On Selfish Behavior in CSMA/CA Networks

Transcription

1 On Selfish Behavior in CSMA/CA Networks Mario Čagalj1 Saurabh Ganeriwal 2 Imad Aad 1 Jean-Pierre Hubaux 1 1 LCA-IC-EPFL 2 NESL-EE-UCLA March 17, IEEE Infocom

2 Introduction CSMA/CA is the most popular MAC paradigm for wireless networks CSMA/CA protocols rely on a (fair) random deferment of packet transmission where nodes control their own random delays CSMA/CA is efficient if nodes follow predefined rules, however, nodes have a rational motive to cheat - IEEE Infocom

3 Cheating pays well 8 7 Cheater Well-behaved cheater Throughput (Mbits/s) well-behaved destination Contention window of Cheater - IEEE Infocom

4 Our goal Study the coexistence of a population of greedy stations Derive the conditions for the stable and optimal functioning of a population of greedy stations - IEEE Infocom

5 Model and assumptions Network of N nodes (transmitters) out of which C are cheaters IEEE protocol MAC layer authentication (no Sybil attack) single collision domain (no hidden terminals) nodes always have packets (of the same size) to transmit Contention window size-based cheating fix W i = W min = W max (no exponential backoff) delay transmissions for d i U {1, 2,..., W i } Cheaters are rational (maximize their throughput r i ) - IEEE Infocom

6 Cheaters utility function U i Derived from Bianchi s model of IEEE each cheater i controls its access probability τ i = 2 W i + 1 cheater i receives the following throughput: U i (τ i, τ i ) r i (τ i, τ i ) = τ i c i 1(τ i ) τ i c i 2 (τ i) + c i 3 (τ i), where τ i (τ 1,..., τ i 1, τ i+1,..., τ N ). - IEEE Infocom

7 Static CSMA/CA game Static games players make their moves independently of other players players play the same move forever A move in the CSMA/CA game corresponds to setting the value of the cheater s contention window W i (that is, τ i ) Solution concept - Nash equilibrium (NE) W = (W 1, W 2,..., W C ) is a NE point if i = 1,..., C, we have: U i (W i, W i ) U i (Ŵi, W i ), Ŵi {1, 2,..., W max } - IEEE Infocom

8 Nash equilibria of the static game Proposition 1. A vector W = (W 1,..., W C ) is a Nash equilibrium if and only if i {1, 2,..., C} s.t. W i = 1. Proposition 2. The static CSMA/CA game admits exactly W C max (W max 1) C Nash equilibria.

9 Nash equilibria of the static game Proposition 1. A vector W = (W 1,..., W C ) is a Nash equilibrium if and only if i {1, 2,..., C} s.t. W i = 1. Proposition 2. The static CSMA/CA game admits exactly W C max (W max 1) C Nash equilibria. Two families of Nash equilibria define D {i : W i = 1, i = 1, 2,..., C} 1st family, D = 1, implying that only one cheater receives a non-null throughput (some allocations are Pareto-optimal!) 2nd family, D > 1, implying r i = 0, for i = 1,..., C (known as the tragedy of the commons) - IEEE Infocom

10 How to avoid undesirable equilibria? Multiple Nash equilibria that are either highly unfair or highly inefficient

11 How to avoid undesirable equilibria? Multiple Nash equilibria that are either highly unfair or highly inefficient To derive a better solution we use Nash bargaining framework solve the following problem max C i=1 (r i r 0 i ) Π 1 : s.t. r R r r 0. where R is a finite set of feasible solutions, and r 0 i max i min i r i, i = 1, 2,..., C is the disagreement point - IEEE Infocom

12 Uniqueness, fairness and optimality Π 1 admits a unique, fair and Pareto-optimal solution W (Nash bargaining solution) 1.2 Throughput (Mbits/s) Nash-equilibrium point Pareto-optimal point Simulations Analytical W* Contention window (W) of greedy stations - IEEE Infocom

13 Dynamic CSMA/CA game W is a desirable solution, but is not a Nash equilibrium i.e., W i > 1, i = 1,..., C

14 Dynamic CSMA/CA game W is a desirable solution, but is not a Nash equilibrium i.e., W i > 1, i = 1,..., C In the model of dynamic games cheaters utility function changes to J i = r i P i, where P i is a penalty function cheaters are reactive - IEEE Infocom

15 Penalty function P i Proposition 3. Define: P i = { pi, if τ i > τ 0, otherwise, where p i / τ i > r i / τ i and τ i < 1, i = 1, 2,..., C. Then, J i = r i P i has a unique maximizer τ. - IEEE Infocom

16 Graphical interpretation Example: P i = k i (τ i τ), with k i > r i / τ i Nash equilibrium point P i r i normalized payoff J i = r i P i i - IEEE Infocom

17 Making W a Nash equilibrium point Let τ = min τ i, (i.e., W = i=1,...,c max W i) i=1,...,c Each player j calculates a penalty p j i given that r i > r j, as follows: to be inflicted on player i j, p j i = r i r j note, r i / τ i > 0 and r j / τ i < 0 p j i / τ i > r i / τ i hence, J i = r i p j i = r j has a unique maximizer τ i = τ j

18 Making W a Nash equilibrium point Let τ = min τ i, (i.e., W = i=1,...,c max W i) i=1,...,c Each player j calculates a penalty p j i given that r i > r j, as follows: to be inflicted on player i j, p j i = r i r j note, r i / τ i > 0 and r j / τ i < 0 p j i / τ i > r i / τ i hence, J i = r i p j i = r j has a unique maximizer τ i = τ j Then, τ i = τ (i.e., W i = W ), i = 1,..., C, is a unique Nash equilibrium. - IEEE Infocom

19 Making W a Nash equilibrium point (cont.) Moving Nash Equilibrium 1.2 Throughput (Mbits/s) Pareto-optimal Nash equilibrium Nash-equilibria Simulations Analytical W* Contention window (W) of greedy stations - IEEE Infocom

20 Implementation of the penalty function Achieved by selective jamming Penalty should result in r i = r j, therefore T jam = (r i /r j 1)T obs Throughput (Mbits/s) Cheater X Other cheaters Time (s) Throughput of cheater X (Mbits/s) With jamming Without jamming Unique maximizer for cheater X Contention window (W X ) of cheater X - IEEE Infocom

21 Adaptive strategy Prescribes to a player what to do when the player is penalized (jammed) Throughput (Mbits/s) Cheater X Other cheaters Contention window size Cheater X 5 Other cheaters Time (s) Time (s) - IEEE Infocom

22 Fully distributed algorithm Evolution of the contention windows and the aggregated cheaters throughput for N = 20, C = 7, the step size γ = Contention window size Ch. X: Distrib. prot. Ch. Y: Distrib. prot. Ch. X, Y: Forced CW Contention window size Our protocol stops at this point 5 Ch. 1-7: Distributed prot. 0 Ch. 1-7: Forced CW Time (s) Aggregated throughput of cheaters (Mbits/s) - IEEE Infocom

23 Conclusions Static CSMA/CA game admits two families of Nash equilibria one family includes equilibria that are also Pareto-optimal Nash bargaining framework is appropriate to study fairness at the MAC layer using this framework we have identified the unique Pareto-optimal point exhibiting efficiency and fairness We have shown how to make this Pareto-optimal point a Nash equilibrium point, by providing a fully distributed algorithm that leads a population of greedy stations to the efficient Paret-optimal Nash equilibrium

24 Conclusions Static CSMA/CA game admits two families of Nash equilibria one family includes equilibria that are also Pareto-optimal Nash bargaining framework is appropriate to study fairness at the MAC layer using this framework we have identified the unique Pareto-optimal point exhibiting efficiency and fairness We have shown how to make this Pareto-optimal point a Nash equilibrium point, by providing a fully distributed algorithm that leads a population of greedy stations to the efficient Paret-optimal Nash equilibrium CSMA/CA game has a high educational value - IEEE Infocom