Performance Analysis of Wireless Powered Communication with Finite/Infinite Energy Storage
PPerformance Analysis of Wireless PoweredCommunication with Finite/Infinite Energy Storage
Rania Morsi, Diomidis S. Michalopoulos, and Robert Schober
Institute of Digital Communications, Friedrich-Alexander-University Erlangen-N¨urnberg (FAU), Germany
Abstract —In this paper, we consider an energy harvesting (EH)node which harvests energy from a radio frequency (RF) signalbroadcasted by an access point (AP) in the downlink (DL). Thenode stores the harvested energy in an energy buffer and usesthe stored energy to transmit data to the AP in the uplink (UL).We consider a simple transmission policy, which accounts for thefact that in practice the EH node may not have knowledge of theEH profile nor of the UL channel state information. In particular,in each time slot, the EH node transmits with either a constantdesired power or a lower power if not enough energy is availablein its energy buffer. For this simple policy, we use the theoryof discrete-time continuous-state Markov chains to analyze thelimiting distribution of the stored energy for finite- and infinite-size energy buffers. Moreover, we take into account imperfectionsof the energy buffer and the circuit power consumption of theEH node. For a Rayleigh fading DL channel, we provide thelimiting distribution of the energy buffer content in closed form.In addition, we analyze the average error rate and the outageprobability of a Rayleigh faded UL channel and show that thediversity order is not affected by the finite capacity of the energybuffer. Our results reveal that the optimal desired transmit powerby the EH node is always less than the average harvested powerand increases with the capacity of the energy buffer.
I. I
NTRODUCTION
The performance of battery-powered wireless communica-tion networks, such as sensor networks, is limited by the life-time of the network nodes. Periodic replacement of the nodes’batteries is costly, inconvenient, and sometimes impossiblewhen the sensor nodes are placed in a hazardous environmentor embedded inside the human body. The lifetime bottle-neck problem of energy-constrained wireless networks thusdemands harvesting energy from renewable energy sources(e.g., solar, wind, thermal, vibration) to ensure a sustainablenetwork operation. The harvested energy can then be usedby the energy harvesting (EH) node to transmit data toits designated receiver. However, the aforementioned energysources are in general intermittent and uncontrollable. Forexample, solar and wind energy are weather dependent and arenot available indoors. In contrast, radio frequency (RF) energyis a viable energy source which is partially controllable andcan be provided on demand to charge low-power devices [1].A common feature of EH communication networks is therandomness of the amount of harvested energy. For instance,solar/wind energy varies throughout the day and the harvestedenergy from an RF signal varies due to time-varying fading.Furthermore, the information signal transmitted by the EHnode encounters also time-varying fading, which introducesanother source of randomness. Therefore, one main objectiveof energy management polices for EH networks is to matchthe energy consumption profile of the EH node to the randomenergy generation profile of the EH source and to the random information channel [1]–[5]. For example, the authors of [1]introduced the concept of energy neutral operation of an EHsystem, where the energy used by the system is always lessthan the energy harvested. Energy neutrality is thus a conditionfor an EH system to operate perpetually. In [2], [3], a harvest-then-transmit protocol is considered for a multiuser systemwith RF wireless power transfer (WPT) in the downlink (DL)and wireless information transfer (WIT) in the uplink (UL),where the users’ sum rate or equal throughput is maximizedon a per-slot basis. In [4], throughput and mean delay optimalenergy neutral policies, which stabilize the data queue of anEH sensor node over an infinite horizon, are proposed ina time-slotted setting. In [5], optimal transmission policiesthat maximize the throughput by a deadline or minimize thetransmission completion time are proposed for an EH nodewith finite energy storage in a continuous time setting.Optimal offline transmission policies typically require non-causal knowledge of energy and channel state information(CSI) at the EH node, whereas optimal online solutions aretypically based on dynamic programming which is compu-tationally intensive even for a small number of transmittedsymbols, see [5] and the references therein. Therefore, theseoptimal policies may not be feasible in practice. For example,typical EH wireless sensor networks are expected to comprisemany small, inexpensive sensors with limited computationalpower and energy storage. In such networks, even causal CSImay not be available at the EH nodes nor at the EH source.Motivated by these practical considerations, in this paper,we consider a simple online transmission policy, where theCSI and the EH profile are not available at the EH node norat the EH source. In particular, an access point (AP) transmitsan RF signal with a constant power in the DL and the EH nodeharvests the received RF energy and uses the stored energy totransmit data to the AP in the UL. In each time slot, the EHnode transmits with either a constant desired power or a lowerpower if not enough energy is available in its energy buffer.We model the stored energy by a discrete-time continuous-state Markov chain and provide its limiting distribution forboth infinite and finite energy storage, when the DL channelis Rayleigh fading. Under this framework, we analyze theaverage error rate (AER) and the outage probability of aRayleigh fading information link. We show that, surprisingly,the diversity order is not affected if the energy storage hasfinite capacity. Furthermore, we show that the optimal desiredUL power of the considered policy is always less than theaverage harvested power and increases with the capacity ofthe energy buffer. The proposed framework also takes intoaccount the system non-idealities such as non-zero circuit a r X i v : . [ c s . I T ] O c t ower consumption and imperfections of the energy buffer.The rest of the paper is organized as follows. Section IIpresents the overall system model. In Sections III and IV,we study the limiting distribution of the stored energy forinfinite- and finite-capacity energy buffers, respectively. InSection V, we analyze the AER and the outage probability ofthe communication link, when both UL and DL channels areRayleigh faded. Numerical and simulation results are providedin Section VI. Finally, Section VII concludes the paper.II. S YSTEM M ODEL
We consider a time-slotted point-to-point single-antenna EHsystem with DL WPT and UL WIT. In particular, the systemconsists of a node with an EH module which captures theRF energy transferred by an AP in the DL and uses theharvested energy to transmit its backlogged data in the UL.The considered system employs frequency-division-duplex,where WPT and WIT take place concurrently on two differentfrequency bands. The AP and the EH node are assumed tohave no instantaneous knowledge of the DL and the UL CSI,respectively, nor of the amount of harvested energy. Next, wedescribe the communication, EH, and storage models as wellas the considered system imperfections.
A. Communication Model
In time slot i (defined as the time interval [ i, i + 1) ), theEH node transmits data to the AP with an UL power given by P UL ( i ) = min( B ( i ) , M ) , (1)where B ( i ) is the residual stored energy at the beginningof time slot i and M is the desired constant UL transmitpower. The transmitted signal encounters a flat block fadingchannel, i.e., the channel remains constant over one time slot,and changes independently from one slot to the next. Thechannel power gain sequence { h UL ( i ) } is a stationary andergodic process with mean Ω UL = E [ h UL ( i )] , where E [ · ] denotes expectation. Additive white Gaussian noise (AWGN)with variance σ n impairs the received signal at the AP. B. EH Model
During the same time slot, the EH node collects X ( i ) unitsof RF energy broadcasted by the AP and stores it in its energybuffer. We assume that the energy replenished in a time slotmay only be used in future time slots. The DL channel isalso assumed to be flat block fading with a stationary andergodic channel power gain sequence { h DL ( i ) } , assumed tobe unknown at the AP, where Ω DL = E [ h DL ( i )] . We adopt theEH receiver model in [6], where the harvested energy in timeslot i is given by X ( i ) = ηP DL h DL ( i ) , where < η < is the RF-to-DC conversion efficiency of the EH moduleand P DL is the constant DL transmit power from the AP.The energy replenishment sequence { X ( i ) } is consequentlyan independent and identically distributed (i.i.d.) stationaryand ergodic process with mean ¯ X = ηP DL Ω DL , probabilitydensity function (pdf) f ( x ) , and complementary cumulativedistribution function (ccdf) ¯ F ( x ) = P ( X ( i ) > x ) , where P ( · ) denotes the probability of an event. The time slot is assumed to be of unit length. Hence, we use the termsenergy and power interchangeably.
C. Storage Model
The harvested energy X ( i ) is stored in an energy buffer,such as a rechargeable battery and/or a supercapacitor [7], witha finite storage capacity of K . The dynamics of the storageprocess { B ( i ) } are given by the storage equation B ( i + 1) = min ( B ( i ) − P UL ( i ) + X ( i ) , K )= min (cid:0) [ B ( i ) − M ] + + X ( i ) , K (cid:1) , (2)where [ x ] + = max( x, . The storage process { B ( i ) } in (2)is a discrete-time Markov chain on a continuous state space S , where S = [0 , K ] for a finite-size energy buffer and S =[0 , ∞ ) for an infinite-size buffer. Remark . Interestingly, our storage model is similar to thedam model proposed by Moran in [8]. In Moran’s model,every year X ( i ) units of water flow into a dam of capacity K and a constant amount of water M is released just beforethe following year. Moran studies the amount of water { Z ( i ) } stored in the dam just after release, which is modeled by thestorage equation Z ( i + 1) = [min( Z ( i ) + X ( i ) , K ) − M ] + ,which for an infinite-capacity dam (i.e., K → ∞ ) reduces to Z ( i + 1) = (cid:40) Z ( i ) + X ( i ) ≤ MZ ( i ) + X ( i ) − M Z ( i ) + X ( i ) > M . (3)The stationary distribution of { Z ( i ) } (if it exists) can be ob-tained e.g. by first defining a new process U ( i ) = Z ( i )+ X ( i ) .Hence, if we add X ( i + 1) to Z ( i + 1) , we get U ( i + 1) = (cid:40) X ( i + 1) U ( i ) ≤ MU ( i ) − M + X ( i + 1) U ( i ) > M , (4)which is identical in distribution to { B ( i ) } in (2) at K → ∞ .Hence, the distribution of { U ( i ) } in Moran’s dam model isidentical to the distribution of { B ( i ) } in our energy buffermodel. Similarly, for a finite storage capacity, { B ( i ) } isequivalent to { min( U ( i ) , K ) } . D. Consideration of Imperfections
We consider imperfections due to the circuit power con-sumption of the EH node and the non-idealities of the energybuffer. In particular, we consider the following imperfections;(a) For the power amplifier of the EH node to transmit anRF power of P UL , it consumes a total power of αP UL , where α > is the power amplifier inefficiency. (b) We assumethat the EH node circuitry consumes a constant power of P ct used mainly for harvesting, processing, and sensing (for EHsensors). (c) Two main imperfections of the energy buffer areconsidered [1]. First, the buffer is assumed to leak a constantamount of energy in each time slot, denoted by P l . Second,we consider the buffer storage inefficiency characterized by < β < , where if X amount of energy is applied at theinput of the buffer, only an amount of βX may be stored.Compared to rechargeable batteries, supercapacitors have highstorage efficiency β , but also high leakage current [7], [1].Define P C as the total constant energy usage in eachtime slot, i.e., P C = P ct + P l . In this case, theenergy buffer dynamics are described by B ( i + 1) = A stationary distribution of a Markov chain is a distribution such that ifthe chain starts with this distribution, it remains in that distribution. in ( B ( i ) − ( P C + αP UL ( i ))+ βX ( i ) , K ) , where the desiredUL transmit power is M . If B ( i ) < P C + αM , then theUL power is reduced to satisfy B ( i ) = P C + αP UL ( i ) ,i.e., P UL ( i ) = ( B ( i ) − P C ) /α which ensures energy neutraloperation . Hence, the storage equation reduces to B ( i + 1) = min (cid:0) [ B ( i ) − ( P C + αM )] + + βX ( i ) , K (cid:1) . (5)Observe that (5) is identical to (2) after replacing M by ˜ M = P C + αM and f ( x ) by ˜ f ( x ) = β f (cid:16) xβ (cid:17) . Thus, in the following,we perform the analysis for an ideal system (i.e., α = 1 , β = 1 ,and P C = 0 ). For a non-ideal system, all the results in SectionsIII-V hold with the aforementioned substitutions.III. I NFINITE -C APACITY E NERGY B UFFER
In this section, we study the energy storage process in (2) foran infinite-capacity energy buffer. We provide conditions forwhich the convergence to a limiting distribution of the buffercontent is either guaranteed or violated. Furthermore, weprovide the limiting distribution of the buffer content in closedform when the EH process { X ( i ) } is i.i.d. exponentiallydistributed, i.e., for a Rayleigh block fading DL channel. Theorem 1.
For the storage process { B ( i ) } in (2) with infinitebuffer size, if M < ¯ X , then { B ( i ) } does not possess astationary distribution. Furthermore, after a finite number oftime slots, P UL ( i ) = M holds almost surely (a.s). Proof.
The proof is provided in Appendix A. (cid:4)
Theorem 2.
For the storage process { B ( i ) } in (2) with infinitebuffer size, if M > ¯ X , then { B ( i ) } is a stationary and ergodicprocess which possesses a unique stationary distribution π thatis absolutely continuous on (0 , ∞ ) . Furthermore, the processconverges in total variation to the limiting distribution π fromany initial distribution. Proof.
The proof is provided in Appendix B. (cid:4)
Theorem 3.
Consider the storage process { B ( i ) } in (2) withinfinite buffer size and M > ¯ X . Let g ( x ) on (0 , ∞ ) be thelimiting pdf of the energy buffer content, then g ( x ) mustsatisfy the following integral equation g ( x ) = f ( x ) M (cid:90) g ( u ) d u + M + x (cid:90) M f ( x − u + M ) g ( u ) d u. (6) Proof.
To understand the integral equation in (6), one may set B ( i ) = u and B ( i + 1) = x , then (2) reads x = (cid:40) X ( i ) u ≤ Mu − M + X ( i ) u > M . (7)Thus, g ( x | u ≤ M ) = f ( x ) and g ( x | u > M ) = f ( x − u + M ) which is non-zero only for a non-negative amount of harvestedenergy, i.e., x − u + M ≥ . These considerations lead to (6).From the analogy between our storage model and Moran’smodel, c.f. Remark 1, (6) is identical to [9, eq. (5)]. (cid:4) Another option to ensure energy neutral operation is to reduce the circuitpower consumption using dynamic voltage scaling or duty cycling [1]. A limiting distribution of a Markov chain is a stationary distribution thatthe chain converges asymptotically to from some initial distribution.
Next, we consider the case when the DL channel is Rayleighblock fading and provide the limiting distribution of the energybuffer content in the following corollary.
Corollary 1.
Consider the storage process in (2) with infinitebuffer size and
M > ¯ X . If the EH process is exponentiallydistributed with pdf f ( x ) = λ e − λx , where λ = X and δ = λM = M ¯ X , then the limiting pdf of the energy buffer contentis g ( x )= − p e px , where p < is given by p = − δ − W ( − δ e − δ ) M and W ( · ) is the Lambert W function of order zero. Proof.
The proof is provided in Appendix C. (cid:4)
IV. F
INITE -C APACITY E NERGY B UFFER
In this section, we first provide the integral equation ofthe stationary distribution of the storage process { B ( i ) } for afinite-size energy buffer and a general i.i.d. EH process. Then,the distribution is provided for a Rayleigh fading DL channel. Theorem 4.
The storage process in (2), with a finite buffersize K , and an EH process { X ( i ) } , which is characterizedby a distribution with an infinite positive tail, is a stationaryand ergodic process which possesses a unique stationarydistribution π that has a density on (0 , K ) and an atom at K . Furthermore, the process converges in total variation tothe limiting distribution π from any initial distribution. Proof.
The proof is provided in Appendix D. (cid:4)
Theorem 5.
Consider the storage process { B ( i ) } in (2), with afinite buffer size K . Let g ( x ) be the limiting pdf of the energybuffer content on (0 , K ) and π ( K ) be the limiting probabilityof a full buffer (i.e., the atom at K ). If f ( x ) and ¯ F ( x ) arerespectively the pdf and the ccdf of { X ( i ) } , then, g ( x ) and π ( K ) must jointly satisfy g ( x ) = f ( x ) M (cid:90) u =0 g ( u ) d u + M + x (cid:90) u = M f ( x − u + M ) g ( u ) d u ≤ x < K − M (8a) f ( x ) M (cid:90) u =0 g ( u ) d u + K (cid:90) u = M f ( x − u + M ) g ( u ) d u + π ( K ) f ( x − K + M ) K − M ≤ x < K (8b) π ( K ) = (cid:34) ¯ F ( K ) M (cid:82) u =0 g ( u ) d u + K (cid:82) u = M ¯ F ( K − u + M ) g ( u ) d u (cid:35) − ¯ F ( M ) , (9)and the unit area condition K (cid:90) g ( u ) d u + π ( K ) = 1 . (10) Proof.
The integral equations in (8), (9) can be understood byadopting the same approach used to prove (6). In particular,if we set B ( i ) = u and B ( i + 1) = x , then (2) reads x = X ( i ) u ≤ M & X ( i ) < Ku − M + X ( i ) u > M & u − M + X ( i ) < KK otherwise . (11)onsider first the continuous part of the distribution, i.e., g ( x ) defined on ≤ x < K given in (8). Eq. (8a) is identical to(6), however, we need to further ensure that the upper limiton u given by M + x (for a non-negative harvested energy)is in the domain of g ( u ) , i.e., M + x < K must hold. Hence,(8a) is valid only for x < K − M (with strict inequality). Forthe rest of the range of x in (8b), i.e., K − M ≤ x < K , theupper limit M + x on u is larger than or equal to K . Thus,the whole range of < u ≤ K contributes to g ( x ) . The range < u < K is covered by the first two integrals in (8b), and u = K is considered in the last term. Finally, at x = K ,the full buffer probability π ( K ) in (9) is obtained similar to(8b). However, rather than considering the pdf at the amountof harvested energy x − [ u − M ] + as in (8b), we consider theccdf ¯ F ( x − [ u − M ] + ) instead (at x = K ). This is because thefull buffer level K is attained when the amount of harvestedenergy is larger than or equal to K − [ u − M ] + , where we sweepover < u ≤ K to obtain (9). This completes the proof. (cid:4) Next, we consider the case when the DL channel is Rayleighblock fading. We provide the exact limiting distribution ofthe energy buffer content in Corollary 2 and an exponentialapproximation of it in Proposition 1.
Corollary 2.
Consider the storage process { B ( i ) } in (2) witha finite buffer size K and an i.i.d. exponentially distributedEH process { X ( i ) } with pdf f ( x ) = λ e − λx , where λ = X and δ = λM , then the limiting pdf g ( x ) of the energy buffercontent and the full buffer probability π ( K ) are given by g ( x ) = π ( K ) λ e − λ ( x − K ) (cid:34) n (cid:88) q =1 e − δq ( q − δq + λ ( x − K )) q − (cid:18) λ ( x − K ) q + δ − (cid:19) (cid:35) , [ K − ( n + 1) M ] + ≤ x < K − nM,n = 0 , . . . , l (cid:48) , (12) and π ( K ) = (cid:40) l − (cid:88) n =0 e nδ (cid:32) e δ − n (cid:88) q =1 (cid:0) δ e − δ (cid:1) q q ! (cid:16) e δ ( q − ( n +1)) q − ( q − n ) q (cid:17)(cid:33) +e lδ (cid:32) e λ ∆ − l (cid:88) q =1 (cid:0) δ e − δ (cid:1) q q ! (cid:18) e λ ∆ (cid:18) q − KM (cid:19) q − ( q − l ) q (cid:19)(cid:33) +1 (cid:41) − , (13) where K = lM + ∆ with l ∈ Z and ≤ ∆ < M . In (12), l (cid:48) is either l (cid:48) = l − if ∆ = 0 or l (cid:48) = l if ∆ (cid:54) = 0 . Proof.
First, we note that the solution of g ( x ) in (12) isobtained in stripes of width M . This is due to the upperintegral limit M + x in (8a), hence the width- M stripes solutionis in fact general for any distribution of the i.i.d. EH process[10]. We derived g ( x ) by induction. In particular, we obtained g ( x ) in the range K − M ≤ x < K from (8b) and (9), aftersetting f ( x ) = λ e − λx and ¯ F ( x ) = e − λx . Then, using (8a) and(8b), we traversed back in sections of width M until x = 0 .Note that the provided solution is general for any K (i.e., K isnot necessarily an integer multiple of M ). The exact derivationis lengthy so we omit it and provide it in the journal version ofthis paper. However, we note that due to the analogy betweenour storage model and Moran’s dam model, c.f. Remark 1, thesolution of g ( x ) is identical to [10, eq. (3.6)] (which is also given by Prabhu in [11, Section 2]). After getting g ( x ) , π ( K ) is obtained by solving (10). (cid:4) Since the exact limiting distribution of the buffer contentprovided in Corollary 2 is quite complicated, we propose anexponential-type approximation which will be used in the AERand the outage probability analysis in Section V.
Proposition 1.
First for notational brevity, we define the n th section of g ( x ) in (12) as g n ( x ) = g ( x ) , [ K − ( n + 1) M ] + ≤ x < K − nM . The limiting distribution of the storage processdescribed in Corollary 2 can be approximated in the range ≤ x < K − n c M by ˜ g ( x ) , where n c is some chosen sectionnumber after which g n ( x ) ≈ ˜ g ( x ) , ∀ n ≥ n c as shown in Fig.1. For ≤ n < n c , ˜ g n ( x ) is given by g n ( x ) in (12) afterreplacing π ( K ) with ˜ π ( K ) , i.e., ˜ g n ( x ) = ˜ π ( K ) π ( K ) g n ( x ) , where ˜ π ( K ) is the approximate full buffer probability that ensuresa unit area of the approximate distribution. The proposedapproximation is tight for K ≥ M and n c ≥ . K − n c M K − M ˜ g ( x ) . . .K − nM ˜ g n ( x ) . . . K − ( n +1) M ˜ g n c − ( x ) K ˜ g ( x )0 Fig. 1. Pdf approximation.
We propose an exponential-type approximation given by ˜ g ( x ) = c e dx , where d and c are given by d = − δ − W j ( − δ e − δ ) M , j = (cid:40) − < δ ≤ δ > , (14) c = ˜ π ( K ) λ e λK l (cid:48) (cid:88) q =1 e − δq ( q − δq − λK ) q − (cid:18) − λKq + δ − (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) Σ , and the approximate atom at K is given by ˜ π ( K ) = (cid:34) + n c − (cid:88) n =0 e nδ (cid:16) e δ − n (cid:88) q =1 (cid:0) δ e − δ (cid:1) q q ! (cid:16) e δ ( q − ( n +1)) q − ( q − n ) q (cid:17)(cid:17)(cid:35) − , (15) where Σ = (cid:40) λ Σ d (cid:0) e d ( K − n c M ) − (cid:1) δ (cid:54) = 1 λ Σ ( K − n c M ) δ = 1 , and W j ( · ) isthe j th order Lambert W function. Proof.
The proof is provided in Appendix E. (cid:4)
V. AER
AND O UTAGE P ROBABILITY A NALYSIS
In this section, we analyze the AER and the outage proba-bility of the communication over the UL channel, when bothUL and DL channels are Rayleigh faded. In the finite-sizebuffer case, we use the approximate pdf ˜ g ( x ) = c e dx given inProposition 1, whereas for an infinite-size buffer with δ > , weuse the exact pdf g ( x ) = − p e px given in Corollary 1. Hence,we show only the results of the finite-size buffer and deducethe latter by setting c = − p and d = p . For an infinite-sizebuffer with δ ≤ , we use P UL ( i ) = M , ∀ i , c.f. Theorem 1. A. AER Analysis
For uncoded transmission, the bit or symbol error rateof many modulation schemes can be expressed as P e ( γ ) = aQ ( √ bγ ) [12], where γ is the instantaneous signal-to-noise-ratio (SNR), Q ( · ) is the Gaussian Q-function, and a , b dependn the modulation scheme used, e.g., for binary phase shiftkeying (BPSK) a = 1 and b = 2 .For an infinite-size buffer with δ ≤ , the AER is given by P e (cid:12)(cid:12) ∞ ,δ ≤ = ∞ (cid:90) aQ (cid:32)(cid:115) bM Ω UL hσ n (cid:33) e − h d h = a (cid:34) − (cid:115) b ¯ γδ b ¯ γδ (cid:35) , (16)where ¯ γ is defined as ¯ γ = Ω UL ¯ X/σ n = Ω UL / (cid:0) λσ n (cid:1) . For afinite-size buffer, the AER is given by P e (cid:12)(cid:12) F = M (cid:90) ∞ (cid:90) aQ (cid:32)(cid:115) bx Ω UL hσ n (cid:33) e − h d h c e dx d x + P M P e (cid:12)(cid:12) ∞ ,δ ≤ , (17)where we define P M = P ( P UL ( i ) = M ) = P ( B ( i ) ≥ M ) = 1 − (cid:82) M c e dx d x = 1 − cλ , where we used λ e dM = λ + d , c.f. Appendix C. The first term in (17) can be sim-plified to ac λ (cid:82) δ e dλ x (cid:16) − (cid:113) b ¯ γx b ¯ γx (cid:17) d x , where (cid:82) δ e dλ x d x = (cid:0) e dM − (cid:1) / ( d/λ ) = 1 . Substituting back in (17), we get P e (cid:12)(cid:12) F = a (cid:34) − (cid:16) − cλ (cid:17) (cid:115) b ¯ γδ b ¯ γδ − cλ δ (cid:90) (cid:115) b ¯ γx b ¯ γx e dλ x d x (cid:35) , (18)where the integral in (18) has finite limits and can be solvednumerically.In order to study the diversity order of the AER in(16) and (18), we consider the high SNR regime, i.e., as ¯ γ → ∞ . We first note that lim y →∞ (cid:16) − (cid:113) y y (cid:17) = y .Hence, the AER in (16) tends asymptotically (denoted by“ (cid:16) ”) to P e (cid:12)(cid:12) ∞ ,δ ≤ (cid:16) a bδ ¯ γ . That is, an infinite-size bufferwith δ ≤ achieves a diversity order of 1 with respectto the AER. For a finite-size buffer, the first term in (17),given by ac λ (cid:82) δ e dλ x (cid:16) − (cid:113) b ¯ γx b ¯ γx (cid:17) d x , tends asymptoticallyto ac λ (cid:82) δ e dλ x b ¯ γx d x , which also has a diversity order of 1.Hence, the diversity order is not affected by the finite capacityof the energy buffer. Therefore, (18) tends asymptotically to P e (cid:12)(cid:12) F (cid:16) ac λb ¯ γ δ (cid:90) e dλ x x d x + (cid:16) − cλ (cid:17) a bδ ¯ γ . (19) B. Outage Probability Analysis
Since the CSI is unknown at the EH node, the nodetransmits data at a constant rate R in bits/(channel use).Therefore, assuming a capacity-achieving code, an outageoccurs whenever R > log (1 + γ ) ⇒ γ < γ thr , where γ is the UL instantaneous SNR and γ thr = 2 R − . Hence, theoutage probability for an infinite-size buffer with δ ≤ is P out (cid:12)(cid:12) ∞ ,δ ≤ = P ( γ < γ thr ) = P (cid:18) M Ω UL hσ n < γ thr (cid:19) = 1 − e − γ thr δ ¯ γ . (20)For a finite-size buffer, the outage probability is given by P out (cid:12)(cid:12) F = M (cid:90) P (cid:18) x Ω UL hσ n < γ thr (cid:19) c e dx d x + P M P out (cid:12)(cid:12) ∞ ,δ ≤ . (21)The first term in (21) reduces to (cid:82) δ P ( x ¯ γh < γ thr ) cλ e dλ x d x = cλ (cid:82) δ (cid:16) − e − γ thr x ¯ γ (cid:17) e dλ x d x , where (cid:82) δ e dλ x d x = 1 . Using P M =1 − cλ and P out (cid:12)(cid:12) ∞ ,δ ≤ in (20), (21) reduces to TABLE IS
YSTEM P ARAMETERS
Parameter ValueAP to EH node distance mAP and EH node antenna gains dBi and dBiDL transmit power P DL = 1 WAP noise figure dBPath loss exponent of DL and UL channels . DL and UL channel models Rayleigh block fadingDL and UL center frequencies
MHz and . GHzPower amplifier inefficiency α = 1 . Storage efficiency β = 0 . Average harvested energy (cid:101) ¯ X = β ¯ X = 10 − JRF-to-DC conversion efficiency η = 0 . Total constant power consumption P C = 0 . µ WStorage capacity K =4 (cid:101) ¯ X , (cid:101) ¯ X , and (cid:101) ¯ X P out (cid:12)(cid:12) F = (cid:16) − e − γ thr ¯ γδ (cid:17) + cλ e − γ thr ¯ γδ − δ (cid:90) e − γ thr ¯ γx e dλ x d x . (22)Using lim y →∞ e − y = 1 − y + o ( y − ) , the outage probabilityin (20) tends asymptotically to P out (cid:12)(cid:12) ∞ ,δ ≤ (cid:16) γ thr δ ¯ γ , i.e., witha diversity order of 1. For a finite-size buffer, the outageprobability in (22) tends asymptotically to P out (cid:12)(cid:12) F (cid:16) (cid:16) − cλ (cid:17) γ thr ¯ γδ + cγ thr λ ¯ γ δ (cid:90) x e dλ x d x. (23)Similar to the AER, the diversity order of the outage probabil-ity is unaffected by the finite capacity of the energy buffer. Wenote that, although for a small buffer size, namely K ≤ M ,the approximate pdf on [0 , M ] is not tight, it can be shownusing the exact pdf in (12) that in this case the diversity orderof the AER and the outage probability is still 1.VI. S IMULATION AND N UMERICAL R ESULTS
In this section, we evaluate the performance of the inves-tigated energy management policy through simulations. Thesimulation parameters are listed in Table I.Fig. 2 shows the AER of the received signal at the AP whenthe EH node transmits a BPSK signal over a bandwidth of BW = 5 MHz. At room temperature (
K), this correspondsto a noise power of − dBm at the AP and an SNR of (cid:101) ¯ γ = Ω UL (cid:101) ¯ X/σ n = 24 . dB, where (cid:101) ¯ X is given in Table I. Wesweep over ˜ δ = ˜ M / (cid:101) ¯ X = 0 . , . . . , . , which corresponds toa desired UL transmit power of M = 0 . µ W , · · · , µ W.The closed-form results shown in Fig. 2 are obtained fromthe expressions in Section V-A, where for a finite-storagecapacity, we use (18), and for an infinite-storage capacity,we use (16) for ˜ δ ≤ and (18) for ˜ δ > , with c = − p and d = p , c.f. Corollary 1. We observe that the closed-form results agree perfectly with the simulated results. Thisemphasizes the tightness of the approximate pdf provided inProposition 1. It is observed that for the considered energymanagement scheme, the optimal ˜ δ , for which the AER is −3 −2 ˜ δ P e K =4 e ¯ XK =7 e ¯ XK =20 e ¯ XK → ∞ optclosed-formSimulated Fig. 2. AER for different buffer sizes and different desired UL transmit power. minimized, is always ≤ and increases with the storagecapacity . As K → ∞ , the optimal ˜ δ → . In other words, forthe considered energy transmission policy, the optimal desiredUL transmit power is higher for larger energy buffers, but itis always less than the average harvested power. Furthermore,our results show that, for a given storage capacity, the optimaldesired UL power decreases with the SNR. This result is notshown here due to space limitations.Fig. 3 shows the outage probability of the UL channel whenthe EH node transmits at a constant rate of . bits/(channeluse), i.e., for γ thr = 5 dB. We sweep over different SNRsof (cid:101) ¯ γ = 10 , . . . , dB, which corresponds to an UL channelbandwidth range of BW = Ω UL (cid:101) ¯ X (cid:101) ¯ γK B T e = 147 MHz , . . . ,
KHz,respectively, where K B is Boltzmann’s constant and T e is theequivalent noise temperature of the AP. For a given SNR (cid:101) ¯ γ and a given buffer size K , the energy management policy isoperated at the optimal ˜ δ for which the outage probabilityis minimized. The closed-form results shown in Fig. 3 areobtained from the expressions in Section V-B, where for afinite-storage capacity, we use (22), and for an infinite-storagecapacity, we use (20) at δ opt = 1 ∀ (cid:101) ¯ γ . Observe that in Fig. 3,the outage probability curves for the different energy buffersizes are parallel. This agrees with our asymptotic analysis inSection V-B, which shows that diversity order is not affectedby an energy storage with finite capacity.VII. C ONCLUSION
In this paper, we considered a simple online energy neutraltransmission policy for an EH node, with finite/infinite energystorage. Using the theory of discrete-time Markov chains ona general state space, we analyzed the limiting distributionof the stored energy in the buffer for a general i.i.d. EHprocess and obtained it in closed form for an exponentialEH process. An exponential-type approximation of the storedcontent distribution is proposed for finite-size buffers andshown to be tight. Our results reveal that the diversity ordersof the AER and the outage probability are not affected by afinite energy storage capacity. Furthermore, for the consideredtransmission scheme, it was shown that the optimal desired We note that while instantaneous CSI knowledge is not required for theadopted transmission protocol, statistical CSI is needed if ˜ δ is to be optimized.
10 15 20 25 30 35 4010 −3 −2 −1 e ¯ γ P o u t K =4 e ¯ XK =7 e ¯ XK =20 e ¯ XK → ∞ closed-formsimulated Fig. 3. Outage probability for different buffer sizes and different SNRs. transmit power of the EH node is always less than the averageharvested power and increases with the storage capacity butdecreases with the SNR.A
PPENDIX A − P ROOF OF T HEOREM K → ∞ and taking the expectation of both sidesof (2), we obtain E [ B ( i + 1)] − E [ B ( i )] = ¯ X − E [ P UL ( i )] . (24)From (1), P UL ( i ) ≤ M, ∀ i ⇒ E [ P UL ( i )] ≤ M , hence from(24), E [ B ( i + 1)] − E [ B ( i )] ≥ ¯ X − M follows. If M < ¯ X , then E [ B ( i + 1)] > E [ B ( i )] (25)must hold. That is, the mean of the process { B ( i ) } changes(increases) with time, and therefore a stationary distribu-tion for { B ( i ) } does not exist. Furthermore, from (25), lim i →∞ E [ B ( i )] = ∞ , i.e., the energy accumulates in thebuffer. Hence, there must be some time slot j , after whichfor i > j , B ( i ) > M a.s. Next, we prove by contradictionthat j must be finite. If P UL ( j ) = B ( j ) < M and j → ∞ ,then lim j →∞ E [ B ( j )] < M which violates lim i →∞ E [ B ( i )] = ∞ .Hence, j must be finite. This completes the proof.A PPENDIX B − P ROOF OF THEOREM { Z ( i ) } in (3) is equivalent to the waiting time of a customer ina GI/G/1 queue [13], where X ( i ) is equivalent to the customerservice time and M is equivalent to the customers’ inter-arrivaltime. Now, our storage process { B ( i ) } in (2) with K → ∞ isequivalent to the process U ( i ) = Z ( i ) + X ( i ) , see (4). Thatis, { B ( i ) } is equivalent to the sojourn time (waiting time plusservice time) of a customer in a GI/G/1 queue. Since { Z ( i ) } and { X ( i ) } are independent and { X ( i ) } is stationary, thenthe steady state behavior of { B ( i ) } is solely governed by thatof { Z ( i ) } . Hence, from [13, Corollary 6.5 and Corollary 6.6], M > ¯ X is a sufficient condition for the process { B ( i ) } topossess a unique stationary distribution to which it convergesin total variation from any initial distribution.A PPENDIX C − P ROOF OF C OROLLARY f ( x ) = λ e − λx in (6) and using δ = λM , we get g ( x ) = λ e − λx M (cid:90) g ( u ) d u + M + x (cid:90) M e − δ e λu g ( u ) d u . (26)hen M > ¯ X , i.e., δ > , we know from Theorem 2 that(26) has a unique solution for g ( x ) . Similar to [9, eq. (11)], wepostulate an exponential-type solution given by g ( x ) = k e px ,then the right hand side of (26) reduces to λ e − λx (cid:104) kp (cid:0) e pM − (cid:1) + k e − δ λ + p (cid:0) e ( λ + p )( M + x ) − e ( λ + p ) M (cid:1)(cid:105) = e − λx (cid:104) λkp (cid:0) e pM − (cid:1) − λkλ + p e pM (cid:105) + kλ e pM λ + p e px ! = k e px . (27)In order for (27) to hold, the coefficient of e px in the secondterm of (27) must be k , which implies λ e pM = λ + p . Thiscondition will also reduce the coefficient of e − λx to zero. From λ e pM = λ + p , p can be obtained using the Lambert W function,i.e., p = (cid:0) − δ − W ( − δ e − δ ) (cid:1) /M , which is < since δ > .Now, k can be obtained from the unit area condition on g ( x ) ,namely, (cid:82) ∞ k e px =1 ⇒ k = − p . This completes the proof.A PPENDIX D − P ROOF OF THEOREM { X ( i ) } has an infinite positive tail, then the state space S contains anatom at K , i.e., the energy level B ( i ) = K is reachable withnon-zero probability. Define the measure φ as φ (0 , K ) = 0 and φ ( { K } ) = 1 , then the process { B ( i ) } is φ -irreducible, see [14,Section 4.2]. Furthermore, { B ( i ) } is also ψ -irreducible with ψ ( A ) = (cid:80) n P n ( K, A )2 − n , where P n ( x, A ) is the probabilitythat the Markov chain moves from energy state x to energy set A in n time steps. The dynamics of { B ( i ) } in (2) ensures thatall energy sets are reachable a.s. from any initial state of thebuffer in a finite mean time. Hence, the chain is positive Harrisrecurrent [14, Proposition 9.1.1], where positive recurrencefollows from [14, Theorem 10.2.2]. Thus, { B ( i ) } possessesa unique stationary distribution π . Finally, with the additionalproperty of { B ( i ) } being aperiodic (i.e., no energy level setsare only revisited after a fixed number of time slots > (period > )), it follows from [14, Theorem 13.3.3] that { B ( i ) } con-verges to the distribution π in total variation from any initialdistribution Γ , i.e., lim n →∞ sup A | (cid:82) Γ( d x ) P n ( x, A ) − π ( A ) | → .This completes the proof.A PPENDIX E − P ROOF OF P ROPOSITION g ( x ) is that although the approximate pdf ˜ g ( x ) istight for most of the range of x (even for n c = 2 ), it is looseat the tail of the distribution (namely for the last two sectionsof the pdf, i.e., n = 0 , ). With ˜ g ( x ) = c e dx , d is obtained inexactly the same manner as p for an infinite-size buffer, c.f.Appendix C. However, unlike in the infinite-size buffer case,the amount of energy in a finite-size buffer with δ ≤ stillconvergences to a limiting distribution, c.f. Theorem 4. Thisexplains the use of the Lambert W function with two differentorders in (14) to consider the two cases of δ ≤ and δ > .Note that d in (14) satisfies d > for δ < (an exponentiallyincreasing ˜ g ( x ) ), d < for δ > (an exponentially decaying ˜ g ( x ) ), and d = 0 for δ = 1 (a nearly uniform distribution).Since ˜ g (0) = c , we obtain c simply from the exact g ( x ) in (12)at x = 0 after replacing π ( K ) by ˜ π ( K ) , i.e., c = ˜ π ( K ) π ( K ) g (0) . Finally, ˜ π ( K ) in (15) guarantees a unit area distribution, i.e., K − n c M (cid:82) ˜ g ( x ) d x + n c − (cid:80) n =0 K − nM (cid:82) [ K − ( n +1) M ] + ˜ g n ( x ) d x + ˜ π ( K ) = 1 .Next, we study the error associated with the proposed approx-imation. As far as the performance analysis is concerned, onlythe pdf in the range [0 , M ] is needed, c.f. Section V. Assuming K = lM , with l ∈ Z for simplicity, then the approximation errorin the range [0 , M ] is given by e ( x ) = g l − ( x ) − ˜ g ( x ) = π ( K ) A ( x ) (cid:34) l − (cid:88) q =0 S q ( x ) − l − (cid:88) q =0 S q (0)e − W j ( − δ e − δ ) xM (cid:34) (cid:80) l − n =0 C n + (cid:80) n c − n =0 C n (cid:35) (cid:35) , (28)where S q ( x ) = ( δ e − δ ) q q ! (cid:16) q + λ ( x − K ) δ (cid:17) q − (cid:16) q + λ ( x − K ) δ − qδ (cid:17) is the summand in (12), A ( x ) = λ e − λ ( x − K ) and C n = e nδ (cid:16) e δ − (cid:80) nq =1 ( δ e − δ ) q q ! 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