random access networks: transmission costs, selsh nodes

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Random Access Networks: Transmission Costs, Selfish Nodes, and Protocol Design Joint work with Clement Yuen and Ran Pang 1

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Page 1: Random Access Networks: Transmission Costs, Selsh Nodes

Random Access Networks:Transmission Costs, Selfish Nodes, and Protocol Design

Joint work with Clement Yuen and Ran Pang

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Page 2: Random Access Networks: Transmission Costs, Selsh Nodes

Random Access Networks:

� Collisions - Stability

� Transmission Cost

� Rate Control?

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Page 3: Random Access Networks: Transmission Costs, Selsh Nodes

Rate Control in Random Access Networks

Questions

� Is transmission cost sufficient to guarantee stability?

� If not, what additional mechanisms are needed?

Answers

� Transmission cost does not guarantee stability

� Pricing mechanism: stability and system performance

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Page 4: Random Access Networks: Transmission Costs, Selsh Nodes

Outline� Non-Cooperative Game - Idealized Situation

– (Symmetric) Nash Equilibrium

– Pricing Mechanism

� Distributed Algorithm

� Model: Slotted Aloha (CSMA/CD)

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Page 5: Random Access Networks: Transmission Costs, Selsh Nodes

Non-Cooperative Game - Slotted Aloha Model

1

� Poisson Arrivals

� Collision - Retransmission

� Probabilistic Retransmissions

� Transmission Cost �

� Infinite set of hosts

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Page 6: Random Access Networks: Transmission Costs, Selsh Nodes

Non-Cooperative Game� Poisson Arrivals with (Aggregated) Rate � ��� � ,� � �

– Packets have different values

– �� �� � ��� �� �

� State � : number of backlogged packets

– whether to accept a new packet

– retransmission probability for backlogged packet

� Strategy � ��� �� �

– � ��� � � � � � ��� � � � ��� � ��� � � � �

– � �� �� � �� �� � �� � � � �

� Strategy � � � � � �

– � � � � � � � � � � � , � � � � � � ��� � �

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Page 7: Random Access Networks: Transmission Costs, Selsh Nodes

Markov Chain Formulation� Nodes are indistinguishable (symmetric strategies)

� Strategy � : Markov chain � � � ��� � � �

� Successful transmission of a backlogged packet for given node:��� �� ��� � � � � � � �� � � ��� �� � �� � � � ��

� Offered load: � � � �� � � � ��� � � � � �

� Instantaneous throughput

� � � � ��� � ��

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Page 8: Random Access Networks: Transmission Costs, Selsh Nodes

Markov Chain Formulation� Cost for successfully transmitting a packet

– new packet: � � � � � �

– backlogged packet: � � � � � �

� Retransmission Probabilities ��

– new packet: � � � � � � �� �– backlogged packet: � � � � � � �� �

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Page 9: Random Access Networks: Transmission Costs, Selsh Nodes

Equilibrium Strategy� Admissible retransmission vector ��

– � � � � � �� � is a random variable, � � �

– set of all admissible retransmission vectors: � � � �

� Admissible strategy �– � � � ��� �

– � � � � � �

� Equilibrium strategy

– � �� � �� � � � � � � � � � �� � , � � �– � � � �� � � � � � �

� Symmetric Nash equilibrium

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Page 10: Random Access Networks: Transmission Costs, Selsh Nodes

Stable Strategy� Stable strategy

– “Expected number of backlogged nodes stays bounded”

� Stable equilibrium strategy

– Single positive recurrent class, and possibly some transientstates

...0 1 2 3

� Questions

– Does a stable equilibrium strategy exist?

– Does a unique stable equilibrium strategy exist?

– What is the performance at a stable equilibrium strategy?

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Page 11: Random Access Networks: Transmission Costs, Selsh Nodes

A particular class of Strategies� Set � � of admissible strategies

– class � � � � � � ��� � �

– � � � ��� � � � � �� � � �� � � � � � �

� Transmitting backlogged packet

– �� �� � �� � � � �� �� � � � � � �

� Cost � � � � � �

– � � � � � �� � � � , � � � �

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Page 12: Random Access Networks: Transmission Costs, Selsh Nodes

Existence of a Equilibrium Strategy � � � �

Proposition 1 There exists a stable equilibrium strategy � � � � if andonly if the following conditions hold

(a) �� � � � �� � ��� � � � � � � ,

(b) � ���� � � � �� � , and

(c) � ��� � � � � .

Idea: The transmission cost � needs to be large enough in order tohave a equilibrium strategy � � � � .

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Page 13: Random Access Networks: Transmission Costs, Selsh Nodes

Existence of a Stable Equilibrium Strategy

Proposition 2 If � is a stable equilibrium strategy then there exists a

� � � such that � � � � .

Interpretation

� If transmission cost � is too small then there does not exist astable equilibrium allocation

� If there exists a stable equilibrium allocation, then there istypically a continuum of stable equilibria (in � ).

� Different values of � lead to different throughput and delay.

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Page 14: Random Access Networks: Transmission Costs, Selsh Nodes

Protocol Design� Need additional mechanism to guarantee stability

� Would like mechanism for choosing �

� Idea: cost � for successfully transmitted packets

� Can choose � and � to

– set throughput/delay (trade-off)

� MAC protocol: choosing �– Pick � in advance ( � � )

– Choose � for throughput/delay

– Determine� � � � (probability of successful retransmission isthe same at all states)

– No node has incentive to deviate (no cheating)

– MAC standard

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Page 15: Random Access Networks: Transmission Costs, Selsh Nodes

Protocol Design� Assumption

– Know � ��� �– Can observe state �

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Page 16: Random Access Networks: Transmission Costs, Selsh Nodes

Protocol Design� Rate Control

– Collision: Increase Price

– Idle: Decrease Price

� Questions

– Stable?

– Operating Point?

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Page 17: Random Access Networks: Transmission Costs, Selsh Nodes

Price Update� Price Signal�

� Aggregated Transmission Rate � � � �

� Collision: Increase Price

� Idle Slot: Decrease Price

� Price Adaptation: � � � , � � �

� � � � �� �� �� ��� � � �� � ��� � � �� �� � � � � � �

� Retransmission Probability:�

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Page 18: Random Access Networks: Transmission Costs, Selsh Nodes

Stability - Markov Chain � � � � � � �

Assumption 1 There exist positive constants � �� � and� �� � such that

� �� ��� � � � � �� � � is strictly decreasing, with � ��� �� � when� � � �� � .

λ (u)

u

Stability: Is the Markov chain positive recurrent?

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Page 19: Random Access Networks: Transmission Costs, Selsh Nodes

Operating Point� Mean Drift of Backlog �

� � � � � � � � � � � � � � � � ���� � � � � � � � �

� Mean Drift of Price�

� � � � � � � � � � � � � � � ��� � � � � � � � �

� Operating Point � ��� � � � �

� � � �� � � � �� � � �� � � � �� �

� Questions

– Does operating point exist?

– Is there a unique operating point?

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Page 20: Random Access Networks: Transmission Costs, Selsh Nodes

Results� System is stable.

� (Under suitable conditions) There exists a unique operatingpoint � �� � � � �

� � � �� � �� �

� We can set � � by choosing � , , � .

– Throughput �� � � �� ���– Backlog ���

– Average Delay � � �� � ��

– ���� � � � � � � � ��

� � ����

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Page 21: Random Access Networks: Transmission Costs, Selsh Nodes

Numerical Results� � � � and � � �� � � � �� � , � � � � � � � �

� � � � , � � , and � � � � � �

� � ��� �� ��� �� � � � � � � ��

� � � � � �

0 50 100 150 2000

50

100

150

200

n

u State Trajectory

0 50 100 1500

1000

2000

3000

4000

5000n Backlog Histogram

� � � � �� � and � � � � � � �

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Page 22: Random Access Networks: Transmission Costs, Selsh Nodes

Summary� Price-Based Rate Control

� Stability

� Performance

� Do not need to know

– State �

– Rate function � ��� �

– Retransmission probability�

� Model

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Page 23: Random Access Networks: Transmission Costs, Selsh Nodes

Delay Differentiation and Dynamic RetransmissionProbabilities

� Delay Differentiation:� � , � � ��� � � ���

� Dynamic Retransmission Probabilities:� � � �

– �� �� � ��� �� �– � ��� � �� � , � � �

– � ��� � ��� � � � �� � � � � � and � � � .

� Delay Differentiation and Dynamic RetransmissionProbabilities

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Page 24: Random Access Networks: Transmission Costs, Selsh Nodes

Dynamic Retransmission Probabilities

Aggregated Arrival Rate � � � �� ��� ��

� �

�� � � � �� �

Retransmission Probability� � � �

� � � �

��� � � � � � �

0 50 100 1500

20

40

60

80

n

u State Trajectory

0 20 40 600

2000

4000

6000

8000n Backlog Histogram

Throughput � � � �� �

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Page 25: Random Access Networks: Transmission Costs, Selsh Nodes

Finite Number of Nodes� Finite Number of Nodes

� ��� ��

��� �

� � ��� ��

� Nodes can have several backlogged packets

� Backlog-Dependant Retransmission Probabilities

� � � � � ��

��

�� � � � � � � � � � � � � �

� � � � otherwise,

� Backlog-Independent Retransmission Probabilities,� � .

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Page 26: Random Access Networks: Transmission Costs, Selsh Nodes

Backlog-Dependant Retransmission Probabilities

Assumption: “Price tends to increase when all nodes are saturatedand retransmit with probability� � � .”

Case Study

Node Bandwidth Delay

1 tolerant tolerant

2 tolerant intolerant

3 intolerant tolerant

4 intolerant intolerant

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Page 27: Random Access Networks: Transmission Costs, Selsh Nodes

10 20 30 40 50 600

5

10

15

20

25

30

35

40

nt

u t0 10 20 30 40 50 60

0

1000

2000

3000

4000

5000

6000

7000

8000

Node � � � � �

1 0.021 186.7

2 0.021 19.9

3 0.206 116.5

4 0.210 11.8

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Page 28: Random Access Networks: Transmission Costs, Selsh Nodes

Backlog-Independent Retransmission Probabilities

Assumption: “Price tends to increase when each nodes has at leastone backlogged packet”

0 50 100 150 2000

200

400

600

800

nt

u t

0 50 100 1500

5000

10000

15000

nNode � � � � �

1 0.006 93.0

2 0.006 85.4

3 0.069 417.6

4 0.068 46.0

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Page 29: Random Access Networks: Transmission Costs, Selsh Nodes

Infinite Node Approximation

−5 0 50.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Thr

ough

put

−5 0 50

500

1000

1500

2000

Del

ay

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Page 30: Random Access Networks: Transmission Costs, Selsh Nodes

Extensions: End-to-End Rate Control� Integration with Price-Based Rate Control for Point-to-Point

Networks

– Marking Scheme by by Athuraliya and Low.

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Page 31: Random Access Networks: Transmission Costs, Selsh Nodes

Conclusions� Random Access Networks with Transmission Costs

� Selfish Nodes

� Price-Based Rate Control

– Operating Point

– Delay and Throughput Differentiation

– End-to-End Rate Control

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Page 32: Random Access Networks: Transmission Costs, Selsh Nodes

Related Work� Selfish Users - Retransmission Probabilities

– MacKenzie and Wicker

� Selfish Users - Experimental

– Altman, El Azouzi, Jimenez

� Cheating

– Raja, Hubaux, Aad

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Page 33: Random Access Networks: Transmission Costs, Selsh Nodes

Related Work� Price-Based Rate Control

– Frank Kelly, Steven Low,.....

� Rate Control and Slotted Aloha

– Kleinrock and Lam

– Mittal and Venetsanopoulos

� TCP over 802.11

– Cali et al.

� Price-Based Rate Control for Random Access Networks

– Jin and Kesidis

– Battiti et al.

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