Base Station Association Game in Multi-cell Wireless Network

1
Base Station Association Game in Multi-cell
Wireless Network

• Libin Jiang, Shyam Parekh, Jean Walrand

2
Agenda

• Base station game introduction
• Equal time allocation analysis
• Equalthroughput allocation analysis
• Generalized situation analysis
• Simulation results
• Conclusion

3
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

4
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

5
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

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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 gt 0
• Skj ck , for all xkj 0 (1)

7
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.

8
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 its concave of z and x
• Hence, subject to the constraints, maximize U

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Utility maximization problem (cont.)

• The KKT condition is
• Hence,

10
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 gt 0
• Skj1 Skj2 for all xkj1 gt 0, xkj2 0
  ..(2)
• Skj Sj

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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 BSs and 2 classes scenario

12
Generalized Situation analysis

• User has its own strictly-concave, increasing
  utility function depends on application
• Tried to examine whether BSs intra-cell
  optimization and user selfish behaviors lead to
  social optimum

13
Generalized Situation analysis (cont.)

• Lemma 1 given any zkj of class k, its users
  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, itll perform its own
  intra-cell optimization, hence, solving
• maxt ?i ? j ui(Rkj ti) s.t. ?i ? j ti zkj

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Lemma 1 proof (cont.)

• Using the previous constraint, define a
  Lagrangian
• L(t,?) ?i ? j ui(Rkj ti) ?(?i ? j ti - zkj
  )
• When the optimal solution is reach, let the
  solved ? be ?kj , and optimal t be t, then
• ui(Rkj tj) ?kj / Rkj
• Let Pi() be inverse of ui(), which is a strctly
  decreasing function
• Recall that Rkj tj Si Pi(?kj / Rkj)

15
Lemma 1 proof (cont.)

• By assumption of small user, at NE, Si would be
  the same whatever BS user i join, and it can be
  said that ?kj / Rkj ak which is a constant
• In term of class, given zkj, total thruput (Ck)
  is
• fixed, to maximize the utility, hence to solve
• max ?i ? k ui(Si) s.t. ?i ? k Si Ck
• Notice that the condition of above are there
• exists a positive constant ßk ui(Si) and
  ?i ? k Si Ck
• By letting ak ßk , conditions meet, this
  implies
• Si Si ,, hence NE max. the class-k utility

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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 )

17
Generalized Situation analysis (cont.)

• Recall that Vk() is related to the ui(Rkj ti) in
  Lemma 1, hence the LM ?kj gives the sensitivity
  of Vk(), thats
• ? Vk(Zk1 ,Zk2 ,,ZkJ )/ ? zkj ?kj if Zkj gt 0
• As the intra-cell optimazation is performed, the
  LM of all classes within BS should be the same,
  hence ?kj ?j
• For BS with no class k users, its price is too
  high to class k, so
• ? Vk(Zk1 ,Zk2 ,,ZkJ )/ ? zkj ?j if Zkj 0

18
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 allocations one, resulting a unique NE

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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 BSs with one user switching

20
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

21
Equal-time allocation
22
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

23
Equal-throughput allocation
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Simulation results

• General concave function
• 2 types
• Type A Log(s), Type B vs
• 50 users for each type
• K 2, J 2, R11 10, R12 20, R21 15, R22
  15
• 2 trials
• Random initial
• BS1 initial

25
General concave function
26
General concave function
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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