• No results found

Fair and Efficient User-Network Association Algorithm for Multi-Technology Wireless Networks

N/A
N/A
Protected

Academic year: 2021

Share "Fair and Efficient User-Network Association Algorithm for Multi-Technology Wireless Networks"

Copied!
5
0
0

Loading.... (view fulltext now)

Full text

(1)

Fair and Efficient User-Network Association

Algorithm for Multi-Technology Wireless Networks

Pierre Coucheney, Corinne Touati and Bruno Gaujal

„

INRIA Rhône-Alpes and LIG, MESCAL project, Grenoble France, {pierre.coucheney, corinne.touati, bruno.gaujal}@imag.fr

Abstract— Recent mobile equipment (as well as the norm IEEE 802.21) offers the possibility for users to switch from one technology to another (vertical handover). This allows flexibility in resource assignments and, consequently, increases the potential throughput allocated to each user. In this paper, we design a fully distributed algorithm based on trial and error mechanisms that exploits the benefits of vertical handover by finding fair and efficient assignment schemes. On the one hand, mobiles gradually update the fraction of data packets they send to each network based on the rewards they receive from the stations. On the other hand, network stations send rewards to each mobile that represent the impact each mobile has on the cell throughput. This reward function is closely related to the concept of marginal cost in the pricing literature. Both the station and the mobile algorithms are simple enough to be implemented in current standard equipment. Based on tools from evolutionary games, potential games and replicator dynamics, we analytically show the convergence of the algorithm to fair and efficient solutions. Moreover, we show that after convergence, each user is connected to a single network cell which avoids costly repeated vertical handovers. To achieve fast convergence, several simple heuristics based on this algorithm are proposed and tested. Indeed, for implementation purposes, the number of iterations should remain in the order of a few tens.

I. INTRODUCTION

The overall wireless market is expected to be served by six or more major technologies (GSM, UMTS, HSDPA, WiFi, WiMAX, LTE). Each technology has its own advantages and drawbacks and none is expected to eliminate the rest. Moreover, radio access equipment is becoming more and more multi-standard, offering the possibility of connecting through two or more technologies concurrently, using norm IEEE 802.21. Switching between networks using different technol-ogy is referred to asvertical handover. This is currently done in UMA, for instance, which gives an absolute priority to WiFi over UMTS whenever a WiFi connection is available. In this paper, in contrast, we address the problem of computing an optimal and fair association through a distributed algorithm. The contributions of the paper are:

- We propose an iterative distributed algorithm with guaran-teed convergence to a Nash equilibrium, based on real-time measurements (as opposed to off-line data).

- Based on tools from potential games, we show that, by appropriately setting up the reward measure, the resulting equilibria can be made Pareto efficient (optimal for the “price of anarchy” [1] and the SDF (Selfish Degradation Factor) [2]), and can correspond to anyα-fair point (defined in cooperative game theory [3]), for arbitrary chosen value of the parameterα. This wide family of fairness criteria includes

„This work was performed at the INRIA -ALU Bell Labs. joint laboratory.

in particular max-min fairness and proportional fairness and can be generalized to cover the Nash Bargaining Solution [4]. - We show that the obtained equilibrium is always pure: after convergence each user is associated to a single technology. - We validate our results through extensive simulations of several implementations of the algorithm in the practical setting of a geographical area covered by a global WiMAX network overlapping with several local WiFi cells.

Evolutionary games [5], [6], or the closely-related pop-ulation games, are based on Darwinian-like dynamics. The evolutionary game literature includes several so-called popula-tion dynamics, which model the evolupopula-tion of each populapopula-tion as time goes by. In our context, a population could be a set of individuals adopting the same strategy (i.e. connected to the same network cell and adopting identical network parameters). Recent work [7] have shown that, considering the so-called replicator dynamics, an appropriate choice of the fitness function (that determines how well a population is adapted to its environment) leads to efficient equilibria. However they do not provide with algorithms that follow the replicator dynamics (and hence converge to the equilibria). Additionally they do not justify the use of evolutionary games. Indeed, such games assume a large number of individuals, each of them having a negligible impact on the environment and the fitness of others. This assumption is not satisfied here, where the number of active users in a given cell is on the order of a few tens. The arrival or departure of a single one of them hence significantly impacts the throughput allocated to others. As the number of players is limited, we are hence dealing with another kind of equilibria, namely the Nash Equilibria.

In the context of load balancing, a few algorithms (see, for instance [8], [9]) have been shown to converge to Nash Equilibria. It has been pointed out that this class of algorithms has similar behavior and convergence properties as replicator dynamics in evolutionary game theory. Yet, such algorithms may converge tomixed strategy Nash equilibria, where each user randomly picks up a decision at each time epoch. Such equilibria are unfortunately not interesting in our case, as they amount to perpetual handover between networks.

In the present paper, we revisit previous works in evolu-tionary games, with additional fairness considerations, while proposing an Nash learning algorithm that can be imple-mented on future mobile equipments. In addition, our work present a novel result which is that our algorithm converges topure(as opposed to mixed) equilibria, preventing undesired repeated handovers between stations. Due to space limitation, the proofs of this work have been omitted. The interested reader should refer to [10] for the full version.

(2)

II. FRAMEWORK ANDMODEL

We present below the model and notations of the paper. A. Interconnection of Heterogeneous wireless networks

We consider a setCof network cells, that can be of various technologies, and a fixed set I of active users. Any of the |I| users1 can connect to a specific subset of these cells and network technologies, depending on her geographical location, wireless equipment and operator subscription. B. User throughput and cell load

By throughput, we refer to the rate of useful information available for a user, in a given network, sometimes also calledgoodputin the literature. It depends on both the user’s own parameters and the ones of others. These parameters include geographical position (interference and attenuation level) as well as wireless card settings (coding schemes, TCP version,...). As done in previous papers [11], we discretize the cells of networks intozonesof identical throughput. This means that users in the same zone will receive the same throughput. We denote by Zc the set of zones in cell c.

The distribution of users and their number in the zones of a network is calledloadof the network (see Fig 1).

A B

Zone separator Zone 2 WiMAX cell

Zone 1

WiFi cell

Fig. 1:Heterogeneous system made of a wide WiMAX cell and several WiFi hot-spots (in grey). As user B (in zone 1) is closer to the WiMAX antenna, she uses a more efficient coding scheme thanA(in zone 2) (eg QAM instead of QPSK). More formally, we suppose that each user has a set of network cells she can connect to, and a specific zone as-sociated to each of them. An admissible choice ai for user

i is a pair ai = (aci, azi) ∈ Ai. The set of all possible

choices is A def= ⊗i∈IAi. The user’s decision is denoted

by Ai

def = (Ac

i, Azi). Then, we denote A the vector of users

decisionsAdef= (Ai)i∈I, and call it an admissibleassociation. Hence, an association is a function from the set of users to the set of possible choices. For each association A, the loadon zonez of network c is denoted by `(c,z)(A), and is

the number of users in this zone using cell c: ∀c ∈ C, z ∈ Zc, `c,z(A)

def

= X

i∈I

δAi,(c,z), with δ the Kronecker delta

(δa,b = 1 if a= b and 0 otherwise). Hence, the load `c on

cellc is a vector of size|Zc|whose components correspond

to the load on a particular zone:`c(A)

def

= (`c,z(A))z∈Zc.

Assumption 1: The throughputtc,zof cellcin zonez is a

function depending only of the vector load`c(A)of cellc.

With this notations, the throughput received by useriwhen she takes decisionAi=ai istai(`aci(A)).

1In the following we use the termusersandmobilesinterchangeably.

III. NASHLEARNINGALGORITHM

In this section, we build an iterative algorithm with guar-anteed convergence to the Nash equilibrium of the system. A. Nash Equilibria (NE): Definitions

Let x be a vector. We denote by xs the sth component

of vector x, and by x−s the set of the other components.

It follows that up to some re-ordering, x = (xs, x−s). We

also introduce eai a vector of size Zaci defined byeai[s] def

= 1 ifs=azi, and0 otherwise.

Definition 1 (Pure strategy NE): An association scheme

A= (Ai)i∈I s.t.∀i, Ai=aiis a pure strategy NE for reward

rif,∀i∈ I,∀a0i=6 ai, rai(`aci(A))>ra0i(`a0ci(A) +ea0i).

Amixedstrategy for useriis the choice of a vector probability

qi over her possible choices. Each qi,ai is the probability

with which she takes actionai ∈ Ai. Equivalently, a mixed

strategy of a user is the choice of the percentage of packets or sessions she sends to each network she has access to. Let Si be the set of strategies for user i: Si = {qi ∈

[0,1]|Ai|s.t.P

ai∈Aiqi,ai = 1}, and Q ∈ S = ⊗i∈ISi the

strategy matrix for all users,Q= (qi)i∈I.

When Ai = ai, we denote by rai(`aci(A)) the reward

received by user i (as for the throughput, it depends on the choices of the other mobiles of the system, reflected in the association vectorA). For mixed strategies,Ri is the random

variable corresponding to the reward of useri(depending on probabilitiesQ). Its mean isEQ[Ri]

def

= E[Ri(A(Q))|Q=Q].

Definition 2 (Mixed strategies NE): A set of |I| probabil-ity vectors qi of size |C|is a NE if for rewardR, ∀i,∀qi0 6=

qi,Eqi,q−i[Ri]>Eqi0,q−i[Ri].

As the number of usersandtheir set of choices (networks) is finite, this is a finite game and it admits (at least one) mixed strategy NE [12]. Ai and Ti are random variables

corresponding respectively to the decision and the throughput of useri(depending on probabilitiesQ). Then, the expected throughput for useriis written EQ[Ti]. (Strictly speaking,T

is a function of the load, itself depending on the strategy Q. For simplicity however, we omit the loadLin the notations.) B. Our Nash Learning Algorithm

A Nash learning algorithm is an iterative algorithm on the strategy set that converges to a NE. Based on [8], we consider Algorithm 1, wherebi(σ)∈Ris the step of useri at timeσ. Since ∀i ∈ I, qi must remain in the strategy space Si, the

step size is constrained by: m 6 bi(σ)rc,z(σ) 6 1, where

m= max ai max(− qi,ai 1−qi,ai ,−1−qi,ai qi,ai ) 60.

Algorithm 1 Learning Algorithm

Initialize arbitrarily vectorsqi(0)for all users

At each time epochσ, foralluseri do

Take decisionˆai

def

= Ai(σ) with probabilityqi(σ)

Receive rewardrˆai(`ˆai(A(σ)))

Update strategy vector:

∀ai∈ Ai, qi,ai(σ+ 1)←qi,ai(σ)

(3)

Eq. 1 determines the update mechanism, which we call system dynamics.

Reward function: Selfish behavior may lead to inefficient use. To circumvent this, we introduce some rewards that are notified to users. Thus, instead of competing for throughput, we consider an algorithm reflecting a non-cooperative game between users that compete for maximizing their rewards. We consider the marginal cost pricing [13], which asserts that each user on a network should pay a tax balancing the loss of throughput caused by her presence. According to the random decisionA, we define the reward function for userias:

rAi(`Aci(A)) =G tai(`Aci(A)) − X j6=i δAc j,Aci G(ui,j(A))−G(vj(A)) , with (2) ui,j(A) =tAj(`Acj(A)−eAi), andvj(A) =tAj(`Acj(A)).

IV. ALGORITHMPROPERTIES

In this section, we study the convergence properties of our Nash learning algorithm.

A. Convergence

Consider b def= supσmaxi∈Ibi(σ), and Qb(θ)

def = Q(σ), ∀θ∈[σb,(σ+1)b)the piecewise-constant interpolation matrix ofQ(σ). Recall that a sequence of random variables(At)t∈R weakly convergesto a random variableAif for any continuous and bounded functionf: E[f(At)]−→E[f(A)]. Then:

Theorem 1: When b→0, the sequence {qb

i,ai(.)} weakly

converges to the solution of a differential equation, belonging to thereplicator dynamicsfamily:

dqi,ai

dθ =qi,ai[fi,ai(Q)−fi(Q)], (3)

withfi,ai(Q)=EQ[Ri|Ai=ai]andfi(Q)=

X a0 i∈Ai qi,a0 ifi,a 0 i(Q).

B. Efficiency and Fairness

As we consider elastic or data traffic, the Quality-of-Service of each user is her experienced throughput. We hence seek at Pareto optimal NE, i.e. matricesQ such that∀Q0 6=Q,∃i∈ I s. t.EQ0[Ti] > EQ[Ti],⇒ ∃j ∈ I,EQ0[Tj] < EQ[Tj]. Further, our NE should not only be Pareto optimal but also

α-fair[3], i.e. satisfies

max Q X i∈I EQ[G(Ti)]withG(x) def = x 1−α 1−α. (4)

Note that in pure strategies, for each mobile i such that

Ai = ai, then EQ[Ti] = tai(`aci(A)). So, we aim at

finding a set of decisions A∗ = (A∗i)i∈I that reaches

maxa∈APi∈IG(tai(`aci(A))).Whenα=0, the

correspond-ing solution is a social optimum. The limit α→1 is the proportional fair point (or Nash Bargaining Solution) and

α→ ∞, the max-min fair point. Parameter α hence gives flexibility in the choice of Pareto optimal points: from fully efficient to perfectly fair allocations.

Consider a dynamics V(Q) = dQ

dθ. We say that Q is a

stationary pointforV ifV(Q) = 0. Also,Qisasymptotically stableif there exists a neighborhoodUofQsuch that:Q(θ)∈ U ⇒Q(θ)−−−→

θ→∞ Q.Then, one can show that the dynamics

(3), (and hence the algorithm whenb→0) can only converge to a NE of the system. Following an idea of [14], we consider the summation of all users’ expectedα-fair throughput:

F(Q) =X

i∈I

X

ai∈Ai

qi,aiEQ[G(Ti)|Ai=ai]. (5)

Then, we have the following proposition:

Proposition 1: F is a potential function for the dynamics (3), i.e. :∀i∈ I,∀ai∈ Ai, fi,ai(Q) =

∂F ∂qi,ai

(Q).

For replicator dynamics (Eq. 3), the potential function F

is a Lyapunov function (it increases along the trajectories) [15], [16]. Hence, the global α-fair throughput converges to an optimal point when the differential equation trajectory approaches a stable state. This is summarized in the following theorem.

Theorem 2: Using reward function (2), the dynamics (3), and hence Algorithm 1 converge to an asymptotically stable state that maximizes the total expected α-fair throughput. C. Pure Strategies

Unlike multi-homing between WiFi systems (see [7], [14]), multi-homing between different technologies (e.g. WiFi and WiMAX) induces several complications: the different tech-nologies may have different delays, packet sizes or coding systems,... and re-constructing the messages sent by the mo-biles may be hazardous. During the convergence phase, each mobile is usingmixed strategies. Yet, studying the stability of equilibrium points of Eq. 3, one can show that our algorithm converges -after a transitional state- to NE in which each user uses a single network, i.e. pure strategy equilibria (full proof available in [10]). From Theorem 2 and the fact that the algorithm converges to a pure equilibrium, we can conclude: Theorem 3: Algorithm 1 with rewards 2 converges to a pure NE which corresponds to theα-fair point of the system.

0 0.2 0.4 0.6 0.8 1 0 500 1000 1500 2000 Time epochσ Probability q1,a 1

Fig. 2:Convergence of the probability values for each of the

5 possible choices of one user.

Consider a typical run of algorithm 1 over a system made of 10 users with 5 choices over 10 networks (Fig. 2). As ∀i,|Ai|= 5, then ∀i,∀ai∈ Ai, qi,ai= 0.2 (initial

equiproba-bility of all choices). As σgrows, all probabilities tend to 0

except1corresponding to the optimal action at the pure NE. V. IMPLEMENTATION ANDVALIDATION

First, notice that each user only needs to know her own reward to update her step size and strategy vector. Second,

(4)

each base station only needs her own load to compute the rewards, hence allowing for a fully distributed algorithm.

In the previous sections, convergence of the algorithm has been shown when the step sizebtends to0. This section shows numerical tests we performed to study possible practical heuristics for the step size computation. Indeed, while the step sizes should be small enough to ensure convergence, larger values speed up convergence.

A. The Different Heuristics for the Steps Each heuristic actually consists of two parts:

A rounding up test: As time increases, the probabilities of choosing each action tends either to0 or 1. To speed up convergence, we consider thresholdsm andM such that:

∀i∈ I,∀ai ∈ Ai,

qi,ai(σ+ 1)←0 if qi,ai(σ)< m qi,ai(σ+ 1)←1 if qi,ai(σ)>1−M.

After rounding up a strategy, the vector is normalized to remain inS. In the tests, m= 0.05andM = 0.3.

A step size computation: : different schemes to compute

bi(σ)are considered:

1) Constant Step Size (CSS): the steps are predefined:∀i∈ I,∀σ, bi(σ) =b. Numerical values areb= 0.01(CSSL, slow

but optimal),b=1(CSSH) and the intermediateb=0.1(CSSM).

2) Constant Update Size (CUS): the steps are such that the changes of probabilities are bounded by a predefined value

∆q (fixed to 0.1 in the experiments): ∀i ∈ I,∀ai ∈ Ai,

abs(qi,ai(σ+ 1)−qi,ai(σ))6∆q.

3) Decreasing Step Size (DSS):potential pure NE are detec-ted during a few large steps iterations and then confirmed or infered by smaller step iterations. We implemented 2 variants: a) DSS-SA: Inspired from simulated annealing, we con-sider cyclic decreasing step size:b= 3/(σmod10).

b) DSS-CSS: A DSS phase to stabilize most users fol-lowed by a constant large step size to speed up convergence of the others:b= 4/σif σ <120andb= 4otherwise. B. System Scenario

We consider a simple scenario of an operator providing subscribers with a service available either through a large WiMAX cell or a series of WiFi access points. For each simulation, a topology is chosen randomly, according to 3

parameters: number of users, of WiFi access points and of possible choices for each user|Ai|. More precisely, for each

user:

• The first choice is the WiMAX cell and one of the8 pos-sible zones (cf Section V-C), picked at random (uniformly).

• All other |Ai| − 1 choices are one of the |C| − 1

cells, picked up according to a uniform law. As explained in Section V-C, we do not consider zones in the WiFi cells.

The strategy vector is initialized with equal probabilities: ∀i∈ I,∀ai ∈ Ai, qi,ai(0) = 1/|Ai|.

C. Throughput of TCP sessions in WLAN and WiMAX To validate our algorithm, we implemented in our simulator the average mean throughputs of TCP connections available in the litterature obtained through fluid approximations.

Equations of throughput in WiFi cells: Based on [17], we consider that the throughput of connectioniis

ti(`c) =

LT CP

`c(TDAT A+TACK + 2TT BO(`c) + 2TW(`c))

where LT CP = 8000 bits, TACK = 1.091 ms, TDAT A =

1.785ms. Then,TW andTT BOare solution of the fixed point

equation given in [17].

WiMAX: We consider a fair sharing of N bC

carriers [18]: if p users are present in the cell, each of them will receiveNbC/psub-carriers. Hence, the throughput of a user in zonez is roughly the fraction1/P

z∈Zc`c,z of

the throughput she would obtain if she were alone in the cell. For a single user, we follow experimental values obtained in [19] for IEEE WiMAX 802.16d for its eight zones:

Modulation QAM64 3/4 QAM64 2/3 QAM16 3/4 QAM16 1/2

TCP goodput 9.58 8.88 6.80 4.50

Modulation QPSK 3/4 QPSK 1/2 BPSK 3/4 BPSK 1/2

TCP goodput 3.37 2.21 1.65 1.08

D. Comparisons between Heuristics

Figure 3(a) displays the performance (in terms of global throughput) obtained by the six heuristics as a function of the total number of users|I|. For a given load, all heuristics have been tested on the same topology to allow a fair comparison. The small constant step size (CSSL withb= 0.01), provides

the best performance. It is even tested optimal for the small values of|I|, up to 20. Most heuristics stay within 10 % of the optimal (except for DSS-CSS whose performance can be poor). Also note that the total capacity of the system is less than 36 (10∗2.6(WiFi)+9.58(WiMAX)) Mbit/s. Thus, the best heuristic is always within 5 % of the optimal. Finally, it should be noted that the medium constant step size (CSSM) with

b= 0.1is always very close to the best (CSSL) and that the

constant update size (CU S) performs better and better when the number of users grows. As for the number of iterations, it varies widely between the different heuristics, even on a logarithmic scale (Figure 3(b)). TheCU S heuristic is a clear winner here (with an average number of iterations never above 80). Meanwhile,CSSL does not always converge within the

limit of 20,000 iterations set in the program. Under high loads, CU S provides the best compromise with very fast convergence and reasonable performance. Under light load, the constant step size of medium size (CSSM) is also an

interesting choice, for its performance is almost optimal and its number of iterations remains below 100.

E. Impact on Fairness

Consider the following scenario: a set of 20 users, each having3available choices among 10 cells. The WiMAX cell is numbered 0 and its 8 zones are numbered from 0 to 7. The set of choices of the users areA=

{{0,1},{8},{1}} {{0,5},{6},{4}} {{0,1},{6},{9}} {{0,2},{2},{6}} {{0,3},{8},{9}} {{0,6},{4},{9}} {{0,7},{3},{6}} {{0,4},{1},{2}} {{0,6},{6},{9}} {{0,5},{3},{4}} {{0,6},{3},{1}} {{0,7},{9},{6}} {{0,3},{8},{1}} {{0,6},{4},{7}} {{0,6},{9},{5}} {{0,0},{6},{5}} {{0,5},{4},{1}} {{0,6},{6},{4}} {{0,3},{3},{4}} {{0,3},{8},{4}}.

The optimal association scheme, for α = 0 (efficient scheme) andα= 2(fair schemes) are respectively:

Aeff = {2,1,2,1,1,1,1,2,2,2,1,1,2,2,2,0,2,1,1,1},

(5)

1 2 3 45 6 Throughput (Mb/s) Load 26 27 28 29 30 31 32 33 34 15 20 25 30 35 40 45 50 35

(a) Average performance

1 23 4 5 6 Number of iterations Load 1 10 100 1000 10000 15 20 25 30 35 40 45 50 100000

(b) Average number of iterations

Fig. 3:Comparison of the heuristics (CU S, DSS−SA, DSS−CSS, CSSL, CSSM andCSSH resp.) under different loads

(with 5% confidence intervals). The induced throughputs are:

Teff= 0.824,1.225,0.824,1.225,1.225,1.225,0.824,1.225,0.824,1.225,

0.824,0.824,0.824,2.245,2.246,9.58, 0.824,1.225,0.824,1.225.

Tfair= 2.22, 1.225,2.22, 1.225,1.125,1.225,1.225,1.225,1.225,1.225,

2.245,1.225,1.225,2.246,1.225,1.225,1.225,1.225,1.125,1.225.

The efficient scheme achieves a total throughput of31.29

Mb/s. The fair scheme suffers a degradation of slightly less than 10%, with a total throughput of28.34Mb/s. Yet a closer look at the figures indicates that the efficient scheme leads to high differences between users (user 1 only obtains a throughput of 0.8 Mb/s while user16is granted 9.58 Mb/s). Meanwhile, in the fair association scheme, all users benefit from throughputs higher1.1Mb/s. As in bandwidth allocation mechanisms in wired systems [4], the parameter α hence allows one to finely tune the compromise between maximum global throughput and fairness between users.

F. Seamless Adaptation to User Arrivals and Departures The association algorithm has to be run at every arrival or departure of a user in a cell. We now discuss the impact of such events. First, typical time scales compare nicely: while arrivals or departures of users in WiMAX or WiFi cells occur every minute or so, the association algorithm converges in less than a second in most cases, when the best heuristic (CUS) for the step sizes is used. Second, the convergence of the algorithm is much faster when the initial state is chosen close to the former optimal solution for all old users and one change occurs (an arrival of a departure). This is illustrated in Fig. 4: in the first phase, 30 users with 3 choices over 10 cells run the algorithm using CUS step updates. Starting with uniform probabilities (1/3,1/3,1/3), convergence occurs after 60 iterations. Upon arrival of a new user, the algorithm is reset the starting points is (1/3,1/3,1/3) for the newcomer and

(1/4,1/4,1/2)for all other users, with probability1/2given to the previously chosen cell. Then, convergence only requires 27 iterations.

VI. CONCLUSION ANDFUTUREWORKS

In this paper, we have designed a distributed algorithm that converges to an optimal (in terms of fairness or efficiency) network association in heterogeneous wireless networks. This opens the way to several interesting future works, such as the implementation of such methods in modern mobile devices in collaboration with Alcatel-Lucent.

Iterations Throughput 20 22 24 26 28 30 32 34 0 10 20 30 40 50 60 70 80 90 Convergence Arrival New convergence

Fig. 4:Convergence speed after arrival of a new user. REFERENCES

[1] E. Koutsoupias and C. Papadimitriou, “Worst-case equilibria,” inProc. of STACS, 1998.

[2] A. Legrand and C. Touati, “Non-cooperative scheduling of multiple bag-of-task appplications,” inProc. of INFOCOM, May 2007. [3] J. Mo and J. Walrand, “Fair end-to-end window-based congestion

con-trol,” inProc. of SPIE, I. Symp. on Voice, Video & Data Comm., 1998. [4] C. Touati, E. Altman, and J. Galtier, “Generalised Nash bargaining solution for banwidth allocation,”Computer Networks, vol. 50, no. 17, pp. 3242–3263, Dec. 2006.

[5] J. W. Weibull,Evolutionary Game Theory. MIT Press, 1995. [6] J. Hofbauer and K. Sigmund, “Evolutionary game dynamics,” Bull.

Amer. Math. Soc., vol. 40, pp. 479–519, 2003.

[7] S. Shakkottai, E. Altman, and A. Kumar, “Multihoming of users to access points in WLANs: A population game perspective,”IEEE J. on Sel. Areas of Comm., vol. 25, no. 6, pp. 1207–1215, 2007.

[8] D. Barth, O. Bournez, O. Boussaton, and J. Cohen, “Distributed learning of Wardrop equilibria,” inProc. of UC, ser. LNCS, vol. 5204, 2008. [9] P. Sastry, V. Phansalkar, and A. Thathachar, “Decentralized learning

of Nash equilibria in multi-person stochastic games with incomplete information,” IEEE Transactions on System, man, and cybernetics, vol. 24, no. 5, pp. 769–777, 1994.

[10] P. Coucheney, C. Touati, and B. Gaujal, “A distributed algorithm for fair and efficient user-network association in multi-technology wireless networks,” INRIA, Tech. Rep., 2008.

[11] M. Coupechoux, J.-M. Kelif, and P. Godlewski, “Network controlled joint radio resource management for heterogeneous networks,” inProc. of IEEE VTC Spring, 2008.

[12] J. F. Nash, “Equilibrium points in n-person games,”Proceeding of the National Academy of Sciences, vol. 36, pp. 48–49, 1950.

[13] R. Cole, Y. Dodis, and T. Roughgarden, “Pricing network edges for heterogeneous selfish users,” inProc. of STOC, 2003.

[14] S. Shakkottai, E. Altman, and A. Kumar, “The case for non-cooperative multihoming of users to access points in IEEE 802.11 WLANs,” in Proc. of INFOCOM, 2005.

[15] D. Monderer and L. S. Shapley, “Potential games,”Games and Eco-nomic Behavior, vol. 14, pp. 124–143, 1996.

[16] W. H. Sandholm, “Potential games with continuous player sets,”Journal of Economic Theory, vol. 24, pp. 81–108, 2001.

[17] D. Miorandi, A. A. Kherani, and E. Altman, “A queueing model for HTTP traffic over IEEE 802.11 WLANs,”IEEE Computer Networks, vol. 50, pp. 63–79, 2006.

[18] C. Tarhini and T. Chahed, “System capacity in OFDMA-based WiMAX,” inProc. of the Int. Conf. on Syst. & Network Comm., 2006. [19] F. Yousaf, K. Daniel, and C. Wietfeld, “Performance evaluation of IEEE 802.16 WiMAX link with respect to higher layer protocols,” inProc. of the Int. Symp. on Wireless Communication Systems, 2007, pp. 180–184.

References

Related documents

Finally, this study delves into the experience of youth and families in the program of 11 Millones de Sueños to understand how the cultural arts, social justice, education,

and Laura Zakaras (eds), Confidentiality, Transparency, and the US Civil Justice System (Oxford University Press 2012) 170-71. It is true that in any event the party with less

This thesis develops a q theory model of housing, where the value of Norwegian housing, q , is defined as the ratio of housing prices to the construction costs of new housing

Dengan memanjatkan puji syukur kehadirat Allah SWT, atas limpahan rahmat dan hidayah-Mu penulis dapat menyajikan tulisan tesis yang berjudul : PENINGKATAN KINERJA

In this paper, we propose a Heterogeneous netwOrk pOlicy enforCement scheme (HOOC), which is an adaptive network policy implementation scheme that will not only support the use of

On behalf of International ski Federation and Kazakhstan ski Association, the Organizing Committee is pleased to invite all member associations of the FIS to participate in the

Analysis of hair samples using microscopical and molecular techniques to ascertain claims of rare animal species.. Zainuddin Z afarina , Sundararajulu

Stratus Technologies, an innovator of industry-standard fault tolerance, has extended the capabili- ties of its innovative x86-based architecture with the availability of Microsoft’s