1 Multi-radio Channel Allocation in Competitive Wireless Networks Mark Felegyhazi, Mario Čagalj, Jean-Pierre Hubaux EPFL, Switzerland IBC’06, Lisbon, Portugal July 4, 2006

2 System model and game

3 Problem formulation  channel allocation in cellular networks  emerging wireless networks (mesh NWs, cognitive radio NWs) – self-organizing – devices can use multiple radios  graph coloring might not be appropriate

4 System model (1/2)  N communicating pairs of devices  sender and receiver are synchronized  C orthogonal channels  single collision domain if they use the same channel  devices have multiple radios  k radios at each device k ≤ C

5 System model (2/2)  N communicating pairs of devices  C orthogonal channels  k radios at each device number of radios by each of the devices in pair i on channel c → example: more radios on one channel ? Intuition:

6 Channel model  channels with the same properties  R(k c ) – total rate on any channel c

7 Multi-radio channel allocation game  selfish users (communicating pairs)  non-cooperative game G – communicating pairs → players – channel allocation → strategy – total rate of the player → utility  strategy:  strategy matrix:  utility:

8 Concepts  Nash equilibrium: The strategy matrix S * defines a Nash Equilibrium (NE), if none of the players can unilaterally change its strategy to increase its utility.  Pareto-optimality: S * is a Pareto-optimal channel allocation, if one cannot improve the utility of any player i without decreasing the utility of at least one other player j. where is the set of strategies for all players except i There is no strategy matrix S’ such that: with strict inequality for at least one player i.

9 Results

10 Non-conflicting case  Fact 1: If, then any channel allocation with is a Nash equilibrium.

11  Lemma 1: Each player should use all of his radios. Full usage of radios p4p4 p4p4

12 Flat channel allocation  Proposition 1: If S * is a Nash equilibrium, then for any channel b and c. Consider two arbitrary channels b and c, where: k b ≥ k c NE candidate: p4p4 p4p4

13 Nash Equilibria (1/2)  Theorem 1: If for any two channels b and c the conditions, hold, then S * is a Nash equilibrium. Consider two arbitrary channels b and c, where: k b ≥ k c Nash Equilibrium: Use one radio per channel. p4p4 p2p2

14 Nash Equilibria (2/2)  Theorem 1: If the following conditions hold: ; ;, then S * is a Nash equilibrium. Consider two arbitrary channels b and c, where: k b ≥ k c Nash Equilibrium: Use more radios on certain channels. channels with the minimum number of radios →

15 Pareto optimality  Theorem 2: If the rate function R(·) is constant, then any Nash equilibrium channel allocation is Pareto-optimal.

16 Centralized algorithm Assign radios to the channels sequentially. p1p1 p1p1 p1p1 p1p1 p2p2 p2p2 p2p2 p2p2 p3p3 p3p3 p3p3 p3p3 p4p4 p4p4 p4p4 p4p4

17 Conclusion  Wireless networks with multi-radio devices  Users of the devices are selfish players  Model using game theory  Results for a Nash equilibrium: – players should use all their radios – flat allocation over channels – Nash equilibria: one radio per channel for each player or unfair NE  NE are Pareto-optimal for the constant rate function  Simple centralized algorithm

18 Future work  General topology networks  Specific network scenarios: mesh networks  The effect of the MAC protocol  Distributed convergence algorithms (to the NE)