Non-Cooperative Multi-Radio Channel Allocation in Wireless Networks Márk Félegyházi*, Mario Čagalj†, Shirin Saeedi Bidokhti*, Jean-Pierre Hubaux* * Ecole Polytechnique Federale de Lausanne (EPFL), Lausanne, Switzerland † University of Split, Croatia Infocom 2007

Márk Félegyházi (EPFL) 2 Problem ► multi-radio devices ► set of available channels How to assign radios to available channels?

Infocom 2007Márk Félegyházi (EPFL) 3 System model (1/3) ► C – set of orthogonal channels (| C | = C) ► N – set of communicating pairs of devices (| N | = N) ► sender controls the communication (sender and receiver are synchronized) ► single collision domain if they use the same channel ► devices have multiple radios ► k radios at each device, k ≤ C

Infocom 2007Márk Félegyházi (EPFL) 4 System model (2/3) ► N communicating pairs of devices ► C orthogonal channels ► k radios at each device number of radios by sender i on channel x → example: Use multiple radios on one channel ? Intuition:

Infocom 2007Márk Félegyházi (EPFL) 5 System model (3/3) ► channels with the same properties ► τ t (k x ) – total throughput on any channel x ► τ(k x ) – throughput per radio

Infocom 2007Márk Félegyházi (EPFL) 6 ► selfish users (communicating pairs) ► non-cooperative game G MRCA – players → senders – strategy → channel allocation – payoff → total throughput ► strategy: ► strategy matrix: ► payoff: Multi-radio channel allocation (MRCA) game

Infocom 2007Márk Félegyházi (EPFL) 7 Game-Theoretic Concepts Nash equilibrium: No player has an incentive to unilaterally deviate. Best response: Best strategy of player i given the strategies of others. Price of anarchy: The ratio between the total payoff of players playing a socially-optimal (max. Pareto-optimal) strategy and a worst Nash equilibrium. Pareto-optimality: The strategy profile s po is Pareto-optimal if: with strict inequality for at least one player i

Infocom 2007Márk Félegyházi (EPFL) 8 Lemma: If S * is a NE in G MRCA, then. Use of all radios Each player should use all of his radios. p4p4 p4p4 Intuition: Player i is always better off deploying unused radios. all channel allocations Lemma

Infocom 2007Márk Félegyházi (EPFL) 9 Proposition: If S * is a NE in G MRCA, then d y,x ≤ 1, for any channel x and y. Load-balancing channel allocation ► Consider two arbitrary channels x and y in C, where k x ≥ k y ► distance: d x,y = k x – k y all channel allocations Lemma Proposition

Infocom 2007Márk Félegyházi (EPFL) 10 Nash equilibria (1/2) Theorem 1: A channel allocation S * is a Nash equilibrium in G MRCA if for all i: ► d x,y ≤ 1 and ► k i,x ≤ 1. p2p2 Nash Equilibrium: p4p4 Use one radio per channel. all channel allocations Lemma Proposition NE type 1 ► Consider two arbitrary channels x and y in C, where k x ≥ k y ► distance: d x,y = k x – k y

Infocom 2007Márk Félegyházi (EPFL) 11 Nash equilibria (2/2) Nash Equilibrium: Theorem 2: A channel allocation S * is a Nash equilibrium in G MRCA if: ► d x,y ≤ 1, ► for any player i who has k i,x ≥ 2, x in C, ► for any player i who has k i,x ≥ 2 and x in C +, k i,y ≥ k i,x – 1, for all y in C – Use multiple radios on certain channels. all channel allocations Lemma Proposition NE type 1 NE type 2 ► Consider two arbitrary channels x and y in C, where k x ≥ k y ► distance: d x,y = k x – k y ► loaded and less loaded channels: C + and C – C+C+ C–C–

Infocom 2007Márk Félegyházi (EPFL) 12 Efficiency (1/2) Corollary: If τ t (k x ) is constant (i.e., ideal TDMA), then any Nash equilibrium channel allocation is Pareto-optimal in G MRCA. Theorem: In G MRCA, the price of anarchy is: where

Infocom 2007Márk Félegyházi (EPFL) 13 Efficiency (2/2) ► In theory, if the total throughput function τ t (k x ) is constant  POA = 1 ► In practice, there are collisions, but τ t (k x ) decreases slowly with k x (due to the RTS/CTS method) G. Bianchi, “Performance Analysis of the IEEE Distributed Coordination Function,” in IEEE Journal on Selected Areas of Communication (JSAC), 18:3, Mar. 2000

Infocom 2007Márk Félegyházi (EPFL) 14 Summary ► wireless networks with multi-radio devices ► users of the devices are selfish players ► G MRCA – multi-radio channel allocation game ► results for a Nash equilibrium: – players should use all their radios – load-balancing channel allocation – two types of Nash equilibria – NE are efficient both in theory and practice ► fairness issues ► coalition-proof equilibria ► algorithms to achieve efficient NE: – centralized algorithm with perfect information – distributed algorithm with imperfect information

Infocom 2007Márk Félegyházi (EPFL) 15 Future work ► general scenario – conjecture: hard ► approximation algorithms ► extend model to mesh networks (multihop communication)

Extensions

Infocom 2007Márk Félegyházi (EPFL) 17 Related work ► Channel allocation – in cellular networks: fixed and dynamic: [Katzela and Naghshineh 1996, Rappaport 2002] – in WLANs [Mishra et al. 2005] – in cognitive radio networks [Zheng and Cao 2005] ► Multi-radio networks – mesh networks [Adya et al. 2004, Alicherry et al. 2005] – cognitive radio [So et al. 2005] ► Competitive medium access – Aloha [MacKenzie and Wicker 2003, Yuen and Marbach 2005] – CSMA/CA [Konorski 2002, Čagalj et al. 2005] – WLAN channel coloring [Halldórsson et al. 2004] – channel allocation in cognitive radio networks [Cao and Zheng 2005, Nie and Comaniciu 2005]

Infocom 2007Márk Félegyházi (EPFL) 18 Fairness Nash equilibria (fair) Nash equilibria (unfair) Theorem: A NE channel allocation S * is max-min fair iff Intuition: This implies equality: u i = u j,  i,j  N

Infocom 2007Márk Félegyházi (EPFL) 19 Centralized algorithm Assign links to the channels sequentially. p1p1 p1p1 p1p1 p1p1 p2p2 p2p2 p2p2 p2p2 p3p3 p3p3 p3p3 p3p3 p4p4 p4p4 p4p4 p4p4

Infocom 2007Márk Félegyházi (EPFL) 20 Convergence to NE (1/3) p1p1 p1p1 N = 5, C = 6, k = 3 p2p2 p2p2 p4p4 p1p1 p3p3 p2p2 p5p5 p4p4 p5p5 p3p3 p3p3 p4p4 p5p5 c1c1 c2c2 c3c3 c4c4 c5c5 c6c6 time p 5 : c 2 →c 5 c 6 →c 4 p 3 : c 2 →c 5 c 6 →c 4 c 1 →c 3 p 2 : c 2 →c 5 p 1 : c 2 →c 5 c 6 →c 4 p 1 : c 4 →c 6 c 5 →c 2 p 4 : idle channels p5p5 p3p3 p2p2 p1p1 p1p1 p4p4 Algorithm with imperfect info: ► move links from “crowded” channels to other randomly chosen channels ► desynchronize the changes ► convergence is not ensured

Infocom 2007Márk Félegyházi (EPFL) 21 Convergence to NE (2/3) Algorithm with imperfect info: ► move links from “crowded” channels to other randomly chosen channels ► desynchronize the changes ► convergence is not ensured Balance: unbalanced (UB): best balance (NE): Efficiency:

Infocom 2007Márk Félegyházi (EPFL) 22 Convergence to NE (3/3) N (# of pairs)10 C (# of channels)8 k (radios per device)3 τ(1) (max. throughput)54 Mbps