Event-Trigger Based Robust-Optimal Control for Energy Harvesting Transmitter

Transcription

1 Evet-Trigger Based Robust-Otimal Cotrol for Eergy Harvestig Trasmitter Yirui Cog, Studet Member, IEEE, ad Xiagyu Zhou, Member, IEEE Secifically, two advaced methods i cyberetics are emloyed i this work. Oe is the robust otimal cotrol [8, whose solutios are largely immue to the system ucerarxiv: v3 [cs.it 2 Nov 206 Abstract This aer studies a olie algorithm for a eergy harvestig trasmitter, where the trasmissio (comletio time is cosidered as the system erformace. Ulike the existig olie algorithms which more or less require the kowledge o the future behavior of the eergy-harvestig rate, we cosider a ractical but sigificatly more challegig sceario where the eergy-harvestig rate is assumed to be totally ukow. Our desig is formulated as a robust-otimal cotrol roblem which aims to otimize the worst-case erformace. The trasmit ower is desiged oly based o the curret battery eergy level ad the data queue legth directly moitored by the trasmitter itself. Secifically, we aly a evet-trigger aroach i which the trasmitter cotiuously moitors the battery eergy ad triggers a evet whe a sigificat chage occurs. Oce a evet is triggered, the trasmit ower is udated accordig to the solutio to the robust-otimal cotrol roblem, which is give i a simle aalytic form. We reset umerical results o the trasmissio time achieved by the roosed desig ad demostrate its robust-otimality. Idex Terms Eergy harvestig, olie algorithm, evet trigger, robust-otimal cotrol, trasmissio-time miimizatio. I. INTRODUCTION A. Motivatio ad Related Work The develomet of eergy harvestig devices has attracted sigificat attetio i recet years with may otetial alicatios i commuicatio etworks for gree ad selfsustaiable commuicatios [. I order to make efficiet use of harvested eergy, both offlie ad olie solutios have bee ivestigated for desigig the otimal trasmissio olicy. Offlie solutios are ossible i highly redictable eviromets where the eergy ad data arrivals i a sufficietly distat future (for commuicatio urose ca be accurately estimated [2 [7. O the other had, olie solutios tyically reduce the deedece o the future kowledge of the eergy ad data arrival rocesses, ad hece are more alicable i ractice. The olie algorithms ca be roughly categorized ito two frameworks as follows: I the first framework, the statistical arameters (e.g., the exectatio of eergy ad data arrival rocesses are kow, ad the desigs of olie trasmissio olicies for eergy harvestig odes are ofte stated as stochastic cotrol roblems. I [8, [9, the eergy ad data arrival rocesses were modeled as statioary ad ergodic stochastic rocesses, where the throughut-otimal ad delay-otimal trasmissio olicies Y. Cog ad X. Zhou are with the Research School of Egieerig, Australia Natioal Uiversity, Australia ( {yirui.cog, xiagyu.zhou}@au.edu.au. were studied. By modelig the eergy arrival rocess as a comoud Poisso rocess, [0 roosed a throughut-otimal trasmissio olicy with a deadlie i the cotiuous-time domai by dyamic rogrammig. I [, the eergy arrival rocess was formulated as first-order statioary Markov model ad the fiite-horizo throughut-otimal trasmissio olicy was derived. Aimig at miimizig the delay, [2 rovided a closed-form desig for the trasmissio olicy which has a multi-level water-fillig structure. Without exlicitly modelig the eergy ad data arrival rocesses, [3 emloyed a uer boud o the log-term data loss ratio ad a threshold o the frequecy of visits to zero battery state to give a ear throughut-otimal trasmissio olicy. Although the majority of the desiged trasmissio olicies aim at either maximizig the throughut or miimizig the delay, other studies also cosidered maximizig the commuicatio reliability [4 or miimizig the eergy cosumtio [5. The secod framework uses arameter-ideedet methodologies. I [6, the eergy ad data arrival rocesses were formulated as time-homogeeous Markov chais without kowig the trasitio matrix, ad Q-learig was alied to erform olie otimizatio o the trasmissio olicy. By Lyauov otimizatio techique combied with the idea of weight erturbatio, [7 roosed a geeric utilitymaximizatio olicy, uder the assumtio that the amout of harvested eergy i each time slot is ideedet ad idetically distributed (i.i.d. but its statistical arameters are totally ukow. Although these arameter-ideedet olicies require less kowledge o the eergy ad data arrival rocesses, the stochastic models of the eergy ad data arrival rocesses still eed to be kow exactly. I ractical sceario, the factors determiig the eergy arrivals are comlex, dyamically chagig ad ofte ukow to the system desiger. It is sometimes eve difficult to come u with accurate models for the eergy-harvestig rate. This leads to a iterestig ad ractical desig roblem: how to desig ad imlemet a olie trasmissio olicy for eergy harvestig odes without imosig ay assumtio o the future behaviors of the eergy-harvestig rate? I this aer, we aim to rovide a aswer to this imortat questio.

2 2 taities. We use it to esure that the system erformace (i.e., the trasmissio time is o worse tha a level (the otimized worst-case erformace, o matter what kid of eergy arrival rocess is imosed. The other method is the evet-trigger based cotrol [20 (or aeriodic cotrol. It ca sigificatly reduce the uecessary comutatios comared to the traditioal eriodic cotrol (i.e., time-slotted cotrol 2. Our aer is mostly related to the recet work i [2 [23, which cosidered the similar assumtios that the future eergy arrival is ukow. I [2 [23, the cometitive aalysis [24 was emloyed to miimize the ga betwee olie ad offlie erformaces. However, miimizig this ga caot directly guaratee a certai system erformace (e.g., otimal worst-case erformace of olie algorithms. Additioally, these recet studies still cosidered time-slotted systems, ad hece, the trasmissio rotocol is udated i every time slot, regardless of the chage i the amout of eergy available. B. Our Cotributios I this work, we study the erformace of a eergyharvestig trasmitter measured by the trasmissio (comletio time, i.e., the time duratio it takes to comlete the trasmissio of a give amout of data. We roose to use evet-trigger based desig to cotrol the trasmit ower without ay kowledge o the future behavior of the eergyharvestig rate. I the cosidered sceario, it is ot ossible to use ay statistics of the trasmissio time i the desig. Hece, we adot the robust-otimal cotrol to miimize the worst-case trasmissio time. Nevertheless, the miimum worst-case trasmissio time may ot always be fiite. Whe the miimum worst-case trasmissio time is ifiite, which haes whe too much data is give to be trasmitted with isufficiet iitial battery eergy, we measure the robustotimality of the trasmit ower desig by lookig at the set of eergy-harvestig rates that result i fiite trasmissio times. The robust-otimal desig esures the largest set of eergyharvestig rates resultig i fiite trasmissio times. The roosed evet-trigger based trasmitter has two buildig blocks for imlemetig the trasmit ower, amely a Evet Detector (ED ad a Trasmissio Plaer (TP. The ED cotiuously moitors the battery eergy ad triggers a ew evet whe it exerieces some sigificat chage sice the last evet. Wheever a evet is triggered, the TP uses the curret kowledge of battery eergy ad data queue to udate the trasmit ower by robust-otimal cotrol. The udated trasmit ower is imlemeted util the ext evet is triggered. To the best of our kowledge, this is the first The term robust otimality origiates from the robust otimizatio (e.g., i [9. It refers to the otimizatio of a objective over a ucertai set of situatios such that the objective is always ot worse tha the otimized level. If all the variables i oe robust otimizatio roblem are fuctios of time, the this roblem becomes a robust-otimal cotrol roblem. 2 Whe desigig commuicatio rotocol for eergy harvestig trasmitters, the trasmissio olicy is desiged accordig to the amout of eergy ad data available. The evet-trigger based cotrol udates the trasmissio olicy oly whe there is a otable chage i the amout of eergy. I cotrast, the traditioal time-slotted cotrol always erforms comutatio to udate the trasmissio olicy at regular time itervals. time that the evet-trigger based desig is imlemeted o eergy harvestig trasmitters. To facilitate the robust-otimal desig, we first give a comrehesive aalysis o the behavior of battery eergy ad data queue i each triggered evet. Secifically, we defie the reachable set, which describes all ossible states (battery eergy ad data queue reachable i oe evet based o the TP s kowledge, ad reflects the relatioshi amog battery eergy, data queue ad trasmit ower. Base o these aalyses, we derive the solutio of the robust-otimal trasmit ower desig, give i a simle aalytic form. C. Paer Orgaizatio ad Notatio I Sectio II, the system model is give ad the evettrigger based trasmissio is itroduced. I Sectio III, the roblem of fidig the robust-otimal trasmit ower desig is defied. We study the roerties of the roosed evet-trigger based system through the reachable set aalysis i Sectio IV. The otimal solutio to the roblem is give i Sectio V. I Sectio VI, simulatio results are show to illustrate the effectiveess of our desig ad corroborate our theoretical results. Fially, coclusio is draw i Sectio VII. Throughout this aer, R +, R +, ad Z + deote the sets of o-egative real umbers, ositive real umbers, ad ositive itegers. Ẋ(t deotes the time derivative of X(t which is a fuctio of time. µ(s is the Lebesgue measure of set S. The restrictio (Page 36 i [25 of fuctio f to domai A is f A. For x R, [x + returs max{x, 0}. II. SYSTEM MODEL AND EVENT-TRIGGER APPROACH A. System Model We cosider a trasmitter-receiver air as show i Fig.. At time t [t 0,, where t 0 is the startig time of the commuicatio, the trasmitter has a battery with eergy E(t R +. The harvested eergy is stored i the battery with eergy-harvestig rate H : [t 0, R +. We assume H, as a fuctio of time, is Lebesgue itegrable over ay subset of R + with fiite measure, ad all such H form the set H. The trasmit ower at time istat t is (t [0, max, which is also Lebesgue itegrable over ay subset of R + with fiite measure, ad max deotes the maximum ower costrait. The, the relatioshi amog battery eergy, eergy-harvestig rate ad trasmit ower is give by a differetial equatio Ė(t = H(t (t, ( where the iitial battery eergy is E(t 0 R +. I this aer, we cosider the trasmissio-time miimizatio roblem, where all the data to be trasmitted is available at t 0. The data queue is Q(t R +, ad the trasmissio rate at t is r(t R +, which is Lebesgue itegrable over ay subset of R + with fiite measure, ad the relatioshi betwee data queue legth ad trasmissio rate is Q(t = r(t, (2 where the iitial data queue is Q(t 0 R +. Equatio (2 meas all the amout of data (equal to Q(t 0 to be trasmitted is available at t 0, ad there is o subsequet data

3 3 H(t Data Queue Q(t E(t Battery Eergy Evet Detector Trasmitter Trigger (t Trasmissio Plaer Fig.. The evet-trigger based eergy harvestig trasmitter. r(t (t Receiver arrival i (t 0,, which is a commoly used assumtio for trasmissio-time miimizatio roblem (see [4 for a examle. The other system assumtios are as follows: The eergy-harvestig rate H is totally ukow. Battery eergy E(t ca be measured at curret time t. The chael is assumed to be static such that the chael caacity is C(t = log 2 ( + (t. Remark. Ulike most existig olie algorithms which somewhat icluded the rior-kowledge o H, we aim to derive a olie algorithm without ay kowledge of H. Additioally, the battery eergy ca be moitored cotiuously. 3 We cosider a simle static chael that is oly affected by AWGN. Note that the static-chael assumtio is widely used i the literature, e.g., [4, [5, [27. Note that the trasmit ower (t ad the trasmissio rate r(t are to be desiged by us, ad i this aer, we assume the chael caacity is achieved, i.e., r(t = C(t = log 2 ( + (t, which meas at each time t the trasmissio rate is a fuctio of trasmit ower, labeled as r((t. Now, the oly variable to be desiged is the maig : t (t. B. Evet-Trigger Based Cotrol As show i Fig., the evet-trigger cotrol relies o a Evet Detector (ED that detects the ecessary chages i the cumulated eergy; ad a Trasmissio Plaer (TP that gives the desig for trasmit ower (t. We give the coditio uder which a evet is triggered. Defiitio (Triggerig Coditio. From a give time istat s, a evet is triggered at t (t > s wheever the followig iequalities is satisfied t s H(τdτ (a = E(t E(s + t where ε R + is the triggerig threshold. s (τdτ ε, (3 I Defiitio, coditio (3 meas that whe the harvested eergy cumulates over a certai level ε, the ED triggers a ew evet. Eve though H(t is ukow, the itegrals of H(t ca be calculated by equality (a i (3, which is derived by 3 The techology for cotiuously moitorig the battery eergy with miimal oeratig eergy cosumtio has bee develoed over the ast decades (e.g., [26. Hece, we assume that this oeratig eergy ca be eglected as comared to the eergy cosumtio for trasmissio. the solutios of (, where E(t is observable. The trasmit ower (τ for τ (s, t is determied by the TP (to be discussed i later art of this subsectio. Recall that the trasmissio is carried out over the etire commuicatio time iterval [t 0,. At the iitial time istat t 0, the ED triggers the start of the trasmissio. The, the ED will start moitorig the system o [t 0, t, where t is the curret time istat. We label the first time istat (after t 0 at which the system satisfies the triggerig coditio i Defiitio by t. After t, the ED will start moitorig the system o (t, t. The ext time istat at which a evet is triggered is labeled as t 2, ad so o. For coveiece, we say that evet starts at t ad fiishes at t +. This comletes our descritio of the ED. Wheever a evet comes, the TP las the trasmit ower to be imlemeted from the curret time istat util the ext evet arrives, ad we aalyze evet without loss of geerality. It is imortat to ote that the TP oly takes ito accout the iformatio available at the begiig of the evet whe laig the trasmit ower. Such iformatio icludes the battery eergy ad data queue legth at t, i.e., E(t ad Q(t. However, ay future chage due to eergy-harvestig rate, i.e., H(t for t > t, caot be take ito accout, simly because H(t for t > t is ukow to the TP at t. Secifically, at the begiig of the th evet, i.e., at the time istat t, the TP records the values of E(t ad Q(t, ad desigs (t to be imlemeted over (t, t + usig ( ad (2 with H(t = 0. This does ot mea the TP eglects the effect from H(t all the time, because H(t determies the arrival time of the ext evet, by the triggerig coditio. I each evet, the TP las the trasmit ower for a fiite time widow after t, ad we call it as the Plaed Trasmit Power (PTP:,ε : (t, t + T [0, max, (4 where the subscrit ε meas the PTP is desiged uder a give triggerig threshold ε, ad the duratio of the time widow T is called as the laed trasmissio time. I (4, the PTP is described as a maig from a time istat to a value of trasmit ower. For ay t (t, t + T,,ε (t is the value of trasmit ower at time t. Thus, a PTP ca be described by two arameters, i.e.,,ε (t ad T. For examle, the PTP,ε (t = si t, t (t, t + is described by air (,ε (t, T = ( si t,. All ossible,ε comose the PTP set P,ε. The desig roblem here is to fid a good maig t,ε (t i (4. Note that T is art of the descritio of a PTP, ad i other words, the desig of,ε effectively icludes the desig of T. We will see i Sectio V that T lays a imortat role i desigig the otimal,ε. Ideally, the PTP should be imlemeted over (t, t + T. However, the TP erforms the desig at t based o the curret iformatio (E(t ad Q(t, ad hece, caot redict the exact value of t +, i.e., it does ot kow whe the ext evet occurs. As a result, the PTP i evet will ot be imlemeted beyod t + because a ew PTP will be laed ad imlemeted after t +. Hece, the actual trasmissio time is mi{t, t + t }. It imlies that whe the ( + th evet comes, the TP will use the ewly desiged PTP for

4 4 evet +, eve if the laed trasmissio for evet is ufiished. Therefore, the actual trasmit ower, deoted by ε is imlemeted iecewise by,ε (t, t + (restrictig,ε to (t, t + for each evet. Secifically, the relatioshi betwee the actual trasmit ower ad the PTP is ε (t =,ε (t, t (t, t +. (5 For a give triggerig threshold ε, all such ε : t ε (t i (5 form the set of all ossible trasmit ower P ε. Recall that ε is Lebesgue itegrable over ay subset of R + with fiite measure. Thus, P ε is a set of o-egative Lebesgue itegrable fuctios over ay subset of R + with fiite measure. To sum u, the evet-trigger cotrol framework is illustrated i Algorithm, where Lies 2 ad 8 are the very art to be desiged i the rest of this aer. Algorithm Evet-Trigger Based Cotrol : Iitial Coditio: t = t 0, t = t 0, E(t = E(t 0, Q(t = Q(t 0, E(t = E(t 0, ad Q(t = Q(t 0. 2: Assumig H(t = 0 for t > t, the TP desig the PTP,ε (see (4; 3: while Q(t > 0 do 4: The ED udates E(t ad Q(t (the udate frequecy is deedet o the chi s clock ad checks the coditio i (3 with s = t ; 5: if Coditio (3 is satisfied the 6: The ED triggers a evet to activate the TP; 7: The TP udates t = t, E(t = E(t, ad Q(t = Q(t; 8: Assumig H(t = 0 for t > t, the TP desig the PTP,ε for this evet; 9: The trasmitter uses the ewly desiged PTP as the trasmit ower, i.e., (t =,ε(t; 0: else : TP is iactive; 2: The trasmitter uses the most recet PTP as the trasmit ower, i.e., (t =,ε(t; 3: ed if 4: ed while III. PROBLEM DESCRIPTION I this aer, we study the trasmissio-time miimizatio roblem uder ukow eergy-harvestig rate H. The trasmissio time is the time set by the trasmitter to clear u the data queue, ad we label it as T ( ε, E(t 0, Q(t 0, H, which is deedet o the trasmit ower desig ε : t ε (t, the iitial battery eergy E(t 0, the iitial data queue Q(t 0, ad the eergy-harvestig rate H. Sice H is totally ukow, give ε, E(t 0, ad Q(t 0, the trasmissio time varies with differet H H. As a result, the trasmissio time is withi the followig rage if T ( ε, E(t 0, Q(t 0, H T ( ε, E(t 0, Q(t 0, H su T ( ε, E(t 0, Q(t 0, H. (6 Note that it is imossible to comute the average (or other statistical roerties of trasmissio time for ay give desig because the statistics of H are totally ukow. Nevertheless, it is ossible to examie the worst-case trasmissio time for ay give desig, i.e., su T ( ε, E(t 0, Q(t 0, H. Thus, our aroach is to fid a desig that achieves the miimum worst-case trasmissio time. I other words, our desig aims to give the best erformace uder the worst-case sceario. Based o this idea, the trasmissio-time-miimizatio roblem is defied i Subroblem. Subroblem (Trasmissio-Time-Miimizatio Problem. Give iitial battery eergy E(t 0, iitial data queue Q(t 0, ad triggerig threshold ε, desig PTP,ε i each evet, with the kowledge of E(t ad Q(t, such that T = if ε P ε su T ( ε, E(t 0, Q(t 0, H, (7 where the actual trasmit ower ε is determied by the PTP,ε as show i (5. Remark 2. Ideed, the idea of defiig Subroblem is borrowed from robust-otimal cotrol [8: I (7, the su oerator returs the worst trasmissio time for a give ε uder its corresodig worst-case eergy-harvestig rate H ε ; while the if oerator reflects our aim of desigig ε whose worst trasmissio time T ( ε, E(t 0, Q(t 0, H ε is the smallest. 4 Oe techical challege i solvig Subroblem is that the worst-case eergy-harvestig rate deeds o the choice of trasmit ower, i.e., for differet ε, the worst-case H ε ca be differet. It is imortat to ote that T is ot always fiite. I the case T =, equatio (7 ca hardly measure the robust otimality o the desiged ε, sice for ay ε the worstcase trasmissio time is always ifiite. Hece, Subroblem is ot sufficiet for describig all scearios ad we eed a differet roblem formulatio to deal with the case of T = as exlaied as follows: For a give ε, there should exist some eergy-harvestig rate H resultig i a fiite trasmissio time, eve though the worst-case H ε may lead to a ifiite trasmissio time. All ossible such eergyharvestig rates form the fiite-trasmissio-time eergy set H f ( ε, E(t 0, Q(t 0 H, defied i Defiitio 2. Defiitio 2 (Fiite-Trasmissio-Time Eergy Set. Give ε, E(t 0, ad Q(t 0, the fiite-trasmissio-time eergy set H f ( ε, E(t 0, Q(t 0 is {H : T ( ε, E(t 0, Q(t 0, H <, H H}. (8 Cosiderig two trasmit ower desigs, deoted by a ε ad b ε, whose worst-case trasmissio times are ifiite, we ca say that a ε is more robust tha b ε if the fiitetrasmissio-time eergy set of a ε (i.e., H f ( a ε, E(t 0, Q(t 0 is larger tha that of b ε (i.e., H f ( b ε, E(t 0, Q(t 0. This is because H f ( a ε, E(t 0, Q(t 0 is more likely to result i a fiite trasmissio time i the actual trasmissio. This motivates us to fid the trasmit ower ε with the largest H f ( ε, E(t 0, Q(t 0 such that ay other H f ( ε, E(t 0, Q(t 0 is its subset, whe T =. Subroblem 2 (Eergy-Set-Maximizatio Problem. Give iitial battery eergy E(t 0, iitial data queue Q(t 0, ad triggerig threshold ε, if T =, desig PTP,ε i each 4 The subscrit ε i H ε highlights the fact that the worst-case H deeds o the give trasmit ower. Although, mathematically H ε (for ay ε might ot exist i H, sice the oerator i (7 is su rather tha max, the existece of the otimal trasmit ower desig ε is give i Theorem.

5 5 evet, with the kowledge of E(t ad Q(t, such that H f ( ε, E(t 0, Q(t 0 H f ( ε, E(t 0, Q(t 0, ε P ε. (9 As a summary of Subroblem ad Subroblem 2, the robust-otimal trasmit ower should: achieve the miimum trasmissio time T uder the worst-case eergy-harvestig rate, if T is fiite; otherwise, esure the largest set of H that results i a fiite trasmissio time, if T is ifiite. Puttig these two subroblems altogether, we defie the Robust- Trasmissio-Time (RTT roblem as follows. Problem (Robust-Trasmissio-Time Problem. Give iitial battery eergy E(t 0, iitial data queue Q(t 0, ad triggerig threshold ε, desig PTP,ε i each evet, with the kowledge of E(t ad Q(t, such that { ε satisfies (7, if T <, ε satisfies (9, if T (0 =, where the relatioshi betwee trasmit ower ε ad the PTP,ε is give i (5. I the rest of this aer, we focus o how to solve the RTT roblem with the evet-trigger based cotrol. IV. REACHABLE SET ANALYSIS Sice our roosed cotrol method is evet-trigger based (see Sectio II-B, i.e., the trasmit ower is iecewise imlemeted i each evet, i this sectio, we aalyze all reachable battery eergy ad data queue (the system states i each evet. We stress that the otimal solutio of the RTT roblem is highly deedet o the structure of the reachable set of battery eergy ad data queue. Recall that durig evet the TP oly takes ito accout the iformatio of battery eergy ad data queue at t ad igores the eergy-harvestig rate H(t for t > t. The battery eergy ad data queue see by the TP behave as {Ẽ (t = E(t t t,ε (τdτ, Q (t = Q(t t ( t r (,ε (τ dτ, where Ẽ(t ad Q (t refer to the dyamics of the battery eergy ad data queue kow by the TP based o its available iformatio (i.e., E(t ad Q(t but ot H(t for t > t, which are distict from the actual battery eergy E(t ad data queue Q(t. 5 Sice i each evet the PTP is desiged by the TP, we should aalyze the roerty of Ẽ(t ad Q (t rather tha E(t ad Q(t. This is because the robust-otimal cotrol of the trasmit ower ca oly be desiged accordig to what the TP kows, i.e., Ẽ (t ad Q (t. Now, we defie the reachable set of our iterest. The reachable set cotais all reachable states ( Ẽ (t, Q (t after imlemetig the PTP over the laed trasmissio time i the th evet, i.e., all reachable ( Ẽ (t + T, Q (t + T. 5 From (3 we kow that the relatioshi betwee Ẽ(t ad E(t is Ẽ (t E(t Ẽ(t + ε for t [t, t +. The relatioshi betwee Q (t ad Q(t is Q (t = Q(t for t [t, t +. Q b o c 2 Fig. 2. Illustratios of reachable set i the Ẽ Q (two-dimesioal regio for evet : oit a deotes (E(t, Q(t, ad b, c as well as d are differet ed oits ( Ẽ, Q i the reachable set R (blue regio. The arrows with umbers ad 2 are two differet aths from a to b. The arrows with umbers 3 ad 4 are the aths from a to c ad d, resectively. Note that the laed trasmissio time T is geerally differet for differet PTPs. Fig. 2 gives a ictorial illustratio of the reachable set: At t = t, the system state is at oit ( Ẽ (t, Q (t (i.e., oit a. After t, the PTP ushes the system state (see by the TP to move alog the arrow (differet PTPs corresod to differet arrows. At t = t +T, the state stos at a oit (e.g., oit b which corresods to ( Ẽ (t + T, Q (t + T. All ( Ẽ (t + T, Q (t + T comose the reachable set (i.e., the shaded area. We defie the reachable set as follows. Defiitio 3 (Reachable Set. From give E(t ad Q(t, the reachable set i th evet is { R = (Ẽ, Q t+t : Ẽ = E(t,ε (τdτ 0, t t+t Q = Q(t r(,ε (τdτ 0, t,ε P },ε, T <, where we use ( Ẽ, Q rather tha (Ẽ (t, Q (t to rereset the oit i R, as ( Ẽ (t + T, Q (t + T ca be the same with differet T, which violates the defiitio of set. I Fig. 2, we ca see that differet PTPs corresod to differet aths or arrows i the figure, which may or may ot arrive at the same ed oit. Eve though Defiitio 3 gives a exressio of the reachable set, it is too abstract ad ot coveiet for desig. To give a more exlicit form of reachable set, we defie the Rate- Power Equilibrium (RPE to hel the subsequet aalysis. Defiitio 4 (Rate-Power Equilibrium (RPE. The rate-ower lie is defied i the r lae (see Fig. 3: 3 4 d r = K, where K = Q(t Q. (2 E(t Ẽ The itersectio of rate-ower lie ad rate fuctio r = log 2 ( + for (0, max (here, is a scalar is called the RPE, ad the corresodig trasmit ower of the RPE a E

6 6 r o r K r K a r r r Kmax e b r Kmi r K max Fig. 3. Rate-ower lie, rate fuctio ad RPE, where r( = log 2 ( + ad K > Kmax > K > K mi > K. For K ad Kmax, o ositive itersectios exist due to the large K. For K, K mi ad K, ositive itersectios exist because of the small K : Poits a as well as b are RPEs, but oit c is ot a RPE due to e > max. We ca see that e decreases as K goig large. is labeled by e. For Ẽ = E(t or Q = Q(t, we defie their e = 0, eve though o RPE exists. From Defiitio 4, we see that ay arbitrary air of values (Ẽ, Q has a corresodig rate-ower lie i the r lae. Because of the cocavity of the rate fuctio, there exists at most oe RPE for a arbitrary ( Ẽ, Q. If the RPE exists, we ca use the followig remark to calculate it. Remark 3. Solvig K e = log 2 ( + e, we have e = K l 2 W ( K l 2 2 K, (3 where W is the real valued Lambert W fuctio [28 i the lower brach (W. Here the RPE is the oit ( e, log 2 ( + e i the r lae. The followig lemma makes the lik betwee the reachable set ad the RPE, which hels us to fid a exlicit exressio of reachable set i order to facilitate trasmit ower desig. Lemma (Criterio o Poits i Reachable Set. (Ẽ, Q R \{(E(t, Q(t } (4 if ad oly if the RPE for ( Ẽ, Q exists. Proof: See Aedix A. Lemma tells that excet for (E(t, Q(t, ay oit i reachable set has a RPE, ad ay oit which has a RPE must be i the reachable set. Based o Lemma, a exlicit exressio of reachable set ca be give. Proositio (Exressio for Reachable Set. Give (E(t, Q(t, the reachable set satisfies R \{(E(t, Q(t }= {(Ẽ, Q :K mi K <K max, 0 Ẽ <E(t, 0 Q } < Q(t, (5 where K is a fuctio of Ẽ ad Q give i (2, ad K mi := r( max r(x, K max := lim max x 0 + x = l 2. (6 c Proof: See Aedix B. Proositio meas that the oit (Ẽ, Q is i the reachable set if ad oly if the corresodig K (i.e., the sloe of the corresodig rate-ower lie is withi a certai rage. As show i Fig. 3, the sloe K decreases as e grows, which imlies: O the oe had, K max is the suremum of the sloe K to have a itersectio betwee the rate-ower lie ad the rate fuctio (i.e., to have a RPE. O the other had, due to the maximum ower costrait, K mi is the miimum sloe to have a RPE. A ictorial illustratio of the reachable set is show i Fig. 4, ad it ca be easily categorized ito three cases, deedig o the relatioshi amog K mi, K max, ad K bal (called the eergy-balaced sloe, where K bal := Q(t 0 E(t 0 = Q(t E(t. (7 To be more secific, Fig. 4(a, Fig. 4(b, ad Fig. 4(c corresod to K bal < K mi, K mi K bal < K max ad K bal K max, resectively. These three cases have imortat hysical meaigs, give i the followig remark. Remark 4 (Categorizatio of Reachable Sets. I Fig. 4(a, E is always greater tha 0, (Ẽ, Q R, which imlies that the battery eergy is abudat. This meas that whe the data queue is cleared, there is still battery eergy remaiig, o matter what PTP is used. I this case, we say that R is eergy-abudat. I Fig. 4(b, origi o is i the reachable set, which meas the data queue ca be cleared by usig all the eergy stored i the battery, ad i this case, we say that R is eergy-balaced. I Fig. 4(c, Q is always greater tha 0, (Ẽ, Q R, which meas the data queue caot be cleared with the available battery eergy, o matter what PTP is emloyed. I this case, we say that R is eergy-scarce. The RPE ot oly hels to shae the reachable set (see Proositio, but also gives the time-otimal PTP. Recall that from oe startig oit (E(t, Q(t, there are multile PTPs that reach the same ed oit (Ẽ, Q (see Fig. 2. These PTPs, however, sed differet amout of laed trasmissio time T. Hece, we eed to fid the time-otimal PTP that has the miimum T for each oit (Ẽ, Q R. For a give startig oit (E(t, Q(t ad a ed oit (Ẽ, Q, the laed trasmissio time T : P,ε R + is a o-egative fuctioal of the PTP, ad the time-otimal PTP has the smallest T (ad we mark the miimum laed trasmissio time as T [Ẽ, Q for ed oit (Ẽ, Q, i.e., T [Ẽ, Q = if T (,ε, (8,ε P,ε [Ẽ, Q where P,ε [Ẽ, Q stads for those PTPs to make the ed oit as ( Ẽ, Q. The followig roositio shows that the trasmit ower i the time-otimal PTP for a give air of startig oit ad ed oit is uique ad remais costat at the value of e (the trasmit ower corresodig to the RPE over the laed trasmissio time T.

7 7 Q a Q d Q h o b L L 3 c L 2 E o L 3 f e L g L 2 E i j o L L 2 L 3 E (a (b (c Fig. 4. Shaes of 3 cases of reachable sets (icludig the startig oits a, d, h, where L, L 2, ad L 3 corresod to lies Q Q(t = K mi (Ẽ E(t, Q Q(t = K max (Ẽ E(t, ad Q Q(t = K bal (Ẽ E(t, resectively. (a Eergy-abudat case (K bal < K mi, oit b is i the reachable set, whereas oit c is ot i the reachable set. (b Eergy-balaced case (K mi K bal < K max, oit e is i the reachable set, while oits f ad g are ot i the reachable set. (c Eergy-scarce case (K bal Kmax, oit j is i the reachable set, but oit i is ot i the reachable set. Proositio 2 (Time-Otimal PTP. ( Ẽ, Q R \{(E(t, Q(t }, the uique time-otimal PTP to achieve T [Ẽ, Q is TIO,ε with arameters ( (,ε (t, T = e, T [Ẽ, Q, (9 where e is the trasmit ower of the corresodig RPE which ca be calculated by (3 i Remark 3, ad the miimum laed trasmissio time is: T [Ẽ, Q = Q(t Q r ( e = E(t Ẽ e. (20 Proof: See Aedix C. With Proositio 2, we ca calculate the time-otimal PTP through (9 to miimize the laed trasmissio time T [Ẽ, Q for ay oit i reachable set excet for (E(t, Q(t. But obviously, the otimal time for (E(t, Q(t is T [E(t, Q(t = 0. Note that differet ed oits (Ẽ, Q corresod to differet T [E(t, Q(t. If the TP wats to clear the data queue with a miimum laed trasmissio time, it is equivalet to cosider the ed oits with Q = 0 ad select oe from them which has the miimum T [Ẽ, Q. This result is give i Corollary. Corollary. If K bal < K mi, we have ( argmi T [Ẽ, Q = E(t Q(t, 0 (Ẽ, Q R, Q K mi =0 whose e (see Proositio 2 is max. If K mi K max, we have, (2 K bal < argmi T [Ẽ, Q = (0, 0, (22 (Ẽ, Q R, Q =0 whose e is labeled as bal bal = K bal l 2 W, which has the followig form: ( l 2 2 Kbal, (23 K bal where K bal is give i (7 ad W is the real valued Lambert W fuctio i the lower brach [28. If K bal K max, we caot fid ay ed oit with Q = 0. Proof: See Aedix D. Remark 5. We claim that Corollary lays a imortat role i the solutio to the RTT roblem, which is show i Theorem. More details ca be foud i the roof of Theorem (see Aedix E. Briefly seakig, the timeotimal PTPs corresodig to e i Corollary give the solutio to the RTT roblem for cases K bal < K mi ad K mi K bal < K max. V. SOLUTION TO RTT PROBLEM AND DISCUSSION ON TRIGGERING CONDITION The aalysis o the reachable set of battery eergy ad data queue i a arbitrary evet as well as the result o timeotimal PTP have eabled us to solve the RTT roblem defied i Problem. I this sectio, firstly, we reset the otimal solutio of the RTT roblem, ad the discuss the effect of the triggerig threshold ε. A. Otimal Solutio to RTT Problem Theorem (Otimal Solutio to RTT Problem. The otimal solutio of RTT roblem is R,ε with the arameters 6 : ( Q(t max, r( max K bal < K mi, ( (,ε (t, T = bal, E(t K bal mi K bal < K max, (0, 0 K bal K max, (24 where bal is give i (23. The corresodig actual trasmit ower imlemeted by R,ε is labeled as R ε. Proof: See Aedix E. 6 Recall that ay,ε i (4 ca be determied by two arameters,ε(t ad T.

8 8 Recall the evet-trigger based framework for trasmissio desig summarized i Algorithm. The solutio to the desig roblem i Lies 2 ad 8 is ow give i Theorem. Remark 6. The structure of R,ε is easy to uderstad. The first row corresods to the eergy-abudat case for R (see Fig. 4(a, ad i this case, the maximum ower is used to trasmit, sice there is eough eergy. Likewise, the secod row stads for the eergy-balaced case (see Fig. 4(b, ad the corresodig trasmit ower is bal, which ca clear the data queue ad use u the battery eergy at the same time. I the third row, the eergy-scarce case (see Fig. 4(c, the trasmitter seds othig, which ca be exlaied as that ay trasmissio i this case would make thigs worse. Remark 7. The worst-case eergy-harvestig rate for the otimal trasmit ower desig R ε exists ad is give by H R ε : t 0, i.e., o eergy arrival i [t 0, (the roof is give i Lemma 2 i Aedix E. We deote such oeergy-arrival case as H o. It should be oted that the worstcase eergy-harvestig rate of ay give trasmissio ower fuctio is ot always H o. Ideed, to determie the worst-case eergy-harvestig rate of a give trasmissio ower desig is difficult i geeral, which is the mai difficulty i solvig the RTT roblem. I Sectio VI, we will show a examle of a trasmit ower desig of which the worst-case eergyharvestig rate is very differet from H o. B. Discussios o the Triggerig Threshold I Sectio V-A, the otimal solutio of the RTT roblem is ivestigated for a give triggerig threshold. A atural questio is that: how does the triggerig threshold ε affect the system behavior (i.e., T ad H f ( R ε, E(t 0, Q(t 0? I this subsectio, we give the corresodig aswers. Firstly, we show that whe T < holds, T is ideedet of the value of ε, which is easy to verify, sice the worst-case H of the robust-otimal solutio R,ε is H o. I this worst case, ε, ε 2 > 0, we have R ε = R ε 2. This is because: For H o, the ext evet would ever be triggered. As a result, the actual trasmit ower is oly deedet o the PTP desiged i evet 0 which is ot affected by the value of the triggerig threshold. Differet from the T < case, for the T = case, H f ( R ε, E(t 0, Q(t 0 is deedet o the value of ε, ad the followig roositio tells that the smaller ε is, the larger H f ( R ε, E(t 0, Q(t 0 will be. Proositio 3. For T =, if ε b is a multile of ε a with multilier z Z + \ {}, i.e., zε a = ε b, the H f ( R ε a, E(t 0, Q(t 0 H f ( R ε b, E(t 0, Q(t 0. Proof: See Aedix F. Proositio 3 imlies that the smaller ε is, the larger the set H f ( R ε, E(t 0, Q(t 0 will be, which meas the more cases of eergy-harvestig rate H ca result i a fiite trasmissio time. However, due to the limited comutatioal resource, we caot make ε arbitrarily small because smaller ε leads to more frequet evet triggers. I ractice, we should balace the comutatioal accuracy ad efficiecy. VI. SIMULATION RESULTS Now, we reset the simulatio results to illustrate the beefit of the roosed trasmissio desigs based o robustotimal cotrol. Sice there are o comarable olie algorithms i the literature, we take the followig three desigs i the evettrigger cotrol framework as examles to comare with our desig. The first is a estimatio-based algorithm, labeled as E ε, which estimates the future eergy-harvestig rate based o the eergy-harvestig rate i the ast. The corresodig PTP E,ε i each evet has arameters: (,ε (t, T = ( Q(t max, r( max K bal < K mi, ( bal E(t +, bal + K mi K bal < K max, (0, 0 K bal K max, (25 where = 0 for = 0, ad = [E(t E(t /(t t for > 0. Comared with the robust-otimal PTP desig i (24, E,ε has the same structure ad the oly differece is the secod lie i (25, where is the additioally allocated ower i every evet. We ca see that is deedet o the average eergy-harvestig rate i the last evet, i.e., [E(t E(t /(t t. The additioally allocated ower takes ito accout the future eergy arrival whose rate is estimated to be the same as that i the last evet. I cotrast, the robust-otimal desig does ot assume ay future eergy arrival withi the curret evet. Therefore, the estimatio-based desig is smarter tha our robust-otimal desig R ε whe H haes to be a statioary rocess, sice the average eergy-harvestig rate [E(t E(t /(t t i the ast ca be a reasoably accurate estimate of the eergyharvestig rate i the future. The secod desig is the modified estimatio-based algorithm, labeled as M ε, which has the same structure (25 to E ε, but with a differet that: = 0 for = 0, ad = 0.25[E(t E(t /(t t for > 0. The modified estimatio-based algorithm M ε is more coservative tha E ε because oly a quarter of the estimated eergy is used. The third desig is the greedy algorithm, labeled by G ε, which simly trasmits data with maximum ower i every evet if E(t > 0 ad Q(t > 0. We reset simulatio results i three scearios. I the first two scearios, the eergy-harvestig rates are determiistic but totally ukow to the trasmitter, ad i the third sceario, the eergy-harvestig rate is a o-statioary stochastic rocess whose statistics are totally ukow to the trasmitter. I the sectio, the uits of all arameters are ormalized, ad we assume t 0 = 0. Sceario. We set the iitial battery eergy as E(0 =, the iitial data queue legth as Q(0 =, the maximum trasmit ower as max = 3, ad the triggerig threshold as ε = The eergy-harvestig rate is chose as H(t = h for t [0, 0.2, ad H(t = h/0 for t [0.2,, where h is selected from [0, 2. The trasmissio time comariso of R 0.05, E 0.05, M 0.05, ad G 0.05 is give i Fig. 5(a, from which we ca observe that:

9 9 The worst trasmissio time of R 0.05 is. The worstcase eergy harvestig rate is H(t with arameter h [0, 0.5, hece icludig H o as a worst case. 2 The worst trasmissio time of E 0.05 i this figure is 3.2. The worst-case eergy harvestig rate is with arameter h = 0.5. Clearly, H o is ot the worst case. 3 The worst trasmissio time of M 0.05 i this figure is.95. The worst-case eergy harvestig rate is with arameter h = 0.5. Clearly, H o is ot the worst case. 4 The worst trasmissio time of G 0.05 is. The worstcase eergy harvestig rate is H o. We ca see that the trasmissio time is guarateed by our desig to be ot greater tha, while the other three desigs ca result i a trasmissio time much larger tha. T (ε,e(t0,q(t0,h T (ε,e(t0,q(t0,h R 0.05 E 0.05 M 0.05 G h R 0.0 E 0.0 M 0.0 G (a R a Fig. 5. Trasmissio time comarisos: (a T < ; (b T =. (b Sceario 2. We set the iitial battery eergy as E(0 = 0.2, the iitial data queue legth as Q(0 =, ad the maximum trasmit ower as max = 3. The eergy-harvestig rate is chose as H(t = a si t, for t [0,, ad H(t = 0 for t [,, where a is selected from [0, 5. The trasmissio time comariso of R 0.0 (ε = 0.0, E 0.0 (ε = 0.0, M 0.0 (ε = 0.0, G 0.0 (ε = 0.0, ad R 0.2 (ε = 0.2 is give i Fig. 5(b. We ca see that T = T ( R ε, E(0, Q(0, H o = (for all ε. Whe ε = 0.0, for R 0.0, the regio of h havig fiite trasmissio time is a [., 5, but for E 0.0, M 0.0, or G 0.0, this regio is much smaller (a [2.7, 5, a [2.6, 5 ad a [2.9, 5, resectively. Hece, if the actual eergy-harvestig rate is with a = 2.5, the robustotimal desig R 0.0 results i a trasmissio time of.33, while E 0.0, M 0.0 ad G 0.0 retur a ifiite trasmissio time. Additioally, sice 0.2 is a multile of 0.0, Proositio 3 tells that H f ( R 0.0, E(0, Q(0 H f ( R 0.2, E(0, Q(0, which is also verified i Fig. 5(b. Sceario 3. We set the iitial battery eergy as E(0 =, the iitial data queue legth as Q(0 =, ad the maximum trasmit ower as max = 3. The eergy-harvestig rate is [ chose as a modified comoud Poisso rocess H(t = N(t i= D i(t +, where {N(t: t 0} is a Poisso rocess with rate λ = 2, ad D i (t is a Gaussia radom variable with mea a si t (a [0, 5 ad variace. The trasmissio time comarisos of R ε, E ε, ad M ε for ε = 0.0, 0.05 are give i Fig. 6. The erformace of the greedy algorithm is ot icluded as it is much worse tha all other algorithms. I Fig. 6(a, the worst-case trasmissio times of E 0.0 ad M 0.0 are 24.6% ad 22.% larger tha that of R 0.0, resectively. Similarly, i Fig. 6(b, the worst-case trasmissio time of R 0.05 is smaller tha those of E 0.05 ad M Comarig Fig. 6(a with Fig. 6(b, we ca see that a larger triggerig threshold actually reduces the worst-case trasmissio time for the estimatio based algorithms E ε ad M ε. This is because H(t is highly o-statioary ad a larger triggerig threshold is less sesitive to the raid ad o-statioary fluctuatio i H(t which results i a better worst-case erformace for the estimatio-based algorithms. For our robust-otimal algorithm R ε, the worst-case trasmissio times are the same for ε = 0.0 ad ε = 0.05, which coicides with our coclusio i Sectio V-B. VII. CONCLUSION We have solved the trasmissio-time miimizatio roblems for a eergy harvestig trasmitter, where the future eergy-harvestig rate is totally ukow. Secifically, our desig is based o two advaced methods i cyberetics: Evet-trigger cotrol: The Evet Detector (ED cotiuously moitors the battery eergy ad triggers a ew evet whe it exerieces some sigificat chage from the last evet. Wheever a evet is trigger, the Trasmissio Plaer (TP uses the curret iformatio of the battery eergy ad the data queue to udate the trasmit ower based o robust-otimal cotrol. The evet-trigger cotrol framework is summarized i Algorithm. Robust-otimal cotrol: It miimizes the worst-case trasmissio time such that the actual trasmissio time is guarateed to be below this level, o matter what eergyharvestig rate is imosed. If the worst-case trasmissio time is always ifiite for all ossible trasmit ower desig, the the robust-otimal cotrol guaratees the largest set of eergy-harvestig rate to have a fiite actual trasmissio time. The robust-otimal trasmit ower desig is give i Theorem. For future work, oe ca adot the aroach used i this work to desig trasmissio rotocols for other objectives, such as throughut maximizatio, with o kowledge o the future behavior of eergy-harvestig rate ad data arrival

10 0 T (ε,e(t0,q(t0,h T (ε,e(t0,q(t0,h R 0.0 E 0.0 M a (a R 0.05 E 0.05 M a Fig. 6. Trasmissio time comarisos: (a ε = 0.0; (b ε = (b rocess. Additioally, it is iterestig to see how the robustotimal solutio erforms usig exerimetal measuremets (e.g., [29 of the eergy-harvestig rate. APPENDIX A PROOF OF LEMMA Necessity. If (4 holds, there exists a t t such that R is ot emty, ad we have Q(t Q t t = r(,ε (τdτ t E(t Ẽ t,ε (τdτ. (26 Accordig to (2, equatio (26 ca be further rewritte as t [K,ε (τ r(,ε (τ dτ = 0. (27 t If o RPE exists, the either K,ε (τ > r(,ε (τ or K,ε (τ < r(,ε (τ holds for [t, t + T, which cotradicts with (27. Therefore, the RPE exists. Sufficiecy. If the RPE exists, we set,ε (t = e as the trasmit ower. Note that K e = r( e, ad hece (4 holds. APPENDIX B PROOF OF PROPOSITION For the simlicity of this roof, we label A = R \{(E(t, Q(t } ad A 2 = {(Ẽ, Q : K mi K < K max, 0 Ẽ < E(t, 0 Q < Q(t }. i A A 2 : (Ẽ(t, Q (t A, the RPE exists accordig to Lemma. Hece, K = r( e / e. Sice r( is strictly cocave for ad r(0 = 0, K = r(e e = r( e r(0 e 0 (28 is strictly decreasig i (0, max. Therefore, K mi K < K max, ad (Ẽ, Q A 2. ii A 2 A : (Ẽ, Q A 2, the RPE exists accordig to (28. Thus, (Ẽ, Q A. To sum u, A = A 2 ad (5 holds. APPENDIX C PROOF OF PROPOSITION 2 Sice r(,ε (t = log 2 ( +,ε (t is strictly icreasig, cocave ad o-egative,,ε (t [0, max, r(,ε (t ca be exressed by r(,ε (t = K e,ε (t + b ɛ(,ε (t, (29 where b is a ositive costat, ad K e is the derivative of r(,ε (t at,ε (t = e, which imlies K e := lim,ε(t e dr(,ε(t/d,ε (t. I (29, ɛ(,ε (t 0, ad ɛ(,ε (t = 0 holds oly whe,ε (t = e. Recall the roof of ecessity i Lemma that if (4 holds, the (27 is satisfied. We break (27 dow ito 3 arts by Lebesgue itegral, Λ(tdt + S Λ(tdt + S 2 Λ(tdt = 0, S 3 (30 where Λ(t = [r(,ε (t K,ε (t, S = {t :,ε (t < e }, S 2 = {t :,ε (t > e } ad S 3 = {t :,ε (t = e }. Let be the average trasmit ower over [t, t +T, ad we have (a = (b = where S,ε (tdt + S 2,ε (tdt + S 3 e dt µ (S + µ (S 2 + µ (S 3 S r (,ε (t dt + S 2 r (,ε (t dt + S 3 r ( e dt K [µ (S + µ (S 2 + µ (S 3 (c = K e + b, K K = S ɛ (,ε (t dt + S 2 ɛ (,ε (t dt K [µ (S + µ (S 2 + µ (S 3 (3 0. (32 I (3, (a reresets the average ower, ad (b is from (30, ad (c is derived by emloyig (29. I (32, the equality holds oly whe µ (S = 0 ad µ (S 2 = 0, i.e., (9 holds 7. From (3, the exlicit form for is = b K K K e b K K e = e. (33 7 Mathematically, formula (9 holds almost everywhere (a.e. i the duratio [t, t +T excet some sub-duratios with zero measures. Practically, sice the trasmit ower caot chage that fast, we exclude the zero-measure cases ad the term a.e. is omitted.

11 Therefore, T = µ (S + µ (S 2 + µ (S 3 = E(t Ẽ E(t Ẽ e = T, (34 where the equality holds if ad oly if (9 holds accordig to (32. Thus, (9 is the time-otimal trasmissio ower. The uiqueess for (9 is obvious, otherwise (34 caot hold. Note that K e = r( e, ad hece (20 is obtaied. APPENDIX D PROOF OF COROLLARY Takig the artial derivative of T [Ẽ, Q i (20 w.r.t. Ẽ, we have T [Ẽ, Q Ẽ [ = Ẽ Emloyig the chai rule, we have e Ẽ = e K = K Ẽ E(t Ẽ e e + e [E(t Ẽ = Ẽ ( e 2. e K (35 K, (36 K E(t Ẽ where e / K is derived by usig the imlicit differetiatio o K e = log 2 ( + e, ad K / Ẽ is obtaied by (2. I (36, K is the derivative of log 2 ( + e, i.e., K := /[l 2( + e. Sice K ad K ca be rewritte as K = log 2( + e 0 e, K log = lim 2 ( + e 0 e e, (37 we kow K > K due to e > > 0. By (36, equatio (35 ca be rewritte as T [Ẽ, Q K (a = Ẽ e (K K > 0, (38 where (a follows from K > K. Thus, with fixed Q, the laed trasmissio time T [Ẽ, Q icreases with Ẽ. This meas that: If K bal < K mi, the miimum T [Ẽ, 0 achieves at oit b i Fig. 4(a, ad we ca calculate the battery eergy of oit b as E(t Q(t /K mi by the equatio of L (see the catio i Fig. 4, which imlies (2 holds. Sice L i Fig. 4(a is with sloe K mi, from Fig. 3 we have e = max. Similarly, if K mi K bal < K max, we ca derive (22 ad (23. For K bal K max, we caot fid ay ed oit with Q = 0, accordig to Fig. 4(c. APPENDIX E PROOF OF THEOREM Before startig, we give two lemmas: Lemma 2 tells that the worst-case eergy-harvestig rate of the desiged PTP R,ε is H o : t 0 (t [t 0, if the reachable set R 0 is ot i eergy-scarce case; Lemma 3 idicates whe T is fiite. Lemma 2. If the iitial battery eergy E(t 0 ad iitial data queue Q(t 0 satisfies Q(t 0 /E(t 0 = K0 bal < K max, the the followig holds su T ( R ε, E(t 0, Q(t 0, H = T ( R ε, E(t 0, Q(t 0, H o <. (39 Proof: Firstly, it ca be easily obtaied that su T ( R ε, E(t 0, Q(t 0, H T ( R ε, E(t 0, Q(t 0, H o. (40 Secodly, for eergy-harvestig rate H o, there are o triggered evets, sice o eergy comes i [t 0,. For H H o, there should be k Z + triggered evets. If k = 0, the the trasmissio time will be the same as that for H o. If k > 0, the i the first triggered evet, the E(t is { E(t 0 (t t 0 max + ε, K bal 0 < K mi, E(t 0 (t t 0 bal 0 + ε, K mi K bal 0 < K max. (4 If such calculated E(t 0, the the trasmissio time is still the same as that for H o. If E(t > 0, the K bal is { Q(t0 (t t 0r( max E(t 0 (t t 0 max+ε Kbal Q(t 0 (t t 0r( bal 0 E(t 0 (t t 0 < bal Kbal 0 +ε 0, K0 bal < K mi, 0, K mi K0 bal < K max. (42 Accordig to Fig. 3 that e decreases w.r.t. K, we have bal 0 bal. Therefore, the trasmissio time becomes smaller. To sum u, H H the followig iequality holds T ( R ε, E(t 0, Q(t 0, H T ( R ε, E(t 0, Q(t 0, H o, (43 which meas su T ( R ε, E(t 0, Q(t 0, H T ( R ε, E(t 0, Q(t 0, H o. (44 Now, combiig iequality (44 with iequality (40 ad oticig that T ( R ε, E(t 0, Q(t 0, H o < (sice the corresodig reachable set for = 0 is eergy-abudat or eergy-balaced, we ca fially derive (39. Lemma 3. T < if ad oly if Q(t 0 /E(t 0 = K bal 0 < K max. Proof: Necessity. By cotraositive, we should rove that: if K0 bal K max, the T =. Uder H o : t 0 (t [t 0,, we have T ( ε, E(t 0, Q(t 0, H o = for ay ε P ε, which imlies T = if ε P ε su T ( ε, E(t 0, Q(t 0, H if T ( ε, E(t 0, Q(t 0, H o =. ε P ε (45 Therefore, T =. Sufficiecy. We rove that if K0 bal < K max, the T <. Firstly, we have T = if ε P ε su T ( ε, E(t 0, Q(t 0, H su T ( R ε, E(t 0, Q(t 0, H, (46

12 2 which combied with K bal 0 < K max ad Lemma 2 imlies that T su T ( R ε, E(t 0, Q(t 0, H T ( R ε, E(t 0, Q(t 0, H o <. (47 Now, we start the roof of Theorem. We divide this roof ito two arts corresodig to T < (see Subroblem ad T = (see Subroblem 2, resectively. i For T <, from Lemma 3, we kow K0 bal < K max holds. Let the otimal solutio be,ε (imlemetig ε ad the corresodig worst-case eergy-harvestig rate be H ε, i.e., T = T ( ε, E(t 0, Q(t 0, H ε. We have T ( ε, E(t 0, Q(t 0, H ε =T su T ( R ε, E(t 0, Q(t 0, H (a = T ( R ε, E(t 0, Q(t 0, H o, (48 where (a follows from Lemma 2, sice for K bal 0 < K max, the worst-case eergy-harvestig rate of R ε is H o. O the other had, we ca derive T ( ε, E(t 0, Q(t 0, H T ( ε, E(t 0, Q(t 0, H o (b T ( R ε, E(t 0, Q(t 0, H o, (49 where (b follows from the fact that R ε is time otimal uder H o. This is because, i this case, evet is ever triggered, ad the PTP is always equal to the actual trasmit ower [see (5, which meas that: for K0 bal < K max, the actual trasmit ower is otimal accordig to Corollary. Therefore, combiig (48 ad (49, we have ε = R ε, i.e.,,ε = R,ε. ii For T =, from Lemma 3 we kow K0 bal K max holds. We will show R ε satisfies (9 i Subroblem 2 by H f ( ε, E(t 0, Q(t 0 H f ( R ε, E(t 0, Q(t 0, ε P ε, (50 or equivaletly, for all ε P ε, H H f ( ε, E(t 0, Q(t 0, H should also be i the set H f ( R ε, E(t 0, Q(t 0. Now, ε P ε, ad H H f ( ε, E(t 0, Q(t 0, we have t0+t t 0 log 2 ( + ε (τdτ = Q(t 0, (5 where T = T ( ε, E(t 0, Q(t 0, H <. Sice fuctio H is Lebesgue itegrable over ay subset of R + with fiite measure, the itegral t 0+T t 0 H(τdτ is fiite, which meas the total umber of triggered evets is also fiite. Assumig evet N is the last triggered evet, we have t0+t ε (τdτ = E(t 0 + Nε, (52 t 0 i which the value ε is the harvested eergy i each evet, sice the itegral of H(t is cotiuous ad the evet is triggered whe t t H(τdτ = ε. Thus, Nε is the total amout of harvested eergy from t 0 to t 0 + T that ca be see from the TP side. If we assume the amout of battery eergy E(t 0 + Nε is available at t 0, the ε is still be a valid trasmit ower desig to clear the data queue Q(t 0, ad the corresodig trasmissio time is also the same, i.e., T = T ( ε, E(t 0 + Nε, Q(t 0, H o. 8. With Corollary, we kow that R ε is timeotimal, i.e., T ( R ε, E(t 0 + Nε, Q(t 0, H o T ( ε, E(t 0 + Nε, Q(t 0, H o <, which imlies Q(t 0 E(t 0 + Nε K max. (53 The, we rove T ( R ε, E(t 0, Q(t 0, H is fiite. I evet 0: If K0 bal = Q(t 0 /E(t 0 K max, the the trasmissio time is fiite, because eve if o eergy arrives i [t 0,, the trasmitter ca still clea u the data queue by (24. 2 If K0 bal = Q(t 0 /E(t 0 K max, the from (24, the trasmit ower is 0 durig [t 0, t. If case 2 haes, the i evet, we have K bal = Q(t 0 /(E(t 0 + ε. Similar to evet 0, if K bal K max, the the trasmissio time is fiite. Otherwise, the trasmit ower is 0 durig [t, t 2, which makes K2 bal = Q(t 0 /(E(t 0 +2ε. Proceedig forward, if case haes, the the trasmissio time is fiite, ad if case 2 haes, the trasmitter trasmit othig. Here, we ca always fid a evet N such that K bal = Q(t 0 /(E(t 0 + ε, if K bal K max. This is guarateed by (53. Now, we have T ( R ε, E(t 0, Q(t 0, H <, ad thus H H f ( R ε, E(t 0, Q(t 0. To sum u, for ay ε P ε, H H f ( ε, E(t 0, Q(t 0, we have H H f ( R ε, E(t 0, Q(t 0, which imlies that H f ( ε, E(t 0, Q(t 0 H f ( R ε, E(t 0, Q(t 0 holds for all ε P ε. APPENDIX F PROOF OF PROPOSITION 3 H H f ( R ε, E(t b 0, Q(t 0, the trasmissio time is T b = T ( R ε, E(t b 0, Q(t 0, H <. The, similar to art ii i the roof of Theorem, we label the last triggered evet umber as N b, ad by (3, the followig holds N b ε b t0+t b t 0 H(τdτ < (N b + ε b. (54 Sice T b is fiite, similar to (53, Q(t 0 /(E(t 0 + N b ε b < K max holds. Notig that zε a = ε b, we have N b ε b = N a ε a, where N a = zn b. Thus, the followig holds Q(t 0 E(t 0 + N a ε a = Q(t 0 E(t 0 + N b ε b < K max, (55 which imlies that for ε a, the trasmissio time T ( R ε a, E(t 0, Q(t 0, H is fiite (similar to art ii i the 8 After movig the future eergy to t 0, the trasmissio time should be T ( ε, E(t 0 + Nε, Q(t 0, H left where H left is a equivalet eergyharvestig rate but ot ecessarily H o. However, sice H left caot trigger ay evet, we have T ( ε, E(t 0 + Nε, Q(t 0, H left = T ( ε, E(t 0 + Nε, Q(t 0, H o