Standard and Quasi-standard Stochastic
Power Control Algorithms
Jie Luo, Sennur Ulukus, Anthony Ephremides
Background
Backgroundand
andMotivation
Motivation
Quasi-standard
Quasi-standardStochastic
StochasticPower
PowerControl
ControlAlgorithms
Algorithms
Power is an important and limited resource in wireless communications. Power
control algorithms that minimize the transmission power while ensuring the quality
of service have been widely studied in the literature. Among them, the standard
power control algorithms are especially popular. However, the original standard
power control algorithms require the perfect knowledge of the interference power,
which can only be estimated from noisy observations in practical systems. When the
perfect interference information is not available, the convergence of the standard
power control algorithms becomes a question mark.
Quasi-standard stochastic interference function
[~
1
3.5
]
1. Mean condition. E I ( p, v ) | p = I ( p ) + g ( p ), I(p) is standard, g(p) is the bias.
(
2. Bias condition. ∃K 3 > 0 , β (n ) ≥ 0 , g ( p(n )) ≤ β (n )K 3 1 + p(n )
)
3. Lipschitz condition. ∃K1 > 0, ∀p1 , p2 , I ( p1 ) − I ( p2 ) ≤ K1 p1 − p2
2
4. Growing condition.
Abstract
Abstract
Simulation
SimulationResults
Results
4
0.6
3
0.4
2.5
p
2
(
~
∃K 2 > 0, E ⎢⎡ I ( p, v ) − I ( p ) − g ( p ) p ⎤⎥ ≤ K 2 1 + p
⎦
⎣
2
0.2
c1-s1 0
2
p3
1.5
2
)
Quasi-standard stochastic power control algorithm
~
p(n + 1) = (1 − α (n )) p(n ) + α (n )I (n )
-0.8
n =0
n=0
Quasi-standard stochastic interference
-1
7
2
< ∞, then the standard stochastic power control
1
2
3
4
5
Number of iterations
4
x 10
6
7
4
x 10
Example 1: Single base station, 4 users,
5-length random signature, random
SIR target. hij=1, α(n)=10/(10000+n)
5000
45
4500
40
Power control with matched filter
35
3500
30
3000
K
∑p
2500
k
Joint power control and blind MMSE
25
k =1
K
∞
∑ α(n ) = ∞, ∑ α(n )
2
< ∞,
n =0
∞
20
1. Positivity. I(p)>0
2. Monotonicity. If p1≥p2 , then I(p1)≥I(p2)
3. Scalability.∀β>1, βI(p)>I(βp)
∑ α(n )β (n ) < ∞ , then quasi-standard stochastic
n =0
5
0
0
x (meter)
Proposition 5:
(
)
sup P p(n ) − p * ≥ ε ≤ K 8α *
control algorithm lim
n →∞
Standard
StandardStochastic
StochasticPower
PowerControl
ControlAlgorithms
Algorithms
Incorrectly designed PCLD
50
40
35
PCMP
K
∑p
k
40
k =1
K 25
PCMMSE
pi bi si
PCLD
15
I ( p1 ) − I ( p2 ) ≤ K1 p1 − p2
(
2
~
∃K 2 > 0, E ⎡⎢ I ( p, v ) − I ( p ) p ⎤⎥ ≤ K 2 1 + p
⎣
⎦
2
)
constraint on the
estimation noise
Standard stochastic power control algorithm
Standard stochastic interference
Comparing the deterministic standard power control algorithm
with the stochastic standard power control algorithm
Estimation noise
(
)
~
p(n + 1) = p(n ) − α (n )( p(n ) − I (n )) + α (n ) I (n ) − I (n )
Stochastic
p(n + 1) = p(n ) − α (n )( p(n ) − I (n ))
Deterministic
User i
2
hij
Base station
of user i
p j bj s j
Incorrectly designed PCMP
20
PCLD Opt
10
K
Deterministic version :
K
matched filter output :
ci zi = ∑ p j hij b j ci s j + ci vi
T
T
j =1
Stochastic version :
Stochastic PC
(
zi = ∑ p j hij b j s j + vi
)
j =1
pi ≥ I i p, ci , ci = arg min I i ( p, ci )
*
~
p(n + 1) = (1 − α (n )) p(n ) + α (n )I (n )
Blind MMSE, start from arbitrary ci and ensure
[
]
ci (n ) = ci (n ) + wi (n ), E wi (n ) | p(n ) ≤ K 5α (n )
*
0
200
400
600
800 1000 1200 1400 1600 1800 2000
Number of iterations
0
0
200
400
600
800 1000 1200 1400 1600 1800 2000
Number of iterations
Example 2: Performance of other quasiExample 2: Example of incorrect
standard stochastic power control
stochastic implementations.
Convergence
of
the
deterministic version does not even
algorithms in example 2.
guarantee a correct behavior of the stochastic version.
User j
T
30
5
hii
0
2
K
k
20
10
[I~( p, v )| p]= I ( p), I(p) is standard.
PCMP Opt
K
∑p
30
k =1
Example:
Example:Joint
JointStochastic
StochasticPC
PCand
andblind
blindMMSE
MMSE
Standard stochastic interference function
60
PCMF
45
*
1000 2000 3000 4000 5000 6000 7000 8000 9000 104
Number of iterations
Example 2: 25 base stations (o), 400 users,
150-length random signature, equal SIR
target. hij=(100/dij)2, α(n)=10/(10000+n)
50
If ∃α * ≥ 0, N > 0, such that α (n ) ≤ α , β (n ) ≤ α , ∀n ≥ N then ∀ε > 0
∃K 8 (ε ) > 0 , such that in the standard and quasi-standard stochastic power
p*=I(p*)
0
0
500 1000 1500 2000 2500 3000 3500 4000 4500 5000
Example 2: 25 base stations (o), 400 users,
150-length random signature, equal SIR
target. hij=(100/dij)2, α(n)=10/(10000+n)
*
α≤1
10
500
Convergence
Convergencein
inprobability:
probability:
One needs to
estimate I(p)
p(n+1)=(1-α)p(n)+αI(p(n))
15
1000
power control algorithm converges to p*, given by p*=I(p*), with probability one.
Where I(p) is a standard interference
function that satisfies
~
p(n + 1) = (1 − α (n )) p(n ) + α (n )I (n )
6
1500
∞
n =0
3. Growing condition.
3
4
5
Number of iterations
2000
If
2. Lipschitz condition. ∃K1 > 0, ∀p1 , p2 ,
2
Example 1: Single base station, 4 users,
5-length random signature, random
SIR target. hij=1, α(n)=10/(10000+n)
y (meter)
∞
Proposition 2:
p ≥ I ( p)
1. Mean condition. E
1
constraint on the
estimation noise
Algorithm converges to p*, given by p*=I(p*), with probability one.
The SINR requirement can be
expressed by
Standard power
control algorithm
-0.6
4000
∞
∑ α(n ) = ∞, ∑ α(n)
If
Standard
StandardPower
PowerControl
ControlAlgorithms
Algorithms
-0.4
p2
0.5
Probability
Probabilityone
oneconvergence:
convergence:
Proposition 1:
-0.2
p4
1
0
We propose a general framework for the stochastic power control (PC) algorithms.
Two types of stochastic PC algorithms are proposed: standard stochastic PC
algorithms where the interference estimator is unbiased, and quasi-standard
stochastic PC algorithms where the interference estimator is only asymptotically
unbiased. It is shown that both algorithms converge to the optimal power vector.
The stochastic versions of several well-known standard PC algorithms are proposed.
They are shown to be either standard or quasi-standard. The algorithms are now
ready for practical implementation.
0.8
p1
2
*
ci
Extended version :
Stochastic PC
blind MMSE,
start from ci(n-1)
Conclusion
Conclusion
We propose a general framework for standard and quasi-standard stochastic power
control algorithms. It is shown that, under certain mild conditions, both algorithms
converge to the optimal solutions. Different types of convergence are shown under
different assumptions on the iteration step size sequence. Several existing stochastic
power control algorithms are studied and several new stochastic power control
algorithms are proposed. We show that these algorithms are either standard or quasistandard. In the examples of quasi-standard stochastic power control algorithm, we
further extend the algorithms so that the stochastic power control and the parameter
optimization can be carried out in parallel. Convergence of the modified algorithms are
verified by computer simulations.