Stochastic Power Control for Wireless Networks via SDE’s:
Probabilistic QoS Measures
C.D. Charalambous ∗
S. M. Djouadi †
S. Z. Denic‡
Abstract
The power control of wireless networks is formulated using a stochastic optimal control
framework, in which the evolution of the channel is described by stochastic differential equations (SDE’s). The latter capture the spatio-temporal variations of the communication link as
well as the randomness. This class of models is more realistic than the static models usually
encountered in the literature. Under this scenario, average and probabilistic Quality of Service
(QoS) measures are introduced to evaluate the performance of any control strategy by using
Chernoff bounds. Moreover, the Chernoff bound is computed explicitly, while the solution of
the stochastic optimal power control is obtained through pathwise optimization. The pathwise
optimization can be solved using linear programming if predictable control strategies are introduced. Finally, if predictable control strategies do not hold, it is shown that the proposed
power control problem reduces to particular convex optimizations.
∗ School of Information Technology and Engineering,
University of Ottawa, Ottawa, Ontario K1N 6N5, Canada, E-
mail: chadcha@site.uottawa.ca. Also Associated Professor with Department of Electrical and Computer Engineering,
University of Cyprus, 75 Kallipoleos Avenue, P.O. Box 20537 Nicosia, 1678, Cyprus, chadcha@ucy.ac.cy . This Work
was Supported by the Natural Science and Engineering Research Council of Canada under an operating Grant.
† Electrical & Computer Engineering Department, University of Tennessee, Knoxville, TN 37996-2100,
djouadi@ece.utk.edu
‡ School of Information Technology and Engineering, University of Ottawa, Ottawa, Ontario K1N 6N5, Canada,
E-mail: sdenic@site.uottawa.ca
1
Introduction
The purpose of the power control (PC) in wireless communication systems is to compensate for
path loss and co-channel interference (in TDMA). In code division multiple access systems the
situation is more complicated due to the near-far effect, and the intra-cell interference. The PC
represents the means of maintaining QoS to every user in the network and it is realized by controlling the power of the transmitted signals with the goal of achieving the desired signal to interference
ratio (SIR) of each user [1, 3, 4]. Most of the research that has been done in this field deals with
static wireless channel models [1, 2, 11, 9, 12]. For instance, approaches using outage probability, dynamic optimization, and joint PC and base station assignment, are reported respectively in
[13, 9, 10, 14, 15].
This paper is different from the above cited references in that it proposes the use of a dynamical
channel model described by stochastic differential equations (SDE’s) and a new power control algorithm (PCA) called Predictable Power Control Strategy (PPCS). In addition it introduces new
probabilistic QoS measures, and upper bounds using Chernoff bounds, which are computed explicitly for both static, and dynamic channels. The correct usage of any power control algorithm
(PCA) requires the use of such channel models that capture both temporal and spatial variations in
the channel. Since very few temporal or even spatial-temporal dynamical models have so far been
proposed in the literature with the application of any PCA, the suggested dynamical model and
the PCA will thus provide a more realistic and efficient optimum control for the wireless fading
channels. The performance of any PCA is usually found by determining the outage probability or
bit error rate (BER) or frame error rate (FER) [9, 13, 16]. Here outage probability is used as a performance measure to evaluate the effectiveness and benefits of the proposed PCA. We introduce
a centralized and a decentralized outage probability together with their corresponding Chernoff
bounds, which are useful for computational purposes.
The Static Deterministic Case. Consider a wireless network with M transmitters and M receivers.
A measure of QoS for each user is often defined through the SIR by [1]
M
∑ pi, subject
(p1 ≥0,...,pM ≥0)
min
to
i=1
∑M
j6=i
pn gnn
≥ γn
p j gn j + ηn
∑M
j=1
pn gnn
≥ γn
p j gn j + ηn
which is equivalent to
M
∑ pi, subject
(p1 ≥0,...,pM ≥0)
min
to
i=1
2
(1.1)
4
where γn =
γn
γn +1 , 0
< γn < 1. Here pi denotes the transmitter power of the ith transmitter, gi j > 0
denotes the channel gain of the transmitter j to the receiver assigned to transmitter i, γi > 0 is the
required SIR and ηi > 0 is the noise power level at the ith receiver, 1 ≤ i, j ≤ M. Define
4
4
4
4
4
p = (p1 , ..., pM ), Γ = diag(γ1 , ..., γM ), GI = diag(g11 , ..., gMM ), G = [gi j ]M×M , η = (η1 , ..., ηM )tr
where diag(·) denotes a diagonal matrix with its argument as diagonal entries, tr stands for matrix
or vector transpose. The matrix representation of the QoS constrains is GI p ≥ Γ(Gp + η). It
is noted in [1, 2] that if the constrains are feasible (e.g., there exists an p ≥ 0 and finite), then
the power vector which satisfies the inequality of the QoS with equality, e.g., (I − ΓG−1
I G)p =
M
ΓG−1
I η minimizes ∑ j=1 pi . Further, the constrains are feasible if and only if the Perron Frobenius
eigenvalue of ΓG−1
I G is less than 1. In this case, people employ power control algorithms, which
converge to the solution of p = ΓG−1
I (Gp + η).
2
Probabilistic QoS Measures
In this section we are interested in deriving certain upper bounds which can be used to evaluate the
performance of PCA. The effectiveness of any PCA is determined by evaluating the interference
or outage probability. Roughly speaking, the outage probability is the probability that a randomly
chosen communication link fails due to excessive signal degradation and/or interference. Outage
probability presents the distribution of the SIR of any link such that the SIR is less than or equal to
the threshold SIR required for an acceptable performance. In this section, we introduce two such
QoS measures which can be used in any control algorithm to determine the failure of meeting the
QoS. These can be used as measures of blocking new transmitters into the network or dropping
existing ones. Define
4
I n (p) =
M
1
∑ p j gn j + ηn − γn pngnn,
n = 1, ..., M
(2.2)
j=1
Decentralized Probabilistic QoS Measure. The probability of failure in achieving the nth QoS
³
´
n
requirement is P I (p) ≥ 0 , and by Chernoff inequality bound
³
´
h
³
´i
P I n (p) ≥ 0 ≤ E exp sn I n (p) , sn > 0
(2.3)
³
´
n
The Chernoff bound is very important because the probabilities P I (p) ≥ 0 are very difficult
to compute when the channel gains are Rayleigh, Ricean, log-normal, or Nakagami. Hence, the
3
Chernoff‘s bound can be used to evaluate the probability of failure to achieve a desired QoS requirement, and to devise the PCA. These bounds can also be used to evaluate the robustness of any
power control algorithm when the incorrect channel gains are used as follows. Suppose, the power
control strategy is determined based on the certain set of probability distribution corresponding to
the gains gi j , and the noises ηi , 1 ≤ i, j ≤ n. If the true distribution of the gains, and noises is
different from the one assumed when the PCA is found, the Chernoff bound (2.3) can be used to
evaluate the true probability of failure. Thus, the Chernoff bound can be used to analyze the robustness of any PCA with respect to incorrect model description. Next, we give a precise definition
of the decentralized probabilistic QoS measure, and we introduce the power control problem.
4
Definition 2.1 (Decentralized Probabilistic QoS) For a given number of transmitters M and −δn =
log αn , αn ∈ [0, 1), 1 ≤ n ≤ M, the nth transmitter’s decentralized QoS requirement is defined by
³
´
h
³
´i
log P I n (p) ≥ 0 ≤ log E exp sn I n (p) ≤ −δn , sn > 0, 1 ≤ n ≤ M.
(2.4)
A consequence of the above definition is that for a given αn ∈ [0, 1), M, the nth mobile user’s
probabilistic QoS requirement is satisfied if there exists a set of power strategies {p j }M
j=1 in some
admissible control set, say, Uad , such that for 1 ≤ n ≤ M,
4
An,M =
where
An,M (sn ) =
n
n
h
³
´i
o [
n
p ∈ Uad ; min log E exp sn I (p) ≤ −δn =
An,M (sn ), (2.5)
sn >0
h
sn >0
³
´i
o
p ∈ Uad ; log E exp sn I n (p) ≤ −δn ,
4
The decentralized probabilistic QoS requirement is satisfied if AM =
(2.6)
TM
m=1 Am,M
is non-empty.
Clearly, (2.4) consists of M inequalities which should be satisfied simultaneously for a given choice
of power strategy {p}M
j=1 . Next, we present an alternative formulation, which is less computationally expensive than the decentralized QoS, by using a single inequality for the QoS.
Centralized Probabilistic QoS Measure. Similarly, the joint probability of failure in achieving
the QoS of all transmitters is defined as follows.
4
Definition 2.2 (Centralized Probabilistic QoS) For a given number of transmitters M and −δ =
log α, α ∈ [0, 1), the centralized almost sure QoS requirement is defined by
³
´
h
³ M
´i
log P I 1 (p) ≥ 0, . . . , I n (p) ≥ 0 ≤ log E exp ∑ sn I n (p) ≤ −δ, sn > 0.
n=1
4
(2.7)
Thus, the centralized QoS requirement is satisfied if there exists a set of control strategies {p j }M
j=1 ∈
Uad , such that (here, we let sn = s∗ , 1 ≤ n ≤ M),
4
AM? =
n
p ∈ Uad
´i
o [
; min
log E exp ∑ s I (p) ≤ −δ =
AM (s? )
?
where AM (s? ) =
h
³
s >0
n
M
? n
(2.8)
s? >0
n=1
h
³ M
´i
o
p ∈ Uad ; log E exp ∑ s? I j (p) ≤ −δ
(2.9)
j=1
Clearly, there are disadvantages as well as advantages in deciding which QoS measure should be
used. The decentralized QoS measure is more attractive because it treats each I n (p) independently.
However, the power control strategy should be chosen to satisfy M inequalities given in (2.4).
On the other hand, the centralized QoS measure requires the solution of a single inequality given
by (2.7). The above definitions enable the designer to optimize with respect to the number of
transmitters, M, thus linking power control and admission control. In addition, these QoS measures
are important in evaluating existing power control algorithms proposed in the literature, to judge
their performance in terms of meeting blocking probabilities which are defined a priori in terms of
the parameters δn , δ. The computation of the Chernoff bound is illustrated in the next section.
3
Stochastic Optimal Control
In this section, we develop a PCA based on a state space model, which capture the spatio-temporal
variations of the wireless channels. In addition, we compute the Chernoff bound explicitly. This
section provides the connection between Sections 1, 2, Chernoff bound, and stochastic control
when the channel is dynamic.
State Space Formulation. Consider a wireless network consisting of M transmitters and M receivers. Assuming a flat fading channel undergoing fast or slow fading [4], the received signal at
the nth base station is
³
´
u
(t)s
(t)
I
(t)
cos(ω
t)
−
Q
(t)
sin(ω
t)
+ dn (t)
c
nj
c
∑ j j nj
M
yn (t) =
(3.10)
j=1
where n = 1, . . . , M. Here Ini and Qni denote the inphase and quadrature components of the channel
connection from transmitter i to base station n, sn is the information signal of transmitter n, un
is control input of transmitter n, which acts as a scaling on the information signal sn , dn is the
channel disturbance or noise at the nth receiver, and ωc is the carrier frequency. The noise dn (t)
5
is assumed to be independent of the channel parameters, and processes. In [6, 7] it is shown
that {In j (t), Qn j (t)}t≥0 are realizable through the multi-dimensional linear stochastic differential
equations of the form

d

XInk (t)
XQnk (t)

 = Ank (t) 

XInk (t)
XQnk (t)
 dt + fnk (t)dt + Gnk (t)dwnk (t),
Ink = Cnk XInk , Qnk = Cnk XQnk
(3.11)
for 1 ≤ n, k ≤ M. Here {wnk (t)}t≥0 is a vector of standard Brownian motions and {XInk (t), XQnk (t)}t≥0
are state vectors, representing the channel processes from transmitter k to receiver n. They represent power path loss associated with the inphase and the quadrature components of the signal as
a function of time, and they model the spatial and time variations of the wireless channels. fnk is
the line of sight component when the inphase, and quadrature components are Ricean distributed.
Specifically, in [8], it is shown that measurement data (provided by the Canadian Communication
Research Center) of Ink , Qnk can be reproduced through Gaussian state space models of the form
(3.11), using parameter identification methods (e.g., the Expectation Maximization algorithm) to
identify the matrices Ank , Gnk , Cnk , and the vector fnk from real noisy measurement data of Ink and
Qnk (see [8]). Next, the received signal is represented in compact form. Define
h
i
h
i
4
4
4
Hnk = skCnk sin(ωct) −skCnk cos(ωct) , Xnk = XInk XQnk , p = (u21 , . . . , u2M ).
The wireless network has a state space representation
dXnk (t) = Ank (t)Xnk (t)dt + fnk (t)dt + Gnk (t)dwnk (t)
(3.12)
M
yn (t) =
∑ u j (t)Hn j (t)Xn j (t) + dn(t)
(3.13)
j=1
for 1 ≤ n, k ≤ M. Equations (3.12) and (3.13) can be represented as follows
dX0 (t) = A0 (t)X0 (t)dt + f0 (t)dt + G0 (t)dw0 (t)
y0 (t) = H0 (u,t)X0 (t) + d0 (t)
(3.14)
(3.15)
where A0 (t), f0 (t), G0 (t), H0 (u,t) are related to the coefficients of (3.12), (3.13) and {w0 (t)}t≥0
is a vector of Brownian motions, which by earlier assumption is independent of {d0 (t)}t≥0 . Let
(Ω, F , P; {Ft }t≥0 ) be a basis filtered probability space on which (3.12), (3.13) are defined. Let H
be a finite-dimensional Hilbert space with norm || · ||. Define
´ 4 n
n
o
³
PF2 0, T ; ℜk = φ(·) = φ(t, ω); 0 ≤ t ≤ T , φ : [0, T ] × Ω → ℜk is Ft − predictable,
6
for each ω ∈ Ω,
Z T
0
¾
||φ(t, ω)|| dt < ∞
2
(3.16)
³
´
Let LF2 0, T ; ℜk denote the Hilbert space of predictable square integrable random processes. The
class of admissible control strategies associated with (3.12), (3.13), denoted by Uad will be either
³
´
³
´
PZ2 0, T ; ℜM or LZ2 0, T ; ℜM , where the filtration {Zt }t≥ denotes the channel information generated by the processes {XInk (t), XQnk (t), sn (t)}t≥0 . Given u ∈ Uad , the processes {Xnk (t), un (t)}t≥0
are called admissible if Xnk (·) ∈ LF2 (0, T ; ℜ` ) are solutions of the stochastic differential equations.
The QoS of each user with respect to power signals over an interval [0, T ] is
RT
pn (t)kHnn Xnn (t)k2 dt
0
≥ γn , 1 ≤ n ≤ M
RT
2
M RT
∑ j=1 0 p j (t)kHn j Xn j (t)k dt + 0 kdn (t)k2 dt
4
where γ =
γn
γn +1 ,
pn (t) = u2n (t), 1 ≤ n ≤ M, or equivalently
n
I0,T
(p)
4
=
Z T
0
2
kHn (u,t)X0 (t)k dt +
Z T
0
kdn (t)k2 dt ≤ 0
(3.17)
where Hn is chosen appropriately.
Computation of Chernoff Bound The Chernoff bound associated with (3.17) subject to (3.14),
(3.15) is computed as follows. By the well known property of expectation
£
¤
£ £
¤¤
n
n
E exp(sn I0,T
(p)) = E E exp(sn I0,T
(p))|X0 (0)
(3.18)
Thus, the left hand side of (3.18) is computed via its right hand side as follows. Define
Z T
¢
¤
¡ n
4 £
n
kdn (t)k2 dt |X0 (t) = x
V (t, x) = E exp sn It,T (p) − sn
(3.19)
Since {dn (t)}t≥0 and the channels are independent, by (3.18), and (3.19)
´
³ Z T
¤¤
£
¤
£
£
n
E exp(sn I0,T
(p)) = E V n (0, X0 )E exp sn
kdn (t)k2 dt |X0 (0)
(3.20)
t
0
Moreover, V n (., .) satisfies the following version of a backward Kolmogorov equation [17]
³
´tr ∂V n (t, x)
∂V n (t, x) 1
∂2V n (t, x) ´ ³
+ Trace G0 Gtr
+
A
x
+
f
0
0
0
∂t
2
∂x2
∂x
2 n
n
+sn kHn (u,t)xk V (t, x) = 0, V (T, x) = 1
(3.21)
It can be shown that (3.21), has a closed form solution of the form V n (t, x) = exp[ 12 xtr Σn (t)x +
xtr mn (t) + 12 gn (t)] where Σn = Σtr
n ≥ 0, mn , and gn satisfy the following equation
tr
tr
Σ̇n + Σn G0 Gtr
0 Σn + A0 Σn + Σn A0 + 2sn Hn Hn = 0, Σn (T ) = 0,
1
tr
ṁn + G0 Gtr
0 Σn mn + Σn f 0 + A0 mn = 0, mn (T ) = 0,
2
tr
ġn + Trace(G0 Gtr
0 Σn ) + 2 f 0 mn = 0, gn (T ) = 0.
7
(3.22)
(3.23)
(3.24)
£
¡ R
¢
¤
Hence, by evaluating V n (., x) at t = 0, and then computing E exp sn 0T kdn (t)k2 dt |X0 (0) , which
s2
is equal to exp[ 2n σ2 T ], when {dn (t)}t≥0 is Gaussian white noise process with intensity σ2 , we deh2
iR
£
¡ n
¢¤
s
(p) = exp 2n σ2 T V n (0, σ) fX0 (0) (σ)dσ, where fσ (.) is the density of X0 (0).
duce E exp sn I0,T
If X0 (0) = x is deterministic, then fX0 (0) (.) is a delta measure concentrated at x, and the previous
i
h2
£
¡ n
¢¤
s
expression reduces to E exp sn I0,T
(p) = exp 2n σ2 T V n (0, x). Thus, the Chernoff bound is
computed explicitly via (3.22)-(3.24), and then minimized over sn ≥ 0.
Pathwise Quality of Service and Predictable Strategies. A natural generalization of the QoS
n (p) ≤ 0, 1 ≤ n ≤ M is now defined by
(1.1) with respect to the system (3.12) and QoS I0,T
n
P1 :
min
p∈Uad
M
∑
Z T
i=1 0
pi (t)dt;
n
I0,T
(p) ≤ 0,
o
{Xnk (t)}t≥0 , 1 ≤ n, k ≤ M
(3.25)
Next, we present a solution of this problem by first introducing the communication meaning of
Predictable Power Control Strategies (PPCS). In wireless communication systems it is practical
to observe and estimate the channel at the base station, and then to communicate the information
to the transmitter to adjust its control input signal {u j (t)}M
j=1 . Since any channel experiences delay, and control is not feasible continuously in time, but only at discrete time instants, we shall
introduce the concept of predictable strategies, which we illustrate as follows. Consider a set of
discrete-time strategies {u(t);t = t1 ,t2 , . . . , T }. The transmitters adjust their t control strategies
M
u(t) = {ui (t)}M
n=1 once the channel information {Ink (t − 1), Qnk (t − 1), sn (t − 1)}n,k=1 is observed
or estimated at the base station, and the information is communicated to the transmitters, which
hold these values during the time interval [t − 1,t). At time t, a new set of channel information {Ink (t), Qnk (t), sn (t)}M
n,k=1 is observed at the base station, and the time t + 1 control strategies
u(t + 1) are computed, communicated to the transmitters, and held constant during the time interval [t,t + 1). Such decision strategies are called predictable. Specifically, we say that a discrete
time signal {ϕ(t);t = 0, 1, 2, ...} is predictable with the respect to a filtration {Zt } if ϕ(t) is Zt−1
measurable. Similarly, a right continuous with left hand limits (corlog) random process {ϕ(t)}t≥0
is called predictable with the respect to a filtration {Zt } if for each ε > 0, the random variable ϕ(t)
is {Zt−ε } measurable for each t. Define
4
gni (tk ,tk + 1) =
Z tk+1
tk
4
kHni Xni (t)k2 dt, ηn (tk ,tk + 1) =
tk
kdn (t)k2 dt, 1 ≤ n, i ≤ M
¢
¤
4£
GI (tk ,tk+1 ) = diag g11 (tk ,tk+1 ), . . . , gMM (tk ,tk+1 ) , G(tk ,tk+1 ) = gni (tk ,tk + 1) M×M
¢tr
4¡
4
η(tk ,tk+1 ) = η1 (tk ,tk + 1), . . . , ηM (tk ,tk + 1) , p(tk+1 ) = (p1 (tk+1 ), . . . , pM (tk+1 ))tr
4
¡
Z tk+1
8
where GI (t), G(t), η(t) denote the same quantities without the integration over [tk ,tk+1 ]. Here
[tk ,tk+1 ] denotes a bit interval or multiple bit intervals, so long as the channel model does not
change significantly, (e.g., [tk ,tk+1 ] is a subset of the time duration of the coherence time of the
channel). Using predictable strategies over [tk ,tk+1 ], the equivalent of (1.1) is
n
−1
min p(tk+1 ) ∑M
i=1 pi (tk+1 ); p(tk+1 ) ≥ ΓGI (tk ,tk+1 )
³
´
o
× G(tk ,tk+1 )p(tk+1 ) + η(tk+1 ) , {Xnk (t)}t∈[tk ,tk+1 ]
(3.26)
which is a linear programming problem in the M ×1 vector of unknowns p(tk+1 ). The optimization
(3.26) gives a time varying optimal power control strategy, which instantaneously compensate
for channel fading. Using predictable control strategies, the following theorem generalizes to the
dynamical case a standard result obtained originally obtained using static models in [1]. It basically
compensate for instantaneous power losses in wireless fading networks.
Theorem 3.1 Consider problem (3.25) during the interval [tk ,tk+1 ] and {u(t)}t≥0
∈P2
Z
³
0, T ; ℜM
´
,
which is predictable with respect to the filtration {Zt }t≥0 , over [tk ,tk+1 ]. The QoS constrains are
feasible if and only if the Perron Frobenius eigenvalue of ΓGI (tk ,tk+1 )−1 G(tk ,tk+1 ) is less than 1.
Moreover, if the inequality Itnk ,tk+1 (p) ≤ 0 is replaced by the equality Itnk ,tk+1 (p) = 0, 1 ≤ n ≤ M, then
the optimal control is
´−1
³
(t
,t
)G(t
,t
)
p(tk+1 ) = I − ΓG−1
× ΓG−1
k k+1
k k+1
I
I (tk ,tk+1 )η(tk ,tk+1 )
(3.27)
Further, dividing by (tk+1 − tk ) and taking the limit from the right, as (tk+1 − tk ) → 0, gives
³
´
GI (t)p(t) ≥ Γ G(t)p(t) + η(t) .
Proof. Follows from the predictability of admissible controls and Section 1.
Clearly, the matrices, and vectors GI , G, and η depend on the paths of the channel during the
interval [tk ,tk+1 ]. The paths of the processes are computed using the state space models (3.14),
(3.15). The optimal control is computed via (3.27), and depends on these paths. The limiting case
(tk ,tk+1 ) → 0, described in the Theorem 3.1, corresponds to the static case described by (1.1).
Remark 3.2 i) Suppose there is only one receiver (e.g.,the double indices are unnecessary). Then
R tk+1
³
´−1
kdn (t)k2 dt
tk
M
?
× R tk+1
by Theorem 3.1 the optimal power is pn (t +1) = γn 1− ∑ j=1 γ j
, 1 ≤ n ≤ M.
kHn Xn (t)k2 dt
tk
p
In simulations one has to replace dn (t) = y(t) − ∑M
p j (t)H j X j (t). Taking the limit from the
j=1
right, as (tk+1 − tk ) → 0, gives
p?n (t) =
³
γn
1 − ∑M
j=1 γ j
kd(t)k2
,1 ≤ n ≤ M
´−1
kHn Xn (t)k2
9
(3.28)
³
´2
where kHn Xn (t)k2 = rn (t) cos(ωct + Φn (t))sn (t) .
4
Generalizations
Without predictable power control strategies, two formulations which follow from the results of
Theorem 3.1 are in terms of convex optimization using linear programming techniques and stochastic control with integral or exponential-of-integral constrains. We formulate both problems.
Convex Optimization and Linear Programming. Consider problem
n
P2 :
M
min
p∈Uad
∑
Z tk+1
pi (t)dt; Itnk ,tk+1 (p) ≤ 0, 0 ≤ pi ≤ pmaxi , {Xnk (t)}t≥0 , 1 ≤ n, k ≤ M
i=1 tk
o
According to the above formulation using predictable strategies this is a convex optimization problem. Also, one can consider any interval [0, T ] as 0 = t0 < t1 < t2 . . . < tk < tk+1 . . .tT = T and
solve the problem over a sequence of intervals. It should be noted that if p(t) is continuous almost
everywhere in the interval [tk , tk+1 ) then the integral
R tk+1
tk
pi (t)dt can be approximated by Riemann
sums as close as desired, and problem P2 reduces to a linear programming problem again.
Stochastic Optimal Control with Integral/Exponential-of-Integral Quadratic Constrains. Consider the problem
n
P3 :
min
p∈Uad
M
Z T
i=1
0
∑E
n
o
4
n
n
pi (t)dt; J0,T
(p) = E I0,T
(p) ≤ 0, 0 ≤ pi ≤ pmaxi ,
o
{Xnk (t)}t≥0 , 1 ≤ n, k ≤ M
If there exists a set of {γn }M
by employing Lagrange multipliers
n=1 such that the QoS are feasible,
n
h R
io
T
M
n (p) we can introduce Lλ (u? , λ) = min
i (p)
λn for each J0,T
E
p
(t)
+
λ
I
, and
i 0,T
p∈Uad ∑i=1
0 i
then solve the problem l(λ? , u? ) = supλ≥0 Lλ (u? , λ). Further, one can show that Lλ (u? , λ) satisfies
a dynamic programming equation of the Hamilton- Jacobi- Bellman type [17]. Similarly, one can
consider the QoS as pointwise constrains (w.p.1) and pursue the problem
n
P4 :
min
p∈Uad
M
Z T
i=1
0
∑E
M
pi (t)dt; kdn (t)k2 + ∑ p j (t)kHn j Xn j (t)k2 −
j=1
o
1
pn (t)kHnn Xnn (t)k2 ≤ 0,
γn
t ∈ [0, T ], 0 ≤ pi ≤ pmaxi , {Xnk (t)}t≥0 , 1 ≤ n, k ≤ M
n (p) is convex in p, and
Note that if pi (t), for i = 1, . . . , M, are convex in the interval [0, T ], I0,T
the admissible set Uad is convex, then problems P3 and P4 are convex optimization problems,
10
since their objective functions are convex, (the expectation E and integral operators are linear,
and sum of positive convex functions is convex), and their constrains convex. Similar Pay-off’s
and constrains could be introduced using the probabilistic bounds of the earlier section. However,
because of the large scale the problem (e.g., 10 transmitters and 10 receivers resulting in 100
channels) any proposed algorithm should provide the power control, in real-time, within the order
of a few microsecond.
5
Conclusion
In this paper, we proposed an optimal PCA based on a dynamical model for short-term fading. The
latter consists of state space equations which capture the spatial and time variations of the wireless
communication links. It is therefore more realistic than the standard static models encountered in
the literature. The optimal PCA is shown to reduce to a simple linear programming problem, if
predictable power control strategies are used. A centralized and decentralized probabilistic QoS
measures are proposed together with their corresponding Chernoff bounds, to assess performance
of power control algorithms. The explicit formula for the Chernoff bound is computed. Finally,
generalizations to power control based on convex optimizations techniques are provided if predictable power control strategies are not assumed.
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