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Traffic-Aware Power Adaptation and Base Station
Sleep Control for Energy-Delay Tradeoffs in Green
Cellular Networks
Jian Wu, Yiqun Wu, Sheng Zhou, Zhisheng Niu
Tsinghua National Laboratory for Information Science and Technology
Dept. of Electronic Engineering, Tsinghua University, Beijing 100084, P.R. China
Abstract—Traffic-aware resource allocation and base station
(BS) sleep control are key methods for energy saving in cellular
networks. In this paper, first, we consider the control problem of
how to adapt transmit power according to flow-level traffic vari-
ations, which leverages the tradeoff between energy consumption
and delay performance. Based on different time scales of traffic
variations, two power adaptation strategies are investigated: load-
aware and queue-aware. The two strategies adapt transmit power
according to flow arrival rate and instantaneous number of
flows, respectively. Optimal solutions are given for both strategies.
Since the optimal solution of the queue-aware strategy has no
explicit form, tight bounds are derived as an approximation.
Simulation results show that the two strategies perform closely
in terms of energy consumption and average delay, while the
queue-aware strategy is better in the tail distribution of delay
and is more robust to system parameter variations. Secondly,
for the load-aware strategy, with more practical concerns like
the total BS energy consumption and BS sleep control taken
into account, the relationship between energy consumption and
delay is explored and energy-optimal rate can be obtained under
certain conditions. Two threshold-based BS sleep strategies are
investigated where the optimal threshold and rate are derived
respectively.
I. INT ROD UC TI ON
The next generation wireless networks are expected to
provide ubiquitous and broadband access to the Internet. The
majority of traffic in wireless networks has been shifting
from mobile voice to mobile data due to the popularity of
the smartphones. The exponentially growing data traffic and
access requirement have triggered vast expansion of network
infrastructures, resulting in dramatically increased energy con-
sumption. It is urgent to focus on the energy-efficient design in
wireless networks from both the environmental and economic
viewpoint.
To deal with the green evolution of wireless networks, many
international research projects have sprung up like EARTH [1]
and GreenTouch [2]. There are also some initial research
efforts that reveal the opportunities and fundamental issues
of green communication. The author in [3] shows there exist
traffic dynamics in cellular networks both in time and spatial
This work was supported in part by the National Basic Research Program of
China (973 Program: No. 2012CB316001) and the Nature Science Foundation
of China (No. 61021001, No.60925002).
PS
BS
User 1
User 2
User n
Queue length
Flow
arrivals Flow
departures
Service rate
(a)
−1 0 1 2 3 4 5 6
−4
−3
−2
−1
0
1
2
x
W(x)
(−1/e,−1)
(b)
Fig. 1. (a) Flow-level model for downlink transmission with traffic dynamics.
One user corresponds to one flow. (b) Lambert Wfunction.
domain and if we can seize the opportunity to trace the
traffic variation and adapt the radio resources in a cell or
the whole cellular networks to it, a great amount of energy
can be saved. Four fundamental trade-offs regarding different
metrics for designing wireless networks are investigated in [4],
which work as a guideline for the green design. Many other
research for the migration to green radio have also been
proposed [5] [6]. In this paper, we study dynamic traffic-aware
power adaptation and BS sleep control, and explore the energy
delay tradeoffs.
Earlier research on power adaptation mainly focus on com-
pensating for the channel fading and controlling interference
rather than reducing energy consumption [7] [8]. Energy-
efficient power control was first explored in [9] where lazy
scheduling was proposed which schedules packet transmis-
sions as slowly as possible to minimize energy consumption
with packet delay constraints. Here we first focus on how
to adapt transmit power according to flow level traffic vari-
ations to achieve energy-saving. Two classes of traffic-aware
power adaptation strategies based on the time scale of traffic
variations are proposed. The “load-aware” strategy bases its
decisions on the flow arrival rate which captures the first-order
statistic characteristics of the traffic load. While the “queue-
aware” strategy takes into account the fluctuation of the queue
length, which is the amount of flows/users in the system in our
problem. Optimal solutions are given for both strategies, and
especially for the queue-aware strategy which is formulated
978-1-4673-0921-9/12/$31.00 ©2012 IEEE
Globecom 2012 - Symposium on Selected Areas in Communications
3171
using markov decision process (MDP) theory, tight bounds
are derived as an approximation.
Besides the transmit power adaptation, BS sleep is incorpo-
rated to save its load-independent static part energy consump-
tion. As pointed out in [10], when we take practical concerns
such as static energy consumption into consideration, the
trade-off relation between energy and delay usually deviates
from the simple monotonic curve [11]. It is important to know
when and how to trade tolerable delay for energy. We explore
this relationship with flow-level dynamics and BS sleep control
taken into account and find the energy-optimal rate when cer-
tain conditions are satisfied. When sleep mode and switching
cost are taken into account, it has been proved that the optimal
sleep strategy has hysteretic structure [12] [13]. Accordingly,
we study two threshold-based sleep strategies that waiting
for deterministic number of users and deterministic period
of vacation time respectively before waking from sleep and
explore their optimal threshold and rate respectively.
The rest of this paper is organized as follows: In Section II
we describe the system model. Section III gives the load-aware
and queue-aware power adaptation strategies. In Section IV,
with BS sleep taken into account, the relationship between
energy and delay is studied, and two threshold-based sleep
strategies are also analyzed. We conclude the whole work in
Section V.
II. SY ST EM MO DE L
We focus on dynamic user populations in a cellular system
where new flows, e.g., file transfers, are initiated at random
and leave the system after being served, which are referred to
as flow-level dynamics [14] as shown in Fig. 1(a), and one
user corresponds to one flow. Assume the total bandwidth is
w. Users arrive according to a Poisson process with rate λ,
and each user requires a random amount of service Lwith
average length E(L) = l.
A. Processor-Sharing Model
The M/G/1 processor-sharing (PS) model is used here [14]
[15]. Given nusers in the system, assume the BS can provide
service at a rate of xn>0units of service per unit time,
and it divides the service rate equally among all users in the
system. That is, when n > 0, each will receive service at a
rate of xn/n per unit time and users depart the system at rate
µ=xn/l. The system framework is shown in Fig. 1(a).
B. Energy-Consumption Model
The BS energy-consumption model proposed in [1] [2] is
adopted. The BS has two modes: active mode and sleep mode.
When BS is in active mode, it consumes the static power
Psand the dynamic part that is proportional to the output
transmit power Pt; while in sleep mode, the BS only consumes
Psleep which is much less than Ps. Specially, when there is a
mode transition, assume a switching energy cost Esw will be
incurred to avoid frequent mode transitions.
Pin =Ps+Pt∆Pactive mode
Psleep sleep mode.(1)
III. TRA FFIC -AWARE POW ER ADA PTATI ON
In this section, we focus on the dynamic part in the BS
energy consumption model: the transmit power Pt.
Using the M/G/1 PS model, the queue length evolves as a
birth-and-death process with arrival rate λand state dependent
service rate xnthat can be chosen from a closed subset Aof
[0,∞). In the problem we described we take
xn=wlog2(1 + γPt(n)), γ =g
N0w.(2)
where g1is the channel gain and N0is the noise density.
Our objective is to minimize average cost over an infinite
planning horizon, where cost has two elements: energy cost
that increases with the power level or service rate chosen, and
delay cost that increases with queue length. For the energy cost
c(xn), it is related to the transmit power needed to induce the
rate xn. For the delay cost dn, we take dn=nfor simplicity.
Due to Little’s Law, we know that the delay cost can just
reflect the delay performance.
A policy is defined as a vector x= (x0, x1, x2, . . .)
with xn∈Afor all n, and the stationary distribution
associated with an ergodic policy xis denoted by p(x) =
(p0(x), p1(x), . . .). Then the long-run average system cost is
z(x)=
∞
n=0
pn(x)[c(xn) + dn]=
∞
n=0
pn(x)[Pt(xn)
β+n].(3)
Here βcontrols the relative relationship between the energy
cost and delay cost.
Recall the Lambert W function that will be used later. It is
defined as [16]
W(z)eW(z)=z, (4)
and z≥ −e−1when W≥ −1, as shown in Fig. 1(b).
To get a foundational comparison results of the two strate-
gies, we make the idealized assumption that in both cases the
rate can be chosen from A= [0,∞).
A. Load-Aware Adaptation Strategy
In the load-aware adaptation strategy, if the flow arrival rate
is given, the service rate does not change with the user number,
that is, x0= 0 and xn=x, n > 0. Using the results for the
M/G/1 PS queue [17], the average queue length E[n] = λl
x−λl
and the busy probability 1−p0=λl
x. The system cost turns to
z=λl
x−λl +λl
x
1
βγ (2 x
w−1).(5)
To minimize the objective, we take dz
dx = 0. The optimal
rate is denoted by x∗
s> λl and is got by solving
W(γβ
e(x∗
s
x∗
s−λl )2−1
e) = ln 2
wx∗
s−1.(6)
1We first study the basic case: users experience homogeneous channels with
gain g. When heterogeneous channel conditions are considered, the multi-class
PS model can be used, which will be discussed in our future work.
3172
The optimal system cost of the load-aware strategy z∗
sis given
below.
z∗
s=λl
x∗
s−λl +λl
x∗
s
1
βγ (2 x∗
s
w−1).(7)
From equation (6) we can get ∂x∗
s
∂λ >0,∂x∗
s
∂l >0and
∂x∗
s
∂γ >0, which means that the optimal load-aware rate x∗
s
is increasing with the traffic load and the channel gain.
B. Queue-Aware Adaptation Strategy
For the queue-aware adaptation strategy, the service rate
will be a function of queue length n. We use the theory of
MDP to formulate this problem. For the dynamic service rate
control problem, a good solution is provided in [18] and we
will use its algorithm to solve our problem here. First we will
recall this algorithm specialized to the case we considered.
Then based on this approach, we give the upper and lower
bound of the optimal control rate to provide some insight into
its structure.
1) Optimal dynamic rate: Using the standard optimality
equation, or Hamilton-Jacobi-Bellman equation for a semi-
Markov decision process with average-cost criterion, the op-
timality equation in this problem is written as follows, where
vnis the minimum expected cost incurred until the next entry
into an arbitrary reference state m≥0, starting in state n,
under a certain revised cost structure, and zis a guess at the
minimum average cost rate z∗.
v0=1
λ(−z)+v1,(8)
vn= inf
x∈A1
λ+x
l
[c(x)+n−z]+ λvn+1
λ+x
l
+
x
lvn−1
λ+x
l, n ≥1.(9)
Then define the relative cost differences un=λ(vn−vn−1),
the optimality equation will be re-expressed as
u1=z, un+1 = sup
x∈A
{z−n−c(x) + x
λl un}.(10)
According to the definition of the function ϕ(u)and ψ(u)
in [18], the two functions in our problem are given below,
where ψ(u)is the minimum value of xthat achieves the
maximum in ϕ(u).
ϕ(u) = sup
x∈A
{x
λl u−c(x)}=uw
λl log2
uwγβ
eλl ln 2 +1
γβ ,(11)
ψ(u) = min{x:x
λl u−c(x) = ϕ(u)}=wlog2
uwγβ
λl ln 2 .(12)
Then it can be obtained
u1=z, un+1 =ϕ(un)−n+z. (13)
Using the stopping criterion given in [18], we can find the
optimal value of z, then uncan be get recursively. The optimal
service rate in state nwill be obtained through
x∗
n=ψ(un).(14)
0 2 4 6 8 10 12
5
5.5
6
6.5
7
7.5
8
8.5 x 107
User number
rate x(n)
lower bound
upper bound
queue−aware
load−aware
(a)
0 5 10 15
5
5.5
6
6.5
7
7.5
8
8.5
9x 107
User number
rate x(n)
lower bound
upper bound
queue−aware
load−aware
(b)
Fig. 2. Comparison between the optimal rate of the two strategies and
the bounds of the queue-aware optimal rate, β= 1, l = 2M B, w =
10M Hz, N0= 10−9, g = 1. (a) λ= 0.5flows/sec, (b) λ= 1flows/sec.
2) Bounds on the optimal dynamic rate: Besides getting
the optimal dynamic control policy numerically, we want to
provide some insight into the structure of the optimal dynamic
rate.
Theorem 1: The optimal service rate x∗
nin state nsatisfies
W(γβ(n−z∗
s) + 1)1
e2−λl
w+ 1w
ln 2 +λl ≤x∗
n≤
minwlog2γβn+1
γβ 2λl
w(e1+Γ(n)−1)
1+Γ(n)+λl ln 2
w
(1+Γ(n))2,
wlog2γβ
ln 2 n+1
γβ 2λl
w(2t∗
−1)
t∗+λl
w
t∗2,where in the lower bound
n≥z∗
s+1
γβ (1 −2λl
w), and in the upper bound Γ(n) =
W((nγβ2−λl
w−1) 1
e), and t∗satisfies Wλlγβ
wt∗21−λl
w−11
e=
t∗ln 2−1.
Its proof is omitted due to space limitations. It can be
observed that both the upper and lower bound are related to
the system load parameters and the user number in the system.
We can see the performance of the bounds from Fig. 2. They
give a good approximation of the optimal rate and is much
tighter when the traffic load is lower. Actually, in the upper
bound the first part which has explicit form plays an important
role.
C. Comparison of Load-Aware and Queue-Aware Strategies
In the section, the comparison between the load-aware and
queue-aware adaptation strategies is given.
Besides the bound performance, Fig. 2 also gives the
optimal rate of the load-aware strategy. Although it widely
differs from the optimal queue-aware rate, it can be observed
later that it still achieves relatively good performance.
3173
0 1 2 3 4 5
0
2
4
6
8
10
12
14
Delay
Power
queue−aware,λ=0.5
queue−aware,λ=1
queue−aware,λ=2
queue−aware,λ=3
queue−aware,λ=4
load−aware,λ=0.5
load−aware,λ=1
load−aware,λ=2
load−aware,λ=3
load−aware,λ=4
(a)
1 2 3 4 5 6 7
0
5
10
λ
Delay
1 2 3 4 5 6 7
0
20
40
60
λ
Power
load−aware,β=1
queue−aware,β=1
load−aware,β=5
queue−aware,β=5
load−aware,β=0.1
queue−aware,β=0.1
load−aware,β=1
queue−aware,β=1
load−aware,β=5
queue−aware,β=5
load−aware,β=0.1
queue−aware,β=0.1
(b)
Fig. 3. l= 2M B, w = 10M Hz , N0= 10−9, g = 1. (a) Tradeoff of
the average delay and transmit power consumption, (b) Comparison of the
average delay and transmit power respectively of the two strategies.
Fig. 3 shows the comparison of the relationship between
transmit power and delay. By calculating the optimal load-
aware and queue-aware rate at different value of β, the
tradeoff between the average delay and transmit power is given
in Fig. 3(a). It is surprising that the load-aware adaptation
strategy almost performs the same as the queue-aware strategy.
For more specific comparison in Fig. 3(b), the queue-aware
strategy has advantages in energy-saving and average delay
performance over the load-aware strategy only when the traffic
load is relatively heavy.
Fig. 4(a) gives the tail distribution of delay. It shows the
probability that delay is greater than a given value. It can
be seen that the queue-aware strategy is better in the tail
distribution of delay than the load-aware strategy. In Fig. 4(b),
we compare the robustness to system parameter variations of
the two strategies. The real arrival rate is λ= 3flows/sec,
when the arrival rate is over-estimated or under-estimated, the
optimal control rate will be calculated according to the mis-
estimated arrival rate. From this figure, it can be seen that the
load-aware strategy greatly deviates from the optimal value,
and the queue-aware strategy has much better robustness.
IV. TRA FFIC -AWARE P OWE R ADAPTATI ON W IT H BS S LE EP
In this section, we will take practical concerns into consid-
eration and use the energy model given in section II-B with
both BS sleep and switching cost taken into account. Since the
performance of the load-aware strategy is almost the same as
the queue-aware strategy from foregoing analysis, in order to
characterize our problem explicitly, we will restrict attention
to load-aware analysis in this part.
0 5 10 15 20 25 30
10−5
10−4
10−3
10−2
10−1
100
t
P(Delay≥ t)
load−aware
queue−aware
(a)
0 1 2 3 4 5 6 7
2
4
6
8
10
12
14
16
18
Estimated λ
Optimal system cost: z
queue−aware
load−aware
(b)
Fig. 4. (a) The tail distribution of delay, λ= 3flows/sec. (b) The robustness
comparison: the real λ= 3flows/sec, and the control rate is solved using the
estimated arrival rate.
The total power consumption consists of three parts as in
equation (15). The first part is the power consumption in active
mode, the second part is the sleep mode power consumption,
and the last part gives the switching energy cost per unit time.
Ptot=pactive(Ps+∆PPt)+psleep Psleep +2Esw /E(Tc),(15)
pactive and psleep is the fraction of time being in active
mode and sleep mode respectively. For the simplicity of the
demonstration later, random variable Tcis defined as the cycle
time to be the sum of two consecutive active period Taand
sleep period Ts.
First we illuminate the basic strategy extended from sec-
tion III-A, then combining the load-aware power adaptation
we analyze two threshold-based BS sleep strategies, “N-based”
and “V-based” strategies, which correspond to “waiting for N
users” and “waiting V deterministic period of vacation time”
before waking from sleep respectively.
A. Basic Strategy
In the basic strategy, assume the BS goes to sleep when
there is no user in the system and returns to active mode as
soon as there is a user arrival. In this situation, Tais the busy
period and Tsis the idle period with E{Ta}= 1/(x/l −λ)
and E{Ts}= 1/λ respectively. We can get E{Tc}= 1/[(1 −
λl/x)λ]. The average delay Dband total power consumption
Ptot
bis given below.
Db=l
x−λl ,(16)
Ptot
b=λl
x(1
γ(2 x
w−1)∆P+Ps
)+(1−λl
x)Psleep+2Esw
(1−λl
x)λ. (17)
3174
10−1 100101
80
90
100
110
120
130
140
average delay[sec]
total power consumption[W]
λ=2,l=2MB
λ=1,l=2MB
λ=1,l=8MB
Fig. 5. Example of Proposition 1, the tradeoff of total power consumption
and average delay, Ps= 100W, Psleep = 30W, Esw = 25J, ∆P= 7 [1].
Exploring the relationship between the total power con-
sumption and the average delay, we get the following propo-
sition, and the proof is omitted due to space limitations.
Proposition 1: 1. The total power consumption Ptot
b(Db)
is monotonously decreasing with the average delay Db, when
either one of the following conditions is satisfied.
i)λ< Ps
−Psleep
2Esw , l ≥w
λln 2 W
γ
∆Pe(Ps−Psleep−2λEsw)−1
e+1
.
ii)λ≥Ps
−Psleep
2Esw .
2. There exists the energy-optimal rate x∗
ewhen the following
condition is satisfied,
i)λ< Ps
−Psleep
2Esw , l < w
λln 2 W
γ
∆Pe(Ps−Psleep−2λEsw)−1
e+1
.
And the the energy-optimal rate is
x∗
e=w
ln 2 W
γ
∆Pe(Ps−Psleep −2λEsw)−1
e+1
.
3. In both of the upper two cases, as delay goes to infinity, the
total power consumption is bounded by 1
γ(2λl
w−1)∆P+Ps.
Remark: The property of the tradeoff line between the
total power consumption and the average delay depends on
the relationship of traffic parameters, system parameters and
power consumption parameters. For the case there exists the
energy-optimal transmit rate, only in the rate region [x∗
e,∞),
delay can be traded off with energy, otherwise, increasing
delay will only cause bad energy performance. Interestingly,
x∗
eis an increasing function of γ, so transmitting faster
when channels are good indeed saves energy. In addition, fast
transmission are beneficial when the gap between static power
consumption Psand sleep mode power consumption Psleep
is high; otherwise large busy probability will consume too
much static energy. As the delay goes to infinity, the bound is
the total power consumption when the system will always be
in active mode with system utilization λl
xgoes to 1. Fig. 5
shows one example of Proposition 1 where the green and
blue line corresponds to 1.(i) and 1.(ii) respectively and the
red line shows case 2 with energy-optimal rate. The energy
consumption parameters of a micro BS in [1] are adopted.
B. Threshold-Based BS Sleep Strategies
1) N-based sleep strategy: Assume the BS goes to sleep
when there is no user in the system and returns to active mode
until the user number increases to Nfrom zero. Using an
extended-Markov-chain given in Fig. 6, the static probability
1111
Fig. 6. The 2-D state transition graph for the N-based sleep strategy.
distribution is given below. Here we define an extended state
space {(i, j) : i= 0, j = 0,1, . . . , N −1; i= 1, j = 1,2, . . .}
such that if i= 0 then jdenotes the number of users in the
system when it’s in sleep mode, and if i= 1 then jcounts
the number of users in the system when it’s in active mode.
P(i, j)=
x−λl
Nx if i= 0;
λl
Nx (1 −(λl
x)j)if i=1,1≤j≤N;
λl
Nx (( λl
x)j−N−(λl
x)j)if i=1, j > N.
(18)
The fraction of time in sleep mode is j=N−1
j=0 P(i=0, j )=1−λl
x,
which is the same with the idle probability in the basic case.
Tsstarts at the moment the BS goes to sleep and last
until Nusers have assembled. The average assembling time
is E{Ts}=N/λ. At the beginning of Tathere are Nusers
in the system, thus E{Ta}=N/(x/l −λ). Then we get the
average delay DNand total power consumption Ptot
N.
DN=1
λ(λl
x−λl +N−1
2)>N−1
2λ,(19)
Ptot
N=λl
x(1
γ(2 x
w
−1)∆P+Ps
)+(1−λl
x)Psleep+2Esw
(1−λl
x)λ
N.(20)
To minimize the objective zN=λDN+Ptot
N/β, we get
the optimal rate x∗and threshold N∗by taking ∂ zN
∂x = 0 and
∂zN
∂N = 0 respectively. x∗can be got by solving equation (21).
W
γβ
∆Pe(x∗
x∗−λl )2
+γ
∆Pe(Ps
−Psleep−2Eswλ
N)−1
e=ln 2
wx∗
−1,
(21)
N∗=4Eswλ
β(1 −λl
x)1/2.(22)
It can be seen from equation (22) that the optimal threshold
N∗is related to the switching cost and the system idle
probability in a square root form, which is consistent with
the result derived by Heymen [12] because Nonly affects the
average delay and the switching power cost in the objective.
The optimal threshold should be an integer, and is the one
chosen from {⌊N∗⌋,⌈N∗⌉} which minimizes zN. Actually,
the basic strategy is a special case of N=1. For Ptot
N(DN),
similar results of Proposition 1 can be obtained.
2) V-based sleep strategy: In practical operation, waiting a
deterministic period of time is preferred due to the convenience
of operation. Assume that once the BS goes to sleep, it will
be asleep for a period of time and then wake up no matter
whether there are users in the system or not. Using the vacation
model given in [19], assume that the vacation duration Vis
a random variable. When deterministic vacation is applied,
E(V)=v, E (V2)= v2. The fraction of time the BS spends on
vacation is pv= (1−λl
x)λv
λv+e−λv . The average cycle time is
3175
0
5
10
x 107
0
5
10
40
60
80
100
120
rate
N−based sleep strategy
threshold N
system cost: zN
0
5
10
x 107
0
5
10
15
40
60
80
100
120
140
rate
V−based sleep strategy
threshold V
system cost: zv
Fig. 7. The system cost of the two threshold-based sleep strategies varying
with the rate and threshold. λ= 0.5flows/sec, β= 1, Ps= 100W, Psleep =
30W, Esw = 25J, ∆P= 7 [1].
E(Tc)=( 1
1−λl/x )(e−λv
λ+v). Then the total power consumption
Ptot
vand delay Dvare obtained as follows.
Dv
=1
λ(λl
x−λl +λ2v2
2(λv+e−λv))>λv2
2(λv+e−λv),(23)
Ptot
v=(1−pv
)(1
γ(2 x
w−1)∆P+Ps
)+pv
Psleep+(1−λl
x)2Eswλ
e−λv
+λv .(24)
To minimize the objective zv=λDv+Ptot
v/β, we get
the optimal rate x∗and threshold v∗by taking ∂zv
∂x = 0 and
∂zv
∂v = 0. For the optimal threshold v∗, we show that zv(v)
assumes its minimum in the interval (0,∞).
zv|v→0→λl
x−λl +1
β1
γ(2 x
w−1)∆P+Ps+2Esw (1−λl
x)λ,
∂zv
∂v |v=0=λ
β(1−λl
x)(Psleep−1
γ(2 x
w−1)∆P−Ps
)<0, zv
|v→∞→∞,
zv(v)is finite at the origin, decreasing at the neighborhood of
zero and goes to infinity as v→ ∞. Hence, there must exist
at least one point v∗∈(0,∞)for which zv(v∗)is minimum.
Actually, v∗can be calculated numerically easily by solving
the following equation (25). Similarly x∗can also be got which
is omitted here.
(λv∗)2
2−(A−1)λv∗−(A−B) = eλv∗(B−(λv∗)2
2),(25)
where A=1
β(1−λl
x)( 1
γ(2 x
w
−1)∆P+Ps
−Psleep )and B=2Eswλ
β(1−
λl
x).
Fig. 7 gives an example of the two threshold-based sleep
strategies where the system cost varying with the rate and
threshold. Practically, based on our analysis of different strate-
gies above such as their total power consumption and delay
performance, with different objectives concerned, different
operation parameters can be designed accordingly.
V. CONCLUSION
In this article, we have studied traffic-aware power adap-
tation and base station sleep control with flow-level traffic
dynamics in green cellular networks. We formulate a total
cost minimization problem that allows for a flexible tradeoff
between energy consumption and flow-level delay perfor-
mance. Load-aware and queue-aware power adaptation strate-
gies are proposed. Especially for the queue-aware strategy,
tight bounds of the optimal solution are given. Simulation
results show that the two strategies perform closely in terms
of energy consumption and average delay, while the queue-
aware strategy is better in the tail distribution of delay and
the robustness to system parameter variations. For the power
adaptation with BS sleep, the explicit tradeoff relationships
between energy and delay are investigated. We find that
sacrificing delay performance cannot always be traded for
energy saving, and there exists energy-optimal rate under
certain conditions. Two threshold-based BS sleep strategies
are analyzed and the optimal threshold and rate are derived
respectively, which are amenable to energy efficient power
adaptation and BS sleep control design. Further analysis and
implementation issues for these sleep strategies are left for the
future work.
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