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 =
Skj
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
solved 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) is
fixed, 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 there
exists a positive constant βk = ui(S#i) and
∑i
# = C
S
єk
i
k
• By letting αk = βk , conditions meet, this implies
S#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
LM 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
inefficient NE
• Intra-cell optimization with users selfish
behaviors 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
investigated in the future