Base Station Association Game in Multi-cell Wireless Network

Libin Jiang, Shyam Parekh, Jean Walrand

Agenda

• Base station game introduction

• Equal time allocation analysis

• Equal–throughput allocation analysis

• Generalized situation analysis

• Simulation results

• Conclusion

Introduction

• Mulit-cell wireless network– E.g. cell phone network– Multi-base stations

• User chooses BS freely

• BS allocate resources to users

• Game-theoretical analyzes the throughput

• Consider downlink only

Assumption

• Simple scheduling policies– Equal time or equal rate

• Concave utility function of user, not unique• No communication between BS for cooperation

of optimization• Continuous population model

– Single user is small

• Allow distributed association in BS• Discrete PHY data rate

Some definitions

• PHY rates to BS j = Rj

• Users in the same class shares same Rj vector, donated as Rkj

• Number of class-k users with BS j = xkj

• Total number of class-k users dk = ∑j xkj

• Throughput of a class-k user with BS j = Sk

j

Equal-time allocation analysis

• Fraction of time of BS j = 1/∑kxkj

• Hence Skj = Rkj / ∑kxkj

• At NE, there is no incentive for any users to switch their BS, a.k.a Wardrop Equilibrium

• By equation, we expect that:

Skj = ck , for all xkj > 0

Skj ≤ ck , for all xkj = 0 ……(1)

Equal-time allocation analysis (cont.)

• There is a unique NE, and it can achieve system-wide proportional fairness

• Proof:

At NE, (1) is satisfied, to achieve the system-wide proportional fairness, tried to solve the utility maximization problem with the individual throughput.

Utility maximization problem

• Max z,x U = ∑k,j xkj log(zkj Rkj / xkj)

s.t. ∑k zkj = 1 for all j

• zkj Rkj / xkj = Skj , thruput of a class-k user

• As a result, U is a utility function of all users and it’s concave of z and x

• Hence, subject to the constraints, maximize U

Utility maximization problem (cont.)

• The KKT condition is:

• Hence,

Equal-thruput allocation analysis

• BS allocate same thruput but different time to user with different PHY rate

• Sj be the thruput to each user in BS j• Time used by a class k user = Sj / Rkj

• Hence, ∑k (Sj / Rkj) xkj = 1• At NE, the condition will be:

Skj1 = Skj2 for all xkj1, xkj2 > 0

Skj1 ≥ Skj2 for all xkj1 > 0, xkj2 = 0 ……..(2)

Skj = Sj

Equal-thruput allocation analysis

• From the above condition, 2 conclusion can be drawn– The individual thruput of all users (all classes)

are the same, hence Sj1 = Sj2

• Proof by contradiction

– There can be infinite number of NE, some of them may not be efficient

• Consider a 2 BS’s and 2 classes scenario

Generalized Situation analysis

• User has it’s own strictly-concave, increasing utility function depends on application

• Tried to examine whether BS’s intra-cell optimization and user selfish behaviors lead to social optimum

Generalized Situation analysis (cont.)

• Lemma 1: given any zkj of class k, its user’s selfish choice will lead to the optimal total utility within class k where opt. total utility = Vk(zk1 ,zk2 ,……,zkJ )

• Proof:

for a particular BS j, it’ll perform it’s own intra-cell optimization, hence, solving

maxt ∑i є j ui(Rkj ti) s.t. ∑i є j ti = zkj

Lemma 1 proof (cont.)

• Using the previous constraint, define a Lagrangian

L(t,λ) = ∑i є j ui(Rkj ti) – λ(∑i є j ti - zkj )• When the optimal solution is reach, let the s

olved λ be λkj , and optimal t be t*, then

u’i(Rkj t*j) = λkj / Rkj

• Let Pi() be inverse of u’i(), which is a strctly decreasing function

• Recall that Rkj t*j = S*i = Pi(λkj / Rkj)

Lemma 1 proof (cont.)

• By assumption of small user, at NE, S*i would be the same whatever BS user i join, and it can be said that λkj / Rkj = αk which is a constant

• In term of class, given zkj, total thruput (Ck) isfixed, to maximize the utility, hence to solve: max ∑i є k ui(Si) s.t. ∑i є k Si = Ck

• Notice that the condition of above are thereexists a positive constant βk = ui(S#

i) and ∑i є k S#

i = Ck

• By letting αk = βk , conditions meet, this impliesS#

i = S*i ,, hence NE max. the class-k utility

Generalized Situation analysis (cont.)

• The NE made by both user and BS is unique and it leads the max. sum of utility of all the users

• Proof:• Consider users reach the NE and the BSs

performed intra-cell optimization, let Zkj be the time allocated, according to Lemma 1, users will reach a max. total utility of Vk(Zk1 ,Zk2 ,……,ZkJ )

Generalized Situation analysis (cont.)

• Recall that Vk() is related to the ui(Rkj ti) in Lemma 1, hence the LM λkj gives the sensitivity of Vk(), that’s

ә Vk(Zk1 ,Zk2 ,……,ZkJ )/ ә zkj = λkj if Zkj > 0• As the intra-cell optimazation is performed, the L

M of all classes within BS should be the same, hence λkj = λj

• For BS with no class k users, it’s price is too high to class k, so

ә Vk(Zk1 ,Zk2 ,……,ZkJ )/ ә zkj ≤λj if Zkj = 0

Generalized Situation analysis (cont.)

• With the above 2 condition, we try to maximize the utility for all class, hence

maxz ∑k Vk(zk1 ,zk2 ,……,zkJ ) s.t. ∑k zkj = 1

• The problem is similar to the problem in equal-time allocation’s one, resulting a unique NE

Generalized Situation analysis (cont.)

• To guarantee the system will converge to unique NE with Vk(zk1 ,zk2 ,……,zkJ ), it can be proven that the total utility will increased if a user switch to another BS which can give a higher thruput

• Proof: consider 2 BS’s with one user switching

Simulation results

• Equal-time allocation– K = 2, J= 2, d1 = 20 ,d2 = 30 , R11 = 10, R12 =

20, R21 = 15, R22 = 15

– Initial random association and BS1 association are tested

Equal-time allocation

Simulation results

• Equal-throughput allocation– K = 2, J= 2, d1 = 20 ,d2 = 30 , R11 = 10, R12 =

1, R21 = 1, R22 = 10

– 3 trials• Initial random association• Class 1 in BS1, class 2 in BS2• Class 2 in BS1, class 1 in BS2

Equal-throughput allocation

Simulation results

• General concave function– 2 types

• Type A: Log(s), Type B: √s

– 50 users for each type

– K = 2, J= 2, R11 = 10, R12 = 20, R21 = 15, R22 = 15

– 2 trials• Random initial• BS1 initial

General concave function

General concave function

Conclusion

• Equal-time allocation results unique NE• Equal-thruput allocation results many NE with in

efficient NE• Intra-cell optimization with users selfish behavior

s results in converging to optimal max. utility NE• Uplink is not considered as it depends heavily on

user activities• Non-concave utility functions are also to be inve

stigated in the future