Low Complexity Power and Scheduling Policies for
Wireless Networks to Provide Quality of Service∗
Satya Kumar V, Vinod Sharma
Department of Electrical Communications Engineering,
Indian Institute of Science, Bangalore
Email: satyakumar@ece.iisc.ernet.in, vinod@ece.iisc.ernet.in
Abstract—We consider optimal power allocation policies for a
single server, multiuser system. The power is consumed in transmission of data only. The transmission channel may experience
multipath fading. We obtain very efficient, low computational
complexity algorithms which minimize power and ensure stability
of the data queues. We also obtain policies when the users
may have mean delay constraints. If the power required is a
linear function of rate then we exploit linearity and obtain linear
programs with low complexity.
Index Terms—Fading channels, Average queue constraint,
Optimal average power policies.
I. INTRODUCTION
Information and communications technology (ICT) industry
is consuming about 3% of world power consumption. This
percentage will further increase as the number of mobile
users is growing at a rapid pace across the globe. Any
reduction in power consumption in the ICT will reduce the
ever increasing carbon footprint from this sector. It will also
reduce the operational cost of the service providers and will
need smaller diesel generators and batteries. Thus one of the
primary challenges for Next Generation Networks (NGN) is
to reduce energy consumption in the cellular network [7]. In
a Base Station (BS) the power is consumed mainly in the air
conditioner (25%), Radio equipment (61%), DC power (11%)
and RF load and Feeder (3%). In this paper we will concentrate
on reducing power in RF transmission, while providing quality
of service (QoS) to the users. This indirectly reduces the power
consumed in the other components also [16].
Goldsmith and Varaiya [10] obtained a power allocation
policy for a single fading link which optimizes the rate. Energy
efficient scheduling under average delay constraint was first
addressed in [4], [5]. Goyal et al. [11] considered the same
problem, proved existence of an optimal policy and obtained
some structural results for the optimal policy. [24] proposed
delay and power optimal policies in a single server queue when
the power used in transmission is a linear function of rate.
[9] proposed an energy efficient scheduling with individual
packet hard delay constraints. [13] presented a policy which
minimizes energy for sending a fixed number of packets under
given delay constraints. Neely [18] proposed a new innovative
algorithm which optimizes power while satisfying an upper
bound on the sum of the average queue lengths of multiple
users. The algorithm needs to learn a parameter. [25] proposed
*Partially funded by a project from ANRC.
a suboptimal policy which minimizes the average power under
average queue constraint when there is an upper bound on
packet loss also. [28] obtained asymptotic lower bounds for
average queue length and average power consumption. Neely
[17] presented an algorithm for a multiuser and multi channel
scenario subject to an upper bound on the sum of average
queue length. To meet the constraints the algorithm needs to
learn two parameters.
In [21] new throughput optimal energy management policies
were developed when the transmitter is powered by energy
harvesting sources. Explicit throughput optimal policies which
also minimize mean delay were also obtained when linear
power rate relation holds. [22] provides efficient MAC policies
in the same setup. [19] implemented an online algorithm by
modifying the value iteration equation and provides QoS by
minimizing the average power for a single user.
A. Problem Statement and our contribution
Our objective is to minimize the average transmit power
while providing QoS, such as stability of the data queue or an
upper bound on the average delay. The later problem is usually
formulated as an Markov Decision Problem (MDP). However,
finding an optimal solution via an MDP is computationally
hard and does not usually provide any insights to the solution.
A few other algorithms provided in literature use learning
algorithms. These can be slow to converge and also do not
provide any insight into the optimal solution. We are not aware
of any algorithm which guarantees an upper bound on mean
delay especially in a multiuser fading environment except via
MDP or learning.
As against the above work, we provide algorithms which are
in closed form or require very low computational complexity.
Our algorithms do not require queue length information or
arrival statistics. These algorithms are optimal in the class
of algorithms requiring only channel gains and provide explicit guarantees on mean delays for individual users. These
algorithms do not require any learning. Furthermore, either
these are probably optimal or shown to be close to optimal
via computations on explicit examples and perform better than
other existing low complexity algorithms.
The paper is organized as follows. Section II presents
the system model. Section III presents our algorithm that
minimizes average power and guarantees stability of the data
queue. Section IV considers the problem when the users also
specifically consider the case, when fj can be approximated
by a linear function,
Rkj = αPkj Hkj ,
Fig. 1.
where αj > 0. This is a good approximation of Shannon
formula (2) at low SNR [24] and also at high bandwidth ([3],
[14]). However as shown in [21], even when (2) cannot be
approximated well by a linear function, (3) can still hold in a
practical systems.We will show that linearity can be exploited
to obtain more efficient algorithms.
First, we consider the system without a delay constraint.
Later on we provide the solution with delay constraints.
System Model.
require mean delay guarantees. Section V provides examples
illustrating these algorithms and compares our results with
other existing well known algorithms. Section VI concludes
this paper.
III. OPTIMAL POLICIES : STABLE SYSTEM
In this section we consider the problem of minimizing
average power
lim sup
II. SYSTEM MODEL
n→∞
We consider a single server, discrete time queue with M
user. A slot is of duration 1. Let Ajk bits arrive in slot k,
representing interval [k, (k + 1)), k ≥ 0 for user j. The data
to be transmitted arrives into the system from higher layers at
the beginning of each slot and placed into an infinite buffer
queue till transmission (Figure 1). Arrival process {Ajk , k ≥
0} is an independent and identically distributed (iid) sequence
with mean E[Aj ]. We will some times also consider the case,
when the arrival process is stationary, ergodic (e.g. modeled
as Markov chain)
We assume that channel gain Hkj is constant over the
duration of slot k and {Hkj , k ≥ 0} is iid, where Hkj ∈
{hj1 , hj2 , ..., hjLj } and 0 < hji < hjl < ∞, for i < l, ∀1 ≤
j ≤ M and hjLj < ∞. We also assume that the channel gain
Hk = (Hk1 , ..., HkM ) is available at the transmitter at time k.
Let P [Hk = h] = p(h).
The i.i.d assumption on {Ajk } and {Hkj } are commonly
made ([1], [6], [27]). The Hkj taking values on a finite set
can be a good approximation for continuous distribution, e.g.,
Rayleigh, Rician, Nakagami, by taking the finite set arbitrarily
large.
Let Rkj bits be transmitted in slot k from user j and let the
power used be Pkj . Let qkj denote the queue length in packets
at time k for user j. Then {qkj } evolves as,
j
qk+1
=
(qkj
+
Ajk
−
Rkj )+ ,
k ≥ 0,
(1)
+
where (x) = max(0, x).
We relate Rkj with Pkj via a general relationship Rkj =
fj (Pkj , Hkj ), where fj is a nondecreasing (in Pkj and Hkj ),
nonnegative function. A common function is, the Shannon
function,
1
log2 (1 + Pkj Hkj /σ 2 ).
(2)
2
This assumes that the channel is Additive White Gaussian
Noise (AWGN) fading channel with noise power σ 2 . We will
Rkj =
(3)
M n−1
1 XX j
E
Pk ,
n j=1
(4)
k=0
subjected to the stability of all the queues, i.e., the queue length
process qk = (qk1 , ..., qkM ) has a stationary distribution. This
could be the QoS if all the users are best effort users in the
Internet.
This problem can be solved via MDP but computing the
optimal solution is too complex to compute for more than
one user. To reduce complexity we ignore the queue length
process in obtaining the optimal policy. We will see that by
obtaining an optimal policy which depends only on the channel
states and the dynamics of the queue we can obtain policies
which are close to the optimal policies, obtainable via MDP.
For simplicity first we obtain the policy for a single user in
Section III-A and will then extend to the multiuser case in
Section III-B. We will call these policies greedy opportunistic
policies.
A. Single User
In this section we remove the user index j. We want to
obtain a power control policy P (h) that minimizes
X
p(hi )P (hi ),
(5)
i
subjected to,
X
p(hi )f (P (hi ), hi ) ≥ E[A] + ,
(6)
i
where > 0 is an arbitrarily small constant. The constraint
(6) is to ensure that the queue is stable.
If f is concave, e.g., Shannon function, then (5)-(6) has
a linear cost function with convex constraints. Instead of
solving for P (h) for each h, we can solve for rate R(h) for
each h and then this becomes an optimization problem with
convex cost function with linear functions. In either case it is
a comparatively easy optimization problem to solve and the
globally optimal solution can be obtained by solving KKT
conditions. Also, it requires only channel statistics, current
channel gain and E[A]. Another important feature of this
policy is that when {(Ak , Hk ), k ≥ 0} is iid then this optimal
policy retains the GI/GI/1 character of the queue. Thus, the
numerous results and approximations available [2] for such
queues will be applicable for this system and can be exploited
to provide QoS guarantees to the wireless system. We will
also see that the optimal policy obtained is very close to
the overall optimal policy via MDP which requires arrival
statistics and queue length information also and has a very
high computational complexity.
If {Ak , Hk } is stationary, ergodic (a generalization very
useful in practice, especially to model voice and video traffic)
then the solution of (5)-(6) again provides the optimal policy
in the class of policies which consider only the channel gain.
If the function f is linear, then a simple optimal closed
form policy is available, in the whole class of policies : serve
all the data in the queue whenever Hk = hL , i.e., clear the
queue whenever the channel gain is maximum. This is the
optimal policy even when {(Ak , Hk )} is stationary, ergodic.
This policy does not require any arrival or channel statistics;
only the current channel gain and consumes average power
= E[A]
αhL . For this optimal policy, the system has several other
desirable features. When {Ak , Hk } is iid (in fact we can allow
{Ak } iid and {Hk } Markov modulated with the modulating
Markov chain irreducible with a finite state space), the epochs
when Hk = hL are regeneration epochs for the process
{qk }. The process has a stationary distribution π. The queue
length process converges in total variation to π exponentially.
Furthermore Eπ [q γ ] < ∞ for any γ ≥ 1 if E[Aγ+1
] < ∞.
1
We also have Eπ [q] = E[A]( p1L − 1).
B. Multiuser
MATLAB). Starting the optimization algorithm with different
initial conditions and picking the best solution provides an
efficient solution.
Any solution obtained this way has the desirable property
that each of the queues behaves as an independent GI/GI/1
queue and hence its performance, e.g., mean delay can be approximately obtained from available GI/GI/1 queue results.
The problem becomes much simpler if we can approximate
fj by linear functions. Then an optimal policy (among all
policies) is : in slot k clear queue j if
Hkj = hjLj .
If there is more than one such queue, pick one among them
with the longest queue. If even then there is more than one
such queue, pick with equal probability any one of them. This
policy does not require any channel or arrival statistics but
only channel gains and queue lengths in the current slots.
Serving the longest queue does not reduce average power;
only mean queue length. If we ignore the queue length and
serve one of the j users satisfying (8), with probability (say
equal), then from the channel statistics we can compute the
probability pj for user j with which its queue is served in
any slot. In addition to being optimal, this remarkably simple
policy has all the desirable properties mentioned above for a
single user :
P E[Aj ]
• The overall average power consumed =
.
j α hj
j
•
•
If there are M > 1 users, the optimal policy, ignoring the
queue length information is obtained as follows. Let the power
needed to transmit R bits in channel state h for user j be
gj (R, h) (obtainable from fj ). Then we seek an optimal policy
which provides rate Rj (h) in channel state h for user j with
probability β(j, h). In a slot only one user will transmit. Thus,
we find Rj (h) and β(j, h) for each channel state vector h and
j that
(8)
•
•
•
Lj
The system has a unique stationary distribution π and
starting from any initial distribution, qk converges in total
variation to π exponentially fast.
0
Eπ [qjγ ] < ∞ whenever E[(Aj1 )γ ] < ∞ for any γ 0 > 1.
Eπ [q j ] = E[Aj ]( p1 − 1), ∀j = 1, ..., M .
j
This policy is optimal for any {(Ak , Hk )} process and
does not require any arrival or channel gain statistics.
It does not require any fragmentation of packets.
IV. OPTIMAL POLICIES : WITH MEAN DELAY
CONSTRAINTS
(7d)
Thus far we have been concerned about minimizing constraints subject to stability of the system. For some real
time and data applications, it will be desirable to have upper
bounds on mean delay. Such constraints can be translated into
mean queue length constraints via Little’s law [2]; the exact
relationship between mean delay and queue length depends
on whether we consider a packet model or fluid model. In the
following we assume that it has been sorted out and finally
we have translated our QoS to the constraint,
X
n
1
lim sup E
qkj ≤ q j , j = 1, ..., M.
(9)
n→∞ n
This is a non-convex optimization problem : the objective
function as well as the constraints are non-convex even when
gi are convex. But if all the functions gi are smooth enough,
e.g., have continuous second order partial derivatives, then
we can obtain locally optimal solutions quite easily (using
Again, this problem can be addressed via MDP but getting
optimal solution, at least for the multiuser case is computationally very demanding. Due to all the desirable properties
we have discussed so far, we limit ourselves to policies which
depend only on the channel state information.
min
M X
X
p(h)gj (Rj (h), h)β(j, h)
(7a)
j=1 h∈H
subject to:
X
Rj (h)p(h)β(j, h) ≥ E[Aj ] + , j = 1, ..., M,
(7b)
h∈H
M
X
β(j, h) ≤ 1, ∀h,
(7c)
j=1
0 ≤ β(j, h) ≤ 1, ∀j, h.
k=1
EECA B−Bernoulli
MaxTrans B−Bernoulli
TOCA B−Bernoulli
Greedy
Total average queue length
4.5
4
3.5
3
2.5
2
1.5
1
0.4
Fig. 2.
0.5
0.6
0.7
Average power
0.8
0.9
Comparison of policies for three users for Bernoulli arrivals.
We consider a policy which selects user j with probability
β(j, h) when the system channel state is h and chooses
to transmit Rj (h) bits. We find β(j, h), Rj (h) that solve
the optimization problem (7a), (7c), (7d) with the additional
constraints (9). Under our policy, left hand side of (9) can be
replaced by Eπ [q j ] where π is the stationary distribution of
{qk } under the given policy. Even though under our policy
each queue is like a GI/GI/1 queue with qkj satisfying
j
qk+1
= (qkj + Ajk − Rkj )+ ,
(10)
a closed form expression for Eπ [q j ] is not available. But
several approximation are available for mean delay of a
GI/GI/1 queue (which can be used here because qkj satisfies
the Lindeley equation [11]) we use [12],
Eπ [q j ] ≈
2
2
(j))
(j) + CR
ρ(j)d(j)E[Aj](CA
.
2(1 − ρ(j))
(11)
where
ρ(j) =
E[Aj ]
2
, CA
(j) =
E[Rj ]
Algorithm (EECA). We have considered three users. The
arrival and channel statistics of the three users are same and
independent of each other. The arrival process {Ak } is a
Bernoulli arrival process with average arrival rate 1 and batch
size 10. The channel gains take values in the set {0.01, 1}
and the respective channel gain probabilities are {0.2, 0.8}.
The sum average queue length vs the corresponding average
powers of our opportunistic greedy policy and the TOCA and
EECA policies of [17] are plotted in Fig. 2. We see that our
greedy policy works much better than TOCA and EECA. In
the figure we have also plotted MaxTrans policy of [20] when
we have included power control and obtained the optimal
policy to provide the required mean delay. EECA becomes
unstable at average power 0.54.
A. Linear Case
If the fi can be taken linear, then we have seen in Section
III that an optimal policy to ensure stability should transmit
data from user j only if its channel gain is hjLj . Since multiple
users may have their best channels in a slot, then a user among
them is picked with a certain probability. Thus eventually
(because {hk } are iid, independent of {Ak }), we obtain an
optimal policy where user j will transmit all of its data in any
given slot with probability pj independently of other system
variables. The queue length process {qkj } is regenerative with
the regeneration epochs as the epochs when the jth queue is
cleared. Let N be the interval between two such epochs. It is
geometrically distributed with parameter pj and independent
of {Ajk }. Thus, for this queue, we can explicitly compute its
mean queue length under stationarity as,
NX
−1
∞
X
qlj |N = k P [N = k]
E
l=0
Eπ q j = = k=1
E[N ]
∞
X
E[Aj1 (k − 1) + ... + Ajk−1 ]P [N = k]
V ar[Aj ]
V ar(Rj )
2
, CR
=
j
2
(E[A ])
(E[Rj ])2
(12)
=
k=1
E[N ]
E[A ](E[N ] − E[N ])
=
2E[N ]
1
= E[Aj ]( j − 1).
pLj
j
2

2
(j))2
2(1 − ρ(j)) (1 − CR
2

(15)

 exp[− 3ρ(j) C 2 (j) + C 2 (j) ], if CR (j) < 1
R
A
d(j) =

(C 2 (j) − 1)

2
If we did not have an upper bound on Eπ q j , then any
 exp[−(1 − ρ(j)) 2 R
],
if
C
(j)
≥
1
R
2 (j)
CR (j) + 4CA
pj > 0 is sufficient to provide stability andminimum
average
(13) power. However, now it is possible that Eπ q j obtained. (15)
is greater than q j for some j (otherwise we have already
X
Rj (h)β(j, h)p(h).
(14) obtained an optimal solution for the current problem). Let
E[(Rj )] =
p̂j be the probability such that using this in (15) provides
h∈H
We use these approximations on the LHS of (9) and equality for j. Now we would like to obtain β(j, h) such that
solve the optimization problem (7). Again it is a non-convex ∀j = 1, ..., M ,
optimization problem and we obtain our solution by getting
locally optimal solutions from several initial conditions and
choosing the best solution.
We compare our results with Neely’s [17] Tradeoff Optimal
Control Algorithm (TOCA) and Energy Efficient Control
β(j, h) > 0,
only if hj = hjLj ,
(16)
β(j, h) = p̂j
(17)
and
X
h
If such a solution does not exist (in particular this will happen
when P (hjLj ) < p̂j , we need to relax condition (16). In
particular, we find β(j, h), the probability we will pick user j
in state h and clear its queue that solves
min
M X
X
p(h)β(j, h)
j=1 h∈H
E[Aj ]
αj hj p̂j
subject to
0 ≤ β(j, h) ≤ 1,
X
β(j, h) = p̂j ,
h
X
β(j, h) ≤ 1.
j
This is a Linear Program (LP) and hence provides a global
optimum with comparatively much lower complexity. If Lj =
L for all j then this LP has M LM variables. It’s complexity
can be further reduced if we consider the Dual LP which has
(M +LM ) variables. The complexity of this LP is much lower
than that of the LP for the MDP because in MDP, queue length
state is also considered.
For a single user, the solution is available in closed form.
We compare our greedy policy with the optimal policy via
MDP. The optimal policy is obtained via policy iteration [8].
We consider the MDP with the average cost that includes
r
transmission power (w(h, r) , αh
) and storage cost βq with a
single state cost being w(h, s) + βq (which can be interpreted
as a Lagrangian relaxation of our original problem [15]). For
this MDP, the new average cost optimality equation is,
Fig. 3.
Single user for Poisson arrivals.
J(q, h) = min{w(h, r) + βq − g ∗ + E[J(Q+ , H + )]}, (18)
r
where r is the action representing data transmission in the
slot, J(q, h) is the optimal relative value function, g ∗ is the
optimal average cost, Q+ = (q − r + A) and H + are the next
slot queue length and channel gain. For our example within 10
iterations of the policy iteration, we obtain the optimal policy.
The β used is such that at the optimal policy the average queue
length is q.
We compare the greedy policy of the last section and the
optimal policy for two examples. In the first example, the
arrival process {Ak } is Poisson with rate 2. The channel gains
take values in the set {0.3, 0.5, 1, 2} and the respective channel
gain probabilities are {0.2, 0.4, 0.3, 0.1}. The average queue
length vs the corresponding average powers of greedy policy
and optimal policy are plotted in Fig. 3. We observe that the
optimal curve and the greedy curve are very close to each
other.
We also present results, when the incoming arrival distribution is Binomial with parameters B(30, 0.1) whose average
arrival rate is 3. The channel statistics are as above. We observe
from Fig. 4. that the average queue length vs average power
for greedy policy and the optimal policy are very close.
Fig. 4.
Single user for Binomial arrivals.
V. CONCLUSIONS
In this paper, we have presented computational low cost
optimal transmit power policies for a wireless system. The
user may experience channel fading and may have an average
delay constraint. Obtaining optimal policies via MDP can
be very computationally complex. Thus we present a greedy
suboptimal policy in closed form which is very close to
optimal and does not require queue length information. Next,
we have extended our greedy policy to a multiuser downlink.
The users may have individual average delay constraints. This
policy is close to optimal and is computationally much simpler.
It can also be used for an uplink.
Our study shows that if we search for optimal policies using
only the channel gains and ignore the queue lengths, we obtain
very efficient policies at very low computational cost which
can also guarantee QOS in a multiuser fading environment.
Sum of Average Queue Lengths
140
TOCA Policy
Opportunistic Greedy Policy
EECA Policy
120
100
80
60
40
20
5
5.2
Fig. 5.
5.4
5.6
Average Power
5.8
6
Five users for Poisson arrivals.
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