Fair and Efficient User-Network Association Algorithm for Multi-Technology Wireless Networks Pierre Coucheney, Corinne Touati, Bruno Gaujal INRIA Alcatel-Lucent, LIG Infocom 2009 Pierre Coucheney (INRIA) Heterogeneous Networks 1 / 14
Motivation: Heterogeneous Wireless Networks To which cell should one associate mobiles A and B? Pierre Coucheney (INRIA) Heterogeneous Networks 2 / 14 Overlapping wireless technologies: LTE, Wifi, Wimax... Recent mobiles are able to connect to several technologies. Benefits: For users: get access to high quality services. For providers: increase global traffic, avoid congestion. LTE B A WiFi WiFi WiMax
Motivation: Heterogeneous Wireless Networks What is efficiency? max throughput users max min users throughput (social optimum). (fairness). Objective function: α-fairness, α [0; ]. max association users G α (throughput), with G α (.) = 1 1 α (.)1 α. Pierre Coucheney (INRIA) Heterogeneous Networks 3 / 14
Motivation: Heterogeneous Wireless Networks What is efficiency? max throughput users max min users throughput (social optimum). (fairness). Objective function: α-fairness, α [0; ]. max association users G α (throughput), with G α (.) = 1 1 α (.)1 α. Problem. Find a mechanism which selects a fair and efficient association of mobiles to cells in a distributed setting. Pierre Coucheney (INRIA) Heterogeneous Networks 3 / 14
Outline and Main Result 1: Model. Model as an association game. Construction of a companion game, which is a potential game. 2: Algorithm. We provide a distributed algorithm following the replicator dynamics and show that: it converges to a pure strategy. converges to a local maximum of the objective function. 3: Experimental results. Simulation of the algorithm. Pierre Coucheney (INRIA) Heterogeneous Networks 4 / 14
Description of the Model Association game. (N, I, U) A set N of users. A set I n of cells for user n. s n choice of user n. l i : load (vector) on cell i: (l i ) n = { 1 if sn = 1 0 otherwise. u n (l i ): throughput of user n given the load on cell i. Pierre Coucheney (INRIA) Heterogeneous Networks Allocation Game 5 / 14
Description of the Model Association game. (N, I, U) Companion game. (N, I, R) α-throughput α-loss of throughput for user m Definition: repercussion utility. r n (l sn ) = G α (u n (l sn )) G α (u m (l sn e n )) G α (u m (l sn )) m n: s m=s n Simple computation made by the BS. Pierre Coucheney (INRIA) Heterogeneous Networks Allocation Game 5 / 14
Property of the Companion Game Association game. Companion game. (N, I, U) (N, I, R) q n,i def = P[s n = i]. q n = (q n,i ) i In : strategy of player n. q is pure if (n, i), q n,i {0, 1}. f n,i (q): expected repercussion utility of user n on cell i. F (q) : global expected α-throughput (objective function). Pierre Coucheney (INRIA) Heterogeneous Networks Allocation Game 6 / 14
Property of the Companion Game Association game. Companion game. (N, I, U) (N, I, R) q n,i def = P[s n = i]. q n = (q n,i ) i In : strategy of player n. q is pure if (n, i), q n,i {0, 1}. f n,i (q): expected repercussion utility of user n on cell i. F (q) : global expected α-throughput (objective function). Proposition. Selfish behavior in companion game leads to a marginal increase of the objective function (F is a potential function). Pierre Coucheney (INRIA) Heterogeneous Networks Allocation Game 6 / 14
Distributed Algorithm Idea: expected utility on i Distributed algorithm on q(t) approximating the replicator dynamics: dq n,i dt ( = q n,i fn,i (q) ) q n,j f n,j (q). j I n lies in [0, 1] average expected utility Replicator dynamics related to potential games: Converges to a set of Nash equilibria. Potential is increasing along trajectories (Lyapunov function). Pierre Coucheney (INRIA) Heterogeneous Networks A Distributed Algorithm 7 / 14
Distributed Algorithm Algorithm: For all n N : Choose initial strategy q n (0). repercussion utility At each time epoch t: Choose sn according to q n (t). Update: q n (t + 1) = q n (t) + ɛ r n (l sn (s)) ( 1 sn=i q n,i (t) ). constant step size { 1 if sn = i 0 otherwise Simple computation for the mobile. Pierre Coucheney (INRIA) Heterogeneous Networks A Distributed Algorithm 8 / 14
Properties of the Algorithm Theorem: The algorithm: 1. converges to a pure strategy Q. 2. Q is a pure Nash equilibrium for the companion game. 3. Q is locally optimal for the objective function. Sketch of the proof: 1. The algorithm is a stochastic approximation of the replicator dynamics. It converges to an asymptotically stable set of RD, which is a face of stationary points. Constant step size ensures convergence to a vertex of the face (pure strategy). 2. Property of the replicator dynamics. 3. The potential function is a Lyapunov function. Q is asymptotically stable if it is a local maximum (existence ensured) of the potential. Pierre Coucheney (INRIA) Heterogeneous Networks A Distributed Algorithm 9 / 14
Convergence to Fixed Association for User n 1 Probability to connect to a cell 0.8 0.6 0.4 0.2 q n,1 q n,2 q n,3 q n,4 q n,5 0 0 500 1000 1500 2000 Time epoch Evolution of one user s strategy that can connect to 5 cells. Pierre Coucheney (INRIA) Heterogeneous Networks Experimental Results 10 / 14
Convergence Speed: Adapt Step Size ɛ 6 heuristics for the choice of ɛ n (t) q n (t + 1) = q n (t) + ɛ n (t) r n (l sn (s)) (1 sn=i q n,i (t)). Throughput (Mb/s) 34 33 32 31 30 29 28 27 26 15 20 25 30 35 40 45 50 System Size Number of iterations 100000 10000 1000 100 10 15 20 25 30 35 40 System Size 45 50 Average performance. (with 5% confidence intervals). Average number of iterations (log. scale). Pierre Coucheney (INRIA) Heterogeneous Networks Experimental Results 11 / 14
Convergence Speed: Dynamic Scenario Throughput (Mb/s) 50 40 30 20 45 10 Arrival 20 10100 10140 0 0 5000 10000 15000 20000 Time Arrivals and departures: evolution of the global throughput with white Gaussian noise. Pierre Coucheney (INRIA) Heterogeneous Networks Experimental Results 12 / 14
Comparison: Arrival Connects to his Best Choice Throughput (Mb/s) 50 algo selfish 40 30 20 50 10 departure 20 12400 12435 0 0 5000 10000 15000 20000 Time Comparison of two policies of association for the same arrival process: the algorithm (algo), and the arrival selects the cell which offers him the best throughput (selfish). Pierre Coucheney (INRIA) Heterogeneous Networks Experimental Results 13 / 14
Conclusion Conclusion Distributed algorithm. Convergence to a locally optimal fixed association. Very simple computation needed on the BS and mobiles. Fast convergence (a few tens) with simple heuristics for the choice of step size ɛ. Future works Analytically study convergence speed. Discuss the relevance of the throughput as a utility function for different kind of applications (e.g. latency). Take into account the variability of signal (fading, shadowing). Pierre Coucheney (INRIA) Heterogeneous Networks Experimental Results 14 / 14