Downlink Power Allocation for Multiclass CDMA Wireless Networks

1
Downlink Power Allocation for Multi-class CDMA
Wireless Networks

• Jang Won Lee, Ravi R. Mazumdar
• and Ness B. Shroff
• School of Electrical and Computer Engineering
• Purdue University
• Represented by Wei Shiou in 2003

2
Outline

• Abstract
• System model and problem description
• Partial-cooperative optimality
• mobile selection algorithm
• power allocation algorithm
• Special case
• the single class of mobiles
• Paper source http//www.ece.purdue.edu/mazum/inf
  ocom02_final.pdf

3
Abstract

• The goal of this paper is to obtain a power
  allocation which maximizes the total system
  utility.
• But natural utility functions for each mobile are
  non-concave.
• The system utility we obtained can be quite close
  to which obtained by the social optimal
  allocation.

4
Part One

• ? ?

5
Assumptions (1/2)

• Let Ui represents the degree of mobile is
  satisfaction of the received QoS, assuming that
• (a) Ui is an increasing function of ?, the SIR of
  mobile i.
• (b) Ui is twice continuously differentiable.
• (c) Ui(0) 0.
• (d) Ui is bounded above.

6
Assumptions (2/2)

• Let Ni presents processing gain, which is
  defined by the ratio of the chip rate to the data
  rate, such that
• (e) For ?i gt 0,
  has at most one solution.
• (f) If
  has one solution for ?i gt 0
• 
  for ?i lt ?i0
• And
  for ?i gt ?i0

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System Model (1/2)

• In general, most utility functions used in wired
  or wireless networks can be represented by three
  functions(types) 9, 12.
• 9 M. Xiao Utility-based power control in
  celuar wireless systems, in Infocom /01/2001
• 12 S. Shenker Fundamental design issues for
  the future Internet, IEEE journal in
  communications /1995
• The three types are sigmoidal-like, concave, and
  convex.

8
System Model (2/2)

• Given the path gain Gi from the BS to the mobile
  i, interference, and noise. We can represent ?i,
  the SIR of mobile i, as follows

9
Basic Formulation

• In order to maximize to total system utility, the
  basic formulation of this problem (A) is
• We call this solution the social optimal power
  allocation.

10
Optimal Power Allocation (1/2)

• Proposition 1 To maximize the total system
  utility, the BS must transmit at its maximum
  power limit, PT
• The proof is trivial and based on the assumption
  (a).

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Optimal Power Allocation (2/2)

• By this property, (A) is equivalent to the
  following problem (B)

12
Global Optimal Solution

• Now we define
• Then, for any ? gt 0, P(?) belongs to Y(?) is a
  global optimal solution of the following
  optimization problem

13
Reference Book (1/4)

• MATHEMATICAL PROGRAMMING Theory and Algorithms
  M.Minoux 1986
• CH 5.2 Sufficient Optimality Conditions
• CH 6.1 Penalty Function Methods
• CH 6.2 Classical Lagrangean Duality
• CH 6.3 Classical Lagrangean Methods

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Reference Book (2/4)

• We shall now study sufficient optimality
  conditions for problems of the type
• And define
• Definition
• Let x belong to S and ? gt 0, we say that
  (x,?) is a saddle-point of L(x, ?)
• if L(x,?) lt L(x,?) for x belongs to S,
• and L(x,?) lt L(x,?) for ? gt 0.

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Reference Book (3/4)

• Characteristic property of saddle-points
• (x,?) is a saddle-point of L(x, ?) if and only
  if
• L(x,?) min (x,?)
• Each gi(x) lt 0
• Each ?i gi(x) 0
• Sufficiency of the saddle-point condition
• If (x,?) is a saddle-point of L(x, ?) then x
  is a global optimum of the primal problem.

16
Reference Book (4/4)

• For any ? gt 0, denote x(?) belongs to Y(?) a
  minimum in x of the function L(x, ?). Then x(?)
  is a global optimum of the (perturbed problem)

17
Good Approximation

• To obtain a good approximation to the solution of
  Problem (A), we will solve the following
  optimization problem (C)

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Distributed Architecture (1/3)

• P(?) solves the first constraint in (C) if and
  only if it solves the following problem (Di)
• Note that the parameters in (Di) are correspond
  only to mobile i.

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Distributed Architecture (2/3)

• We decompose (C) as the mobile problem (Di) for
  each mobile i and the following base station
  problem (E)
• In this distributed architecture, each mobile i
  solves (Di) independently one another and the BS
  solves (E).

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Distributed Architecture (3/3)

• Stage 1 mobile selection algorithm
• Each mobile i solves problem (Di)
• To select the mobiles which will be allocated
  positive power.
• Stage 2 power allocation algorithm
• The BS solves problem (E)
• To allocate positive power optimally among the
  participants.

21
Price Scheme

• We can view ? as the price per unit power,
  which is decided by BS, each mobile i tries to
  maximize its net utility by solving (Di).
• Revenue in Microeconomics is defined as PQ
  minus the C.
• Linear pricing scheme
• The total cost for power unit price the
  amount of allocated power.

22
Partial-cooperative Property

• The problem we discuss can be interpreted as a
  non-cooperative M-person game with dynamic
  pricing (linear pricing scheme).
• partial-cooperative property
• The BS needs to coordinate mobiles in the mobile
  selection algorithm
• And to make the unselected mobiles not
  participate in the power allocation algorithm
• Therefore, this problem is a so-called
  partial-cooperative M-person game with dynamic
  pricing.

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Part Two

• ? ?

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Mobile Selection (1/5)

• Lemma 2 By assumptions (b) and (e), Ui function
  is one of the three types.
• For brief, the proof is omitted here.
• We define Pi0 as

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Mobile Selection (2/5)

• Lemma 3 If
  ,
• then Pi(?)0, Pi(?)PT, or Ui(?i(Pi(?))) is in
  the concave region.
• The proof is omitted here.
• This lemma tells us that if the utility function
  Ui is a convex function for 0ltPltPT, mobile i
  always requests a power level of 0 or PT.

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Mobile Selection (3/5)

• Proposition 2 There exists a unique ?imax for
  mobile i such that
• and for ? gt ?max, Pi(?) 0.
• The proof is obtained by discussing 3 types of Ui
  function.

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Mobile Selection (4/5)

• Each mobile i can calculate ?imax as follows

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Mobile Selection (5/5)

• By the previous results, we summarize the
  properties of Pi(?) as
• Pi(?) is discontinuous at ? ?max ,if Ui is a
  convex function or a sigmoidal-like function.
• Pi(?) is continuous, if Ui is a concave function.
• Pi(?) is positive and a decreasing function of ?
  for ?imin lt ? lt ?imax
• Pi(?) is zero for ? gt ?imax
• Pi(?) is PT for ? lt ?imin

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Mobile Selection Algorithm

• The BS broadcasts its PT
• Each mobile i reports its ?imax to the BS
• BS set k 1.
• The BS broadcasts price ?kmax .
• Each mobile i reports its power request Pi(?kmax
  ) to the BS.
• If k1 and Pi(?kmax )PT, select from mobile 1
  to k-1 and stop.else if k1 and Pi(?kmax )ltPT go
  to (9), else go to (7).
• If , select from 1 to k-1 and stop.
• If and , select from
  1 to k-1 and stop.
• Set kk1. If kltM go to (4), else select from 1
  to k-1 and stop

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Power Allocation (1/2)

• Suppose that mobiles i, i1,2,k-1 are selected
  and ?1max gt ?2max gt . gt ?k-1max. Then, the BS
  problem (E) can be rewritten as problem (F)
• Where
• ?min ?kmax, if stop in (7) in the mobile
  selection algorithm.
• ?min 0, if stop in (8) or (9) in the mobile
  selection algorithm.
• ?max ?k-1max, if stop in (7) or (9) in the
  mobile selection algorithm.
• ?max ?kmax, if stop in (8) in the mobile
  selection algorithm.

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Power Allocation (2/2)

• Theorem 2 If a power allocation P(?)(P1(?),
  P2(?),., Pk-1(?)) is a solution of problem (F)
  and (Di), it is a global optimal solution of the
  following optimization problem
• Therefore, the solution above is a Pareto optimal
  power allocation.

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Pareto Optimal

• Definition A power allocation vector, P (P1,
  P2,. ,PM) is called a Pareto optimal power
  allocation vector, if there is no other power
  allocation vector, P (P1, P2, .,PM) such that
  Ui(?i(P)) gt Ui(?i(P)), for all i 1,2,,M and
  Uj(?j(P)) gt Uj(?j(P)) for some other utility
  function j.

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Nash Equilibrium

• Nash equilibrium
• ??B???,?A????????,???A???,B???????,???????????
  Nash equilibrium.
• Pareto-optimal
• ???????????????????????,??????????,????????????.

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Power Allocation Algorithm

• Set left(1) ?min , right(1) ?max and n 1.
• The BS broadcasts the price ?(n)
  0.5(left(n)right(n)) to all selected mobiles.
• Each mobile i reports its power request Pi(?(n))
  to the BS.
• If , stop.
• If , set left(n1) ?(n),
  right(n1) right(n), else set left(n1)
  left(n), right(n1) ?(n).
• n n1 and go to (2).

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Part Three

• ? ?

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Minimum SIR Requirement (1/2)

• We introduce a minimum SIR requirement, ?imin for
  each mobile i. problem (B) can be modified as
  problem (I)

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Minimum SIR Requirement (2/2)

• The system feasibility can be maintained by the
  call admission control and the scheduling. (I) is
  equivalent ot the following problem (J)

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Single Class of Mobiles

• The case single class of mobiles is a special
  case of this algorithm.
• In the homogeneous mobile case, mobiles are
  selected with descending order of ?imax by this
  algorithm.
• The partial-cooperative optimal selection
  excludes the mobiles which obtain relatively low
  utility by the social optimal selection.

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Comparison
40
Conclusions

• We adopted a utility based framework to maximize
  the total system utility.
• The proposed algorithm can be implemented in a
  distributed way using a partial-cooperative
  M-person power allocation game with dynamic
  pricing.
• It provides a partial-cooperative optimal power
  allocation which is Pareto optimal and a good
  approximation of the social optimal power
  allocation.

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Any Discussion ?
42
Future Works

• ???????????
• ???????
• ????????????????????????
• ????? Lagrange Multiplier Problem????????????

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Thanks for Your Participation

• Wei Shiou
• Graduate Institute
• Information Management Department
• National Taiwan University
• r91044_at_im.ntu.edu.tw