Random Access Networks: Transmission Costs, Selfish Nodes, and Protocol Design Joint work with Clement Yuen and Ran Pang 1
Random Access Networks:Transmission Costs, Selfish Nodes, and Protocol Design
Joint work with Clement Yuen and Ran Pang
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Random Access Networks:
� Collisions - Stability
� Transmission Cost
� Rate Control?
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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
3
Outline� Non-Cooperative Game - Idealized Situation
– (Symmetric) Nash Equilibrium
– Pricing Mechanism
� Distributed Algorithm
� Model: Slotted Aloha (CSMA/CD)
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Non-Cooperative Game - Slotted Aloha Model
1
� Poisson Arrivals
� Collision - Retransmission
� Probabilistic Retransmissions
� Transmission Cost �
� Infinite set of hosts
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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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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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Markov Chain Formulation� Cost for successfully transmitting a packet
– new packet: � � � � � �
– backlogged packet: � � � � � �
� Retransmission Probabilities ��
– new packet: � � � � � � �� �– backlogged packet: � � � � � � �� �
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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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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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A particular class of Strategies� Set � � of admissible strategies
– class � � � � � � ��� � �
– � � � ��� � � � � �� � � �� � � � � � �
� Transmitting backlogged packet
– �� �� � �� � � � �� �� � � � � � �
� Cost � � � � � �
– � � � � � �� � � � , � � � �
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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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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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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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Protocol Design� Assumption
– Know � ��� �– Can observe state �
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Protocol Design� Rate Control
– Collision: Increase Price
– Idle: Decrease Price
� Questions
– Stable?
– Operating Point?
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Price Update� Price Signal�
� Aggregated Transmission Rate � � � �
� Collision: Increase Price
� Idle Slot: Decrease Price
� Price Adaptation: � � � , � � �
� � � � �� �� �� ��� � � �� � ��� � � �� �� � � � � � �
�
� Retransmission Probability:�
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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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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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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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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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Summary� Price-Based Rate Control
� Stability
� Performance
� Do not need to know
– State �
– Rate function � ��� �
– Retransmission probability�
� Model
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Delay Differentiation and Dynamic RetransmissionProbabilities
� Delay Differentiation:� � , � � ��� � � ���
� Dynamic Retransmission Probabilities:� � � �
– �� �� � ��� �� �– � ��� � �� � , � � �
– � ��� � ��� � � � �� � � � � � and � � � .
� Delay Differentiation and Dynamic RetransmissionProbabilities
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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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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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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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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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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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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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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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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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Related Work� Selfish Users - Retransmission Probabilities
– MacKenzie and Wicker
� Selfish Users - Experimental
– Altman, El Azouzi, Jimenez
� Cheating
– Raja, Hubaux, Aad
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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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