Grid Energy Consumption and QoS Tradeoff in Hybrid Energy Supply Wireless Networks

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by  Yuyi Mao

1
Grid Energy Consumption and QoS Tradeoff in Hybrid
Energy Supply Wireless Networks
Yuyi Mao, Student Member, IEEE, Jun Zhang, Senior Member, IEEE, and Khaled B. Letaief, Fellow, IEEE
Abstract—Hybridenergysupply(HES)wirelessnetworkshave radio signal, wind energy, etc [3], and thus it is becoming
recently emerged as a new paradigm to enable green networks, an indispensable solution for green communication systems
which are powered by both the electric grid and harvested
[4]. By introducing EH techniques to the next-generation
renewable energy. In this paper, we will investigate two critical
6 but conflicting design objectives of HES networks, i.e., the grid communication systems, it is estimated that a 20% CO2
1 energy consumption and quality of service (QoS). Minimizing emission reduction can be achieved [5].
0 grid energy consumption by utilizing the harvested energy will Dueto the intermittentnatureofthe EH processes,conven-
2 make the network environmentally friendly, but the achievable tionalcommunicationprotocolscannottakefulladvantagesof
QoS may be degraded due to the intermittent nature of energy
n the harvested energy. Consequently, transmission policies for
harvesting.Toinvestigatethetradeoffbetweenthesetwoaspects,
a communicationsystemswithEHcapabilitieshavebeenwidely
J we introduce the total service cost as the performance metric,
which is the weighted sum of the grid energy cost and the QoS investigated. The optimal transmission policy for point-to-
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degradation cost. Base station assignment and power control is pointEHsystemswithoff-linesideinformation(SI),including
2
adopted as the main strategy to minimize the total service cost, channel SI (CSI) and energy SI (ESI), was studied in [6],

while both cases with non-causal and causal side information
] which was shown to have a directionalwater-filling structure.
T are considered. With non-causal side information, a Greedy
The investigation has been later extended to scenarios with
Assignment algorithm with low complexity and near-optimal
I
. performance is proposed. With causal side information, the morepracticalassumptionsonSI[7]–[9],andtoothersystems,
s
design problem is formulated as a discrete Markov decision including broadcast channels [10] and multiple access chan-
c
[ problem. Interestingsolution structures are derived, which shall nels[11],aswellascooperativecommunicationsystems[12]–
help to develop an efficient monotone backward induction algo-
[16]. Moreover,jointenergy-bandwidthallocationpoliciesfor
1 rithm.Tofurtherreducecomplexity, aLook-Aheadpolicyanda
multi-user EH systems have been recently proposed in [17],
v Threshold-based Heuristic policy are also proposed. Simulation
5 resultsshallvalidatetheeffectivenessoftheproposedalgorithms [18].
6 and demonstrate the unique grid energy consumption and QoS Because of the innate characteristic of EH systems, it
8 tradeoff in HES networks. is difficult to maintain stable communication performance.
7
IndexTerms—Greencommunications,energyharvesting(EH), To get full benefits of the renewable energy sources while
0
hybrid energy supply(HES), power control, base station assign- guaranteeing the quality of service (QoS) requirement, the
.
1 ment, Markov decision process (MDP), QoS. electric grid can be retained as a backup energy source. As
0
a result,recently,hybridenergysupply(HES)communication
6
systems have attracted significant attentions, where EH and
1 I. INTRODUCTION
: electric grid coexist. The initial investigation started from
v A. Related Works and Motivations
point-to-point HES systems. Recursive expressions for the
i
X WITH the explosion of mobile data traffic in wireless powergrid energyconsumptionin HES systems were derived
r ecosystems, the energy consumption of the Informa- in [19]. Given the grid energy budget, a two-stage water-
a
tionandCommunicationTechnology(ICT)industryisbecom- fillingalgorithmwasproposedforthroughputmaximizationin
ing an economic and ecological issue. It is estimated that the [20], where the CSI and ESI was assumed to be non-causally
annual carbon emission and electric grid power consumption known at the transmitter. In [21], power allocation strategies
oftheICTindustrywillreachupto235Mto[1]and414TWh for weighted energy cost minimization were proposed. For
[2],respectively,in2020.Thus,itiscrucialtoseekalternative HESorthogonalfrequencydivisionmultipleaccess(OFDMA)
environmentallyfriendlyenergysourcesforwirelessnetworks. systems, a power and sub-carrier allocation policy to maxi-
Energy harvesting (EH) is a promising candidate, which can mize energy-efficiencywas developed in [22]. More recently,
capture ambient recyclable energy, including solar radiation, there have been substantialeffortson HES wireless networks.
Targeting on saving the grid energy consumption, the green
ManuscriptreceivedApril1,2015;revisedAugust4,2015andNovember energyutilizationoptimizationproblemwassolvedin[23].In
11, 2015; accepted January 16, 2016. The associate editor coordinating the
[24],jointoptimizationofsleepcontrolandresourceallocation
review of this paper and approving it for publication was Dr. Federico
Boccardi. This work was supported by the Hong Kong Research Grants for HES networks was investigated. For HES heterogeneous
Council underGrantNo.610212. networks, a green energy and latency aware user association
The authors are with the Department of Electronic and Computer Engi-
scheme was proposed in [25] to minimize the grid energy
neering, theHongKongUniversity ofScience andTechnology, Clear Water
Bay,Kowloon, HongKong(e-mail: {ymaoac, eejzhang, [email protected]}). consumption while achieving the required QoS. Besides, in
Khaled B. Letaief is also with Hamad bin Khalifa University, Doha, Qatar coordinated multi-point (CoMP) HES systems, joint energy
(e-mail: [email protected]).
cooperation and communication cooperation was proposed
This work was presented in part at IEEE Wireless Communications and
Networking Conference (WCNC),NewOrleans,LA,Mar.2015. in [26]. In addition, a low-complexity online base station
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assignment and power control algorithm based on Lyapunov tradeoff. This is innovative compared to existing works
optimization was developed in [27]. on HES systems, which aimed at minimizing the grid
Most of the existing works on HES systems assume that energy consumption under given QoS requirements.
both the harvested energy and grid energy are available at 3) We investigate the total service cost minimization
the same transmitter. However, considering practical imple- (TSCM) problem assuming either non-causal or causal
mentation issues, e.g. the high deployment cost of the power side information, which provides a viable approach to
cable and the increasing number of base stations (BSs), it balancethe gridenergyconsumptionandtheachievable
is difficult to connect all the BSs to the electric grid [28]. QoS, i.e., the number of dropped packets. Specifically:
Consequently,in futureHESnetworks,there willbea portion
• With non-causal CSI and ESI, i.e., the off-line
ofBSspoweredpurelybytheEHdevices,i.e.,withoutelectric
setting, the TSCM problemturnsoutto be a highly
cable connection [29]. Motivated by the greenness brought
difficultmixedintegernonlinearprogrammingprob-
by EH techniques, together with the constraints on the power
lem. By exploitingthe problem structure, we trans-
cable deployment, in this paper, we will investigate how to
form the TSCM problem into an equivalent zero-
coordinate a grid-powered BS (GP-BS) and an EH-BS to
oneintegerprogrammingproblem,which,however,
serve a mobile user. The number of transmitted data packets
is still NP-hard. Thus, a low-complexity Greedy
isusedasameasureoftheachievableQoS.Insuchnetworks,
Assignment algorithm is proposed, which is sub-
minimizing the grid energy consumption at the GP-BS and
optimal in general but achieves a close-to-optimal
maximizingthe QoSaretwo importantbutconflictingtargets.
performance. For two special cases, we show that
That is, in order to achieve a better QoS, more grid energy
the greedy algorithm is actually optimal. This not
will be consumed, and vice versa. Consequently, to balance
only offers key intuitions on developing the online
the grid energy consumption and the degradation of QoS,
algorithms with causal SI, but also facilitates the
i.e., the number of dropped data packets, it is of fundamental
evaluationoftheonlinealgorithmswithouttheneed
importance to investigate their tradeoff.
of calling for high-complexity solvers.
Different from previous studies with both grid energy and
• With causal CSI and ESI, i.e., the online setting,
harvestedenergyat the same BS, new design challengesarise
we formulate the TSCM problem into a discrete
intheHESnetworkconsideredinthispaper.Inparticular,both
Markov decision process (MDP) problem. Interest-
BS assignment, i.e., whether the GP-BS or the EH-BS should
ing monotone structures of the optimal policy are
be selected to serve the mobile user, and BS power control,
derived,whichcoincidewiththeintuitionsobtained
i.e., how to determine the transmit power of the picked BS in
from the Greedy Assignment algorithm and help to
each block, need to be optimized. Compared to conventional
develop an efficient monotone backward induction
wireless networks, where users are normally assigned to their
algorithm. In order to further reduce complexity, a
nearest BSs [30], the major design challenge in HES wireless
Look-AheadpolicyandaThreshold-basedHeuristic
networkscomesfromthesporadicenergyarrivalattheEH-BS.
policy are also proposed.
Thus, the BS assignment should depend on both the channel
4) Extensivesimulationresultsare providedtovalidatethe
condition and the EH profile. In addition, since the EH-BS
effectiveness of the proposed policies. In particular, we
does not have a grid energy supply, efficient utilization of
will show that the proposed Greedy Assignment algo-
the harvested energy should adapt to conditions of multiple
rithm achieves a near-optimal performance for the off-
channelsfromboththeEH-BSandGP-BS,whichcomplicates
linecase,whiletheThreshold-basedHeuristicpolicycan
the design compared to systems with co-located grid energy
approachthe discrete MDP solution for the onlinecase.
and harvested energy.
Moreover,thefundamentalgridenergyconsumptionand
QoS tradeoff in such networks is illustrated. Based on
B. Contributions the tradeoff curves, for a given QoS requirement, the
In this paper, we will study the grid energy consumption operatingpointtominimizethegridenergyconsumption
and QoS tradeoff in HES wireless networks. Our major can be determined via adjusting the weighting factor.
contributions are summarized as follows: Our results reveal that intelligent energy management
schemes that fully exploit the available SI play an
1) We consider an HES wireless network with a GP-
essential role in HES wireless networks.
BS and an EH-BS, which is motivated by practical
implementation considerations. Such a network model
has not been considered in previous works on HES
systems, which only assumed a single BS with co- C. Organization
located grid energy and harvested energy [20]–[26].
2) Duetothedifferenceinthenetworkmodel,thebasicde- Therestofthispaperisorganizedasfollows.We introduce
sign principle is different from previous studies. In par- the system model in Section II. In Section III, the TSCM
ticular,weproposeanovelperformancemetric,namely, problem with non-causal SI is formulated, for which an
thetotalservicecost,tooptimizetheHESnetwork.This efficient algorithm is developed. In Section IV, the case with
new metric helps to reveal a new design aspect of the causalSIisstudied.SimulationresultsarepresentedinSection
HESnetwork,i.e.,thegridenergyconsumptionandQoS V and we will conclude this paper in Section VI.
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II. SYSTEMMODEL The EH process is modeled as successive energy packet
A. Network Model and Energy Model arrivals, i.e., at the beginning of the i-th block, an energy
packet with E Joule arrives at the EH-BS and is available
H,i
for transmission from then on. Without loss of generality,
we assume E ’s are independent and identically distributed
H,i
(i.i.d.) within [0,E ] with probability density function(PDF)
m
f (e) and average harvesting power P . A battery with
EH avg
sufficiently large capacity is installed at the EH-BS. Since
the energy packets cannot be used before their arrivals, the
(cid:3)(cid:1)(cid:0)(cid:2) following energy causality constraint should be satisfied2:
(cid:11)(cid:12)(cid:13)(cid:14)(cid:15)(cid:16)(cid:17)(cid:18)(cid:16)(cid:19)
(cid:9)(cid:10)(cid:6)(cid:7)(cid:8)
i i
p τ ≤ E ,∀i∈N. (2)
H,l H,n
l=1 n=1
X X
(cid:4)(cid:5)(cid:6)(cid:7)(cid:8) Both path loss and small-scale fading are considered. The
channelsareassumedtobei.i.d.blockfading.Denoteγ and
G,i
Fig.1. AnHESnetworkwithanEH-BS,aGP-BSandamobileuser. γH,i as the small-scale fading channel gains of the GP-BS to
userchannel(calledG-channel)andtheEH-BStouserchannel
As shown in Fig. 1, we consider an HES wireless network (called H-channel) in the i-th block, respectively. Thus, the
with an EH-BS, a GP-BS and a mobile user. The GP-BS, channel gains of the G-channel and H-channel are given by
which can be a macro-BS to maintain network coverage, is h =g d−θγ and h =g d−θγ , respectively, where
G,i 0 G G,i H,i 0 H H,i
poweredby the electric grid and its maximumtransmitpower g is the path loss constant, θ is the path loss exponent, d
0 G
is denoted as pmax. The EH-BS is equipped with an EH and d are the distances from the GP-BS to the user and
G H
componentand powered purely by the harvested energy, with from the EH-BS to the user, respectively. In the i-th block,
themaximumtransmitpoweraspmax.ThisEH-BSmayserve when the GP-BS is assigned to serve the user, the amount
H
asasmallcelltoincreasesystemcapacity[29].ThesetwoBSs of bits that can be transmitted is given by r(p ,h ).
G,i G,i
willcoordinateto servetheconsidereduser. Suchdeployment Similarly, when the EH-BS is selected, r(p ,h ) bits can
H,i H,i
of HES networks is practically attractive for occasions where be delivered. Here, r(p,h) = τW log 1+hσ−2p is the
2
the electric grid is not easily accessible. We consider an N- Shannon-Hartley formula, where W is the system bandwidth
(cid:0) (cid:1)
blockdownlinktransmissionframewithblocklengthτ,where and σ2 is the noise variance at the receiver.
the user can be servedby atmostone BS in each block1. The
block index set is denoted as N ={1,2,...,N}.
B. Performance Metric
We assume a packet with R bits arriving at the beginning
In most previous studies on HES communication systems,
ofeachblock,anditmaybetransmittedviatheEH-BSorthe
minimizing the grid energy consumption while guaranteeing
GP-BSinthefollowingblock.Itmayalsohappenthatneither
certainQoSrequirementisconsideredasafundamentaldesign
of the BSs will serve the user. The number of transmitted
objective[20],[21],[25],[36].However,withpracticalconsid-
packetsis usedtomeasurethe achievableQoS[35],while BS
erations as mentioned before, it may happen that some data
assignment and power control is adopted to optimize the sys-
packets could not be transmitted, i.e., the QoS requirement
tem.TheGP-BSwill collectalltherequiredsideinformation,
cannot be satisfied. For instance, the EH-BS may not have
and make the transmission decision. In this paper, the delay
enough energy while the channel from the GP-BS is in deep
incurred by signalling and decision making is neglected for
fading.Inthiscase,neitheroftheBSsiscapableofdelivering
simplicity. Denote I ∈ {0,1}, with j ∈ {G,H,D}, as the
j,i
the data packet. In some real-time applications, this packet
BS assignment indicators in the i-th block, where I = 1
G,i
may indeed be dropped, while for other applications, this
(I = 1) indicates that the GP-BS (EH-BS) is assigned to
H,i
will increase the transmission delay. To take this aspect into
deliver the packet, and I = 1 represents the event that
D,i
consideration, we assume that a unit of cost will be induced
neitheroftheBSsisselected,i.e.,thepacketisdropped.These
once a packet is dropped. On the other hand, due to the
indicators are subject to the following operation constraint:
intermittentandsporadicnature ofthe EH processofthe EH-
IG,i+IH,i+ID,i =1,∀i∈N. (1) BS, the GP-BS needs to serve the user from time to time,
which incurs one unit of cost of grid energy consumptionper
In addition, the transmit power of the GP-BS and EH-BS in
Joule. Hence, it is desirable to minimize both costs from the
thei-thblockisdenotedasp andp , respectively.Inthis
G,i H,i
systemdesignperspective,which,unfortunately,isimpossible
work, energy consumed for purposesother than transmission,
ingeneral.Thus,thetradeoffbetweentheseconflictingtargets
e.g. cooling and baseband signal processing, is neglected.
is of particular interest. To characterize this tradeoff, we
1In this paper, we will adopt BS assignment to optimize the network,
i.e., the user is served by one of the BSs at each time, which is a widely- 2One interesting extension is to incorporate the wireless backhaul energy
usedtransmissionschemeforcellular networks [30],[31].Inprinciple, joint consumption at the EH-BS, which can be handled if the backhaul energy
transmission schemes, i.e.,theBSs transmit simultaneously, can beusedfor consumptionateachblockisknown.Inthiscase,‘pH,l’in(2)canberegarded
HESnetworks[32].However,itwillcausesignificantfeedbackoverheadand asthetotalpowerconsumptionofboththedownlinktransmissionandwireless
havemorestringent requirement onsynchronization [33],[34]. backhaul inthel-thblock.
4
introducethe totalservice cost overthe N-blocktransmission form of the multi-dimensional Knapsack problem [37], as
frame as the performance metric, which is the weighted sum shown in Lemma 1.
of thegrid energycostandthe packetdropcost, with weights Lemma 1: P can be equivalently transformed into the
1
w andw , respectively.Specifically,thetotalservicecostis following IP problem:
G D
defined as
N
N Pˆ :min (1−α )c (9)
1 n n
TSC, wGpG,iτ +wDID,i, (3) αi n=1
X
i=1 i i
X
wherewGpG,iτ andwDID,i aretheweightedgridenergycost s.t. αkpiHn,vkτ ≤ EH,l,∀i∈N (10)
and the packet drop cost in the i-th block, respectively. It is Xk=1 Xl=1
worthwhileto note that, althoughthe costs of per packetdrop αipiHn,vi ≤pmHax,∀i∈N (11)
and per Joule of grid energy consumption are assumed to be α ∈{0,1},∀i∈N, (12)
i
normalized in this paper, we can also handle non-normalized
values by embodying them into the weights wD and wG, where piHn,vi is the H-channel inversion power, i.e., piHn,vi =
respectively. 2WRτ −1 σ2h−H1,i. {ci} is a series of cost parameters given
Remark 1: Byadjustingtheweightofthegridenergycost, (cid:16)by (cid:17)
w , and the weight of the packet drop cost, w , we can
G D
achieve different grid energy consumption and QoS tradeoffs c = wD, piGn,vi >min{pmGax,wD(wGτ)−1} (13)
in such networks. More specifically, when wG ≫ wD, the i (wGpiGn,viτ, piGn,vi ≤min{pmGax,wD(wGτ)−1},
network is grid energy sensitive. On the other hand, when
wD ≫ wG, the network emphasizes more on the successful where piGn,vi is the G-channel inversion power, i.e., piGn,vi =
packet delivery, i.e., addresses more on QoS. 2WRτ −1 σ2h−G1,i. Given a solution {αi} to Pˆ1, the corre-
Inthefollowingtwosections,wewillinvestigatetheTSCM s(cid:16)ponding s(cid:17)olution to P1 is given as
problems under non-causal and causal SI settings, which will
reveal the tradeoff between the grid energy consumption and IH,i =αi
the achievable QoS. I =(1−α )1{pinv ≤κ}
G,i i G,i (14)
I =1−I −I
D,i G,i H,i
III. TOTALSERVICE COST MINIMIZATION WITH p =I pinv,j ∈{G,H},
j,i j,i j,i
NON-CAUSAL SIDE INFORMATION
whereκ,min{pmax,w (w τ)−1}and1{·}istheindicator
In this section, we will first solve the off-line TSCM G D G
function.Furthermore,if {α } isoptimalfor Pˆ , {I }, {p }
problem, i.e., assuming {E }, {h } and {h } are non- i 1 i H,i
H,i G,i H,i
and {p } are optimal for P .
causally known. This will provideinsights for designing such G,i 1
Proof: See Appendix A.
systems, as well as serving as a performanceupperboundfor
the online case. Lemma 1 shows that transmitting the packet with the
Denote I , [I ,I ,I ]. The off-line TSCM problem channelinversionpowerwillnotaffecttheoptimality.Tosolve
is then formiulatedG,ais H,i D,i Pˆ1, we only need to decide when to assign the EH-BS, i.e.,
{α }. If the EH-BS is not assigned to serve the user in a
i
N
block, whether to drop the packet or to transmit it with the
P : min w p τ +w I (4)
1 Ii,pj,in=1 G G,n D D,n GP-BS only depends on the relationship between piGn,vi, pmGax,
X w and w . These problem structures help reduce the size
s.t. (1) and (2) (5) G D
of the optimizationvariables.However,Pˆ is still an NP-hard
1
Ij,ir(pj,i,hj,i)≥(1−ID,i)R,∀i∈N (6) problem, as shown in the following corollary.
j∈{XG,H} Corollary 1: Pˆ1 is an NP-hard problem.
p ≤pmax,∀i∈N,j ∈{G,H} (7) Proof: Suppose there is a polynomial-time algorithm to
j,i j
solve Pˆ , then there exists a polynomial-time algorithm to
I ∈{0,1},∀i∈N,j ∈{G,H,D}. (8) 1
j,i
solve the following Knapsack problem:
In P , the objective is the total service cost over the N-block
1 N
frame. (6) is the QoS constraint, i.e., when it is decided to
min (1−y )d
n n
transmit the packet in the i-th block, the channel capacity yi
n=1
should be greater than the packet size. (7) and (8) are the X
N
peak power constraint and the zero-one indicator constraint,
s.t. y f ≤e
k k
respectively.
k=1
X
In general, P1 is a mixed integer nonlinear programming yi ∈{0,1},∀i∈N,
(MINLP) problem with continuous variables p together
j,i
with binary variables I , which is very difficult to solve. By withe,d ,f ≥0,∀i∈N,whichisaspecialcaseofPˆ when
i i i 1
exploiting the problem structure, we manage to transform P E =0,∀i∈N \{1}. Since the Knapsack problem is NP-
1 H,i
into an integer programming(IP) problem, which is a special hard [38], i.e., there is no polynomial-time algorithm for the
5
optimal solution, the assumption is violated. Therefore, Pˆ is explicitly written as
1
also an NP-hard problem.
N =
feas
Though commercial solvers, e.g. MOSEK [39], can be
n n
appliedtocomputetheoptimalsolutionforPˆ1,theyareknown i i∈NHc,piHn,vi≤pmHax, α′kpiHn,vkτ≤ EH,l,∀n∈N ,
to have a worst-case exponential complexity and cannot pro- (cid:26) (cid:12) k=1 l=1 (cid:27)
vide insights on the obtained solution. Thus, low-complexity (cid:12)(cid:12) X X (15)
(cid:12)
algorithmsare needed.To assist thealgorithmdesign,we first where α′ = α+e and I = 1{N 6= ∅}. In each loop,
i feas
provide the following property of the optimal solution.
the block from N with the highest value of Ξ c ,pinv is
feas i H,i
Proposition 1: If {αi} is optimal to Pˆ1, then there exists selected as the H-block. Key properties of the GA algorithm
no j >i, such that αi =1, αj =0, ci <cj and piHn,vi ≥piHn,vj. are listed as follows: (cid:0) (cid:1)
1) As an H-block is selected in each while loop, the GA
Proof: Suppose {α } is optimal and there exists j > i, algorithm will be terminated within at most N loops.
i
such that α = 1, α = 0, c < c and pinv ≥ pinv. It is In each loop, the block with the maximum value of the
i j i j H,i H,j
always possible to construct a new solution by setting α = metric function will be identified as an H-block and the
i
0 and α = 1. With the constructed solution, the value of blocks that are still feasible will be found. Thus, the
j
the objective function will decrease by c − c . Hence, we complexity of the GA algorithm is O N2 .
j i
can conclude that {α } is not optimal, which contradicts the 2) FromextensivesimulationresultsprovidedinSectionV,
i
(cid:0) (cid:1)
assumption. the GA algorithm achieves near-optimal performance.
We call the block in which the EH-BS is assigned to serve 3) When hH,i =hH or hG,i =hG,∀i∈N, i.e., whenone
the user as an H-block, i.e., α = 1. From Proposition 1, ofthechannelsisconstant,theGAalgorithmisoptimal.
i
the H-blocks should have higher values of c and h . The The proof is given in Appendix B.4
i H,i
intuitionisthat,we canreducemoreservicecostwitha given The off-line formulation requires full SI, which is too
amountofharvestedenergy.Inthefollowing,wewillpropose restrictive and thus impractical. In the next section, we will
the Greedy Assignment (GA) algorithm to solve Pˆ , which turnourattentiontotheonlinecase,whereSIisonlycausally
1
identifiestheH-blocksgreedilyaccordingtoametricfunction known. As will be seen latter, the intuition obtained from the
Ξ(p,q) : R2 → R , which is non-decreasing with p for a GA algorithm,i.e.,theEH-BSshouldservethe userwhenthe
+ +
given q, and non-increasing with q for a given p. With the H-channelconditionis goodwhile the G-channelconditionis
assistance ofthismetricfunction,thesolutionobtainedbythe poor, matches the structural properties of the optimal online
GAalgorithmwillsatisfyProposition1automatically.TheGA solution and helps to develop a low-complexity suboptimal
algorithm is summarized in Algorithm 1. online policy.
IV. TOTALSERVICE COST MINIMIZATION WITH CAUSAL
Algorithm 1 Greedy Assignment Algorithm.
SIDE INFORMATION
Input: EH ={EH,i}, c={ci}, {piHn,vi}, pmHax, τ and N In this section, we focus on the online case, where the EH
Output: α={α }, and cGA
i Σ profile and channel states are causally known. Consequently,
1: Initialize α=0, NHc ={1,2,...,N}, Nfeas =∅; weareinterestedinminimizingtheexpectedtotalservicecost.
2: [I,Nfeas]=find feas EH,α,{piHn,vi},pmHax,τ,NHc ; Thus, the online TSCM problem can be formulated as
3: While I ==1 do
45:: pαic=k mα,+suecmh;that m(cid:0)=argi∈mNafexasΞ(cid:0)ci,piHn,vi(cid:1); (cid:1) P2 :πm∈iΠnEπ"nN=1(1−αn)cn#, (16)
76:: [NI,HNc f=easN]=Hcfi\n{dmf}e;as EH,α,{piHn,vi},pmHax,τ,NHc ; lwatheerr)e, πΠ dreenporetesesnatsfethaesibsleetXpooflicayll(wfehaiscibhlewipllolbiceiessp,eccifieids
8: End while n
9: cGA =(1−α)Tc; (cid:0) (cid:1) definedin (13) and the expectationis overall the randomness
10: RΣeturn α={αi}, cGΣA. ibnevoolbvteadin.eIdnpbryinficnipitlee-,htohreizoopntimdyanlasmoliuctipornogorfatmhimspinrgob(lDemP)caaln-
gorithms,which,however,sufferfromthecurseofdimension-
ality. To avoid the high computational complexity of the DP
In Algorithm 13, we initialize α by setting all blocks not
solutions,wewillreformulatetheonlineoptimizationproblem
to be H-blocks. N is the set of blocks that have not yet
Hc as a discrete Markovdecisionprocess(MDP) problem,where
been selected as H-blocks. The function find feas(·) finds
thesystemisdescribedbyafinitenumberofstatesincontrast
the block from N , which is feasible if it is selected as an
Hc to DP.
H-block given the current α, and returns the feasible block
index set Nfeas together with an indicator I, which can be 4Although the proposed low-complexity GA algorithm is only optimal in
some special cases, i.e., it is not guaranteed to be a performance upper
bound, its close-to-optimal performance (which can be observed from the
simulationresultsinSectionV)demonstratesitscapabilitytobenchmarkthe
3y={yi}represents anN×1columnvector,em isabasicvectorwith performance achieved by the online algorithms without the need of calling
them-thentryequalto1,and∅istheemptyset. forhigh-complexity commercial solvers.
6
A. A Discrete MDP Approach wherethe expectationis overall possiblestate sequencess=
Denote the battery capacity as Bm. The MDP formulation {s1,...,sN}inducedbyπ.Theoptimalpolicyandtheoptimal
consists of the following components: expected total service cost can be expressed as
as1s)i,Syisstreemprsetsaetnet:eTdhbeysaysttreimplestt,ait.ee.i,n the i-thblock,denoted π∗ =argπm∈iΠncπΣ(s1) and c∗Σ(s1)=πm∈iΠncπΣ(s1), (22)
s ,hǫ ,γ ,γ i, (17) respectively. The cost-to-go function in the i-th block, which
i i G,i H,i
is the sum of the expected total service cost from the i-th
where ǫ is the battery energy state. We partition the battery
i block to the last block, is given by
capacity into M intervals with equal lengths, i.e., [0,B ] =
m uπ(s )=
[0,Bm/M)∪...∪[(M −1)Bm/M,Bm], and each of them i i
is an energy state represented by its mid-value, i.e., ǫi ∈ c(si,dπi (si))+ p(s′|si,dπi (si))uπi+1(s′), i<N
{chBamnn/e2lMg,a.in..i,s(2qMuan−tiz1e)dBimnt/o2MK}.noTnh-oevesmrlaapllp-isncgaleintfeardvianlgs c(sN,dπN(sN)),Ps′ i=N.
using the equi-probable steady state method [40], and each (23)
interval is a channel state represented by the correspond- 
According to the Principle of Optimality, the optimal cost-to-
ing mean value H , i.e., γ ,γ ∈ {H ,...,H } and
k G,i H,i 1 K go functionsshouldsatisfy the following Bellman’s equations
Pr{γ =H }=1/K.
j,i k [41]:
2)Actionspace:Theactioninthe MDPproblemis α , and
i
the action space is {0,1}. The allowable action space As , is u∗i (si)=
i
the feasible action set in the state s , i.e.,
i min c(s ,α )+ p(s′|s ,α )u∗ (s′) , i<N
i i i i i+1
Asi =({{00,}1,}, optiHnh,vier>wimsei,n{ǫi/τ,pmHax} (18) ααNi∈m∈AiAnssiN(cid:26)c(sN,αN),Ps′ (cid:27) i=N.
which indicates that the EH-BS can transmit only when the  (24)
harvestedenergyissufficientandthepeakpowerconstraintis The optimal policy π∗ of this MDP problem can be obtained
not violated as well. by solving (24) recursively with the backward induction al-
3) State transition probability: The state transition proba- gorithm (BIA) [41], in which, the cost-to-go functions are
bility, denoted as p(si+1|si,αi), is the probability that the evaluatedover all possible states in each block. Next, we will
system willbein state si+1 in the (i+1)-thblock,giventhat derive interesting monotone structures of π∗, which will help
in the i-th block it was in state si and action αi was taken. us design computationally efficient algorithms by reducing
Since the i.i.d. block fading channel is quantized into equal such evaluations.Firstly, π∗ is monotoneoverthe small-scale
probability states, we have fading of the G-channel, as specified in Proposition 2.
p(si+1|si,αi) Proposition 2: Foragivenbatteryenergystateǫi,iftheH-
=Pr{γ =H }·Pr{γ =H }·p(ǫ |s ,α ) channel is in state γH,i, then the optimal policy is monotone
G,i+1 m H,i+1 n i+1 i i
over γ . Specifically, the EH-BS will be assigned to serve
=K−2·p(ǫ |s ,α ),∀m,n=1,··· ,K, G,i
i+1 i i the user, i.e., α =1, only whenγ ≤Γ (ǫ ,γ ), where
i G,i G,i i H,i
(19)
Γ (ǫ ,γ ) is the threshold G-channel state.
G,i i H,i
where p(ǫ |s ,α ) is the energy state transition prob- Proof: Suppose γ+ ≥ γ− , and denote s+ =
i+1 i i G,i G,i i
ability. The energy state evolves according to ǫi+1 = hǫi,γG+,i,γH,iiands−i =hǫi,γG−,i,γH,ii.ToproveProposition
Q ǫi−αipiHn,viτ +EH,i+1 , where 2, it suffices to show if d∗i s+i = 1, then d∗i s−i = 1.
For i = N, this result is straightforward. For i < N, since
Q(cid:0)(ε) (cid:1) d∗ s+ =1, according to (2(cid:0)4), (cid:1) (cid:0) (cid:1)
Mmin{ε,B } B (20) i i
m m
= 2min +1,M −1 · ,
(cid:18) (cid:26)(cid:22) Bm (cid:23) (cid:27) (cid:19) 2M (cid:0) (cid:1) c s+i ,1 + p s′|s+i ,1 u∗i+1(s′)
and ⌊x⌋ is the floor function. Q(ε) is a quantization func- (cid:0) (cid:1) Xs′ (cid:0) (cid:1) (25)
tion mapping ε to the battery energy states, which is non- ≤c s+,0 + p s′|s+,0 u∗ (s′).
i i i+1
decreasing over ε. s′
(cid:0) (cid:1) X (cid:0) (cid:1)
4)Costfunction,decisionruleandpolicy:Thecostfunction Since c s+,1 = c s−,1 = 0, c s+,0 ≤ c s−,0 and
isc(si,αi)=(1−αi)ci,whereci dependsonsi asgivenby p s|s+,αi =p s|s−,iα ,∀α,s, i i
(13).Thedecisionruleisamappingfromthestatespacetothe i (cid:0) (cid:1) i(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
caicstiioonnsrpualecse,foi.re.e,adcih:bslio→ck,αi.ie..A, πpo=lic{ydππ1 (iss1a)s,e.q..u,ednπNce(soNfd)e}-. (cid:0) c(cid:1)(cid:0)s−i ,(cid:0)1(cid:1)+Xs′(cid:1)p(cid:0)s′|s−i ,1(cid:1)u∗i+1(s′) (26)
The set of all admissible policy is denoted as Π. If the state ≤c s−,0 + p s′|s−,0 u∗ (s′),
in the first block is s , the expected total service cost can be i i i+1
1 s′
written as (cid:0) (cid:1) X (cid:0) (cid:1)
i.e., d∗ s− =1, which ends the proof.
N i i
cπ(s )=Eπ c(s ,dπ(s )) s , (21) Thefollowingtwolemmaswillhelptoderivethemonotone
Σ 1 s "n=1 n n n (cid:12) 1# structur(cid:0)e of(cid:1)π∗ over the H-channel state γH,i.
X (cid:12)
(cid:12)
(cid:12)
7
Lemma 2: Suppose f(ǫ) is a non-increasing function of since EH-BStransmitswith lowerpowerwhenγ is higher.
H,i
the energy state ǫ. If 0 ≤ u ≤ u ≤ ǫ , then Hence,
1 2 0
p(ǫ |ǫ ,u )f(ǫ ) ≤ p(ǫ |ǫ ,u )f(ǫ ), where
ǫ1 1 0 1 1 ǫ2 2 0 2 2 c s+,1 +K−2 p ǫ′|s+,1 uˆ∗ (ǫ′)
p(ǫ |ǫ ,u )istheprobabilitythatthebatterywillbeinenergy i i i+1
n 0 n
sPtateǫn inthenextblockgivenPthecurrentenergystateǫ0 and (cid:0) (cid:1) Xǫ′ (cid:0) (cid:1) (31)
the amountof harvested energy(un) consumedin the current ≤c s+i ,0 +K−2 p ǫ′|s+i ,0 uˆ∗i+1(ǫ′),
block (n=1,2). (cid:0) (cid:1) Xǫ′ (cid:0) (cid:1)
Proof: The proof can be obtained by rewriting i.e., d∗ s+ =1, which ends the proof.
i i
ǫnp(ǫn|ǫ0,un)f(ǫn) as Acco(cid:0)rdin(cid:1)g to Proposition 2 and Proposition 3, if
d∗(hǫ ,γ ,γ i) = 1, then d∗ hǫ ,γ′ ,γ′ i =
P i i G,i H,i i i G,i H,i
f (e)f(Q(ǫ −u +e))de (27) 1,∀γ′ ≤γ ,γ′ ≥γ . These propositionsalso indicate
EH 0 n G,i G,i H,i H,i (cid:0) (cid:1)
Ze that,itwouldbemorebeneficialtoassigntheEH-BStoserve
forn=1,2andutilizingthenon-decreasingpropertyofQ(ε). the user when the H-channelis in a good condition while the
G-channelispoor.Interestingly,thisconclusioncoincideswith
Define uˆ∗(ǫ) , u∗(hǫ,γ ,γ i), which can be the intuitions we found while developing the GA algorithm.
i i G,i H,i
γG,i,γH,i That is, we can save more service cost with less harvested
regarded as the normaPlized expected total service cost from
energyif theH-blocksare identifiedsequentiallyaccordingto
the i-th block given the battery is in energy state ǫ, and the
the metric function Ξ c ,pinv , which uncovers the inherent
future expected total service cost can be represented in terms i H,i
characteristics of HES wireless networks.
of uˆ∗ (ǫ) as (cid:0) (cid:1)
i+1 Inordertodevelopthealgorithmfortheoptimalpolicy,we
also need the following properties of the optimal cost-to-go
p(s′|s ,α )u∗ (s′)
i i i+1 function.
Xs′ Proposition 4: ∀i ∈ N, u∗(s ) obeys the following prop-
=K−2 p(ǫ′|s ,α ) u∗ (hǫ′,γ′ ,γ′ i) i i
i i i+1 G H (28) erties:
Xǫ′ γGX′ ,γH′ 1) u∗(hǫ ,γ ,γ i) = u∗(hǫ ,Γ (ǫ ,γ ),γ i),
i i G,i H,i i i G,i i H,i H,i
=K−2 p(ǫ′|si,αi)uˆ∗i+1(ǫ′). ∀γG,i ≤ΓG,i(ǫi,γH,i).
Xǫ′ 2) u∗i (hǫi,γG,i,γH,ii) = u∗i hǫi,γG,i,Γ˜H,i(ǫi,γG,i)i ,
The following lemma shall demonstrate that a higher energy ∀γ ≤ Γ˜ (ǫ ,γ ),(cid:16)where Γ˜ (ǫ ,γ ) (cid:17),
H,i H,i i G,i H,i i G,i
state leads to a lower expected total service cost.
max{H |s.t.H <Γ (ǫ ,γ )}.
k k H,i i G,i
Lemma 3: ∀i∈N, uˆ∗(ǫ) is non-increasing with ǫ.
i Proof: The proof can be obtained with the monotone
Proof: See Appendix C.
structures derived in Proposition 2 and Proposition 3. Details
With the assistance of Lemma 2 and Lemma 3, we can
are omitted for brevity.
obtain the monotone structure of π∗ over the H-channelstate
By exploiting the monotone structures, we propose a
γ as shown in the following proposition.
H,i monotone backward induction algorithm (MBIA) with the
Proposition 3: Foragivenbatteryenergystateǫ ,iftheG-
i main idea illustrated via an example in Fig. 2. Given
channel is in state γ , then the optimal policy is monotone
G,i the block index i and the battery energy state ǫ , as-
i
over γ . Specifically, the EH-BS will be assigned to serve
H,i sume that we traverse the K2 channel states hγ ,γ i
G,i H,i
the user,i.e., α =1, onlywhen γ ≥Γ (ǫ ,γ ), where
i H,i H,i i G,i with an order as hH ,H i, hH ,H i,...,hH ,H i, hH ,H i,
5 5 4 5 1 5 5 4
Γ (ǫ ,γ ) is the threshold H-channel state.
H,i i G,i hH ,H i,...,hH ,H i,hH ,H i.TheBIAevaluatesthecost-
Proof: Suppose γ+ ≥ γ− , and denote s+ = 4 4 2 1 1 1
H,i H,i i to-go function over all 25 channel states as shown in Fig.
hǫ ,γ ,γ+ iands− =hǫ ,γ ,γ− i.ToproveProposition
i G,i H,i i i G,i H,i 2(a). In the MBIA, as shown in Fig. 2(b) after evaluating the
3, it suffices to show if d∗i s−i = 1, then d∗i s+i = 1. cost-to-go function in state hH5,H5i, in which the optimal
For i = N, the result is straightforward. For i < N, since decision is α∗ = 1, there is no need to go through the states
d∗i s−i =1, according to (2(cid:0)4), (cid:1) (cid:0) (cid:1) with a loweri G-channel gain, i.e., the next evaluation will
be in state hH ,H i. Since α∗ = 0 in state hH ,H i, i.e.,
(cid:0) c(cid:1)s−,1 +K−2 p ǫ′|s−,1 uˆ∗ (ǫ′) 5 4 i 5 4
i i i+1 Γ (ǫ ,H )=H >H , there is no need to go through the
H,i i 5 5 4
(cid:0) (cid:1) Xǫ′ (cid:0) (cid:1) states with a lower H-channelgain. Thus, after the evaluation
≤c s−i ,0 +K−2 p ǫ′|s−i ,0 uˆ∗i+1(ǫ′) (29) at hH4,H4i, the MBIA will go to hH4,H3i directly. Such
ǫ′ procedures carry on until the optimal actions in all states are
(cid:0) (cid:1) X (cid:0) (cid:1)
(=a)c s+,0 +K−2 p ǫ′|s+,0 uˆ∗ (ǫ′). determined, and eventually, only 9 evaluations of the cost-to-
i i i+1
go function are needed.
ǫ′
(cid:0) (cid:1) X (cid:0) (cid:1) Details of the MBIA are given in Algorithm 2, where the
(a)isbecausethecostparameterc isirrelevantwithγ and
i H,i majordifferencescomparedtotheconventionalBIAlieonthe
there is no harvested energy consumption when α = 0. By
i procedure of checking the threshold states, i.e., from Line 9
combining Lemma 2 and Lemma 3, we have
to 16. Based on Propositions 2, 3 and 4, once the threshold
p ǫ′|s+,1 uˆ∗ (ǫ′)≤ p ǫ′|s−,1 uˆ∗ (ǫ′), (30) states are found, the optimal action for the states with a
i i+1 i i+1
lower G-channel (H-channel) gain will be determined as 1
ǫ′ ǫ′
X (cid:0) (cid:1) X (cid:0) (cid:1)
8
(0). Meanwhile, the value of the optimal cost-to-go function Algorithm 2 Monotone Backward Induction Algorithm.
u∗(s ) in these remaining states will be assigned the same Input: B , w , w , g , θ, σ2, d , d , {H }, pmax, pmax,
i i m G D 0 H G k G H
as that of their corresponding threshold states. Once π∗ is τ, M, K and N
obtained, it acts as a look-up table. During data transmission, Output: π
N×M×K×K
based on the system state, the operation can be determined 1: Initialize u∗ = +∞×1N×M×K×K,π = 0N×M×K×K,
immediately by referring to the associated entry in such a t=N;
table. 2: While t≥1 do
3: For m=1 to M do
g g
(cid:29)(cid:28) (cid:21)(cid:20) (cid:23)(cid:24) (cid:21)(cid:25) (cid:23)(cid:22) (cid:31)(cid:30) 76 /. 12 /3 10 98: 4: k1 =K,k2 =K;
(cid:27)(cid:26) $, $- %, %- , 54 F 5: While k1 ≥1 and k2 ≥1 do
(cid:23)(cid:21)(cid:24)(cid:20) $$(* %%)+ %%(* )+ (* 1/2. BE C D 76:: s=pu(hs∗t(:′m2|ms:k,2α−,k)11u)∗Bm/2=M,H·k12{,mtHi6=nk1αNi;}c(s;,α) +
(cid:21)(cid:25) $$ %’ %$ ’ $ /3 @ A 8: Pm′,k1′,k2′ αopt t+=1:m′:k2′:k1′argminα(cid:8)c(cid:9)(s,α) +
g(cid:23)(cid:22) $% %& %% & % g10 > ? m′,k1′,k2′ p(s′|s,α)u∗t+1:m′:k2′:k1′ ·1{t6=N(cid:8)} ;
Fig".2!#. Illustrat(iao)nBoIfAthemonotonestruct<ur;=es,K=(5b)(TMheBcIAoloredstatesare 11019::: P Ifuπα∗tto::pmmt::kk=::kk=11 ==1 tu1h,∗te:∀mnk:k≤2:kk12,;∀k <k2; (cid:9)
ww(dii∗itthh(ssdoi∗il)i(d=s-lii)0n)e=).(d1o.tTdhasehn-luinmeb)ecriricnleesaacrhesthtaetethirsesthheoldtrasvtaetressewoirtdherd.∗iT(hsei)st=ate1s 111342::: Elsuke1∗t:m=:kk21:k−=1u; ∗t:m:k2:k1,∀k <k1;
15: k2 =k2−1;
Remark 2: Although we have discovered the monotone
16: End if
structures of π∗ over the channel gains, unfortunately, such
17: End while
structures do not hold over the battery energy level ǫ . In
i 18: End for
other words, as M increases, the complexity of the MBIA
19: t=t−1;
may still be high. Consequently, it is important to choose a
20: End while
suitable resolution for the battery energy state to balance the
computational complexity and the achievable performance.
B. Heuristic Online Policies currentblock,withoututilizingtheG-channelSI.Ontheother
hand, the performance of the Look-Ahead policy is closely
Though the MBIA algorithm accelerates the conventional
related to the state quantization levels, i.e., the computational
BIA, as the size of the state space grows, its execution time
complexity.IntuitionssuggestthattheEH-BSprefersto serve
becomes unacceptable. Moreover, the memory requirement
the user when the H-channelis in a good condition while the
increases in order to store π∗. To further reduce complexity,
G-channelisindeepfading.Also,ahigherbatteryenergystate
in this subsection, we propose a Look-Ahead policy and
may further stimulate the EH-BS to transmit more packets.
a Threshold-based Heuristic policy. First, we introduce a
Thus, in the Threshold-based Heuristic policy, for i < N, if
Greedy-Transmit policy as the performance benchmark.
pinv ≤min{ǫ /τ,pmax}, αTH is determined by
1) Greedy-Transmit Policy: The Greedy-Transmit policy H,i i H i
always takes a higher priority to use the harvested energy.
αTH =1 ǫ Ξ c ,pinv ≥ζP τΞ(λ ,λ ) . (32)
In each block, if the available harvested energy can afford i i i H,i avg 1 2
to transmit the packet and the peak power constraint is not
(cid:8) (cid:0) (cid:1) (cid:9)
violated, the EH-BS will be assigned to serve the user, i.e., Otherwise, αTiH = 0. For i =N, αTiH = 1{piHn,vi ≤ min{ǫi/
αGi T =1{piHn,vi ≤min{ǫi/τ,pmHax}},∀i∈N. τ,pmHax}}. In (32), ζ is a scaling factor that can be obtained
2) Look-Ahead Policy: The Look-Aheadpolicy is a simpli- and optimized in prior, and λ1, λ2 are constants given by
fied version of the discrete MDP approach, the idea of which λ1 =E{ci}andλ2 =E{piHn,vi|piHn,vi ≤pmHax},respectively.For
has been previously adopted for online algorithms designs Rayleigh fading channels, they can be simplified as
[16],[42].IntheMDPapproach,inthei-thblock,weneedto
considerthe randomnessofthe futureN−iblocks.However, λ1 =wD 1−e−AG/κ +wGτAGE1(AG/κ), (33)
intheLook-Aheadpolicy,weonlyconsidertherandomnessin
(cid:16) (cid:17)
the nextblock,whichlowersthe complexityatthe expenseof
pweerfocarmnasnimcepldyegprlaudgatiionnN. To=ob2taiinntthhee MLoBoIkA-A. hDeuardinpgodliactya, λ2 =AHE1(AH/pmHax)·eAH/pmHax, (34)
transmission, if i < N, according to the system state, the
actionαLAisdeterminedbythecorrespondingentryinπLA . where Aj = µj 2WRτ −1 σ2g0−1dθj, µj = E[γj,i],j ∈
i 1,:,:,:
If i=N, αLA =1{pinv ≤min{ǫ /τ,pmax}}. {G,H},andE (x(cid:16))= +∞(cid:17)e−t·dt.Indeed,thispolicyobeys
i H,i i H 1 x t
3) Threshold-based Heuristic Policy: The Greedy-Transmit Proposition 2 and Proposition 3, i.e., it is a threshold-based
R
policy always tends to minimize the service cost in the policy.
9
C. Possible Extensions
0.06
Inthissubsection,possibleextensionsoftheobtainedresults Greedy Transmit
Thres. Heuristic (z*)
will be discussed. Although we focus on a single-user HES Look Ahead
0.05 MBIA (M=25)
network in this paper, our results provide valuable guidelines
MBIA (M=100)
for designing more general networks. Specifically, for HES MBIA (M=400)
GA Algorithm
networks where the EH-BS and the GP-BS coordinate to 0.04 Off−line Optimal
serve multipleusers with OFDMA5, the GA algorithmcan be Cost
tOdhier(erNectlKiysa(apNptrl+aiendsKmf)oi)tr,tphwoewhneeorrenc-KcoanusistsraaltiShneItsfncouernmaablrlieorthwoeiftuhusescerosrmsa.ptlSeeixnaiccthye Total Service 0.03
0.02
BS, the feasibility of the sum transmit power should be
checked in addition to the energy causality constraint in each
loop of the GA algorithm. For the causal SI scenario, the 0.01
MDP-based algorithms are computationallyintractable due to
highcomplexity.However,monotonestructuresoftheoptimal 0
10 20 30 40 50 60 70
onlinesolutionstillhold.Thus,thelow-complexityThreshold- dH (m)
based Heuristic policy developed in Section IV-B can be Fig. 3. Total service cost vs. dH, wD = 0.01, Pavg = 20 mW and
modified for multiple users. Specifically, in the i-th block, dH+dG=80m.
αTH (theuserindexisomittedforbrevity)canbedetermined
i
accordingto(32)foreachuser,andtheEH-BSservestheusers
with the largest values of Ξ ci,piHn,vi and αTiH = 1 given 0.08
Greedy Transmit
the available energy and transmit power constraints. Sample Thres. Heuristic (z=10)
(cid:0) (cid:1)
simulationresultsforatwo-userHESnetworkwillbeprovided Look Ahead
0.07 MBIA (M=25)
intheSectionV.ForHESnetworkswithmultipleEH-BSs,the MBIA (M=100)
MBIA (M=400)
ideaofcentralenergyqueuecanbeappliedtoformanalliance GA Algorithm
oftheEH-BSs[44].By adoptingtheproposedalgorithms,the 0.06 Off−line Optimal
aEsHsi-gBnSesd,BcaSnfobreedaectherumsienr,edi..e.,eithertheGP-BSoroneofthe Service Cost0.05
It is worthwhile to note that, as the network size grows, Total
the amount of SI and the number of decision variables grow 0.04
accordingly, and the coupling among different BSs/users in
resourceallocationbecomesmoresevere,bothofwhichmake
0.03
the optimization highly difficult. As a result, the extensions
of the algorithms dedicated for single-user networks may
0.02
not offer a complete solution, but they can still serve as a 10 15 20 25 30
P (mW)
benchmark for future investigations. avg
Fig.4. Totalservicecostvs.EHpower,wD =0.01,dH =30m,dG=50
m.
V. SIMULATION RESULTS
Inthissection,weevaluatetheperformanceoftheproposed
algorithmsand show the unique grid energyconsumptionand
service cost in Fig. 3 by assuming that the user is located
QoStradeoffinHESwirelessnetworks.Inthesimulations,we
on the line between the two BSs. In the Threshold-based
set w = 1, R = 50 Kbits, N = 50, τ = 1 ms, σ2 = −97.5
G Heuristic policy, ζ∗ is obtained via simulation within the
dBm, W = 10 MHz, g = −40 dB, θ = 4, pmax = 0.5
0 H candidate set [0:0.5:200]. We see that, when the user is
W and pmax = 2 W. Without loss of generality, we assume
G near the EH-BS, the total service cost approaches zero, since
f (e)=1/E ,e∈[0,E ]andγ ,γ areexponentially
EH m m G,i H,i with low path loss the EH-BS can deliver most of the data
distributed with mean 0 dB. In the GA algorithm, Ξ(p,q) =
packets without incurring any cost. As the user moves closer
p/q is adopted. We use K =25, B = NE in the MBIA.
m m to the GP-BS, we see that the total service cost decreases,
TheGreedy-Transmitpolicyisusedasabaselinefortheonline
which is due to the small path loss of the G-channel, and
setting, while exhaustive search is used to find the optimal
thus the GP-BS is able to transmit the packets with little
solution of the off-line case for comparison.
grid energy consumption. In Fig. 4, we fix d = 30 m,
H
d = 50 m and show the total service cost versus the
G
A. Total Service Cost Minimization
average EH power of different policies. In accordance with
We startwiththeinvestigationoftheimpactofthedistance theintuition,thetotalservicecostdecreasesasP increases.
avg
from the EH-BS to the user, denoted as d , on the total As for performance comparison between different schemes,
H
for the off-line case, from both Fig. 3 and Fig. 4, we see that
5Each user is assumed to be allocated with one sub-carrier with equal
the proposed GA algorithm enjoys near-optimalperformance.
bandwidth. The system performance can be further optimized by adopting
bandwidthallocation techniques [17],[18],[22],whichisbeyondourscope. For the online setting, the MBIA with M = 25,100,400
10
are evaluated6, and the total service cost of M = 100 is
25
significantly lower than that of M = 25. On the other hand,
the performance gain from M=100 to 400 is negligible, 555555000000
which indicates that M = 100 is a viable choice. Moreover,
all the proposed policies achieve a noticeable improvement me (mJ)20 444444000000
ccoMaohnnmodly=icpMoea2uro5et=fp,deζb4tru,of0tto0htahr.emlesposbreotphnpeecorhfsLomeordmaoTrkskh-cArpleoohslsehiecaoydlt.dopS-btouhalreispceyMrdisaBHinnIedgAulytrhwi,sewittiMhcithpMBoaIlAip=cryow1npi0oteh0rt Energy Consumption per Fra1105 Pack drop ratio Grid GTLMeohnrBeoreeIekArs gd A.(yy MHh cTee=orau1andr0nsis0sutm)mici pt(zti=on10) 232323232323000000000000Packet Drop Ratio (%)
0.2 Greedy Transmit Grid 5 GOAff− Alinlgeo rOitphtmimal
Thres. Heuristic (z=10) 111111000000
GA Algorithm
0.18 Off−line Optimal
0 000000
−−−−−−−4444444 −−−−−−−3333333 −−−−−−−2222222 −−−−−−−1111111 0000000 1111111 2222222
0.16 log w
D
Service Cost0.14 dFGig.=6.50Gmri,dPenavegrgy=c2o0nsmumWp.tion&packetdropratiovs.wD,dH =30m,
Total 0.12
energy consumption. For instance, assuming 96% successful
0.1
packet transmission is required, we may choose the Look-
Aheadpolicywithw =10−0.5andthegridenergyconsump-
0.08 D
tionwillbearound17.5mJ.WecanalsoadopttheThreshold-
based Heuristic policy with w = 10−2, which will enjoy a
0.06 D
10 15 20 25 30
P (mW) lowercomplexity,butconsumeslightly moregridenergy,i.e.,
avg
18.2 mJ. In addition, the off-line policies are more competent
Fig.5. Totalservicecostvs.EHpower(twousers),wD =0.01,bothusers tosuppressboththegridenergyconsumptionandpacketdrop
are30mfromtheEH-BSanddG=50mfromtheGP-BS.
ratio compared to the online policies. This is because of the
availability of full SI that helps to identify the optimal H-
In Fig. 5, we show the total service cost performance in a
blocks.Interestingly,whenw isrelativelylarge,e.g.>10−2,
two-userHESnetwork,wherebothusersared =30mfrom D
H the grid energy consumed by the proposed online policies is
the EH-BS and d = 50 m from the GP-BS. It can be seen
G significantlygreaterthanthatbythebenchmark.Thereasonis
that the GA algorithm still achieves competitive performance
that, with causalSI, the optimalpolicyshouldavoidthe event
compared to the off-line optimal algorithm. Also, the low-
ofpacketdrop.Thus,inordertoretainmoreharvestedenergy
complexity Threshold-based Heuristic policy substantially re-
forthe blockswith poorG-channelcondition,the GP-BS will
ducesthetotalservicecostcomparedtothebenchmarkpolicy,
transmit more packets when the H-channel is in deep fading,
and its performance is comparable to those achieved by the
which leads to higher grid energy consumption.
off-line algorithms. These observations are similar with those
from Fig. 4, which indicates the potential of extending our
55
findings to more general networks. Off−line Optimal
50 wDfi 0 GA Algorithm
MBIA (M=100)
45 Look Ahead
B. Grid Energy Consumption and QoS Tradeoff Thres. Heuristic (z=10)
40 Greedy Transmit
In this subsection, we will demonstrate the grid energy wD decreases Without EH−BS
cshoonwsumthpetigornidaenndeQrgoySctornadsuemofpftiinonHaEnSdwpaircekleetssdrnoeptwroatrikos,.iW.e.e, Ratio (%)3305 Achievable Region
thepercentageofthedroppedpackets,underdifferentpolicies Drop 25
icnonFsiugm. 6p.tiWoneasneedtQhaotS, btyraaddejoufsftsinagrewDac,hdieifvfeedre.nItngrpidarteinceurlgary, Packet 20 wD increases
15
when w is small, the QoS requirement is loose, so the
D
network tends to save grid energy by dropping more packets. 10
However, as wD gets larger, the grid energy consumption 5
increasesand the packetdrop ratio decreases, which indicates
0
0 5 10 15 20 25
that the network addresses more on QoS. Hence, given the Grid Energy Consumption per Frame (mJ)
QoSrequirement,w canbedeterminedtominimizethegrid
D
Fig.7. Packetdropratioandgridenergyconsumptiontradeoff regions.
6ThenumberofpossiblestatesintheMDPis1.25×107whenM =400.
ThankstoMBIA,wecanobtaintheoptimalpolicywithinfewhours. InFig.7,weplotthegridenergyconsumption-QoStradeoff