-
1Fundamentals of Heterogeneous Cellular Networkswith Energy
Harvesting
Harpreet S. Dhillon, Ying Li, Pavan Nuggehalli, Zhouyue Pi, and
Jeffrey G. Andrews
AbstractWe develop a new tractable model for K-tier
hetero-geneous cellular networks (HetNets), where each base station
(BS)is powered solely by a self-contained energy harvesting
module.The BSs across tiers differ in terms of the energy
harvestingrate, energy storage capacity, transmit power and
deploymentdensity. Since a BS may not always have enough energy, it
mayneed to be kept OFF and allowed to recharge while nearby
usersare served by neighboring BSs that are ON. We show that
thefraction of time a kth tier BS can be kept ON, termed
availabilityk, is a fundamental metric of interest. Using tools
from randomwalk theory, fixed point analysis and stochastic
geometry, wecharacterize the set of K-tuples (1, 2, . . . K),
termed theavailability region, that is achievable by general
uncoordinatedoperational strategies, where the decision to toggle
the currentON/OFF state of a BS is taken independently of the other
BSs.If the availability vector corresponding to the optimal
systemperformance, e.g., in terms of rate, lies in this
availability region,there is no performance loss due to the
presence of unreliableenergy sources. As a part of our analysis, we
model the temporaldynamics of the energy level at each BS as a
birth-death process,derive the energy utilization rate, and use
hitting/stopping timeanalysis to prove that there exists a
fundamental limit on k thatcannot be surpassed by any uncoordinated
strategy.
I. INTRODUCTION
The possibility of having a self-powered BS is becomingrealistic
due to several parallel trends. First, BSs are beingdeployed
ever-more densely and opportunistically to meetthe increasing
capacity demand [2]. The new types of BSs,collectively called small
cells, cover much smaller areas andhence require significantly
smaller transmit powers comparedto the conventional macrocells.
Second, due to the increasinglybursty nature of traffic, the loads
on the BSs will experiencemassive variation in space and time [3].
In dense deployments,this means that many BSs can, in principle, be
turned OFFmost of the time and only be requested to wake up
intermit-tently based on the traffic demand. Third, energy
harvestingtechniques, such as solar power, are becoming
cost-effectivecompared to the conventional sources [4]. This is
partly due tothe technological improvements and partly due to the
marketforces, such as increasing taxes on conventional power
sources,and subsidies and regulatory pressure for greener
techniques.
H. S. Dhillon and J. G. Andrews are with the Wireless
Networkingand Communications Group (WNCG), The University of Texas
at Austin,TX (email: [email protected] and
[email protected]). Y. Li andZ. Pi are with Samsung Research
America, Richardson, TX (email: {yli2,zpi}@sta.samsung.com). P.
Nuggehalli is with the Mobile and Wireless Groupof Broadcom
Corporation, Sunnyvale, CA (email: [email protected]).
This work was done while the first author was with Samsung
ResearchAmerica, Richardson, TX. A part of this paper is accepted
for presentation atIEEE Globecom 2013 in Atlanta, GA [1].
Manuscript updated: July 22,2013.
Fourth, high-speed wireless backhaul is rapidly becoming
areality for small cells, which eliminates the need for otherwired
connections [5]. Therefore, being able to avoid theconstraint of
requiring a wired power connection is even moreattractive, since it
would open up entire new categories of low-cost drop and play
deployments, especially of small cells.
A. Related Work
The randomness in the energy availability at a
transmitterdemands significant rethinking of conventional wireless
com-munication systems. There are three main directions takenin the
literature to address this challenge, which we orderbelow in terms
of complexity and realism. The first considers arelatively simple
setup consisting of single full-buffer isolatedlink, and study
optimal transmission strategies under a givenenergy arrival process
[6][8]. The effect of data arrivals canbe additionally incorporated
by considering two consecutivequeues at the transmitter, one for
the data and the other forthe energy arrivals [9], [10].
Second, a natural extension of an isolated link, consid-ers a
broadcast channel, where a single isolated transmitterserves
multiple users. Again one can assume full-buffer atthe transmitter
so that the transmission strategies need tobe adapted only to the
energy arrival process, e.g., in [11].More realistically, one can
relax the full-buffer assumption toexplicitly consider data
arrivals as discussed above for the iso-lated link, and optimize
various metrics, e.g., minimize packettransmission delay [12], or
maximize system throughput [13].
The third and least investigated direction is to consider
mul-tiple self-powered transmitters, which significantly
generalizesthe above two directions. Generally speaking, the main
goal isto adapt transmission schemes based on the energy and
loadvariations across both time and space. While some progresshas
been recently made in advancing the understanding ofmobile ad-hoc
networks (MANETs) with self-powered nodes,see [14], [15] and
references therein, our understanding ofcellular networks in a
similar setting is severely limited. Thisis partly due to the fact
that conventional cellular networksconsisted of big macro BSs that
required fairly high power,and it made little sense to study them
in the context of energyharvesting. As discussed earlier, this is
not the case with aHetNet, which may support drop and play
deployments, es-pecially of small cells, in the future.
Comprehensive modelingand analysis of this setup is the main focus
of this paper.
To capture key characteristics of HetNets, such as
hetero-geneity in infrastructure, and increasing uncertainty in
BSlocations, we consider a general K-tier cellular network with
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2K different classes of BSs, where the BS locations of eachtier
are sampled from an independent Poisson Point Process(PPP). This
model was proposed for HetNets in [16], [17],with various
extensions and generalizations in [18][20]. Themodel, although
simple, has been validated as reasonablesince then both by
empirical evidence [21] and theoreticalarguments [22]. Due to its
realism and tractability, it hasbecome an accepted model for
HetNets, see [23] for a detailedsurvey.
B. Contributions
Tractable and general system model. We propose a generalsystem
model consisting of K classes of self-powered BSs,which may differ
in terms of the transit power, deploymentdensity, energy harvesting
rate and energy storage capacity.Due to the uncertainty in the
energy availability, a BS mayneed to be kept OFF and allowed to
accumulate enough energybefore it starts serving its users again.
In the meanwhile, itsload is transferred to the neighboring BSs
that are ON. Thus,at any given time a BS can be in either of the
two operationalstates: ON or OFF. In this paper, we focus on
uncoordinatedoperational strategies, where the operational state of
each BSis toggled independently of the other BSs. For tractability,
weassume that the network operates on two time scales: i) longtime
scale, over which the decision to turn a BS ON or OFF istaken, and
ii) short time scale, over which the scheduling andcell selection
decisions are taken. As discussed in Section III,this distinction
facilitates analysis in two ways: a) it allows usto assume that the
operational states of the BSs are static overshort time scale, and
b) it allows us to consider the averageeffects of cell selection
over long time scale.
Availability region. We show that the fraction of time a kth
tier BS can be kept in the ON state, termed the availability
k,is a key metric for self-powered cellular networks. Using
toolsfrom random walk theory, fixed point analysis, and
stochasticgeometry, we characterize the set of K-tuples (1, 2, . .
. K),termed the availability region, that are achievable with aset
of general uncoordinated strategies. Our analysis involvesmodeling
the temporal dynamics of the energy level at eachBS as a
birth-death process, deriving energy utilization ratefor each class
of BSs using stochastic geometry, and usinghitting/stopping time
analysis for a Markov process to provethat there exists a
fundamental limit on the availabilities {k},which cannot be
surpassed by any uncoordinated strategy. Wealso construct an
achievable scheme that achieves this upperlimit on availability for
each class of BSs.
Notion of optimality for self-powered HetNets. The
charac-terization of exact availability region lends a natural
notion ofoptimality to self-powered HetNets. Our analysis
concretelydemonstrates that if the K-tuple (1, 2, . . . , K)
correspond-ing to the optimal performance of the network, e.g., in
terms ofdownlink rate, lies in the availability region, the
performanceof the HetNet with energy harvesting is fundamentally
thesame as the one with reliable energy sources. Using
recentresults for downlink rate distribution in HetNets [24], [25],
wealso show that it is not always optimal from downlink data
rateperspective to operate the network in the regime
corresponding
TABLE INOTATION SUMMARY
Notation DescriptionK Set of indices for BS tiers, i.e., K = {1,
2, . . . ,K}
k , k; Independent PPP modeling locations of kth tierBSs, its
density; set of all BSs, i.e., = kKk
u;u An independent PPP modeling user locations,density of
users
k; k; Nk Energy harvesting rate, utilization rate, and
energystorage capacity of a kth tier BS
k;R Availability of kth tier BSs; availability region(a)k ,
(a)k ;
(a) Independent PPP modeling the kth tier BSs that areavailable,
its density (a)k = kk; all available
BSs, i.e., (a) = kK(a)kPk Downlink transmit power of a kth tier
BS to each
user in each resource blockhk;Xk; Small scale fading gain hk
exp(1); large scale
shadowing gain (general distribution) from a kthtier BS; path
loss exponent
x(z)k , x
(z) Candidate serving BS in (a)k for user at z u,serving BS for
z u
Pc; Coverage probability; target SIRRc; T Rate coverage; target
rate
to the maximum availabilities, i.e., it may be preferable to
keepa certain fraction of BSs OFF despite having enough energy.
II. SYSTEM MODEL
A. System Setup and Key Assumptions
We consider a K-tier cellular network consisting of Kdifferent
classes of BSs. For notational simplicity, defineK = {1, 2, . . .
,K}. The locations of the BSs of the kth tierare modeled by an
independent PPP k of density k. EachBS has an energy harvesting
module and an energy storagemodule, which is its sole source of
energy. The BSs acrosstiers may differ both in terms of how fast
they harvest energy,i.e., the energy harvesting rate k joules/sec,
and how muchenergy they can store, i.e., the energy storage
capacity (orbattery capacity) Nk joules. We assume that the
normalizationof k and Nk is such that each user requires 1 joule of
energyper sec. This assumption can be easily relaxed to
incorporateusers requiring more energy under sufficient
randomization,but this case is not in the scope of this paper. For
resourceallocation, we assume an orthogonal partitioning of
resources,e.g., time-frequency resource blocks in orthogonal
frequencydivision multiple access (OFDMA), where each resource
blockis allocated to a single user. Due to orthogonal
resourceallocation, there is no intra-cell interference. Note that
a usercan be allocated multiple resource blocks as discussed in
detailin the sequel. We further assume that a kth tier BS transmits
toeach user with a fixed power Pk in each resource block, whichmay
depend upon the energy harvesting parameters, althoughwe do not
study this dependence in this paper. The target SIR is the same for
all the tiers.
The energy arrival process at a kth tier BS is modeled as
aPoisson process with mean k. Since most energy harvestingmodules
contain smaller sub-modules, each harvesting energyindependently,
e.g., small solar cells in a solar panel, thenet energy harvested
can be modeled as a Binomial process,which approaches the Poisson
process when the number of
-
3sub-modules grow large. Interestingly, this model has
beenvalidated using empirical measurements for a variety of
energyharvesting modules [26]. Since the energy arrivals are
randomand the energy storage capacities are finite, there is
someuncertainty associated with whether the BS has enough energyto
serve users at a particular time. Under such a constraint, itis
required that some of the BSs be kept OFF and allowed torecharge
while their load is handled by the neighboring BSsthat are ON.
Besides, as discussed in the sequel, it may alsobe preferable to
keep a BS OFF despite having enough energy.Therefore, a BS can be
in either of the two operational states:ON or OFF. The decision to
toggle the operational state fromone to another is taken by the
operational strategies that canbe broadly categorized into two
classes.
Uncoordinated: In this class of strategies, the decision
totoggle the operational state, i.e., turn a BS ON or OFF, istaken
by the BS independently of the operational states of theother BSs.
For example, a BS may decide to turn OFF if itscurrent energy level
reaches below a certain predefined leveland turn back ON after
harvesting enough energy. The BS mayadditionally consider the time
for which it is in the currentstate while making the decision. For
instance, a BS may starta timer whenever the state is toggled and
may decide to toggleit back when the timer expires or the energy
level reaches acertain minimum value, whichever occurs first. This
class willbe the main focus of this paper.
Coordinated: In this class of strategies, the decision totoggle
the state of a particular BS is dependent upon the statesof the
other BSs. For example, the BSs may be partitioned intosmall
clusters where only a few BSs in each cluster are turnedON. The
decision may be taken by some central entity basedon the current
load offered to the network. This is useful inthe cases where the
load varies by orders of magnitude acrosstime, e.g., due to diurnal
variation. A small fraction of BSs isenough to handle smaller load,
with the provision of turningmore ON as the load gradually
increases. In addition to theload, other factors such as network
topology and interferenceamong BSs may also affect the
decision.
For tractability, we define the following two time scales
overwhich the network is assumed to operate.
Definition 1 (Time scales). The scheduling and cell associ-ation
decisions are assumed to be taken over a time scalethat is of the
order of the scheduling block duration. We termthis time scale as a
short time scale. On the other hand, theoperational policies that
toggle the operational state of a BSare assumed to be defined on a
much longer time scale. Wewill henceforth term this time scale as a
long time scale.
As discussed in the sequel, this distinction is the key
totractability because of two reasons: i) it allows us to assumethe
energy states of the BSs to be static over short time scales,and
ii) it allows us to consider the average effects of cellselection
while determining the energy utilization rates overlong time
scales. Due to uncertainty in the energy availabilityor due to the
optimality of a given performance metric, e.g.,downlink rate, all
the BSs in the network may not alwaysbe available to serve users.
This is made precise by definingavailability of a BS as
follows.
Definition 2 (Availability). A BS is said to be available if
itis in the ON state as a part of the operational policy and
hasenough energy to serve at least one user, i.e., has at least
oneunit of energy. The probability that a BS of tier k is available
isdenoted by k, which may be different for each tier of BSs dueto
the differences in the capabilities of the energy harvestingmodules
and the load served. For notational simplicity, wedenote the set of
availabilities for the K tiers by {k}.
For uncoordinated strategies, it is reasonable to assumethat the
current operational state (ON or OFF) of a BS isindependent of the
other BSs, especially since the energyharvesting processes are
assumed to be independent acrossthe BSs. The coupling in the
transmission of various BSsthat arises due to interference and
mobility is ignored. Underthis independence assumption, the set of
ON BSs of the kth
tier form a PPP (a)k with density (a)k = kk. This results
from the fact that the independent thinning of a PPP leadsto a
PPP with appropriately scaled density [27]. As will beevident from
the availability analysis in the next section, thisabstraction is
the key that makes this model tractable and leadsto meaningful
insights.
B. Propagation and Cell Selection Models
For this discussion it is sufficient to consider only the
BSsthat are available, i.e., the ones that are in the ON state.
Fornotational ease, define (a) = kK(a)k . The user locationsare
assumed to be drawn from an independent PPP u ofdensity u. More
sophisticated non-uniform user distributionmodels [28] can also be
considered but are not in the scopeof this paper. The received
power at a user located at z ufrom a kth tier BS placed at xk (a)k
in a given resourceblock is
P (z, xk) = Pkh(z)kxkX (z)kxkxk z, (1)
where h(z)kxk exp(1) models Rayleigh fading, X(z)kxk
modelslarge scale shadowing, and xk z represents
standardpower-law path loss with exponent , for the wireless
channelfrom xk (a)k to z u. Note that since h(z)kxk and X
(z)kxk
are both independent of the locations xk and z, we will dropxk
and z from the subscript and superscript, respectively, anddenote
the two random variables by hk and Xk, whenever thelocations are
clear from the context.
For cell selection, we assume that each user connects to theBS
that provides the highest long term received power, i.e.,small
scale fading gain h(z)kx does not affect cell selection. Fora
cleaner exposition, we denote the location of the candidatekth tier
serving BS for z u by x(z)k (a)k , which is
x(z)k = arg max
x(a)kPkX (z)xk x z. (2)
A user z u now selects one of these K candidate servingBSs based
on the average received signal power, i.e., thelocation of the
serving BS x(z) {x(z)k } is
x(z) = arg maxx{x(z)k }
PkX (z)kx x z. (3)
-
4Nk Nk 1 0
k
k
k
k
k
k
Fig. 1. Birth-death process modeling the temporal dynamics of
the energyavailable at a kth tier BS.
Owing to the displacement theorem for PPPs [29], any gen-eral
distribution of Xk can be handled in the downlink analysisof a
typical user as long as E
[X 2k
]< . This is formally
discussed in detail in [24], [30]. The most common assumptionfor
large scale shadowing distribution is lognormal, whereXk = 10
Xk10 such that Xk N (mk, 2k), where mk and
k are respectively the mean and standard deviation in dBof the
shadowing channel power. For lognormal distribution,E[X 2k
]= exp
(ln 10
5mk +
12
(ln 10
5k
)2), which can be
easily derived using moment generating function (MGF) ofGaussian
distribution [24]. The fractional moment is clearlyfinite if both
the mean and standard deviation of the normalrandom variable Xk are
finite. For this system model, we nowstudy the availabilities of
different classes of BSs.
III. AVAILABILITY ANALYSIS
The first challenge in studying the model introduced in
theprevious section lies in characterizing how the energy
availableat the BS changes over time. Without loss of generality,
weindex the energy states of a kth tier BS as 0, 1, . . . , Nk,
andmodel the temporal dynamics as a continuous time Markovchain
(CTMC), in particular a birth-death process, as shown inFig. 1.
When the BS is ON, the energy increases according tothe energy
harvesting rate and decreases at a rate that dependsupon the number
of users served by that BS. When the BS isOFF, it does not serve
any users and hence the birth-deathprocess reduces to a birth-only
process. We now derive aclosed form expression for the rate k at
which the energyis utilized at a typical kth tier BS.
A. Modeling Energy Utilization Rate
Before modeling the energy utilization rate, there are
twonoteworthy points. First, if a BS is not available, the
loadoriginating from its original area of coverage is directed to
thenearby BSs that are available, thus increasing their
effectiveload. Equivalently, the coverage areas of the BSs that
areavailable get expanded to cover for the BSs that are
notavailable, as shown in Fig. 2. The second one is related to
thecontrol channel coverage and given in the following
remark.Recall that control channel coverage Pc is the probability
thatthe received signal-to-interference-ratio (SIR) is greater
thanthe predefined minimum SIR needed to establish a connectionwith
the BS. Thus the users that are not in control channelcoverage
cannot enter the network and hence cannot accessthe data channels.
Therefore, these users do not account forany additional energy
expenditure at the BS.
Fig. 2. Coverage regions for a two-tier energy harvesting
cellular network(averaged over shadowing). The unavailable BSs are
denoted by hollowcircles. The thin lines form coverage regions for
the baseline case assumingall the BSs were available.
Remark 1 (Control channel coverage). The control channelcoverage
Pc is independent of the densities of the BSs in
aninterference-limited network when the target SIR is the samefor
all tiers [17], [18], [24]. While this result will be familiarto
those exposed to recent coverage probability analysis
usingstochastic geometry, it is not directly required in this
sectionexcept the interpretation that the density of users
effectivelyserved by the network is independent of the effective
densitiesof the BSs and hence independent of {k}. We will
validatethis claim in Section III-E.
Assuming fixed energy expenditure for control signaling,only the
users that are in control channel coverage will resultin additional
energy expenditure at the BS. As remarked above,the density of such
users is Pcu. Each user is assumed torequire 1 joule of energy per
sec such that the net energyutilization process at each BS can be
modeled as a Poissonprocess with mean defined by the average number
of usersit serves. It should be noted that the assumption of 1
jouleenergy requirement is without any loss of generality and
ismade to simplify the notation. To find the average number ofusers
served by a typical BS of each class, we first need todefine its
service region whose statistics such as its area will,in general,
be different for different classes of BSs due to thedifferences in
the transmit powers as evident from Fig. 2. Theservice region can
be formally defined as follows.
Definition 3 (Service region). The service region Ak(xk) R2 of
the kth-tier BS located at xk (a)k is Ak(xk) ={
z R2 : xk = arg maxx{x(z)j }
PjX (z)j x z,
where x(z)j = arg max
x(a)jPjX (z)j x z
}. (4)
We now derive the average area of the service region of a
-
5typical BS of each tier in the following Lemma.
Lemma 1 (Average area of the service region). The averagearea of
the service region of a kth tier typical BS is given by
E[|Ak|] =E[X 2k
]P
2
kKj=1 jjE
[X 2j
]P
2j
. (5)
Proof: The proof follows from the definition of theservice area
using basic ideas from Palm calculus and is givenin Appendix A.
Using the expression for average area, the average numberof
users served by a typical BS of kth tier, equivalently theenergy
utilization rate, is now given by the following corollary.
Corollary 1 (Energy utilization rate). The energy
utilizationrate, i.e., the number of units of energy required per
second,at a typical BS of kth tier is given by
k = PcuE[|Ak|] =PcuE
[X 2k
]P
2
kKj=1 jjE
[X 2j
]P
2j
, (6)
where recall that Pc denotes the coverage probability, which
isindependent of the availabilities and will be calculated laterin
this section and is given by (35).
Remark 2 (Invariance to shadowing distribution). From (6),note
that the energy utilization rate k is invariant to theshadowing
distribution of all the tiers if E
[X 2j
]= E
[X 2k
],
for all j, k K. For lognormal shadowing, this correspondsto the
case when mj = mk and j = k, for all j, k K.
It should be noted that the availabilities of various tiersare
still unknown and even if all the system parameters aregiven, it is
still not possible to determine the energy utilizationrate from the
above expression. This will lead to fixed pointexpressions in terms
of availabilities, which is the main focusof the rest of this
section. It is also worth mentioning that theenergy utilization
rate derived above is just for the service ofthe active users.
There are some other components of energyusage, e.g., control
channel signaling and backhaul that are notmodeled. While we can
incorporate their effect in the currentmodel by assuming fixed
energy expenditure and deducting itdirectly from the energy arrival
rate, a more formal treatmentof these components is left for future
work.
B. Availabilities for a Simple Operational Strategy
After deriving the energy utilization rate in Corollary 1
andrecalling that the energy harvesting rate is k, we can, in
prin-ciple, derive BS availabilities for a variety of
uncoordinatedoperational strategies. We begin by looking at a very
simplestrategy in which a BS is said to be available when it is
notin energy state 0, i.e., it has at least one unit of energy.
Asshown later in this section, this strategy is of
fundamentalimportance in characterizing the availability region for
the setof general uncoordinated strategies. The availability of a
kth
tier BS under this strategy can be derived directly from the
stationary distribution of the birth-death process as
k = 1
1 kk1
(kk
)Nk+1 (7)
= 1
1
kKj=1 jjE
[X
2j
]P
2j
PcuE[X
2k
]P
2k
1kKj=1 jjE[X 2j ]P 2j
PcuE[X
2k
]P
2k
Nk+1. (8)
Interestingly we get a set of K fixed point equations in termsof
availabilities, one for each tier. Clearly k 0, k K, is a trivial
solution for this set of fixed point equations.However, this means
that none of the BSs is available forservice, which physically
means that the users are in outageif there is no other, in
particular positive, solution for the set offixed point equations.
We will formalize this notion of outage,resulting from energy
unavailability, later in this section. Dueto the form of these
equations, it is not possible to deriveclosed form expressions for
the positive solution(s) of {k}.However, it is possible to
establish a necessary and sufficientcondition for the existence and
uniqueness of a non-trivialpositive solution. Before establishing
this result, we show thatthe function of {k} on the right hand side
of (8) satisfiescertain key properties. For notational simplicity,
we call thisfunction corresponding to kth tier as gk : RK R,
usingwhich the set of fixed point equations given by (8) can
beexpressed in vector form as
12...K
=
g1(1, 2, . . . , K)g2(1, 2, . . . , K)
...gK(1, 2, . . . , K)
= (1, 2, . . . , K),(9)
where we further define function : RK RK for simplicityof
notation. Our first goal is to study the properties of functiongk :
RK R, which can be rewritten as
gk(x) = 1
1Kj=1 ajxj1
(Kj=1 ajxj
)N , (10)
where x RK , N > 1, and ak R+ for all k K. Therelevant
properties are summarized in the following Lemma.
Lemma 2 (Properties). The function gk(x) : RK R definedby (10)
satisfies following properties for all ak > 0, k K:
1) gk(x) is an element-wise increasing function of x.2) gk(x) is
concave, i.e., it is a concave function of xk R
for all k K.Proof: The proof is given in Appendix B.
Lemma 2 can be easily extended to the function : RK RK to show
that it is also a monotonically increasing andconcave function. The
conditions for existence and uniquenessof the fixed point for such
functions can be characterized by
-
6specializing Tarskis theorem [31] for concave functions.
Theresult is stated below. To the best of the knowledge of
theauthors, it first appeared in [32, Theorem 3]. Since the proofis
given in [32], it is skipped here.
Theorem 1 (Fixed point for increasing concave functions).Suppose
: Rn Rn is an increasing and strictly concavefunction satisfying
the following two properties:
1) (0) 0, (a) > a for some a Rn+,2) (b) < b for some b
> a.
Then has a unique positive fixed point.
Before deriving the main result about the existence
anduniqueness of positive solution for the set of fixed
pointequations (8), for cleaner exposition we state the
followingintermediate result that establishes equivalence between
anenergy conservation principle and a key set of conditions.
Lemma 3 (Equivalence). For k > 0,k, the following setsof
conditions are equivalent, i.e., (11) (12)
kKj=1
jjE[X 2j
]P
2j
kPcuE[X 2k
]P
2
k
> 1,k K (11)
Kk=1
kk > uPc, (12)
where (12) is simply the energy conservation principle, i.e.,the
net energy harvested by all the tiers should be greaterthan the
effective energy required by all the users.
Proof: The proof is given in Appendix C.Using Theorem 1 and
Lemma 3, we now derive the main
result of this subsection.
Theorem 2. The necessary and sufficient condition for
theexistence of a positive solution k > 0, k K for thesystem of
fixed point equations given by (8) is
Kk=1
kk > uPc. (13)
Proof: For sufficiency, it is enough to show that the
givencondition is sufficient for the function : RK RK definedby (9)
to satisfy both the properties listed in Theorem 1.Further, it is
enough to show this for each element functiongk : RK R of . For k
6= 0, the function gk, as a functionof k can be expressed as
gk(k) = 1(
1 kk1 (kk)Nk+1
), (14)
where
k =kKj=1 jjE
[X 2j
]P
2j
kPcuE[X 2k
]P
2
k
. (15)
Note that the function gk(k) < 1 for finite Nk. Now settingb,
as defined in Theorem 1, equal to 1, it is enough to findconditions
under which a < b such that gk(a) > a. Sincegk(k) = 0 for k
0, for the existence of a such that
gk(a) > a it is enough to show that g(k) > 1 for k
0.Furthermore, it is easy to show that g(k) = k for k 0,which leads
to the condition k > 1 for the existence of a asdefined above.
This leads to the following set of inequalitiesfor 1 k K:
kKj=1 jjE
[X 2j
]P
2j
kPcuE[X 2k
]P
2
k
> 1. (16)
From Lemma 3, this set of conditions is the same as (13)and
hence proves that (13) is a sufficient condition for theexistence
and uniqueness of the positive solution for {k}.
To show that the given condition is also necessary, weconstruct
a simple counter example. Let K = 1 and dropall the subscripts
denoting the indices of tiers for notationalsimplicity. The fixed
point equation for this simple setup is
= 1
1 Pcu1
(Pcu
)N = g(), (17)
It is easy to show that g() does not have a positive fixed
pointwhen < Pcu, which proves that the given condition (13)is
also necessary.
The existence and uniqueness of the positive solution for theBS
availabilities {k} will play a crucial role in establishingthe
availability region later in this section. The unique
positivesolution for {k} can be computed easily using
fixed-pointiteration. Before concluding this section, it is
important toformalize some key ideas.
Remark 3 (Energy outage). From Theorem 2, it is clear thatthe
total energy harvested by the HetNet must be greaterthan the total
energy demand to guarantee a positive solutionfor the
availabilities {k}. However, if this condition is notmet, the
system may drop a certain fraction of users toensure that the
resulting density of users u is such thatKk=1 kk >
uPc. The rest of the users are said to be
in outage due to energy unavailability, or in short
energyoutage. The probability of a user being in energy outage
is
Oe = 1 u
u 1
Kk=1 kkuPc
, (18)
where the lower bound is strictly positive ifKk=1 kk uPc.
For
cleaner exposition, it is useful to define an
over-provisioningfactor as the ratio of total energy harvested in
the networkand the effective energy demand, i.e.,
=
Kk=1 kkuPc
> 1. (19)
So far we focused on a particular strategy, where a BS issaid to
be available if it is not in the 0 energy state, i.e.,it has at
least one unit of energy. In the next subsection,we develop tools
to study availabilities for any general un-coordinated strategy
using stopping/hitting time analysis. Ouranalysis will concretely
demonstrate that the simple strategydiscussed above maximizes the
BS availabilities over the spaceof general uncoordinated
strategies. Extending these resultsfurther, we will characterize
the availability region that isachievable by the set of general
uncoordinated strategies.
C. Availabilities for any General Uncoordinated Strategy
We first focus on a general set of strategies{Sk(Nkmin, Nkc)} in
which a BS toggles its state basedsolely on its current energy
level, i.e., a kth tier BS togglesto OFF state when its energy
level reaches some level Nkminand toggles back to ON state when the
energy level reachessome predefined cutoff value Nkc > Nkmin as
shown inFig. 3. Although not required for this analysis, it
shouldbe noted that the cutoff value Nkc can be changed by
thenetwork if necessary on an even larger time scale than thetime
scale over which the BSs are turned ON/OFF. Now notethat for the
proposed model, the strategies {Sk(Nkmin, Nkc)}with energy storage
capacity Nk and {Sk(0, Nkc Nkmin)}
with energy storage capacity Nk Nkmin, are equivalentbecause
when the BS is turned OFF at a non-zero energylevel Nkmin in the
first set of strategies, it effectively reducesthe energy storage
capacity to Nk Nkmin. Therefore,without any loss of generality we
fix Nkmin = 0 (for alltiers) and denote this strategy by Sk(Nkc)
for notationalsimplicity. For this strategy, we denote the time for
which akth tier BS is in the ON state after it toggles from the
OFFstate by Jk1(Nkc) and the time for which it remains in theOFF
state after toggling from the ON state by Jk2(Nkc). Thecutoff value
in the arguments will be dropped for notationalsimplicity wherever
appropriate. The cycles of ON and OFFtimes go on as shown in Fig.
3. It is worth highlightingthat both Jk1 and Jk2 are in general
random variables dueto the randomness involved in both the energy
availabilityand its utilization, e.g., Jk1 can be formally
expressed asJk1(Nkc) = inf{t : Ek(t) = 0|Ek(0) = Nkc}, where
Ek(t)denotes the energy level of a kth tier BS at time t. For
thissetup, the availabilities depend only on the means of Jk1
andJk2 as shown in the following Lemma.
Lemma 4 (Availability). The availability of a kth tier BS forany
operational strategy can be expressed as
k =E[Jk1 ]
E[Jk1 ] + E[Jk2 ]=
1
1 +E[Jk2 ]E[Jk1 ]
, (20)
where E[Jk1 ] is the mean time a BS spends in the ON stateand
E[Jk2 ] is the mean time it spends in the OFF state.
Proof: For a particular realization, let {J (i)k1 } and
{J(i)k2}
be the sequences of ON and OFF times, respectively, with ibeing
the index of the ON-OFF cycle. The availability cannow be expressed
as the fraction of time a BS spends in theON state, which leads
to
k = limn
ni=1 J
(i)k1n
i=1 J(i)k1
+ni=1 J
(i)k2
. (21)
The proof follows by dividing both the numerator and
thedenominator by n and invoking the law of large numbers.
To set up a fixed point equation similar to (8) for the
strategySk(Nkc), we need closed form expressions for the mean
ONtime E[Jk1 ] and the mean OFF time E[Jk2 ]. Note that theOFF time
for Sk(Nkc) is simply the time required to harvestNkc units of
energy, which is the sum of Nkc exponentiallydistributed random
variables, each with mean 1/k. Therefore,
E[Jk2 ] =Nkck k = 1
1 + NkckE[Jk1 ]. (22)
To derive E[Jk1 ], we first define the generator matrix for
thebirth-death process corresponding to a kth tier BS as Ak =
k k 0 0 0k k k k 0 00 k k k 0 0...
.... . .
0 0 0 k k
, (23)where the states are ordered in the ascending order of
theenergy levels, i.e., the first column corresponds to the
energy
-
8level 0. To complete the derivation, we need the
followingtechnical result. Please refer to Proposition 5.7.2 of
[33] for amore general version of this result and its proof.
Lemma 5 (Mean hitting time). If the embedded discreteMarkov
chain of the CTMC is irreducible then the mean timeto hit energy
level 0 (state 1) starting from energy level i (statei+ 1) is
E[Jk1(i)] =(
(Bk)1 1)
(i), (24)
where 1 is a column vector of all 1s, and Bk is a (Nk 1)(Nk 1)
sub-matrix of Ak obtained by deleting first row andcolumn of
Ak.
For Ak given by (23), we can derive a closed form ex-pression
for each element of (Bk)1 after some algebraicmanipulations. The
(i, j)th element can be expressed as
(Bk)1 (i, j) = 1jk
min(i,j)n=1
jnk n1k . (25)
Now substituting (25) back in (24) gives us the mean ON timefor
any strategy Sk(Nkc), which when substituted in (22) givesa fixed
point equation in {k} similar to (8), as illustratedbelow for the
two policies of interest.
1) Policy 1 (Sk(1)): In this policy, each BS serves usersuntil
it depletes all its energy after which it toggles to OFFstate. It
toggles back to ON state after it has harvested oneunit of energy.
Using (24) and (25), the mean ON time E[Jk1]for this policy can be
expressed as
E[Jk1 ] =1
k
1(kk
)Nk1
(kk
) , (26)which when substituted into (22) leads to
k = 11 kk
1(kk
)Nk+1 , (27)which is the same fixed point equation as (8). This
establishesan equivalence between this policy and the one studied
in theprevious subsection. In particular, this policy is an
achievablestrategy to achieve the same availabilities as the ones
possiblewith the strategy studied in the previous subsection.
2) Policy 2 (Sk(Nk)): As in the above policy, each BSserves
users until it depletes all its energy after which it togglesto OFF
state. Under this policy, the BS waits in the OFF stateuntil it
harvests Nk units of energy, i.e., its energy storagemodule is
completely charged. Using (24) and (25), E[Jk1 ]can be expressed
as
E[Jk1 ] =1
k kkk
1(kk
)Nk1
(kk
) Nkk k , (28)
which can be substituted in (22) to derive the fixed
pointequation for this policy.
While policy 1 will be of fundamental importance in
es-tablishing the availability region, we will also consider
policy
2 at several places to highlight key points. We now provethe
following theorem, which establishes a fundamental upperlimit on
the availabilities of various types of BSs that cannot besurpassed
by any uncoordinated strategy. Please note that al-though we have
discussed only energy-based uncoordinatedstrategies so far, the
general set of uncoordinated strategiesalso additionally includes
timer-based, and the combination ofenergy and timer-based
strategies. This is taken into accountin the proof of the following
theorem.
Theorem 3. For a given K tier network, the availabilities ofall
the classes of BSs are jointly maximized over the space ofgeneral
uncoordinated strategies if each tier follows strategySk(1). The
availabilities are strictly lower if any one or moretiers follow
Sk(i), i > 1, with a non-zero probability.
Proof: From (22), note that the availability for a kth tierBS is
maximized if E[Jk1(Nkc)]/Nkc is maximized. Using(24) and (25), it
is straightforward to show that
arg max1iNk
E[Jk1(i)]i
= 1. (29)
The proof now follows from the fact that if any tier
followsstrategy Sk(i) (i > 1) with a non-zero probability, its
availabil-ity will be strictly lower than that of Sk(1), which
increasesthe effective load on other tiers and hence decreases
theiravailabilities, as discussed in Remark 5. Therefore, to
jointlymaximize the availabilities of all the tiers, each tier has
tofollow Sk(1).
Now note that any strategy that is fully or partly basedon a
timer can be thought of as an arbitrary combination ofSk(i), where
i > 1 with some non-zero probability. Hence theavailabilities
for such strategies are strictly lower than Sk(1).
Using this result we now characterize the availability regionfor
the set of general uncoordinated strategies.
D. Availability Region
We begin this subsection by formally defining the availabil-ity
region as follows.
Definition 5 (Availability region). Let R(UC) RK be the setof
availabilities (1, 2, . . . , K) RK that are achievable bya given
uncoordinated strategy S(UC). The availability regionis now defined
as
R = S(UC)R(UC), (30)where the union is over all possible
uncoordinated strategies.
From Theorem 3 we know that the availabilities of all thetiers
are jointly maximized if they all follow strategy Sk(1).For
notational ease, we define these maximum availabilities bymax =
(max1 ,
max2 . . .
maxK ). This provides a trivial upper
bound on the availability region as follows
R { RK : k maxk , k K}, (31)which is simply an orthotope in RK .
Our goal now is tocharacterize the exact availability region as a
function ofkey system parameters. As a by product, we will show
that
-
9the upper bound given by (31) is rather loose. For
cleanerexposition, we will refer to Fig. 4, which depicts the
exactavailability region for a two-tier setup along with the
boundgiven by (31). Before stating the main result, denote
byk({j}\k) the maximum availability achievable for the kthtier BSs,
given the availabilities of the other K 1 tiers. It isclearly a
function of (1, . . . k1, k+1, . . . , K). Followingthe notation
introduced in (9), k({j} \ k) (denoted by kfor notational
simplicity) can be expressed as
k = gk(1, . . . k, . . . , K), (32)
where k is the solution to the fixed point equation given
theavailabilities of the other K1 tiers. Recall that while
definingk in terms of gk, we used Theorem 3, where we provedthat
strategy Sk(1) maximizes availability for any given tierand also
leads to the same set of fixed point equations asgiven by (9). In
Fig. 4, the solid line denotes 2(1), and thedotted line denotes
1(2). We remark on the achievability ofthe availabilities
corresponding to these lines for k maxk ,k K below.Remark 6
(Achievability of 1(2) and 1(2)). To showthat for 1 max1 , all the
points on 2(1) are achievable,consider point G = (G1 ,
G2 ) in Fig. 4. Given
G2 , the
maximum possible availability for first tier corresponds topoint
H on 1(
G2 ), which further corresponds to strategy
S(1). Clearly G1 1(G2 ), and hence achievable by
someuncoordinated strategy. One option is to time share betweenS(1)
and a fixed timer that keeps a BS OFF despite havingenergy to serve
users. The timer can be appropriately adjustedsuch that the
effective availability is G1 . Likewise, all thepoints on 1(2) are
also achievable. Clearly, this constructioneasily extends to
general K tiers.
Using these insights, we now derive the exact availabilityregion
for the set of uncoordinated strategies in the
followingtheorem.
Theorem 4 (Availability region). The availability region forthe
set of general uncoordinated strategies is
R = { RK : k k({j} \ k), k K}. (33)Proof: To show that R defined
by (33) is in fact the
availability region, it is enough to show that R isachievable
and / R is not achievable. For ease of exposition,we refer to Fig.
4 and prove for K = 2, with the understandingthat all the arguments
trivially extend to general K. To showthat R is achievable,
consider point E in Fig. 4. Thispoint is achievable by time sharing
between strategies thatachieve availabilities corresponding either
to points A and Bor C and D, which are all achievable as argued in
Remark 6.This clearly shows that there are numerous different ways
withwhich R is achievable. To show that the point / R isnot
achievable, consider point F = (F1 ,
F2 ) in Fig. 4. Note
that given F1 , the maximum availability possible for secondtier
is constrained by the corresponding value 2(
F1 ) on the
solid curve. Since F2 > 2(
F1 ), it contradicts the fact that
2(F1 ) is the maximum possible availability for second tier
given F1 . Hence point F is not achievable.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Availability of first tier (1)
Avai
labi
lity o
f sec
ond
tier (
2)
A
B
CD E
F G H
Fig. 4. Availability region for a two-tier HetNet. The upper
bound and theexact availability regions are respectively
highlighted in light and dark shades.Setup: = 4,K = 2, N1 = 10, N2
= 8, = 1.1, 1 = 2, 2 = 1, 2 =101, m1 = m2, 1 = 2.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Availability of first tier (1)
Avai
labi
lity o
f sec
ond
tier (
2)
Fig. 5. Availability region for a two-tier HetNet is denoted by
lightlyshaded region. The availability region when one of the tiers
is constrainedto use Sk(Nk) is denoted by the dark shade. Setup: =
4,K = 2, N1 =20, N2 = 15, = 1.1, 1 = 15, 2 = 5, 2 = 101, m1 = m2, 1
= 2.
Remark 7 (Effect of constraining the set of strategies onR).
Recall that k given by (32) and used in defining theavailability
region R corresponds to fixed point solution forstrategy S(1). In
principle, it is possible to restrict one of tiersto follow a
particular strategy by defining k as the fixed pointsolution for
that strategy. For instance, we could define k asa solution to the
fixed point equation corresponding to strategySk(Nk). Clearly, all
the points R will not be achievable inthis setup. For a two tier
setup, we plot the availability regionfor this case in Fig. 5,
along with the availability region Rdefined by Theorem 4. Note that
as expected the set of pointsachievable under this constrained
setup is strictly containedin the availability region defined by
Theorem 4.
-
10
We conclude this subsection with two remarks about theoptimality
of the availability region.
Remark 8 (Higher availability is not always better). It isnot
always optimal in terms of certain performance metricsto operate
the network in the regime corresponding to themaximum
availabilities. We will validate this in Section IVin terms of the
downlink rate. Interestingly, a similar idea,although applicable at
a much smaller time scale, of intention-ally making a macrocell
unavailable on certain sub-framescan be used to improve downlink
data rate by offloadingmore users to the small cells. This concept
is called almostblank sub-frames (ABS) and was introduced as a part
ofenhanced inter-cell interference coordination (eICIC) in 3GPPLTE
release 10 [34]. While this is an interesting analogy, thetwo
concepts are not exactly the same because in addition tothe
differences in the time scales, ABS additionally
assumescoordination across BSs.
Remark 9 (Notion of optimality). The performance of aHetNet with
energy harvesting is fundamentally the sameas the one with a
reliable energy source if for the givenperformance metric, the
optimal availabilities lie in theavailability region, i.e., R. For
example, if correspondsto point E in Fig. 4, the HetNet despite
having unreliableenergy source will achieve optimal performance. On
theother hand, if is, say, point F in Fig. 4, there will be
someperformance loss due to unreliability in energy
availability.
We now study the coverage probability and downlink ratein the
following subsection, which will be useful in the nextsection to
demonstrate the above ideas about optimality.
E. Coverage Probability and Downlink Rate
We now study the effect of BS availabilities {k} on thedownlink
performance at small time scale. As described inSection II, the
availabilities change on a much longer timescale and hence the
operational states of the BSs can beconsidered static over small
time scale. Therefore, for thisdiscussion it is enough to consider
the set of available BSs(a). For downlink analysis, we focus on a
typical userassumed to be located at the origin, which is made
possibleby Skivnyaks theorem [35]. Assuming full-buffer model
forinter-cell interference [17], i.e., all the interfering BSs in
(a)
are always active, the SIR at a typical user when it connectsto
a BS located at x (a)k is
SIR(x) =Pkh
(0)kxX (0)kx x
jKz(a)j \{x}
Pjh(0)jz X (0)jz z
. (34)
Using tools developed in [24], Theorem 1 of [18] can beeasily
extended to derive the coverage probability under thegeneral cell
selection model of this paper, which additionallyincorporates the
effect of shadowing. Since the extension isstraightforward, the
proof is skipped.
Theorem 5 (Coverage). The coverage probability is
Pc = P(SIR(x(0)) > ) =1
1 + F(, ) , (35)
where
F(, ) =(
2
2)
2F1
[1, 1 2
, 2 2
,
], (36)
and 2F1[a, b, c, z] =(c)
(b)(cb) 1
0tb1(1t)cb1
(1tz)a dt denotesGauss hypergeometric function.
Clearly, the coverage probability for
interference-limitedHetNets is independent of the densities of the
available BSs,and hence of the availabilities {k}. This validates
Remark 1.However, it is not necessarily so in the case of downlink
ratedistribution, which we discuss next. Assuming equal
resourceallocation across all the users served by a BS, the
complimen-tary cumulative distribution function (CCDF) of rate R
(inbps/Hz) achieved by a typical user, termed rate coverage Rc,is
calculated in [24] for the same cell selection model as thispaper.
Assuming the typical user connects to a kth tier BS,R can be
expressed as R = 1k log(1 + SIR(x(0)), wherek is the number of
users served by the kth tier BS to whichthe typical user is
connected. The approach of [24] includesapproximating the
distribution of k and assuming it to beindependent of SIR(x(0)) to
derive an accurate approximationof R. With two minor modifications,
i.e., the density of kthtier active BSs is kk, and the effective
density of activeusers is Pcu, the result of [24] can be easily
extended to thecurrent setup and is given in the following theorem.
For proofand other related details, please refer to Section III of
[24].This result will be useful in demonstrating the fact that
theoptimal downlink performance may not always correspond tothe
regime of maximum availabilities.
Theorem 6 (Rate CCDF). The CCDF of downlink rate R (inbps/Hz) or
rate coverage Rc is
P(R > T ) =n0
1
1 + F (n+1, )Kk=1
kkE[X 2k
]P
2
kjK
jjE[X 2j
]P
2j
3.53.5
n!
(n+ 4.5)
(3.5)
(PcuPkkk
)n(3.5 +
PcuPkkk
)(n+4.5)where n+1 = 2T (n+1) 1 and
Pk =kkE
[X 2k
]P
2
kjK jjE
[X 2j
]P
2j
. (37)
Remark 10 (Invariance to shadowing distribution). FromTheorem 6,
we note that the rate coverage is invariant tothe shadowing
distribution when E
[X 2j
]= E
[X 2k
], for all
j, k K. This is similar to the observations made in Remark
2.
IV. NUMERICAL RESULTS AND DISCUSSION
Since most of the analytical results discussed in this paperare
self-explanatory, we will focus only on the most importanttrends
and insights in this section. For conciseness, we assumelognormal
shadowing for each tier with the same mean mdB and standard
deviation dB. Recall that both the energyutilization and the rate
distribution results are invariant toshadowing under this
assumption, as discussed in Remarks 2
-
11
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Availability of first tier (1)
Avai
labi
lity o
f sec
ond
tier (
2)
N=1
N=10
N=100N=1000
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Availability of first tier (1)
Avai
labi
lity o
f sec
ond
tier (
2)
N=1
N=10
N=100N=1000
Fig. 6. (first) Availabilities region for various values of
energy storagecapacity N , where N1 = N2 = N . (second) One of the
tiers constrainedto use strategy Sk(N). Setup: = 4,K = 2, = 1.1, P
= [1, 0.1], 1 =10, 2 = 3, 2 = 101.
and 10. We begin by discussing the effect of battery capacityon
the availability region.
A. Effect of Battery Capacity on Availability Region
We consider a two tier HetNet and plot its availability
regionfor various values of the capacity of the energy storage
module,i.e., battery capacity, in the first subplot of Fig 6. For
ease ofexposition, we assume that the storage capacities of the BSs
ofthe two tiers are the same. As expected, the availability regionR
increases with the increase in battery capacity. Interestingly,it
is however not possible to achieve all the points in thesquare [0,
1] [0, 1] even by increasing the battery capacityinfinitely. The
maximum availability region is a function ofover-provisioning
factor , which is set to 1.1 for this result.Additionally, we note
that the maximum availabilities for boththe tiers approach unity
even at modest battery levels. Werepeat the same experiment for the
case when one of thetiers is constrained to use the strategy Sk(N)
and present the
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Availability of first tier (1)
Avai
labi
lity o
f sec
ond
tier (
2)
= 1.01
= 1.25
= 1.50
= 1.75
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Availability of first tier (1)
Avai
labi
lity o
f sec
ond
tier (
2)
= 1.01
= 1.25
= 1.50
= 1.75
Fig. 7. (first) Availabilities region for various values of .
(second) One ofthe tiers is constrained to use strategy Sk(N).
Setup: = 4,K = 2, N1 =20, N2 = 5, 1 = 10, 2 = 3, 2 = 101.
results in the second subplot of Fig. 6. Recall that this
casewas discussed in Remark 7. We observe that for the samebattery
capacity N , the achievable region is smaller in this casecompared
to Fig. 6, which is consistent with the observationsmade in Section
III. The difference is especially prominent forsmaller values of
battery capacity N .
B. Effect of Over-Provisioning Factor on Availability Region
We now study the effect of the over-provisioning factor onthe
availability region in the first subplot of Fig. 7. Recall thatthe
over-provisioning factor is the ratio of the net energyharvested
per unit area per unit time and the net energy utilizedper unit
area per unit time. The first and foremost observationis that
unlike increasing battery capacity, the availability regionexpands
by increasing and will cover the complete square[0, 1] [0, 1] for
sufficiently large . Also note that the beyonda certain value of ,
the availability of a tier may be non-zeroeven if the
availabilities of the other tiers are zero. This is thecase when
that tier harvests enough energy on its own to serve
-
12
all the load offered to the network, i.e., kk > Pcu. As inthe
previous subsection, we now repeat this experiment underthe
constraint that one of the tiers follows strategy Sk(Nk) andpresent
the results in the second subplot of Fig. 7. As expected,the
availability region is considerably smaller in this case.
C. Rate coverageUsing rate coverage, given by Theorem 6, we
demonstrate
that it may not always be optimal to operate the network in
theregime corresponding to maximum availabilities. We plot
ratecoverage as a function of (1, 2) for two different setups
inFig. 8. In both the cases, we note that it is strictly
suboptimalto operate at the point (1, 2) = (1, 1). Furthermore, as
thesecond tier density is increased, it is optimal to keep
firsttier BSs OFF more often. As expected, the rate coverage
alsoincreases with the increase of second tier density. This
exampleadditionally motivates the need for the exact
characterizationof for various metrics of interest, which forms a
concreteline of future work. Once the optimal for a given metricis
known, the system designers can, in principle, design theenergy
harvesting modules such that R. In such a case,the HetNet with
energy harvesting will have fundamentallyoptimal performance, i.e.,
the same performance as theHetNet with reliable energy sources.
V. CONCLUSIONSIn this paper, we have developed a comprehensive
frame-
work to study HetNets, where each BS is powered solelyby its
energy harvesting module. Developing novel toolswith foundations in
random walk theory, fixed point analysisand stochastic geometry, we
quantified the uncertainty in BSavailability due to the finite
battery capacity and inherentrandomness in energy harvesting. We
further characterizedthe availability region for a set of general
uncoordinated BSoperational strategies. This provides a fundamental
character-ization of the regimes under which the HetNets with
energyharvesting modules are fundamentally optimal, i.e., have
thesame performance as the ones with reliable energy sources.
This work has many extensions. From modeling perspective,it is
important to incorporate more accurate energy expendituremodels
taking into account energy spent on backhaul andcontrol signaling,
and model the energy required to transmit apacket to a user as a
function of the SIR. It is also importantto extend the developed
framework to study coordinatedstrategies. From optimality
perspective, it is important tocharacterize the optimal
availabilities as a function of keysystem parameters for various
metrics, such as the downlinkrate, so that the energy harvesting
modules can be accordinglydesigned. From physical layer
perspective, more sophisticatedtransmission techniques, such as
MIMO, should be taken intoaccount, e.g., using tools developed in
[36], [37]. From cellularperspective, it is desirable to consider
the effect of unreliableenergy sources on uplink, e.g., using tools
from [38].
APPENDIXA. Proof of Lemma 1
Denote by Pxk(a)k
() and Exk(a)k
[], the conditional (Palm)probability and conditional
expectation, conditioned on xk
00.2
0.40.6
0.81
00.2
0.40.6
0.810
0.02
0.04
0.06
0.08
0.1
0.12
0.14
Availability 2Availability 1
Rat
e co
vera
ge
00.2
0.40.6
0.81
00.2
0.40.6
0.810
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Availability 2Availability 1
Rat
e co
vera
ge
Fig. 8. Rate coverage as a function of 1 and 2. Setup: = 4,K =
2, P =[1, 0.01], T = 0.1, u = 1001. (first) 2 = 21, (second) 2 =
201.Note that = [1 1] is not optimal in both the cases.
(a)k . Please refer to [29], [35], [39] for details on Palm
calculus. Before we derive the average area, note that for
givenrealizations of the BS locations and the channel gains, the
areaof the service region of the kth tier BS is
|Ak(xk)| =R2
jK
x(a)j
1
(PkX (z)kxk z
PjX (z)jx z
)dz.
(38)
The average service area can now be expressed as
E[|Ak(xk)|] (a)= EExk(a)k
[|Ak(xk)|] (39)(b)= EE0
(a)k
[|Ak(0)|] (c)= EE(a)k [|Ak(0)|], (40)
where (a) follows by distributing the expectation over thepoint
process (a)k and the rest of the randomness, (b) followsfrom the
stationarity of the homogeneous PPP, and (c) followsfrom Slivnyaks
theorem [35]. Substituting the expressionfor |Ak(0)| in (40) and
distributing the expectation acrossvarious random quantities, we
can express the average area
-
13
as E[|Ak(xk)|] =
EXkR2
jK
E(a)j
x(a)j
EXj1(PkXkz
PjXjx z
)dz,
(41)
where the expectations over point processes (a)j and shad-owing
gains Xj can be moved inside respective product termsdue to
independence, and superscript on X (z)k and X (z)j areremoved for
notational simplicity. The expectation over pointprocess (a)j can
be evaluated using the probability generatingfunctional (PGFL)
[35], which simplifies the average areaexpression to
EXkR2
jK
ejjEXj
R2 1
(PkXkz 0, (49)
which completes the proof for the monotonicity property. Toshow
that the function is concave, we need to show that thedouble
derivative with respect to x is negative, which is
g(x) = a2N(ax)N2 1 ax(1 (ax)N )3 (
(N 1)(1 (ax)N+1)1 ax
ax(N + 1)(1 (ax)N1)1 ax
),
(50)
where the term inside the bracket is positive except at x = 1a
,where it has a minima and takes value 0. As in the case ofthe
first derivative, it is easy to show using LHopitals rulethat the
limit at this point is
limx 1a
g(x) = a2N2 16N
< 0, (51)
which shows that the function is strictly concave for all x
R.This completes the proof.
C. Proof of Lemma 3
For the proof of (11) (12), take the denominator of (11)to the
right hand side of inequality and multiply both sides byk to
get
kk
Kj=1
jjE[X 2j
]P
2j > kkPcuE
[X 2k
]P
2
k ,k K.
Now add all the K inequalities, i.e., sum both sides fromk = 1
to K, which leads to (12) and hence completes half ofthe proof. For
the proof of (11) (12), multiply both sidesof (12) by
Kj=1 jjE
[X 2j
]P
2j to get
Kk=1
kk
Kj=1
jjE[X 2j
]P
2j >
Kk=1
PcukkE[X 2k
]P
2j .
(52)
Rearranging the terms we get
Kk=1
k
kKj=1 jjE[X 2j
]P
2j
PcukE[X 2k
]P
2j
1 > 0. (53)
Since k is arbitrary, for the above condition to always hold,we
need the term inside the bracket to be positive for allk K. This
set of conditions is the same as (11) and hencecompletes the
proof.
-
14
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I IntroductionI-A Related WorkI-B Contributions
II System ModelII-A System Setup and Key AssumptionsII-B
Propagation and Cell Selection Models
III Availability AnalysisIII-A Modeling Energy Utilization
RateIII-B Availabilities for a Simple Operational StrategyIII-C
Availabilities for any General Uncoordinated StrategyIII-C1 Policy
1 (Sk(1))III-C2 Policy 2 (Sk(Nk))
III-D Availability RegionIII-E Coverage Probability and Downlink
Rate
IV Numerical Results and DiscussionIV-A Effect of Battery
Capacity on Availability RegionIV-B Effect of Over-Provisioning
Factor on Availability RegionIV-C Rate coverage
V ConclusionsAppendixA Proof of Lemma 1B Proof of Lemma 2C Proof
of Lemma 3
References
