Embed Size (px)
Citation preview
arX
iv:1
009.
0569
v3 [
cs.N
I] 5
Sep
201
21
Basic Performance Limits and Tradeoffs in EnergyHarvesting Sensor Nodes with Finite Data and
Energy StorageRahul Srivastava,Member, IEEEand Can Emre Koksal,Member, IEEE
Abstract—As many sensor network applications require de-ployment in remote and hard-to-reach areas, it is critical toensure that such networks are capable of operating unattendedfor long durations. Consequently, the concept of using nodes withenergy replenishment capabilities has been gaining popularity.However, new techniques and protocols must be developed tomaximize the performance of sensor networks with energyreplenishment. Here, we analyze limits of the performance ofsensor nodes with limited energy, being replenished at a variablerate. We provide a simple localized energy management schemethat achieves a performance close to that with an unlimitedenergy source, and at the same time keeps the probability ofcomplete battery discharge low. Based on the insights developed,we address the problem of energy management for energy-replenishing nodes with finite battery and finite data buffercapacities. To this end, we give an energy management schemethat achieves the optimal utility asymptotically while keepingboth the battery discharge and data loss probabilities low.
I. I NTRODUCTION
Advances in wireless networking combined with data acqui-sition have enabled us to remotely sense our environment [1],[2]. As these applications may require deployment in hard-to-reach areas, it is critical to ensure that such networks arecapable of operating with full autonomy for long durations.The lack of a continuous power source in most scenarios andthe limited lifetime of batteries have hindered the deploymentof such networks. However, developments in renewable energysources [3]–[8] suggest that it is feasible for sensor networksto operate unattended for extended periods. These renewablesources of energy typically provide energy replenishment at arate that could be variable and dependent on the surround-ings. Examples include, self-powered sensors that rely onharvesting strain and vibration energies from their workingenvironment [4], as well as sensors with solar cells [5]–[7].
In this paper, we analyze thelimits of the performanceofnetworks comprised of sensor nodes with limited energy, beingreplenished at a variable rate. We provide a simple localizedenergy management schemethat achieves a performance, closeto the optimal scheme that has access to an unlimited energyreservoir. Indeed, we show that, if the performance can bemeasured by a general utility function of the energy, under
Rahul Srivastava is with the Wireless Connectivity Group, BroadcomCorporation, Sunnyvale, CA (e-mail: [email protected]).
Can Emre Koksal is with the Department of Electrical and ComputerEngineering, The Ohio State University, Columbus, OH (e-mail: [email protected]).
This work was in part supported by NSF Grants CNS 0831919, CCF0916664, and CNS 1054738.
mild assumptions on the replenishment process, it is possibleto observe a polynomial decay for the probability of completebattery discharge, and at the same time achieve aΘ
((logM)2
M2
)
convergence to the optimal achievable utility1. HereM is thetotal capacity of the energy source. Based on the insightsdeveloped, we address the problem of energy management inthe presence of a finite data buffer. We modify our basic energymanagement scheme to achieve aΘ
((logK)2
K2
)
convergence tothe maximum utility achievable by a scheme that has access toan infinite data and energy buffers. HereK is the data buffersize. In addition, this scheme achieves an exponential decaywith M for the battery discharge probability and a polynomialdecay withK for the data loss probability. To evaluate thesedecay rates, the main tools we use are thelarge deviationstheoryandstochastic process limits.
The added dimension of renewable energy makes the prob-lem of energy management in sensor networks substantiallydifferent from its non-replenishment counterpart. For nodeswith replenishment, conservative energy expenditure may leadto missed recharging opportunities due to battery capacitylimitations. On the other hand, aggressive usage of energymay cause battery outages that leads to lack of coverage orconnectivity for certain time periods. Thus, new techniquesmust be developed to balance these seemingly contradictorygoals to maximize performance. Here, ourmain goalwill beto identify the performance limits of sensor nodes with energyreplenishment and provide guidelines to approach these limits.
Many fundamental wireless communication and networkingproblems can be stated asutility maximization problems,subject to energy constraints. The utility function can be thethroughput (e.g., in energy efficient routing), the probabilityof detection of an intruder (e.g., in coverage) or the networklifetime (e.g., in sleep-wake scheduling) or the achievablerate of reliable transmission in basic wireless communication.These problems have been mainly addressed for stations withunlimited and/or non-replenishing energy stores. Here, weaddress the problem of maximizing a utility function of thedata transmission rate in the presence of energy replenishment.The solution of the optimization problem requires stochasticoptimization techniques involving high computational over-heads that might be unsuitable for sensor nodes. Consequently,
1The following notations will be used to compare rates of convergence:an = O(bn) if an goes to zero at least as fast asbn; an = o(bn) if angoes to zero strictly faster thanbn; an = Θ(bn) if an andbn go to zero atthe same rate;an = Ω(bn) if an goes to zero no faster thanbn.
2
PSfrag replacements
r(t) eS(t)
MB(t) 0Fig. 1. Energy store with a replenishment rater(t).
we will focus our attention on simple localized solutions thatachieve near-optimal or asymptotically optimal performance.We use tools from large deviations theory and stochasticprocess limits to find closed-form expressions for the dataloss and the battery discharge probabilities. These techniquesallow us to analyze our schemes under mild assumptions onthe battery charging and data arrival processes.
There have been recent works that have studied differ-ent problems in networks with energy replenishment. Kar,et. al., [9] proposed an activation scheme for rechargeablesensors that maximizes the network-level utility of sensingnetworks. The utility function in [9] depends on the number ofactive sensors. Gatzianas, et. al., [10] used back pressurepoli-cies to maximize the network flow of information in networkswith energy replenishment. While [9], [10] look at the totalsystem utility, we will focus on the analyzing node-level per-formance leading to localized energy management schemes.Liu, et. al., [11] derived a battery control scheme similar to theone described in this work. In addition to providing strongerconvergence results than the one in [11] with sole batterycontrol, we also consider the effect of a finite data buffer inthispaper. Ozel and Ulukus [12] evaluated the Gaussian channelcapacity in the energy harvesting scenario and showed that thecapacity is unchanged for a class of replenishment process.Kansal, et. al., [13] introduced the concept of energy neutraloperation, wherein the energy consumed by a node is lessthan or equal to the energy harvested. Vigorito, et. al., [14]extended the idea of energy neutral operation to propose analgorithm that attempts to keep the battery state close to a fixedlevel and at the same time stabilizes the duty cycle in order tomaximize system performance. Sharma, et. al., [15] proposeda throughput optimal energy management scheme for energyharvesting nodes. Ho and Zhang [16] solved the problem ofoptimal energy allocation in energy harvesting nodes usingdynamic programming techniques. However, [13]–[16] do notcontain an analytical evaluation of the battery discharge or thedata loss probabilities for their energy management schemes.
The outline of this paper is as follows. We first state thegeneral form of the utility maximization problem in SectionIIand show ways to achieve the maximum achievable utility withreplenishing sources. In Section III we add a finite buffer totheproblem and study energy management schemes that achieveoptimal utility asymptotically while keeping the probabilitiesof battery discharge and data loss low. We numerically eval-uate the performance of our energy management schemes inSection IV. We wrap up with conclusions in Section V.
II. A CHIEVING MAXIMUM UTILITY WITH A
FINITE-BATTERY CONSTRAINT
A. System Model and Problem Statement
Fig. 1 shows the energy store (or the battery) of a node. Thetotal capacity of this battery isM units of energy. We denotethe total available energy in the battery asB(t), wheret is the
discrete time index. The battery replenishes at a rater(t). Theprocessr(t), t ≥ 1 is assumed to be an ergodic stochasticprocess with a long term meanlimτ→∞
1τ
∑τt=1 r(t)
a.s.→ µ. An
energy management schemeS draws energy from this batteryat a rateeS(t) to achieve certain tasks. The success of thenode in achieving these tasks is measured in terms of a utilityfunctionU(eS(t)) of the consumed energyeS(t). We assumeU(e) to be a concave, non-decreasing2 and analytic functionof e over e ≥ 0. We define the time average utility,
US(τ) =1
τ
τ∑
t=1
U(eS(t)). (1)
We consider the optimization problem in which a nodetries to maximize its long-term average utility,US =lim infτ→∞ US(τ), subject to battery constraints:
maxeS(t), t≥1
US (2)
subject to B(t) = minM,B(t− 1) + r(t) − eS(t− 1)
and eS(t) ≤ B(t).
One approach to solving this optimization problem is by usingMarkov decision process (MDP) techniques. Since solvingMDPs is computationally intensive, these methods may notbe suitable for computationally-limited sensor nodes. Con-sequently, we seek schemes that are easy to implement andyet achieve close to optimal performance. The next lemmagives an upper bound for the asymptotic time-average utilityachieved over all ergodic energy management policies.
Lemma 1. Let US∗
be the solution to Problem (2). Then,US∗
≤ U(µ).
The proof of this lemma, given in Appendix A, usesJensen’s inequality and conservation of energy arguments.Lemma 1 tells us that for any ergodic energy managementschemeS, US ≤ U(µ). With an unlimited energy reservoir(i.e.,M = ∞) and average energy replenishment rateµ, if oneuseseS(t) = µ for all t ≥ 1, this upper bound can be achieved.However, ifM < ∞, achievingUS = U(µ) using this simplescheme is not possible. Indeed, due to finite energy storageand variability inr(t), B(t) will occasionally get dischargedcompletely. At such instances,eS(t) has to be set to0, whichwill reduce the time-average utility. The question we answeris, “how close can the average utilityUS get to the upperbound asymptotically, asM → ∞, while keeping the long-term battery discharge rate low?”
B. An Asymptotically Optimal Energy Management Scheme
In this section we show that there is a trade-off betweenachieving maximum utility and keeping the discharge rate low.First, we make some weak assumptions on the replenishmentprocessr(t), which we will be using throughout this paper. Inparticular, we assume that the asymptotic semi-invariant log
2Note that, in many practical scenarios, it is reasonable to assume that theutility function is non-decreasing and concave, due to diminishing returns forincreasing power.
3
PSfrag replacementsU(eS(t))
eS(t)µ
U+
U−
δB
δB
Fig. 2. With schemeB, utility alternates betweenU+ andU−.
moment generating function,
Λr(s) = limτ→∞
1
τlogE
[
exp
(
s
τ∑
t=1
r(t)
)]
, (3)
of r(t) exists for s ∈ (−∞, smax), for some smax >0. We also assume that the asymptotic varianceσ2
r ,
limτ→∞1τ var(
∑τt=1 r(t)) of r(t) exists3. Note that, in prac-
tice, the recharging process is not necessarily stationary. Whilethis assumption does allow the possibility that the statistics ofr(t) has variations (e.g., due to clouds and the solar powerat different times of the day), it rules out the possibility oflong-range dependencies inr(t).
From the discussion in previous section, we can infer thatby choosing a battery drift, defined asr(t) − e(t − 1), thatgoes to zero with increasing battery size, one might achievealong-term average utility that is close toU(µ) asM increases.However, smaller drift away from the empty battery stateimplies a more frequent occurrence of the complete batterydischarge event. In the following theorem, we quantify thistradeoff between the achievable utility and the battery dis-charge rate, asymptotically in the large battery regime. Inthisregime, the battery sizeM is large enough for the variationsin r(t) to average out nicely over the time scale thatB(t)changes significantly. Consequently, we now define the long-term battery discharge rate as the probability of discharge, i.e.,pdischarge(M) , limτ→∞
1τ
∑∞t=1 I
B0 (t), where the indicator
variableIB0 (t) = 1 if B(t) = 0 and is identical to0 otherwise.
Next, we show that one can achieve a battery dischargeprobability that exhibits a polynomial decay of arbitraryorder with the battery size, and at the same time achievesa utility that approaches the maximum achievable utility as(logM)2/M2.
Theorem 1. Consider any continous, concave, non-decreasing, and analytic utility functionU(e(t)) over the non-
negative real line such that∣∣∣∂2U(e)∂e2
∣∣∣ < ∞ for all e > 0.
Given anyβ ≥ 2, there exists an energy management schemeB such that the associated battery discharge probability
pBdischarge(M) = Θ(M−β) andU(µ)− UB = Θ
((logMM
)2)
.
We give a brief sketch of the proof, details of which canbe found in Appendix B. Our proof is constructive as we
3Examples of valid processes include the following. 1) Any i.i.d. processwith a sample distribution that has finite moments of all orders; 2) AllGaussian processes with an autocovariance function that has a finite integral;3) The process obtained by adding a deterministic periodic function of time(to mimic the daily cycles of solar radiation) to the aforementioned processesin 1), 2).
show a strategy that achieves the asymptotic convergence ratesgiven in Theorem 1. Our scheme is motivated by the buffercontrol strategy introduced in [17] to achieve the near-optimaldistortion for variable rate lossy compression. Consider theallocation schemeB in which
eB(t) =
minµ− δB, B(t), B(t) < M/2
µ+ δB, B(t) ≥ M/2, (4)
for someδB > 0. As shown in Fig. 2, the instantaneous utilityassociated with SchemeB alternates betweenU− and U+,depending on the battery state. By choosingδB1 = βσ2
rlogMM
for someβ ≥ 2, we show that long-term maximum utilityU(µ) can be achieved asymptotically while achieving decay,as a polynomial of arbitrarily high order, for the batterydischarge probability. We note that while the order of thepolynomial decayβ can be made arbitrarily large, it comes atthe expense of slower convergence (by some constant factor)to the maximum utility.
Here, we illustrated that with a simple scheme, it is possibleto achieve desirable scaling laws for the performance of agiven task, under the assumption that the asymptotic momentgenerating function of the replenishment process exists. Toillustrate the theorem we consider a specific example.
Example 1Achievable Rate in a Gaussian Channel:We study thebasic limits of point to point communication with finite butreplenishing energy stores. For simplicity, we consider thestatic Gaussian channel. At timet, the transmitter transmits acomplex valued block (vector of symbols)X(t) of unit powerand the receiver receivesY(t). We have,
Y(t) = hX(t) +W(t), (5)
where the channel gainh is a complex constant andW(t) isadditive Gaussian noise with sample varianceN0. We definethe channel SNR asγ , |h|2/N0. The maximum amountof data that could be reliably communicated [18] over thischannel with an amount of energye(t) at time t is:
C(e(t)) = log2 (1 + e(t)γ) bits/channel use, (6)
assuming the block size is long enough so that sufficientaveraging of additive noise is possible. Thus, the rate at whichreliable communication can be achieved at a given block is aconcave non-decreasing function of the transmit power andit can be viewed as our utility function. Consequently, usinga constant powerµ, the maximum utility ofC = C(µ) canbe achieved, which is the famous Gaussian channel capacityresult. Clearly, the capacity is possibly achievable, onlyif theenergy store is infinite.
With an energy store that is not capable of providing powerat a constant rate (e.g., an energy replenishing battery), onemay observe outages due to occurences of complete dischargeat times. Thus, for such stores, it is not possible to achievetheaforementioned Gaussian channel capacity. However, we canshow that, using our simple energy management scheme, onecan achieve an average rate that converges to the capacity atan outage probability that converges to zero asymptotically as
4
M → ∞. We assume that each time slot is large enough forsufficiently long code blocks to be formed.
We simply substituteU(·) with C(·) in Eq. (1) to getthe relevant optimization problem. With an unlimited energystore (M = ∞) of limited average powerµ, the maximumachievable long term average rate is identical to the channelcapacity, i.e.,C(µ) = log2(1+µγ) bits/channel use. By usingthe energy management schemeB given in Eq. (4), an averagerateCB can be achieved such thatC(µ)−CB = Θ
((logM)2
M2
)
while the battery discharge (i.e., the outage) probabilityfol-lows pBdischarge(M) = Θ(M−β) for any givenβ ≥ 2.
C. Basic Limits of Energy Management Schemes
To understand the strength of Theorem 1, we note that it isnot trivial to achieve decaying discharge probability and max-imum utility with increasing battery size. In fact, an ergodic4
energy management scheme cannot achieve exponential decayin discharge probability and convergence (even asymptotically)to the maximum average utility function simultaneously. Weformalize this statement in the following theorem.
Theorem 2. Consider any continous, concave and non-decreasing utility functionU(·). If an ergodic energy manage-ment schemeS has a discharge probabilitypSdischarge(M) =Θ(exp(−αcM)) for some constantαc > 0, then the timeaverage utility,US , for SchemeS satisfiesU(µ)−US = Ω(1).
The proof of this theorem is provided in Appendix C andit is similar to that of Theorem 1. We applylarge deviationstechnique to the net drift of the battery process to find thedecay rate ofpSdischarge(M) with M . Jensen’s inequality is thenused to lower bound the difference betweenU(µ) and US .
So far, we have shown how to maximize a concave non-decreasing utility function subject to battery constraints. Atevery point in time, one should choose a power level as close tothe replenishment rate as the battery constraints allow andthisway one can asymptotically achieve a performance very closeto that with unlimited energy stores. The main limitation ofthisapproach is that it may not be feasible for some applicationsin practice. For instance in many sensor network applications,data is stored in finite buffers for transmission. Since schemeB does not adapt to the buffer state, this may lead to datalosses. To overcome these limitations, in the next section,we investigate energy management schemes with buffer andbattery constraints.
III. A CHIEVING MAXIMUM UTILITY WITH FINITE
BUFFER AND BATTERY CONSTRAINTS
A. System Model and Problem Statement
In this section, we extend the problem introduced in Sec-tion II to the case when data packets arrive at a node andare kept in a finite buffer before transmission. Hence, thetask is to transmit packets arriving at the data buffer withoutdropping them due to exceeding the buffer capacity. We defineQ(t) as the data queue state at timet, and the data buffer
4An ergodic energy management schemeeS(t) is the one that satisfieslimτ→∞
1τ
∑τt=1 e
S(t) = E[
eS(t)]
−5 −4 −3 −2 −1 0 1−25
−20
−15
−10
−5
0
5
drift in queue state
drift
in b
atte
ry s
tate
Fig. 3. Possible drift directions for(Q(t), B(t)) for a Gaussian channel ofchannel SNR0 dB. Here, at timet, r(t) = 0, a(t) = 0.
size is K < ∞. The data arrival processa(t), representsthe amount of data (in bits) arriving at the data buffer in thetime slot t. The processa(t), t ≥ 1 is an ergodic processindependent of the energy replenishment processr(t), t ≥ 1andE [a(τ)] = λ. We assume that the processa(t) has a finiteasymptotic varianceσ2
a = limτ→∞1τ var(
∑τt=1 a(t)). The
energy replenishment model is the same as used previously.We useC(·) as given in Eq. (6) as the rate-power function(continuous, concave, non-decreasing, and analytic) for thewireless channel and assume that data is served at that rateas a function of the consumed energye(t) at time t. Wealso assume thatλ < C(µ). Without this condition, thereexists no joint energy and data buffer control policy that cansimultaneously keep the long-term battery discharge and dataloss rates arbitrarily low asymptotically, asK,M → ∞.
The objective of an efficient energy management schemein this case is to maximize the average utility function of thedata transmitted subject to battery and data buffer constraints:
maxe(t), t≥1
lim infτ→∞
1
τ
τ∑
t=1
UD(C(e(t))) (7)
subject to B(t) = minM,B(t− 1) + r(t) − e(t− 1),
Q(t) = minK,Q(t− 1) + a(t)− C(e(t− 1)),
0 ≤ e(t) ≤ B(t) and C(e(t)) ≤ Q(t).
Here UD(C(e)) is a non-decreasing, concave, and analyticutility gained by transmittingC(e) bits. Sinceλ < C(µ),we know thatUD(λ) is an upper bound on the achievablelong-term utility with any energy management scheme. Thisstatement can be proved using Jensen’s inequality, followingidentical steps as the proof of Lemma 1 and we skip it toavoid repetition.
B. An Asymptotically Optimal Energy Management Scheme
Solution of Problem (7) jointly controls the data queue stateand the battery state to avoid energy outage and data overflowwhile maximizing the utility. The main complexity in such anapproach stems from the fact that the drifts ofQ(t) andB(t)are dependent. In Fig. 3 we illustrate the connection betweenthe service rate and the energy consumed at a time slott foran Gaussian channel with SNR0 dB anda(t) = 0, r(t) = 0.For instance, to provide 3 units of service, the node needs toconsume∼ 18 units of energy.
With this dependence, a critical factor one needs to takeinto consideration is the relative “size” of the data bufferwith respect to the battery. In the sequel, we assume alarge
5
battery regime, which implies that, within the duration thatsome change occurs inB(t), Q(t) may fluctuate significantly.Technically, for an Gaussian channel with an SNRγ, thisassumption impliesM ≫ 1
γ (2λ − 1)K, i.e., the total amount
of energy in the battery is much larger than that required toserve a full data buffer worth of packets. In the subsequentasymptotic results, in which bothK,M → ∞, the largebattery regime implies the following. For all sequences ofvalues,Kn,Mn, where both sequence goes to∞ asn → ∞,we assumeKn/Mn → 0 asn → ∞.
Intuitively, in large battery regime, an energy control algo-rithm should give “priority” to adjusting the queue state toachieve a high performance. Consequently, it should choosee(t) such that the drift ofQ(t) is always toward a desiredqueue state even though this may cause battery drift to benegative. Since battery size is large, such temporary negativedrifts are expected to affect the battery discharge rate onlyminimally. With these observations, we state the followingtheorem, which indeed verifies our intuition. This theoremshows an asymptotic tradeoff between the achieved utility andthe long-term rates of discharge and data loss asK → ∞.In this regime, the data buffer size is large enough for thevariations in a(t) to average out over the time scale thatQ(t) changes significantly. Consequently, we now define thelong-term data loss rate as the data loss probability, i.e.,pQloss(K) , limτ→∞
1τ
∑τt=1 I
QK(t), where the indicator vari-
ableIQK(t) = 1 if Q(t) = K and is identical to0 otherwise.
Theorem 3. Consider any non-decreasing concave utilityfunctionUD(·) such that
∣∣∣∂2UD(C(e))
∂e2
∣∣∣ < ∞ for all e > 0 and
a rate-power functionC(·), both of which are analytic in thenon-negative real line. For anyλ < C(µ), there exists someβ > 0 for which an energy management schemeQ achieves adata loss probabilitypQloss(K) = O(K−β), a battery dischargeprobabilitypQdischarge(M) = O(exp(−αQM)) for someαQ > 0
and a utility that satisfiesUD(λ)− UQ = Θ(
(logK)2
K2
)
underthe large battery regime.
Theorem 3 states that it is possible to have an exponentialdecay (withM ) for the battery discharge probability and apolynomial decay (withK) for the data loss probability andat the same time achieve a time average utility that approachesthe upper bound on the achievable long-term utility,UD(λ), as(logK)2/K2. Note thatUD(λ) can only be achieved with aninfinite battery and data buffer sizes. We provide an outlineforthe proof, a full version of which can be found in Appendix D.The proof is constructive as we first present schemeQ, andthen derive the performance metrics for this scheme.
Consider the energy management schemeQ, where
eQ(t) =
minµ− δ(r)1 , B(t), Q(t) ≥ K/2
minµ− δ(r)2 , B(t), Q(t) < K/2
, (8)
and the driftsδ(r)1 andδ(r)2 are chosen to satisfy the relationship
C(µ− δ(r)1 )− λ = λ− C(µ− δ
(r)2 ) = βQσ
2a
logK
K, (9)
whereβQ is constant greater than 2. From Fig. 4, we notethat this choice of energy drifts correspond to a queue driftof
PSfrag replacements
Rate
Power
δ(a)
δ(a)λ
µ
δ(r)1
δ(r)2
QueueEncoder/
Transmitter
FSMCDecoder/
Receiver
Feedback
Delay d
Fig. 4. Relationship betweenδ(a), δ(r)1 andδ(r)2 .
PSfrag replacements
00
M M
(K,M)
K/2K/2 KK
δ(r)1δ
(r)2
δ(a)δ(a)
Queue
QueueState
QueueState
BatteryState
BatteryState
QueueDrifts
BatteryDrifts
Fig. 5. Drifts of the data queue and battery state for schemeQ.
|δ(a)| = βQσ2alogKK , toward the stateK/2, regardless of the
queue stateQ(t). The queue and battery drifts with schemeQare illustrated in Fig. 5. We observe that even though schemeQ regulates the data queue to a desired state (i.e.,K/2), thebattery is always regulated towards full state (i.e.,M ). Stateequation forQ(t) is given by,
Q(t+ 1)
=
minK,Q(t) + a(t)− λ− δ(a), Q(t) ≥ K/2
max0, Q(t) + a(t)− λ+ δ(a), Q(t) < K/2, (10)
and the state equation forB(t) is given by,
B(t+ 1)
=
minM,B(t) + r(t) − µ+ δ(r)1 +, Q(t) ≥ K/2
minM,B(t) + r(t) − µ+ δ(r)2 +, Q(t) < K/2
,
(11)
wherea+ = max0, a.The main challenge in the proof of Theorem 3 is the
coupling of the two queues. More specifically, the battery driftin a particular time slot depends on the data queue state, whicheliminates the possibility of the application of large deviationtechniques for calculatingpQdischarge(M) andpQloss(K) difficult.Indeed, closed-form analysis of the stationary distribution forthe state of the two-dimensional finite queueing processesis not possible except in some special cases given in [19].Since our model does not fall in that category, the steady stateprobabilities of loss and discharge cannot be derived in closedform. To show the desired order results for SchemeQ, wetransform the problem in two steps as follows:(T1) We remove the upper and lower boundaries for the databuffer and the battery respectively, and allowQ(t), t ≥0 to take on values in the entire[0,∞) region andB(t), t ≥ 0 to take on values in the entire(−∞,M ]region. Then, under schemeQ as given in (8), we definepQoverflow(K) , limτ→∞ P (Q(τ) > K) and pQunderflow(M) ,
6
limτ→∞ P (B(τ) < 0). Using Theorems 1 and 2 in [20], onecan see thatpQloss(K) = O(K−β) for someβ > 0 if andonly if there exists someβQ > 0 such thatpQoverflow(K) =O(K−βQ). Similarly, one can obtain from Theorem 1 and 2in [20] that pQdischarge(M) = O(exp(−αQM)) if and only if
pQunderflow(M) = O(exp(−αQM)). Thus, it suffices to showthe desired scaling laws for the aforementioned unboundedqueue state and battery state processes.(T2) Next, we construct a sequence of arrival rates such thatλ ↑ C(µ). In this limiting regime, from Eq. (9),δ(r)1 , δ
(r)2 ↓ 0
and henceδ(a) ↓ 0, for which we also use a sequence ofK values that increase to∞ to satisfy the second equalityin Eq. (9). As a result, both the battery and the data queuewill operate in theheavy traffic limit. We denote the dataqueue state and the battery state processes in the associateddiffusion limitwith Q(t) andB(t), respectively. Note that, theprobabilities for overshooting the boundaries calculatedfor theassociated diffusion limits,P (Q(t) ≥ K) and P (B(t) ≤ 0)are identical toP (Q(t) ≥ K) andP (B(t) ≤ 0), respectivelyin the heavy traffic limit (Chapter 5 [21]). Furthermore, sincethe heavy traffic limit poses a worst case for the probabilitiesunder consideration, the order results of the form O(·) shownin the heavy traffic limit hold for allλ < C(µ).
However, it is still not straightforward to calculate theassociated probabilities in the diffusion limit, since neitherQ(t), nor B(t) will yield a Brownian motion(BM), due tostate-dependent variable drifts. To that end, we define upper-half queue state process,Qu(t), as the queue state processwhen the state is aboveK/2. This process is formed by takingthe sample path ofQ(t) and putting the segments for whichQ(t) > K/2 in a sequence, next to each other (as will beillustrated in Fig. 12). Thus,Qu(t) ≥ K/2 for all t with prob-ability 1. Now, one can see thatlimτ→∞ P (Qu(τ) ≥ K) ≥pQoverflow(K). Thus if we prove thatlimτ→∞ P (Qu(τ) ≥ K) =O(K−βQ) for some βQ > 0, then it is also true thatpQoverflow(K) = O(K−βQ) for that βQ. The good news is that,since Qu(t) has a constant drift, under the diffusion limit,it will have a Brownian analogueQu(t), which means thatthe calculation of the desired probability is easy. Using theproperties of BM, we show in Appendix D that, indeed withSchemeQ, limτ→∞ P (Qu(τ) ≥ K) = O(K−βQ). To achievethat, we first define a unit reward every timeQu(t) goes aboveK. Using renewal-reward theory, we find
pQoverflow(K) =δ(a)
2
σ2a
exp
(
−δ(a)K
σ2a
)
. (12)
By substitutingδ(a) = βQσ2alogKK , we have the desired scaling
law for the queue overflow probability.Similarly, we denote the diffusion limit of the battery pro-
cessB(t) by B(t), which does not constitute a Brownian mo-tion, due to its state-dependent drift. To show the exponentialdecay rate for the undershoot probability forB(t), define a BMthat lower bounds any given sample path ofB(t). We showthat limτ→∞ P (Bl(τ) ≤ 0) scales as O(exp(−αQM)), whichimplies the same scaling law for the underflow probability ofB(t). Finally, proof for the convergence of the time averageutility follows the same line of argument to that for Theorem1.
PSfrag replacements
Rate
Power
δ(a)δ(a)
λ
C(eE)
µeE
δ(r)
δ(r)
Fig. 6. Relation betweenδ(a) and δ(r).
C. Exploring Tradeoffs Between Battery Discharge and BufferOverflow Probabilities
So far, we focused on achieving performance that was closeto the optimal while keeping the probabilities of dischargeanddata loss low. In this section we look at quantifying tradeoffbetween the probabilities of battery discharge and data loss.
Theorem 4. For a channel with a rate-power functionC(·)that is continous atµ, there exists an energy managementschemeE that simultaneously achieves
limM→∞
limλ↑C(µ)
1
Mδ(r)log pEunderflow(M) = −
2
σ2r
, (13)
limK→∞
limλ↑C(µ)
1
Kδ(a)log pEoverflow(K) = −
2
σ2a
, (14)
whereδ(r) = ν(µ− C−1(λ)
), δ(a) = C(µ−δ(r))−λ for any
ν ∈ (0, 1).
The proof of this theorem (given in Appendix E) is construc-tive. We consider a energy management schemeE , where,
eE(t) = minµ− δ(r), B(t), (15)
for all t, with δ(r) = ν(µ− C−1(λ)
)for someν ∈ (0, 1).
The mean drifts for the battery state and the data queue stateare given byδ(r) and C(µ − δ(r)) − λ, respectively. In thelimiting regimeλ ↑ C(µ), δ(r) ↓ 0 and henceC(µ − δ(r)) −λ ↓ 0. As a result, both the battery and the data queue willoperate in the heavy traffic limit and we can apply the diffusionlimits on these processes to get the required probability results.Fig. 6 illustrates the relationship betweenδ(a) and δ(r). Anyincrease inδ(r) would lead to a corresponding decrease inδ(a).Sinceδ(r) is proportional to the discharge probability decayexponent andδ(a) is proportional to the data loss probabilitydecay exponent, we will observe the given tradeoff.
Theorem 4 shows that, in the heavy traffic limit, we observean exponential decay for both the battery underflow and thebuffer overflow probabilities, with the battery size and thedatabuffer size respectively. However, one can also see that, thereis a tradeoff in the decay exponents of these two probabilies.More specifically, by varyingδ(r), it is possible to increase(or decrease) the decay exponent for the data loss probability.However this will result a proportional decrease (or increase)in the decay exponent for the battery discharge probability.
IV. N UMERICAL EVALUATION
Our theorems illustrate tradeoffs for energy managementschemes in the buffer and battery size asymptotic regimes
7
and showed optimality of some simple energy managementschemes. In this section, we conduct simulations to evalu-ate the performance of those schemes in the presence ofa finite battery and a finite data buffer. We construct theenergy replenishment processr(t) using the real solar radi-ation measurements collected at the Solar Radiation ResearchLaboratory [22]. The data set used is the global horizontalradiation or the total solar radiation using a Precision SpectralPyranometer. We use data from January 1999 to July 2010collected at 1 minute intervals.
In our simulations, we chose a battery with storage capacityin the range of10-103 J. For the replenishment, we considereda 10 cm2 solar panel with1% overall efficiency to get the long-term average of the energy replenishment processµ = 1.92mW. We used the Gaussian channel capacity as the utilityfunctionU(e) = log(1+γe), where channel SNRγ = |h|2/N0
was defined in Example 1. Here, we take|h|2 = d−κ, wheredis the distance between the transmitter and the receiver andκ isthe path loss exponent. In the simulations, we letN0 = −70dBm, d = 51.8 m, and κ = 3.5, which gives us a meanchannel SNR of12.83 dB. Fig. 7(a) shows a sample of thesolar irradiation process over a 48 hour period.
A. Battery Constraints with Infinitely Backlogged Buffer
In Fig. 7, we revisit the energy management schemeBdiscussed in Example 1 for an infinitely backlogged databuffer. The communication channel is Gaussian and we choosethe polynomial decay exponentβ = 2. To illustrate The-orems 1 and 2 simultaneously, we define an energy man-agement schemeC, which allocates a power strictly lessthan the average replenishment rate. In particular,eC(t) =minB(t), µ−c+, wherec is a constant. From Theorems 1and 2 we know that policyC achieves an exponential decayfor discharge probability compared to the quadratic decayfor schemeB. On the other hand, policyC can not achievethe maximum utility while the utility achieved by schemeBshould approach maximum utility as(logM)2/M2. Fig. 7(b)plots the battery discharge rate as a function of the batterysize. As expected, policyC performs better than schemeB. However, the advantage of using policyB is evidentin Fig. 7(c), which compares the normalized time averageutilities, UC
U(µ) and UB
U(µ) , achieved by each scheme. It can beseen that, for the choice of parameters used in this simulation,schemeB achieves the maximum utilityU(µ) for a batterysize of 250 J, whereas the schemeC does not achieve themaximum utility even asymptotically.
B. Buffer and Battery Constraints
Fig. 8 compares the performance of energy managementschemes when both battery and buffer constraints are present.We simulate the data arrival process by generating a Markov-modulated Poisson process with meanλ = 4.12 bits per timeslot. We use a two-state Markov chain to generate a burstydata arrival process. One state of the Markov chain generatesa Poisson random variable with mean 10 bits and the otherstate generates a Poisson random variable with mean 1. The
data utility function is chosen asUD(x) = log2(1 + 10x
λ
). As
previously,µ = 1.92 mW and we chooseβ = 10.To compare with our scheme, we also simulated the perfor-
mances of Throughput-Optimal (TO) and modified TO (MTO)policies policy given in Eq. (4) and Eq. (6) of [15] respectively.The energy allocation by TO policy, which is an instance ofpolicy C, is specified as:
eTO(t) = minB(t), µ− ǫ, (16)
whereǫ is a constant such thatC(µ− ǫ) > λ; and the energyallocation for MTO policy is given by:
eMTO(t) =minC−1(Q(t)), B(t),
0.99(µ+ 0.001[B(t)− 0.1Q(t))+]. (17)
In Fig. 8(a), we fix the buffer size to104 bits and plot thebattery discharge rate as a function of the battery sizeM . Thedischarge rates should decay exponentially for both schemes.However, the decay exponent for schemeQ is larger than thedecay exponent for the TO scheme. Indeed, Theorem 4 showsthat, in the heavy traffic limit, the decay exponent for thedischarge probability is proportional to the drift of the batterystate. Thus, for small values ofµ−C−1(λ), the decay exponentfor schemeQ is approximately proportional toµ − C−1(λ),while the decay exponent for the TO scheme is approximatelyproportional toǫ < µ − C−1(λ), sinceǫ is the drift of thebattery state as given in Eq. (16).
In Fig. 8(b), we plot the data loss rate as a function of thebuffer size while keeping the battery size fixed at103 J. Weobserve that the loss rate for the TO scheme decays faster thanthat for schemeQ. This trend is expected as the TO schemeshould have an exponential decay compared to a quadraticdecay for schemeQ. Achieving an exponential decay in thedata loss rate comes at the cost of reduced average utilityfor the TO scheme. On the other hand, the MTO scheme isdesigned to achieve a low average data queue length. As aresult, we observe that the data loss rate of the MTO schemeis lower than the other two schemes. However, the MTOscheme pays the price of slightly higher battery dischargerate. Fig. 8(c) compares the convergence of the time averageutilities to the maximum utility function for the two schemes.We observe that schemeQ converges to the maximum utilityfor larger buffer sizes (∼ 5000 bits). On the other hand, TOand MTO schemes do not achieve the optimal utility.
Fig. 9 compares the performance of energy managementschemes with increasing traffic intensity. We define trafficintensity asρ , λ
C(µ) = λlog2(1+γµ) . We fix the buffer
length at2000 bits and battery capacity is set at100 J. InFig. 9(a) we observe that the discharge rate increases withtraffic intensity. For values ofρ = 0.87, schemeQ performsalmost an order of magnitude better than the TO scheme interms of the discharge rate. For traffic intensities close tounity the schemeQ degenerates to the TO scheme and theirperformances converge. Fig. 9(b) shows that the data lossrates for both schemes also increases with increasing trafficintensity. Similar to the discharge rate, for values ofρ = 0.87,the loss rate for schemeQ is almost an order of magnitudelower than that for the TO scheme. This can be explained
8
0 500 1000 1500 2000 25000
100
200
300
400
500
600
700
Time, t (minutes)
Tota
lSola
rR
adia
tion
(W/m
2)
(a) Sample of the replenishment processr(t) overa 48 hour period.
10 100 100010
−3
10−2
10−1
100
Battery Size, M (J)
Bat
tery
Dis
char
geR
ate
Scheme C
Scheme B
(b) Battery discharge rate scaling with battery sizeM .
10 100 10000.84
0.86
0.88
0.9
0.92
0.94
0.96
0.98
1
1.02
Battery Size, M (J)
Nor
mal
ized
Tim
eA
ver
age
Uti
lity
Maximum Utility (U (µ))
Scheme C (UC)Scheme B (UB)
(c) Time average utility scaling with battery sizeM .
Fig. 7. Performance evaluation for the Gaussian channel example.
10 100 100010
−4
10−3
10−2
10−1
100
Battery Size, M (J)
Bat
tery
Dis
char
geR
ate
TO Scheme
MTO Scheme
Scheme Q
(a) Battery discharge rate scaling with battery sizeM .
102
103
104
10−6
10−5
10−4
10−3
10−2
10−1
100
Buffer Size, K (bits)
Dat
aLos
sR
ate
TO Scheme
MTO Scheme
Scheme Q
(b) Data loss rate scaling with buffer sizeK.
102
103
104
0.84
0.86
0.88
0.9
0.92
0.94
0.96
0.98
1
1.02
Buffer Size, K (bits)
Nor
mal
ized
Tim
eA
ver
age
Uti
lity
Maximum Utility (UD(µ))
TO Scheme (UT O
D)
MTO Scheme (UMT O
D)
Scheme Q (UQD
)
(c) Time average utility scaling with buffer sizeK.
Fig. 8. Performance evaluation for energy management schemes under buffer and battery constraints.
0.8 0.85 0.9 0.95 110
−4
10−3
10−2
10−1
Traffic Intensity, ρ
Bat
tery
Dis
char
geR
ate
TO Scheme
Scheme Q
(a) Battery discharge rate scaling with traffic inten-sity ρ.
0.8 0.85 0.9 0.95 110
−4
10−3
10−2
10−1
Traffic Intensity, ρ
Dat
aLos
sR
ate
TO Scheme
Scheme Q
(b) Data loss rate scaling with traffic intensityρ.
0.5 0.6 0.7 0.8 0.9 10.86
0.88
0.9
0.92
0.94
0.96
0.98
1
1.02
Traffic Intensity, ρ
Nor
mal
ized
Tim
eA
ver
age
Uti
lity
Maximum Utility (UD(µ))
TO Scheme(UT O
D)
Scheme Q (UQD
)
(c) Time average utility scaling with traffic intensityρ.
Fig. 9. Performance evaluation of energy management schemes under buffer and battery constraints with increasing traffic intensities.
by the significantly higher discharge rate for the TO schemeleading to severe performance degradation. Finally, we observein Fig. 9(c) that the TO scheme achieves a low average utilityat low traffic intensities (ρ < 0.87). This could be due to thecombination of the data buffer getting cleared very often andconcavity of the utility function leading to a lower averageutility. On the other hand, schemeQ regulates the buffer levelto a non-empty level that ensures that it has data to transmitin most time slots. Asρ → 1, the performances of the twoenergy management schemes degrade highly. This is a directconsequence of increasing battery discharge and data loss ratesleading to sub-optimal performance.
C. Trade-offs Between Buffer Overflow and Battery DischargeProbabilities
In Fig. 10, we evaluate the trade-off between battery over-flow and buffer underflow given in Theorem 4. We use thedata arrival and energy replenishment process used previ-ously. Fig 10(a) illustrates that in order to increase the decayexponent for the battery underflow probability, the energymanagement scheme has to decrease the exponent for thebuffer overflow probability. We choose three operating pointson this curve and evaluate the battery underflow and bufferoverflow scaling for these points. As we go from operatingpoint 1 to 3, the buffer underflow decay exponent increases andthe battery underflow decay exponent decreases. In Fig. 10(b),
9
0 0.05 0.1 0.150
0.01
0.02
0.03
0.04
0.05
0.06
X: 0.03669Y: 0.0402
−E
2lo
gpov
erflow
(K)
−E1 log punderflow(M)
X: 0.07338Y: 0.0273
X: 0.1101Y: 0.01392
(a) Trade-off between battery discharge and data
loss probabilities whereE1 =σ2r
2Mµ, E2 =
σ2a
2Kλ.
102
103
104
10−6
10−5
10−4
10−3
10−2
10−1
100
Buffer Size, K (bits)
Dat
aLos
sR
ate
Operating Point 1 (X = 0.1101, Y = 0.0139)
Operating Point 2 (X = 0.0734, Y = 0.0273)
Operating Point 3 (X = 0.0367, Y = 0.0402)
(b) Data loss rate scaling for different operatingpoints.
10 100 1000
10−4
10−3
10−2
10−1
100
Battery Size, M (J)
Bat
tery
Dis
char
geR
ate
Operating Point 1 (X = 0.1101, Y = 0.0139)
Operating Point 2 (X = 0.0734, Y = 0.0273)
Operating Point 3 (X = 0.0367, Y = 0.0402)
(c) Battery discharge rate scaling for different op-erating points.
Fig. 10. Performance evaluation of energy management schemes under buffer and battery constraints with increasing traffic intensities. HereX =σ2r
2Mµ(loge pdischarge(M)) andY =
σ2a
2Kλ(loge poverflow(K)).
we observe that the quickest decay for the loss rate is foroperating point 1. In Fig. 10(c), as expected, we see theopposite effect wherein the discharge rate decays fastest forthe operating point 3.
V. CONCLUSIONS
In this paper, we studied the basic limits and associatedtradeoffs for energy management schemes in energy replen-ishing sensor networks. We showed that it is possible toobserve a polynomial decay of arbitrary order for the dischargeprobability with increased battery sizeM , and at the sametime achieveΘ((logM)2/M2) convergence to the maximumachievable utility using a simple energy management scheme.We showed the strength of this result by showing that it isnot possible to simultaneously observe an exponential decayfor the discharge probability and achieve maximum utility.With the insights drawn, we addressed the problem of energymanagement with buffer and battery constraints. We showedthat with a finite data buffer of sizeK, in addition to achievingΘ((logK)2/K2) convergence to the optimum utility, it ispossible to achieve a polynomial decay for the data lossprobability and exponential decay for the the battery dischargeprobability using a simple energy management scheme.
To analyze the buffer and battery processes we made useof large deviations theory and diffusion approximations. Themain advantage of using these tools in our work is that itallows analytical tractability while keeping the system modelfairly general in nature. Finally, we numerically illustrated theperformance of the our simple energy management schemesalong with that of another existing scheme, and demonstratedthat our scheme can perform up to an order of magnitude betterin terms of outage probabilities while achieving the maximumutility asymptotically.
APPENDIX APROOF OFLEMMA 1
To prove this lemma, we first use the finite form of Jensen’sinequality to establish,
1
τ
τ∑
t=1
U(eS(t)) ≤ U
(
1
τ
τ∑
t=1
eS(t)
)
.
Since this inequality holds for any finiteτ , passing the limitτ → ∞, the inequality is preserved,
lim infτ→∞
1
τ
τ∑
t=1
U(eS(t)) ≤ lim infτ→∞
U
(
1
τ
τ∑
t=1
eS(t)
)
= U
(
lim infτ→∞
1
τ
τ∑
t=1
eS(t)
)
, (18)
where (18) follows sinceU(·) is a continuous function [23].From conservation of energy, we have
lim infτ→∞
1
τ
τ∑
t=1
eS(t) ≤ lim infτ→∞
1
τ
τ∑
t=1
r(t) = limτ→∞
1
τ
τ∑
t=1
r(t),
(19)
sinceM < ∞. Combining Eqs. (18) and (19), we have therequired result,
lim infτ→∞
1
τ
τ∑
t=1
U(eS(t)) = US ≤ U(µ). (20)
APPENDIX BPROOF OFTHEOREM 1
In this appendix, we prove that the energy managementschemeB achieves the scaling properties given in Theorem 1.First, consider a general form of SchemeB:
eB(t) =
µ− δ−, B(t) ≤ M/2
µ+ δ+, B(t) > M/2, (21)
for some pairδ−, δ+, that will be chosen later. We will showthat the desired solution involvesδ− = δ+ = δB.
Depending on whether the battery state is less than (or morethan) half full, the expected drift of the battery state becomespositive (or negative). GivenB(t) ≤ M/2, the asymptoticsemi-invariant log-moment generating function of the batterystate drift,d−(t) , r(t) − (µ− δ−), is
Λd−(s) = limτ→∞
1
τlogE
[
exp
(
s
τ∑
t=1
d−(t)
)]
= Λr(s)− s(µ− δ−). (22)
10
WhereΛr(s) is given by Eq. (3). Lets∗d− be the negative root5
of Λd−(s), i.e.,Λd−(s∗d−) = Λr(s∗d−)−s∗d−(µ−δ−) = 0. Also
asδ− → 0, s∗d− → 0.Before we prove Theorem 1, we state and prove the follow-
ing lemmas. Lemma 2 gives the rate of decay of the probabilityof battery discharge with respect to the battery sizeM for theSchemeB. Lemma 3 expresses the rate decay exponents∗d−
for schemeB in terms of the asymptotic variance of energyreplenishment processr(t).
Lemma 2. The probability of battery discharge underSchemeB with battery sizeM follows pBdischarge(M) =
Θ(
exp(
s∗d−
M
2
))
, wheres∗d− is the negative root ofµd−(s).
Proof: Fix a constantA > 0 and decompose the time lineinto intervals, such that each interval is of length⌈M
2A⌉ andthe ith interval ends at time slotti = i⌈M
2A⌉. Assume that thesystem has been active sincet = −∞. We defineEi as theevent that the battery is empty at the end of time slot 0 and thelast time the battery was half full (i.e.,M/2) is some instantduring the interval−i =
[−(i+ 1)⌈M
2A⌉+ 1,−i⌈M2A⌉]. The
event of an empty battery at time slot 0 can be decomposedas a union of eventsEi,
pBdischarge(M) =∞∑
i=0
P (Ei) . (23)
A necessary condition for eventEi to occur is,
0∑
t=−(i+1)⌈ M2A ⌉+1
(eB(t)− r(t)
)>
M
2. (24)
Using Chernoff’s bound, for anyθi ≥ 0,
P
0∑
t=−(i+1)⌈ M2A ⌉+1
(eB(t)− r(t)) >M
2
≤ E
exp
θi
0∑
t=−(i+1)⌈ M2A ⌉+1
(eB(t)− r(t))
exp
(
−θiM
2
)
= E
exp
−θi
0∑
t=−(i+1)⌈ M2A ⌉+1
r(t)
× exp
(
θi(i+ 1)
⌈M
2A
⌉
(µ− δ−)
)
exp
(
−θiM
2
)
= exp
(
−M
2
[
θi
(
1−i+ 1
A(µ− δ−)
)
−i+ 1
AΛr(−θi)
+ ǫi(M, θi)
])
, (25)
whereǫi(M, θi) → 0 asM → ∞.
5Note that Λd−(0) = 0 and∂Λ
d−(s)
∂s
∣
∣
∣
∣
s=0
=
limT→∞1T
∑Tt=1 E
[
d−(t)]
= δ− > 0. Consequentlys∗d−
< 0will exist.
PSfrag replacements
0
iJ
γ + iβ
γ
fi(θ)
0
slope=E[Xi] = −δ(a)1
infr≤r∗bufferµ(r)
−µ∗r∗buffer
(0)
Fig. 11. A geometric proof for the existence ofJ and δ > 0 such that forevery i > J , fi(θ) > γ + iδ.
In order to find the tightest bound for eachi, we chooseθ∗i ≥ 0 to maximize,
fi(θ) , θ
(
1−i+ 1
A(µ− δ−)
)
−i+ 1
AΛr(−θ), (26)
over all θ > 0 and let γ = infi≥0 supθ≥0 fi(θ) =infi≥0 fi(θ
∗i ). We can rewritefi(θ) as,
fi(θ) = θ −µ− δ−
Aθ −
Λr(−θ)
A− i
((µ− δ−)θ + Λr(−θ)
A
)
.
Since limτ→∞1τ
∑τt=1 E [r(t)] = µ > µ − δ−, the function
(µ− δ−)θ+Λr(−θ) has a negative slope atθ = 0. Hence, wecan choose someθ > 0, such that(µ− δ−)θ + Λr(−θ) < 0.This implies that there exists aJ and aβ > 0 such that forevery i > J ,
fi(θ) > γ + iβ (27)
as illustrated in Fig. 11. Returning to Eq. (23),
pBdischarge(M) =
∞∑
i=0
P (Ei)
≤∞∑
i=0
P
0∑
k=−(i+1)⌈ M2A ⌉+1
(eB(k)− r(k)) >M
2
≤
J∑
i=0
exp
(
−M
2[fi(θ
∗i ) + ǫi(M, θ∗i )]
)
+
∞∑
i=J+1
exp
(
−M
2
[
fi(θ) + ǫi(M, θ)])
≤
J∑
i=0
exp
(
−M
2
[
γ + min0≤i≤J
ǫ(M, θ∗i )
])
+
∞∑
i=J+1
exp
(
−M
2
[
γ + iβ + infi>J
ǫi(M, θ)
])
= exp
(
−M
2γ
)[
(J + 1) exp
(
min0≤i≤J
ǫ(M, θ∗i )
)
+exp
(
−M2
(
(J + 1)β + infi>J ǫi(M, θ)))
1− exp(−βM
2
)
]
. (28)
11
From Eq. (28),lim supM→∞2M log pBdischarge(M) ≤ −γ. Since
this inequality holds for anyA > 0, we letA → ∞ as follows:
lim supM→∞
2
Mlog pBdischarge(M)
≤ − infi≥0
supθ≥0
[
θ
(
1−i
A(µ− δ)
)
−i
AΛr(−θ)
]
= − infT≥0
supθ≥0
[θ (1− T (µ− δ))− T Λr(−θ)
]
= − infT≥0
T supθ≥0
[
−θ
(
µ− δ −1
T
)
− Λr(−θ)
]
. (29)
Next, we find the lower bound. For someT ≥ 0, a sufficientcondition for the battery to be empty at some time slot in theinterval [−⌈TM/2⌉, 0] is that,
0∑
t=−⌈TM2 ⌉+1
(eB(t)− r(t)) > M. (30)
We can lower boundpBdischarge(M) using the union bound,
P
(
B(t) = 0 within somet ∈
[
−
⌈TM
2
⌉
, 0
])
= P
0⋃
t=−⌈TM/2⌉
B(t) = 0
≤
0∑
t=−⌈TM/2⌉
P (B(t) = 0)
=
⌈TM
2
⌉
pBdischarge(M). (31)
We also have
P
0∑
t=−⌈TM2 ⌉+1
(eB(t)− r(t)) >M
2
= P
0∑
t=−⌈ TM2 ⌉+1
(µ− δ− − r(t)) >M
2
. (32)
We define, ZM,T , 2TM
∑0k=−⌈ TM
2 ⌉+1 (µ− δ− − r(k)).Consequently,
P
0∑
t=−⌈ TM2 ⌉+1
(µ− δ− − r(t)) >M
2
= P
(
ZM,T >1
T
)
.
Now, limM→∞ E [ZM,T ] = −δ− < 0 < 1T for all T > 0.
Applying the Gartner-Ellis Theorem, we get,
limM→∞
2
MlogP
(
ZM,T >1
T
)
= − sups≥0
[1
Ts− s (µ− δ) + T Λr
(
−s
T
)]
= −T sups≥0
[
−s
T
(
µ− δ −1
T
)
− Λr
(
−s
T
)]
= −T supθ≥0
[
−θ
(
µ− δ −1
T
)
− Λr(−θ)
]
. (33)
Combining Eqs. (31) and (33), we have,
lim infM→∞
2
Mlog pBdischarge(M)
≥ − infT≥0
T supθ≥0
[
−θ
(
µ− δ −1
T
)
− Λr(−θ)
]
. (34)
From Eqs. (29) and (34) we have,
limM→∞
2
Mlog pBdischarge(M)
= − infT≥0
T supθ≥0
[
−θ
(
µ− δ −1
T
)
− Λr(−θ)
]
= s∗d− . (35)
This gives uspBdischarge(M) = Θ(exp
(s∗d−
M2
)).
Lemma 3. The asymptotic variance ofr(t), σ2r ,
limT→∞1T var
(∑T
t=1 r(t))
satisfies
∂s∗d−
∂δ−
∣∣∣∣δ−=0
= −2
σ2r
(36)
Proof: First, we defineΛ(n)d− (0) =
∂nΛd−
(s)
∂sn
∣∣∣s=0
. The
Taylor series expansion ofΛd−(s∗d−) abouts = 0 gives,
0 = Λd−(s∗) =
∞∑
n=0
Λ(n)d− (0)
(s∗d−)n
n!
= Λd−(0)︸ ︷︷ ︸
=0
+Λ(1)d−(0)s
∗d− + Λ
(2)d−(0)
(s∗d−)2
2!+ · · ·
=
∞∑
n=1
Λ(n)r (0)
(s∗d−)n
n!− (µ− δ−)s∗d−
= µs∗d− +
∞∑
n=2
Λ(n)r (0)
(s∗d−)n
n!− (µ− δ)s∗d− .
Rearranging the terms, we have∞∑
n=2
Λ(n)r (0)
(s∗d−)n−1
n!= −δ−. (37)
Differentiating with respect toδ−, we have,
∂s∗d−
∂δ−
∞∑
n=2
Λ(n)r (0)
(n− 1)(s∗d−)n−2
n!= −1.
As δ− → 0, s∗d− → 0 the above expression reduces to,
∂s∗d−
∂δ−
∣∣∣∣δ−=0
Λ(2)r (0)
1
2= −1. (38)
SubstitutingΛ(2)r (0) = σ2
r in Eq. (38), we have the requiredresult.
Lemma 3 implies∂s∗
d−
∂δ− = − 2σ2r+ o(δ−) and hence,
s∗d− = −2
σ2r
δ− + o(δ−), (39)
where o(δ−)/δ− → 0 asδ− → 0.Substituting this in Eq. (35), we have,
pBdischarge(M) = Θ
(
exp
[(
−2
σ2r
δ− + o(δ−))
M
2
])
. (40)
By choosing δ− = α logMM and α = βσ2
r we havepBdischarge(M) = Θ(M−β).
Next we show that withδ+ = α logMM the scheme achieves
an average utilityUB such thatU(µ) − UB = Θ(
(logM)2
M2
)
.
12
The instantaneous utilityU(eS(t)) is zero with anΘ(M−β)probability. For the remaining time, the utility alternates be-tweenU+ andU− as illustrated in Fig. 2. Noting thatU(·) isan analytic function on the non-negative real line, the Taylorseries expansion of the utility function aboutµ will be,
U+ = U(µ) + U (1)(µ)δ+ + U (2)(µ)(δ+)2 + o((δ+)2),
and,
U− = U(µ)− U (1)(µ)δ− + U (2)(µ)(δ−)2 + o((δ−)2).
We defineρ+ as the fraction of time thatB(t) > M/2 andρ− = 1 − ρ+ as the fraction of time thatB(t) ≤ M/2. Theaverage utilityUB can be written as,
UB = ρ+U+ + (ρ− − pBdischarge(M))U−
= U(µ) + U (1)(µ)(ρ+δ+ − ρ−δ−) + Θ
((logM)2
M2
)
,
(41)
where Eq. (41) follows from the fact thatδ−, δ+ = α logMM
andpBdischarge(M) = Θ(M−β) whereβ ≥ 2.From conservation of energy, the replenishment energy is
consumed completely except for the amount lost due to batteryoverflows. Thus,
ρ+(µ+ δ+) + (ρ− − pBdischarge(M))(µ − δ−)
= µ(1 − pBoverflow(M)), (42)
wherepBoverflow(M) is the probability of the battery being fullunder the energy management schemeB. By a trivial extensionof Lemmas 2 and 3, it can be shown thatpBoverflow(M) =Θ(M−β
). We can simplify Eq. (42) as,
ρ+δ+ − ρ−δ− = Θ(M−β
). (43)
By substituting Eq. (43) in the first-order term of Eq. (41),we observe that the schemeB achievesU(µ) − UB =
Θ(
(logM)2
M2
)
. Choosing δB = δ+ = α logMM in Eq. (4)
completes the proof of Theorem 1.
APPENDIX CPROOF OFTHEOREM 2
Consider any ergodic energy management schemeS thatuseseS(t) units of energy in the time slott. Note that schemeS can be deterministic or randomized. The asymptotic semi-invariant log moment generating function of the net batterydrift dS(t) , eS(t)−r(t) is given byΛdS (s). First, we state alemma that gives the discharge probability scaling for schemeS.
Lemma 4. The probability of battery discharge underSchemeS with battery sizeM follows pSdischarge(M) =Θ(exp(−s∗dSM)), wheres∗dS is the positive root ofµdS (s).
Proof: This lemma gives the rate of decay of the proba-bility of complete discharge with respect to the battery sizeM .To prove this result, we first find anupper boundfor the re-quired probability. We fix a constantA > 0 and decompose thetime line into intervals, such that each interval is of length ⌈M
A ⌉and theith interval ends at time slotti = i⌈M
A ⌉. Next, define
Ei as the event that the battery is empty at the end of time slot0 and the last time the battery was full (i.e.,M ) is some timeduring the interval−i =
[−(i+ 1)⌈M
A ⌉+ 1,−i⌈MA ⌉]. The
event of an empty battery at time slot 0 can be decomposedas a union of eventsEi,
pSdischarge(M) =
∞∑
i=0
P (Ei) (44)
A necessary condition for eventEi to occur is,
0∑
t=−(i+1)⌈MA
⌉+1
(eS(t)− r(t)
)> M. (45)
Using Chernoff’s bound, for anyθi ≥ 0,
P
0∑
t=−(i+1)⌈MA
⌉+1
dS(t) > M
≤ E
exp
θi
0∑
t=−(i+1)⌈MA
⌉+1
dS(t)
exp (−θiM)
= exp
(
−M
(
θi −i+ 1
AΛdS (θi) + ǫi(M, θi)
))
, (46)
whereǫi(M, θi) → 0 asM → ∞.
In order to find the tightest bound for eachi, we chooseθ∗i ≥ 0 to maximizefi(θ) , θ − i+1
A ΛdS (θ), and defineγ =infi≥0 supθ≥0 fi(θ). We can rewritefi(θ) as,
fi(θ) = θ −ΛdS (θ)
A− i
ΛdS(θ)
A.
Since limτ→∞ E[dS(τ)
]< 0, the functionΛdS (θ) has a
negative slope atθ = 0. Hence, we can choose someθ > 0,such thatΛdS (θ) < 0. This implies that there exists aJ anda β > 0 such that for everyi > J (see Fig. 11 for a graphicalproof),
fi(θ) > γ + iβ. (47)
13
Returning to Eq. (44),
pSdischarge(M) =
∞∑
i=0
P (Ei)
≤∞∑
i=0
P
0∑
t=−(i+1)⌈MA
⌉+1
dS(t) > M
≤
J∑
i=0
exp (−M (fi(θ∗i ) + ǫi(M, θ∗i )))
+
∞∑
i=J+1
exp(
−M(
fi(θ) + ǫi(M, θ)))
≤
J∑
i=0
exp
(
−M
(
γ + min0≤i≤J
ǫ(M, θ∗i )
))
+
∞∑
i=J+1
exp
(
−M
(
γ + iβ + infi>J
ǫi(M, θ)
))
= exp (−Mγ)
[
(J + 1) exp
(
min0≤i≤J
ǫ(M, θ∗i )
)
+exp
(
−M(
(J + 1)β + infi>J ǫi(M, θ)))
1− exp (−βM)
]
. (48)
As M → ∞,
lim supM→∞
1
Mlog pSdischarge(M) ≤ −γ
(49)
Since this inequality holds for anyA > 0, we letA → ∞ andget,
lim supM→∞
1
Mlog pSdischarge(M) ≤ − inf
i≥0supθ≥0
[
θ −i
AΛdS (θ)
]
= − infT≥0
supθ≥0
[θ − T ΛdS (θ)
]
= − infT≥0
T supθ≥0
[θ
T− ΛdS (θ)
]
.
(50)
Next, we find thelower bound. For someT ≥ 0, a sufficientcondition for the battery to be empty at some time slot in theinterval [−⌈TM⌉, 0] is that,
0∑
t=−⌈TM⌉+1
(eS(t)− r(t)) > M. (51)
We can lower boundpSdischarge(M) using the union bound,
pSdischarge(M)⌈TM⌉
≥ P (battery is empty in some slot during[−⌈TM⌉, 0]) .(52)
We define,ZM,T , 1TM
∑0t=−⌈TM⌉+1 d
S(t). Consequently,
P
0∑
t=−⌈TM⌉+1
dS(t) > M
= P
(
ZM,T >1
T
)
.
Now, limM→∞ E [ZM,T ] < 0 < 1T for all T > 0. Applying
the Gartner-Ellis Theorem, we get,
limM→∞
1
MlogP
(
ZM,T >1
T
)
= − sups≥0
[ s
T− ΛZ (s)
]
= −T sups≥0
[ s
T 2− ΛdS
( s
T
)]
= −T supθ≥0
[θ
T− ΛdS (θ)
]
,
(53)
where ΛZ(s) is the asymptotic semi-invariant log-momentgenerating function ofZM,T . By noting that Eq. (53) holdsfor all T ≥ 0 and combining it with Eq. (52), we have,
lim infM→∞
1
Mlog pSdischarge(M) ≥ − inf
T≥0T sup
θ≥0
[θ
T− ΛdS (θ)
]
.
(54)
From Eqs. (50) and (54) we have,
limM→∞
1
Mlog pSdischarge(M) = − inf
T≥0T sup
θ≥0
[θ
T− ΛdS (θ)
]
= −s∗dS . (55)
This gives uspSdischarge(M) = Θ(exp(−s∗dSM)).Note thats∗dS > 0 exists6 if and only if E
[dS(t)
]< 0.
Therefore, forpSdischarge(M) to decay exponentially withM ,we require,
E[eS(t)
]< E [r(t)] = µ. (56)
On the other hand, ifE[dS(t)
]≥ 0, there exists no rates > 0
at which the battery discharge probability decays exponentiallywith M , i.e.,pSdischarge(M) = Ω(exp(−sM)) for all s > 0. Bysubstitutingαc = s∗dS in Lemma 4, we get the required scalinglaw pSdischarge(M) = Θ(exp(−αcM)).
The difference between the utilities is given by,
U(µ)− US = U(µ)− lim infτ→∞
1
τ
τ∑
t=1
U(eS(t))
(a)
≥ U(µ)− U
(
lim infτ→∞
1
τ
τ∑
t=1
eS(t)
)
(b)= U(µ)− U
(E[eS(t)
]) (c)= Ω(1). (57)
Where(a) follows from Eq. (18),(b) is due to the ergodicityof eS(t) and(c) follows from Eq. (56) and the factU(·) is anincreasing function. This completes the proof for Theorem 2.
APPENDIX DPROOF OFTHEOREM 3
As discussed in Section III-B, our proof is constructive. Weuse the energy management schemeQ given in Eq. (8).
We find the individual probabilities in the following lemmas.
Lemma 5. For the energy management schemeQ, given anyβQ ≥ 2, pQoverflow(K) = O(K−βQ).
Proof: First, consider a processQu(t) that is formed by
6SinceΛdS (0) and∂Λ
dS(s)
∂s
∣
∣
∣
∣
s=0
= limτ→∞ E[
dS(τ)]
< 0, s∗dS
> 0
will exist.
14
PSfrag replacements
Q(t)
K2
K2
Qu(t)
t t
fi(θ)
0slope=E[Xi] = −δ
(a)1
infr≤r∗bufferµ(r)
−µ∗r∗buffer
(0)
Fig. 12. A graphical representation of the relationship betweenQ(t) andQu(t).
splicing together the intervals during which the processQ(t)is above the buffer stateK/2. Fig. 12 illustrates a sample pathof the processQ(t) and the correspondingQu(t). We denotethe diffusion limit of Q(t) and Qu(t) by Q(t) and Qu(t),respectively. WhileQ(t) is not a Brownian motion due to itsstate-dependent drift,Qu(t) will be a Brownian motion in thelarge-battery regime, with reflections atK/2 and unboundedfrom above as detailed in Step (T1). We assume the startingstate ofQu(t) to beQu(0) = K/2. Note that due to the strongMarkovian property of a Brownian motion [24], the instantsT u
n , n = 1, 2, . . . at which the processQu(t) returns tostateK/2 (i.e., Qu(T
ui ) = K/2) is probabilistically equal to
the starting state. Hence we can study these renewal epochs7
to obtain steady state properties for the data queue process.If we define a unit reward (i.e.,R(t) = 1) for every timet
that the processQu(t) > K then,
limt→∞
P (Qu(t) > K) = limt→∞
E [R(t)] . (58)
From renewal-reward theory [24] we can write,
limt→∞
E [R(t)] =E [Rn]
E [X ], (59)
where E [Rn] is the expected award accumulated in onerenewal period, andE [X ] is the expected length of the renewalperiod. To get the correct expression forlimt→∞ E [R(t)], weneed to write the expressions forE [Rn] andE [X ] carefully.We defineE [X(ǫ)] as the expected time for processQu(t) toreturn toK/2 given that it starts atK/2 + ǫ. The expressionfor E [X(ǫ)] is given by [25],
E [X(ǫ)] =ǫ
δ(a). (60)
Similarily, we defineE [Rn(ǫ)] as the probability of reachingK beforeK/2 starting atK/2+ ǫ. Passing the limitǫ ↓ 0 willgive the expected reward accumulated in one renewal period.Applying the expression for this probability from [25],
E [Rn(ǫ)] =exp
(2δ(a)
σ2aǫ)
− 1
exp(
2δ(a)
σ2a
K2
)
− 1=
2δ(a)
σ2aǫ+ o(ǫ)
exp(
2δ(a)
σ2a
K2
)
− 1. (61)
Dividing Eqs. (61) by (60) and passing the limitǫ ↓ 0, wehave,
limt→∞
E [R(t)] = limǫ↓0
(2δ(a)
σ2a
+ o(ǫ)ǫ
)
δ(a)
exp(
2δ(a)
σ2a
K2
)
− 1. (62)
7If we assume the starting state to beQu(0) 6= K/2, we can simplyconsider the process to be a delayed renewal process. The steady stateproperties in the resulting analysis will not change.
Evaluating the limit ǫ ↓ 0, and noting that the overflowprobability for the processQu(t) will be an upper bound onthe overflow probability of processQ(t), for largeK we have,
pQoverflow(K) ≤ limt→∞
P (Qu(t) > K)
=2δ(a)
2
σ2a
exp
(
−δ(a)K
σ2a
)
. (63)
By choosingδ(a) = βQσ2alogKK , we have,
pQoverflow(K) ≤ βQ2σ2
a
(logK
K
)2
exp (−βQ logK)
= O(K−βQ
). (64)
Note that Lemma 5 implies that given anyβQ > 2, thereexists an energy management schemeQ that achieves anoverflow probabilitypQoverflow(K) = O(K−βQ). Following thediscussion in Step (T1), this implies that for someβ > 0,pQloss(K) = O(K−β).
Lemma 6. For the energy management schemeQ,limλ↑C(µ) limM→∞
1M(µ−C−1(λ)) log(p
Qunderflow(M)) ≤ − 2
σ2r,
whereC−1(·) is the inverse of the analytic rate-power functionC(·).
Proof: Recall thatB(t) is not a Brownian motion, butmerely the process obtained by applying the diffusion limitonthe battery processB(t). In order to evaluatepQunderflow(M) =limt→∞ P (B(t) < 0), for schemeQ, we introduce a newenergy management schemeQl, which allocates an amount ofenergy identical to:eQl(t) = µ− δ
(r)2 for all t. Further, even
whenQ(t) = 0, SchemeQl uses up energyµ−δ(r)2 to transmit
dummy bits. Thus, the drift of the associated battery state isa constant, completely independent of the queue state. Hence,the diffusion limit for the associated battery process yields areflected Brownian motion with a single barrier atM (recallthat the lower barrier was removed). We denote this Brownianlimit by Bl(t). For the same energy replenishment processr(t), t ≥ 0, the net battery drift is defined asdS(t) , r(t)−eS(t), for a given schemeS ∈ Q,Ql. We know for allsample paths that, the net drifts satisfy:
dQ(t) ≥ dQl(t) (65)
for all t. The battery underflow probability for schemeQl isgiven by pQl
underflow(M) = limt→∞ P (Bl(t) < 0). It followsfrom Eq. (65) that,
pQunderflow(M) ≤ pQl
underflow(M). (66)
The Brownian limitBl(t) is an exponentially distributed ran-dom variable, and the underflow probability is given by [26],
pQl
underflow(M) = exp
(
−2δ
(r)2
σ2r
M
)
, (67)
Note that, from the Taylor series expansion of the analyticrate-power functionC(·), we have,
δ(r)2 = µ−
(
C−1(λ) + (C−1(λ))(1)δ(a) + o(δ(a)))
. (68)
15
Since δ(a) = βQσ2alogKK → 0 as K → ∞, we have
limK→∞ δ(r)2 = µ − C−1(λ) Using this observation and
combining Eqs. (66) and (67) we have,
limM→∞
limλ↑C(µ)
1
M(µ− C−1(λ))log(pQunderflow(M)) ≤ −
2
σ2r
,
(69)
completing the proof. Note that, we haveαQ = 2(µ−C−1(λ))σ2r
in the heavy traffic limit.Finally, we focus on the average utilityUQ
D of the energymanagement schemeQ. The instantaneous utility will be zerowhen the queue is empty or when the battery is discharged.Since pQdischarge(M) = O(exp(−αQM)), the contribution ofthe discharge term can be ignored, since it is a 0-probabilityevent under the large battery regime. For the schemeQ, thecalculation of the average utility becomes exactly the sameproblem as that for the energy management schemeB, whichwas analyzed in Appendix B. Here, we replaceB(t) with Q(t)and the battery sizeM with the data buffer sizeK to get,
UD(λ) − UQD = Θ
((logK)2
K2
)
. (70)
We repeat the proof here for completeness. Noting thatUD(·) is an analytic function on the non-negative real line,the Taylor series expansion of the utility function aboutλ willbe,
U+D = UD(µ) + U
(1)D (µ)δ(a) + U
(2)D (µ)(δ(a))2 + o((δ(a))2),
and,
U−D = UD(µ)− U
(1)D (µ)δ(a) + U (2)(µ)(δ(a))2 + o((δ(a))2).
We defineρ+ as the fraction of time thatQ(t) > K/2 andρ− = 1 − ρ+ as the fraction of time thatQ(t) ≤ K/2. Theaverage utilityUQ
D can be written as,
UQD = ρ+U+
D + (ρ− − pQempty(K))U−D
= UD(λ) + U(1)D (λ)(ρ+δ(a) − ρ−δ(a)) + Θ
((logK)2
K2
)
,
(71)
where pQempty(K) is the probability of the data buffer beingempty under the schemeQ. Eq. (71) follows from the fact thatδ(a) = βQσ
2alogKK , while Lemmas 2 and 3 yieldpQempty(K) =
Θ(K−βQ) whereβQ ≥ 2.The average utilityUQ
D can be written as,
UQD =
1
2
(
UD(λ+ δ(a)) + UD(λ − δ(a)))
(1 − pQloss(K))
= UD(λ) + U(2)D (λ)(δ(a))2 + o
(
(δ(a))2)
(72)
= UD(λ) + Θ
(
β2Q
(logK)2
K2
)
, (73)
where Eq. (72) follows sincepQloss(K) = O(K−βQ) forsome βQ ≥ 2 and UD(·) is an analytic function on thenon-negative real line, and Eq. (73) comes from choosingδ(a) = βQσ
2alogKK . This completes the proof of Theorem 3.
From conservation of data, the incoming data is transmittedcompletely except for the amount lost due to data bufferoverflows. Thus,
ρ+(λ+ δ(a)) + (ρ− − pQempty(K))(λ− δ(a))
= λ(1 − pQloss(K)). (74)
By an extension of Lemmas 2 and 3, it can be shown thatpQloss(K) = Θ
(K−βQ
). We can simplify Eq. (74) as,
ρ+δ(a) − ρ−δ(a) = Θ(K−βQ
). (75)
By substituting Eq. (75) in the first-order term of Eq. (71),we observe that the schemeQ achievesUD(µ) − UB
D =
Θ(
(logK)2
K2
)
. This completes the proof of Theorem 3.
APPENDIX EPROOF FORTHEOREM 4
The proof of this theorem is constructive. We use the energymanagement schemeE , given in Eq. 15. With this scheme, themean drifts for the battery state and the data queue state aregiven byδ(r) = ν
(µ− C−1(λ)
)andδ(a) = C(µ− δ(r))−λ,
respectively. Since the limiting regime for the load isλ ↑C(µ), δ(r) ↓ 0 and henceC(µ − δ(r)) − λ ↓ 0. As a result,both the battery and the data queue will operate in the heavytraffic limit and we can apply the diffusion limits on theseprocesses to obtain the required probability decay exponents.
In the diffusion limit [21], [26], the decay rate for thedischarge probability,pEunderflow(M) = limt→∞ P (B(t) < 0),for this energy management scheme can be calculated as,
limM→∞
limλ↑C(µ)
1
Mδ(r)log pEunderflow(M) = −
2
σ2r
, (76)
where the decay exponent is found via the direct applicationof Theorem 7.1 in Chapter 10 of [26]. With an identicalapproach, by applying the diffusion limit to the data bufferprocess and substitutingδ(a) = C(µ−δ(r))−λ, we can find thedecay rate for the buffer overflow probability,pEoverflow(K) =limt→∞ P (Q(t) > K) as,
limK→∞
limλ↑C(µ)
1
Kδ(a)log pEoverflow(K) = −
2
σ2a
. (77)
ACKNOWLEDGMENT
We thank the associate editor, Professor Lachlan Andrewfor his valuable feedback and guidance throughout the reviewprocess.
REFERENCES
[1] C. Hartung, R. Han, C. Seielstad, and S. Holbrook, “FireWxNet: AMulti-Tiered Portable Wireless System for Monitoring Weather Condi-tions in Wildland Fire Environments,” inProc. MobiSys’06, June 2006,pp. 28–41.
[2] K. Martinez, R. Ong, and J. Hart, “Glacsweb: A Sensor Network forHostile Environments,” inProc. IEEE SECON’04, Santa Clara, CA,October 2004, pp. 81– 87.
[3] A. Kansal and M. B. Srivastava, “An Environmental EnergyHarvestingFramework for Sensor Networks,” inProc. ISLPED’03, August 2003,pp. 481–486.
16
[4] S. Meninger, J. O. Mur-Miranda, R. Amirtharajah, A. Chandrakasan,and J. H. Lang, “Vibration-to-electric energy conversion,” IEEE Trans.VLSI Syst., vol. 9, no. 1, pp. 64–76, Feb. 2001.
[5] J. A. Paradiso and M. Feldmeier, “A Compact, Wireless, Self-PoweredPushbutton Controller,” inProc. Ubicomp’01. Springer-Verlag, October2001, pp. 299–304.
[6] J. M. Rabaey, M. J. Ammer, J. L. da Silva, D. Patel, and S. Roundy,“PicoRadio Supports Ad Hoc Ultra-Low Power Wireless Networking,”IEEE Computer, vol. 33, no. 7, pp. 42–48, 2000.
[7] W. Weber, “Ambient Intelligence: Industrial Research on a VisionaryConcept,” inProc. ISLPED’03, August 2003, pp. 247–251.
[8] V. Raghunathan, A. Kansal, J. Hsu, J. Friedman, and M. B. Srivastava,“Design Considerations for Solar Energy Harvesting Wireless EmbeddedSystems,” inProc. IEEE IPSN’05, Los Angeles, USA, April 2005, pp.457 – 462.
[9] K. Kar, A. Krishnamurthy, and N. Jaggi, “Dynamic Node Activationin Networks of Rechargeable Sensors,”IEEE/ACM Trans. Networking,vol. 14, no. 1, pp. 15–26, Feb. 2006.
[10] M. Gatzianas, L. Georgiadis, and L. Tassiulas, “Control of wirelessnetworks with rechargeable batteries,”IEEE Trans. Wireless Commun.,vol. 9, no. 2, pp. 581–593, Feb. 2010.
[11] R.-S. Liu, P. Sinha, and C. E. Koksal, “Joint Energy Management andResource Allocation in Rechargeable Sensor Networks,” inProc. IEEEINFOCOM’10, San Diego, USA, March 2010.
[12] O. Ozel and S. Ulukus, “Information-Theoretic Analysis of an EnergyHarvesting Communication System,” inProc. IEEE PIMRC’10, Instan-bul, Turkey, September 2010, pp. 330–335.
[13] A. Kansal, J. Hsu, S. Zahedi, and M. B. Srivastava, “Power Managementin Energy Harvesting Sensor Networks,”ACM Trans. Embed. Comput.Syst., vol. 6, no. 4, p. 32, September 2007.
[14] C. M. Vigorito, D. Ganesan, and A. G. Barto, “Adaptive Control of DutyCycling in Energy-Harvesting Wireless Sensor Networks,” in Proc. IEEESECON’07, San Diego, CA, June 2007, pp. 21–30.
[15] V. Sharma, U. Mukherji, V. Joseph, and S. Gupta, “Optimal energymanagement policies for energy harvesting sensor nodes,”IEEE Trans.Wireless Commun., vol. 9, no. 4, pp. 1326–1336, April 2010.
[16] C. K. Ho and R. Zhang, “Optimal Energy Allocation for Wireless Com-munications Powered by Energy Harvesters,” inProc. IEEE ISIT’10,June 2010, pp. 2368–2372.
[17] D. Tse, R. G. Gallager, and J. N. Tsitsiklis, “Optimal Buffer Controlfor Variable-Rate Lossy Compression,” inProc. 31st Allerton Conf.Commun. Control and Comput., Allerton, USA, September 1993.
[18] T. M. Cover and J. A. Thomas,Elements of Information Theory, 1st ed.John Wiley and Sons, 1999.
[19] J. M. Harrison and R. J. Williams, “Multidimensional reflected brownianmotions having exponential stationary distributions,”Ann. Prob., vol. 15,no. 1, pp. 115–137, January 1987.
[20] H. S. Kim and N. B. Shroff, “On the Asymptotic Relationship betweenthe Overflow Probability and the Loss Ratio,”Adv. Appl. Prob., vol. 33,no. 4, pp. 810–835, December 2001.
[21] W. Whitt, Stochastic-Process Limits, 1st ed. Springer, 2002.[22] Measurement and Instrumentation Data Center. National Renewable
Energy Laboratory. [Online]. Available: http://www.nrel.gov/midc/[23] W. Rudin,Principles of Mathematical Analysis, 3rd ed. McGraw-Hill,
1976.[24] J. M. Harrison,Brownian Motion and Stochastic Flow Systems, 1st ed.
John Wiley and Sons, 1985.[25] S. M. Ross,Stochastic Processes, 2nd ed. Wiley, 1995.[26] S. Asmussen,Applied Probability and Queues, 2nd ed. Springer, 2003.
Rahul Srivastava received the B.Tech. degree inelectrical engineering from the Indian Institute ofTechnology, Madras, India, the M.S. degree in elec-trical and computer engineering from Rice Uni-versity, Houston, USA, and the Ph.D. degree inelectrical and computer engineering from the OhioState University, Columbus, USA in 2002, 2005,and 2010 respectively. He is currently working asa Staff Scientist in the Wireless Connectivity Groupat the Broadcom Corporation, Sunnyvale, USA. Hisresearch interests include wireless communications,
communication networks, stochastic processes, and optimization theory.
Can Emre Koksal received the B.S. degree in elec-trical engineering from the Middle East TechnicalUniversity, Ankara, Turkey, in 1996, and the S.M.and Ph.D. degrees from the Massachusetts Instituteof Technology (MIT), Cambridge, in 1998 and 2002,respectively, in electrical engineering and computerscience. He was a Postdoctoral Fellow in the Net-works and Mobile Systems Group in the ComputerScience and Artificial Intelligence Laboratory, MIT,until 2003 and a Senior Researcher jointly in theLaboratory for Computer Communications and the
Laboratory for Information Theory at EPFL, Lausanne, Switzerland, until2006. Since then, he has been an Assistant Professor in the Electrical andComputer Engineering Department, Ohio State University, Columbus, Ohio.His general areas of interest are wireless communication, communicationnetworks, information theory, stochastic processes, and financial economics.
He is the recipient of the National Science Foundation CAREER Award in2011, the OSU College of Engineering Lumley Research Award in 2011, andthe co-recipient of an HP Labs - Innovation Research Award in2011.
