 5, , 𝐿 = 6, , 𝐿 = 𝐿 = 10

24

Future Research: Priority Users and Multiple Channels

• Multiple access network with two groups of users.
• Single transmission option.
• 𝐿 homogenous high-priority users and 𝐿 homogenous low-priority users.
• Model a general channel using real channel parameter set 𝐷 and the virtual channel parameter set 𝐷 .
• There exists a virtual packet in each time slot.
• Define 𝛿 =
• .

𝑞 𝑞 Extend the result to network with multiple user groups with different priority levels network with multi channels

Future Research Problem Two User Groups

Preliminary Ideas

Step 1

• Define the theoretical channel contention measure as a function of 𝛿 and 𝑂

to be decreasing in 𝑂 .

• Estimate the number of low-priority users by comparing the theoretical channel contention measure

with the real channel contention measure. Step 2

• Develop a distributed MAC algorithm to guarantee the system converges to a unique equilibrium.
• Prove its Convergence and uniqueness of the equilibrium.

Step 3

• Investigate the impact of a mismatched 𝛿 on location of system equilibrium.

Steps

25

Thanks