Towards Achieving Perpetual Operation in Rechargeable Sensor Networks

Transcription

1 Towards Achevng Perpetual Operaton n Rechargeable Sensor Networks Ren-Shou Lu, Student member, IEEE, Prasun Snha, Member, IEEE, Can Emre Koksal, Member, IEEE Abstract Energy harvestng sensor platforms have opened up a new dmenson to the desgn of network protocols. In order to sustan the network operaton, the energy consumpton rate cannot be hgher than the energy harvestng rate, otherwse, sensor nodes wll eventually deplete ther batteres. In contrast to tradtonal network resource allocaton problems where the resources are statc, tme varatons n rechargng rate presents a new challenge. In ths paper, we frst explore the performance of an effcent dual decomposton and subgradent method based algorthm, called QuckFx, for computng the data samplng rate and routes when a DAG routng structure s gven. Then, we analytcally study the key propertes of the optmal DAG(s) and propose a mechansm for constructng a DAG that can support hgh network utlty. Moreover, fluctuatons n rechargng can happen at a faster tme-scale than the convergence tme of the tradtonal approach. Ths leads to battery outage and overflow scenaros, that are both undesrable due to mssed samples and lost energy harvestng opportuntes respectvely. To address such dynamcs, a local algorthm, called SnapIt, s desgned to adapt the samplng rate wth the objectve of mantanng the battery at a target level. Our evaluatons usng the TOSSIM smulator show that QuckFx and SnapIt workng n tandem can track the nstantaneous optmum network utlty whle mantanng the battery at a target level. When compared wth IFRC, a backpressure-based approach, our soluton mproves the total data rate by % on the average whle sgnfcantly mprovng the network utlty. I. INTRODUCTION In varous applcaton scenaros, energy can be harvested from the surroundng envronment to recharge the batteres and extend the network s lfetme. Many forms of energy such as solar, wnd, water flow, thermal and vbraton are beng explored for drvng sensor network platforms [], [], [3], []. Such sensor platforms have opened up a new dmenson to the desgn of networkng protocols. However, the strength of harvested energy s a functon of varous statc parameters, such as the specfcatons and orentaton of the solar panel; and tme-varyng parameters, such as the weather and the season. The protocol needs to adapt to such dynamcs to avod runnng out of battery. Ths s especally needed for envronmental montorng applcatons as they often requre perodc data collecton from all nodes n the network. Our objectve n ths paper s to desgn a dstrbuted and adaptve soluton that jontly computes the data collecton rates for each node, fnds the routes and schedules transmssons based on Ren-Shou Lu and Prasun Snha are wth the Department of Computer Scence and Engneerng, The Oho State Unversty, Columbus, OH 3 USA. E-mal: {rslu,prasun}@cse.oho-state.edu Can Emre Koksal s wth the Department of Electrcal and Computer Engneerng, The Oho State Unversty, Columbus, OH 3 USA. E-mal: koksal@ece.osu.edu Ths work was n part presented n Infocom. nterference constrants and energy replenshment rates, such that the network utlty can be maxmzed whle mantanng perpetual operaton of the network. Smlar to the standard flow control and resource allocaton (e.g. [], []) approaches, we formulate the problem as a constraned optmzaton problem. In our formulaton, we have an energy conservaton constrant nvolvng the replenshment rates, along wth the standard flow conservaton and nterference constrants. The fundamental dfference n our problem s that, our energy conservaton constrant s tme-varyng, due to the tme-varyng nature of energy replenshment,.e., our problem s dynamc, and thus, t has a dfferent soluton n each pont n tme. The standard mplementatons of the dual decomposton method for resource allocaton nvolves a large number of teratons, each of whch ncurs a hgh overhead due to the necessary message exchange between neghborng nodes. In a statc settng, the convergence tme s less of an ssue, snce the soluton of the problem s fxed. However, wth tme-varyng replenshment rate, we have a tme-varyng resource allocaton problem. To that end, the frst major queston we answer n ths paper s, how closely can we track that optmal pont by usng dstrbuted teratons to solve the Lagrangan dual problem? We show (n Secton IV) that, n a network wth an arbtrary topology, the tme scale of such solutons s too slow to follow the varatons n the replenshment rate for the optmal pont to be tracked suffcently closely. Consequently, we focus on well-structured networks wth an underlyng drected acyclc network graph (DAG). We explot the DAG structure to develop an effcent synchronous message passng scheme, QuckFx, motvated by general soluton structure of dynamc programs. We show that the convergence tme of QuckFx s suffcently small to track the optmal soluton farly closely. However, the maxmum achevable network utlty and data collecton rates can vary sgnfcantly when dfferent DAGs are used. Thus, t s mportant that a good DAG s constructed so that the network utlty can be maxmzed. In ths paper, we also address ths problem by analytcally studyng the key propertes of the optmal DAG(s). Based on our observatons, a mechansm for constructng a DAG that can support hgh network utlty s proposed. The second major queston we address s, how does the fnteness of the total amount of energy storage n each node affect the performance of jont energy management and resource allocaton? If one accounts for the fnte batteres n the orgnal problem formulaton, the soluton nvolves complcated Markov decson processes, whch does not lead to better understandng of the dynamcs of the system. Instead, we propose a smple novel adaptve localzed algorthm,

2 SnapIt, that operates above QuckFx, to slghtly modfy the optmal samplng rate provded by QuckFx, based on local battery states. We show that usng SnapIt along wth QuckFx has two mportant consequences. Frstly, the fracton of tme for whch a node s down reduces sgnfcantly due to a complete battery dscharge reduces sgnfcantly (to n many nstances as shown n our evaluatons). Secondly, due to the short response tme of SnapIt, the jont algorthm s capable of respondng to the changes n replenshment rate much more quckly. Hence, the overall network utlty also ncreases wth SnapIt. In summary, the key contrbutons of ths paper are as follows: Based on a gven DAG routng structure, we develop a decomposton and subgradent based dstrbuted soluton, called QuckFx, that can effcently track nstantaneous optmal samplng rates and routes n the presence of a tme-varyng rechargng rates. Pror works based on such technques have mostly focused on statc scenaros. The key propertes of the optmal DAG(s) are analytcally studed and a mechansm for constructng a DAG that can support hgh network utlty s also developed. For perpetual operaton n the presence of rapd fluctuatons n the rechargng rates, we present a new local algorthm, called SnapIt, that attempts to mantan the battery at a certan target level whle stochastcally mantanng a hgh network utlty. Smulatons usng TOSSIM.x [7] show that the algorthms workng n tandem can lead to an effcent soluton for smultaneously achevng proportonal farness and perpetual operaton. It s also shown to sgnfcantly outperform IFRC, a backpressure-based protocol. II. RELATED WORK Rate Control and Routng: Routng n energy harvestng sensor network s explored n [8]. The authors proposed two heurstc solutons. The frst soluton s a localzed protocol that allows the source node and exactly one ntermedate node to devate from the shortest path and opportunstcally forward packets to a solar-powered neghborng node. On the other hand, the second soluton, wth the assstance of the snk, chooses the shortest path wth the mnmum number of nodes that run solely on the batteres. In another work [9], an onlne routng algorthm whch seeks to maxmze the number of accepted flows s proposed. In contrast to these works, we consder jont computaton of routes and rates wth a dfferent objectve of far data collecton and perpetual operaton. Lexcographcally maxmum rate assgnment and routng for perpetual data collecton has been studed n []. It extends the framework n [], and presents a centralzed as well as a dstrbuted algorthm. The centralzed algorthm s proven to gve the optmal lexcographc rate assgnment. However, the dstrbuted algorthm can only reach the optmum f the routng paths are predetermned, and the network s a tree rooted at the snk. Furthermore, the dynamcs of rechargng profles s not consdered. The exact rechargng profle for each day must be known and t s senstve to the ntal battery levels. In contrast, our dstrbuted soluton does not requre any knowledge of the future rechargng rates. Furthermore, our dstrbuted soluton can work on a more generalzed structure than a tree. All of these factors result n a more practcal algorthm n comparson to the algorthm proposed n []. Other pror works on rate control n WSNs nclude [], [3], [], [], [], [7], [8]. Although some of these works [], [], [7] also am to acheve farness n WSNs, none of them consder energy replenshment. In ths paper, we reengneered IFRC [] to consder rechargng capabltes of modern sensor platforms and use t as a benchmark. Dual-Decomposton Technque for Optmzng Network Performance: The dual decomposton technque has been appled n many works to maxmze the network utlzaton. In [], the utlty functon, as ours, s defned as the summaton of the log of the rate acheved by each flow n the network. However, n ther formulaton, the route for each flow s fxed. The authors propose both a prmal and a dual-based algorthm. Both algorthms tune the transmsson attempt probablty of each node to maxmze the utlty functon. In contrast to [], a general utlty functon s consdered n []. It also apples the dual decomposton technque to develop a dstrbuted algorthm. However, the algorthm not only addresses rate assgnment, but also routng. Maxmn farness of rate assgnment problem n WSNs s studed n [9]. Ther proposed algorthm s also based on the dual decomposton technque. In ther model, each sensor transmts wth a certan probablty. Thus, after Lagrangan relaxaton, the dual functon s not concave. To overcome ths problem, they convexfy ther problem and apply the Gauss-Sedel method to solve the problem. Although all the above approaches allow nodes n the network selftune certan parameters to maxmze the network utlzaton, none of them consder energy constrants, not to menton rechargng. III. NETWORK MODEL Ths secton outlnes the network model and the problem formulaton. We consder a statc sensor network represented as G = (N,L), where N s the set of sensor nodes ncludng the snk node, s, and L s the set of drected lnks. We assume that a DAG (drected acyclc graph) rooted at the snk s constructed over the network for data collecton. A mechansm for constructng a DAG that supports hgh network utlty wll be explored n Secton V. For a gven DAG routng structure, the problem s formulated as a convex optmzaton problem. Ths formulaton not only provdes clear system desgn gudelnes but also allows us to desgn an effcent sgnalng scheme. Before presentng the problem formulaton, we ntroduce some of the key notatons used n the paper. The notatons and ther correspondng semantcs are also lsted n Table I for reference. For each sensor node N, A denotes ts ancestors n the DAG,.e., the nodes that are on some path(s) from node to the snk s. Conversely, D denotes the descendants. Also, C and P are the chldren and parent nodes of node respectvely. The amount of energy consumed n sensng the

3 3 TABLE I NOTATIONS Symbol Semantcs D The set of descendants of node A The set of ascendants of node C The set of chldren nodes of node P The set of parent nodes of node B (t) Battery state of node at tme t M Battery capacty of node π (e) Estmated average rechargng rate of node n epoch e ρ (t) The nstantaneous rechargng rate of node at tme t τ Epoch length r Samplng rate of node λ (sn) Energy cost for samplng at node λ (tx) j Average energy cost for TX over lnk (, j) λ (rx) Energy cost for RX at node f j The amount of capacty allocated on lnk (, j) w j The fractonal amount of node s traffc that passes lnk (, j) z j (w) A functon of w, that represents the fractonal amount of s traffc that passes j Π The feasble regon of lnk capacty varables f j W The feasble regon of routng varables w j µ Lagrange multpler for energy conservaton constrant at node υ Lagrange multpler for flow balance constrant at node α The constant step sze used n the subgradent method envronment at node s represented by λ (sn). We assume that the the expected number of retransmssons over a long tme s known for each lnk. Thus, λ (tx) j represents the average energy consumpton for delverng a packet over lnk (, j) and λ (rx) represents the energy cost for recevng a packet at node. These parameters are the energy costs of the system. In partcular, we consder slotted-tme system and the tme durng a day s broken nto multple tme ntervals called epochs. The length of each epoch s τ slots. We use π (e) to represent the average (long term) energy replenshment rate of node n epoch e, whle ρ (t) s used to represent the real nstantaneous (short term) energy replenshment rate of[ node n tme slot t. For each epoch e, we defne π (e) = τ E eτ ] t=(e )τ+ ρ (t). We assume that π (e) can be estmated by each node wth hgh accuracy. Estmatng π (e) s beyond the scope of ths paper. Moreover, we defne w j to be the fractonal outgong traffc of that passes through a parent j and z j (w) to be the fractonal outgong traffc of that passes through an ancestor j, where w s the vector of w j,j. Thus, w j =, j / P and z j (w) =, j / A. The recursve relaton between the two varables s gven below. z j (w) = k P w k z kj (w) () to acheve proportonal farness []. Thus, we formulate our problem as an convex optmzaton problem as shown below (the proof of convexty can be found n Appendx A): P e : max r,f j,w j π (e) λ (sn) r + z j (w)λ (rx) j D log(r ) s.t. () r j + λ (tx) j f j (3) j P f j r + z j (w)r j () j P j D r R +, f j Π, w j W The frst and the second constrants are the energy conservaton and flow balance constrants, respectvely. The flow balance constrant states that the sum of allocated capacty for each outgong lnk should be greater than the total amount of traffc gong through each node, ncludng ts own data. Besdes these two constrants, the amount of capacty allocated on each lnk must be n the feasble capacty regon Π. One can observe that ths problem s dynamc, snce the energy conservaton constrant nvolves the tme varyng replenshment rate π (e). Thus, the the feasble regon and the optmum soluton dffer n each epoch. One can vew ths dynamc problem as a sequence of statc problems. The standard method to solve each statc problem nvolves the applcaton of the dual decomposton and the subgradent methods. However, the mplementaton of these solutons n the network nvolves a large overhead due to message exchange between neghborng nodes. Consequently, the convergence tme becomes an mportant ssue. To that end, we ntroduce QuckFx, whch, n each teraton of the subgradent method, explots the specal structure of DAG to form an effcent control message exchange scheme. Ths scheme s motvated by the general soluton structure of a dynamc program. QuckFx s based on the herarchcal decomposton approach as the startng pont. By relaxng the energy conservaton and flow balance constrants n Problem P e, we get the partal dual functon q(µ,υ) as follows: q(µ, υ) = max r,f j,w j X log(r ) + X (e) λ (sn) r X z j (w)λ (rx) r j X j:(,j) D + X X X f j r z j (w)r j A j P j:(,j) D s.t. r R +, f j Π, w j W λ (tx) j f j j P A IV. A CROSS-LAYER APPROACH TO DYNAMIC RESOURCE ALLOCATION In ths secton, we present an outlne of the dual based cross-layer framework for dstrbutedly computng the rates and routes n each epoch. We defne the utlty functon U (r ) for node to be log(r ), where r s the samplng rate of node. Our goal s to maxmze the sum of the utlty functons U (r ) = log r, whch s strctly concave and known The dual problem s therefore: mn q(µ,υ) () µ,υ The dual problem n () can be decomposed herarchcally [], [], [3] as follows. The top level dual master problem s responsble for solvng the dual problem. Snce the dual functon s not dfferentable, the subgradent method [3], []

4 s appled to teratvely update the Lagrange Multplers µ and υ at each node usng (). The notaton [.] + means projecton to the postve orphan of the real lne R. ( µ k+ = [µ k α π (e) λ (sn) r z j (w)λ (rx) r j )] + λ (tx) j f j j D j P [ ( υ k+ = υ k α f j r )] + z j (w)r j () j P j D Each tme the Lagrange Multplers are updated, a prmal decomposton s performed to decouple the couplng varables f j, such that when f j are fxed, the problem can be further decomposed nto two layers of subproblems. The upper level optmzaton problem can then be further decomposed nto many smaller subproblems, one for each, as shown n (7): max r,w j log(r ) µ λ (sn) r + z j (w)λ (rx) j D υ r + z j (w)r j j D s.t. r R +, w j W (7) At the lower level, we have the optmzaton problem shown n (8), whch s n charge of updatng the couplng varables f j max f j X (υ λ (tx) j µ )f j (,j) L s.t. f j Π (8) The lower level optmzaton problem n (8) s equvalent to the maxmum weght matchng problem. We assume the node exclusve nterference model as n []. Thus, t can be solved n polynomal tme. However, n order to solve the maxmum weght matchng problem n a dstrbuted fashon, we utlze the heurstc algorthm n []. Whle applyng the algorthm, nstead of the queue dfference between neghborng nodes, we use the combned energy and queue state of a node (υ µ ) to modulate the MAC contenton wndow sze, when a node attempts to transmt a packet over the lnk (,j). Note that n the upper level optmzaton problem, the f j and υ varables are fxed, and snce the objectve functon s strctly concave n (r, z(w)), t admts a unque maxmzer as shown n (9). In the followng dscusson, we refer to λ (sn) µ + υ n (9) as node s local prce, and R = λ (tx) j z j (w) j µ j + υ j ) as ts aggregate prce. j A r = λ (sn) µ + υ + z j (w ) j µ j + υ j ) j A r j (9) Computed utlty QuckFx Standard dual-based algorthm Iteratons Fg.. The convergence tme comparson between QuckFx and the standard dual-based algorthm that s derved from the standard node-centrc flow balance formulaton. Proposton. The aggregate prce R can be recursvely computed as R = ( ) w j j µ j + υ j ) + R j () j P Proposton. The aggregate traffc T of node can be recursvely computed as T = k C w k (r k + T k ) () Observe that f a node wants to maxmze ts rate, t should fnd the path(s) such that ts aggregate prce R s mnmzed,.e., t s a jont routng and rate control problem. Snce our formulaton utlzes the DAG structure, ths allows a node to calculate ts aggregate prce recursvely from those of ts parents as stated n Proposton. Furthermore, Proposton mples that a node should select the parent wth the mnmum sum of local and aggregate prces as ts relay node n the DAG. Ths motvates the followng dstrbuted routng and rate control algorthm. Each node collects the local and aggregate prces from all ts parents and selects the parent(s) wth the mnmum sum of the local and aggregate prces as the relay node(s) n the DAG. Then, each node uses () to calculate ts own aggregate prce and then apples (9) to determne ts rate. Havng determned the local rate and the outgong lnk(s) to use, a node dstrbutes ts local and aggregate prces to ts chldren, so that the chldren nodes can determne ther routes and rates. Ths process contnues untl the leaf nodes are reached. Now, startng from the leaf nodes, each node reports ts aggregate traffc to ts parents, so that the parent nodes can have the needed nformaton to update ther local prces. Aggregate traffc T of node s the total amount of traffc generated by the descendants that goes through node. Smlar to the computaton of the aggregate prce, a node can compute ts aggregate traffc recursvely usng (). The proof of Propostons and can be found n Appendx A. Fgure compares the convergence tme of QuckFx wth the standard dual-based algorthm that s derved from the standard node-centrc flow balance formulaton. To focus on the rate of convergence, a fxed rechargng rate π s used for each node N. Here, both QuckFx and the standard dual-based algorthm are run over a DAG of 7 nodes. The mprovement n convergence rate wth QuckFx s apparent.

5 Algorthm : QuckFx: Dstrbuted routng, rate control and schedulng algorthm for energy-harvestng sensor networks based on our proposed DAG formulaton whle true do /* Phase I: */ Each node locally adjusts the Lagrange Mulplers µ, υ usng () The snk ntates a new teraton by broadcastng ts local and aggregate prces, whch are both. Non-snk (sensor) nodes wat untl the local and aggregate prces of all the parents have been collected. Once the prces from all the parents have been collected, select the lnk(s) wth the mnmum sum of the local and aggregate prces. If multple lnks are selected, equally splt the flows among them. Compute aggregate prce usng () Broadcast the local and aggregate prces to the chldren nodes. Compute the maxmum rate usng (9) /* Phase II: */ Once a leaf node has determned ts rate and outgong lnks n Phase I, t reports ts local rate to ts parent nodes of the selected lnks. Non-leaf nodes must collect the aggregate traffc from all ts chldren nodes on the DAG. After that, a non-leaf node computes ts aggregate traffc usng () and report t to all the parent nodes. Inform MAC layer of newly selected lnk(s) and ther weght(s) υ λ (tx) j µ Start applyng the newly computed rate and routng paths However, we defer the convergence rate analyss of QuckFx algorthm for future work. V. DAG CONSTRUCTION To utlze the QuckFx algorthm presented n Secton IV, a DAG routng structure must be present n the network. However, there are many ways to construct a DAG that carres data from the sensor nodes to the snk and the maxmum achevable network utlty and data collecton rates vary when dfferent DAGs are used. Thus, t s mportant that a good DAG s used so that the network utlty s maxmzed. However, t s dffcult to characterze the optmal DAG wthout computng the rate assgnment and flows n the network. So we start wth analytcally studyng a few key propertes of the optmal DAG(s). Then, based on our observatons, a heurstc algorthm s proposed to construct a DAG over the underlyng network. Before presentng our observatons, we frst gve a few defntons used n the followng dscusson. Defnton. Gven a DAG A rooted at the snk and ts correspondng optmal soluton to the network utlty optmzaton problem, we call the nodes, whose energy conservaton constrants are actve n the optmal soluton, the bottleneck nodes, whch s represented by B A n the followng dscusson. For each node B A, F s used to denote the set of nodes whose flow or subflow passes through node. Note that F s a subset of D, the set of descendants of node n the DAG. If F, we call node a non-leaf bottleneck node. Defnton. A DAG s sad to be optmal f the network utlty that can be acheved by usng only those lnks n the DAG s maxmum. Wth the above defntons, we have the followng observatons (Note that, for ease of presentaton, we assume that all the nodes have equal sensng, RX, and TX costs n ths secton): Lemma V.. Let G denote a network and A denote a DAG n network G. Let r be the rate of a non-leaf bottleneck node B A and F D denote the set of nodes whose flow or subflow passes through node (. If there ) exsts a path j k n G, where j F, k B A {s} \F, r k > r, and no ntermedate node on the path s a bottleneck node, then, the DAG A s not optmal. Proof: Snce, there s no ntermedate bottleneck nodes on the path j k, the network utlty can be mproved by reducng r k, redrectng some traffc generated by node j to node k through the path j k, and ncreasng r. The amount of ncrement (decrement) that can be made to r (r k ) s lmted by a few factors. Frstly, (r k r ) because f r k becomes smaller than r, then we wll start losng more than we can gan n network utlty by reducng r k and ncreasng r. Secondly, suppose f j r j s the amount of node j s traffc that passes through node. If we reduce r k by, then the amount of traffc f j that can be redrected from node j to k s at most f j = ρ f j where ρ = λ(sn) +λ (tx) λ (rx) +λ (tx). f j cannot be greater than f j because t s mpossble to reduce node s load further by redrectng more traffc to node k. Thus, we get f j ρ Lastly, let δ be the mnmum unused rechargng capacty of all the non-bottleneck nodes on path j k. Then, δ because the maxmum amount of extra traffc (λ (tx) +λ (rx) ) ρ that can be carred by non-bottleneck nodes on path j k s δ at most. In summary, λ (tx) +λ (rx) { = mn f j, (r k r ), f j ρ, } δ (λ (tx) + λ (rx) ) ρ, () Note that path j k must contan some lnk(s) not used n the DAG A, as otherwse, we can mprove the network utlty usng the aforementoned steps and ths mples that the hypothess (the soluton s optmal) s not true. After addng these lnks, the overall mprovement n network utlty s We take r s =, where s s the snk.

6 log(r + ) + log(r k ) log(r ) log(r k ) =log (r ( + )(r k ) = log + (r ) k r ) r r k r r k > (3) Ths completes the proof. The mplcaton of Lemma V. s that, f a DAG s optmal, every path from any sensor node to the snk n the network, must contan at least one bottleneck node. If we follow these paths, the set of bottleneck nodes, that we frst encounter on the paths, must have equal rates. Ths property allows us to determne whether a DAG s optmal and dentfy a lnk that can be used to mprove the network utlty when t s added to the DAG. The detals about how ths lemma s used wll be dscussed later when we present our DAG constructon algorthm. To further nvestgate the propertes of the optmal DAG(s), we assume that the energy cost for sensng the envronment s at most equal to the energy cost for recevng the sensed data on the rado,.e. λ (sn) λ (rx). Ths s a reasonable assumpton as sensng cost s typcally much lower than TX and RX cost n most sensor platforms. Ths assumpton leads to the followng lemma. Lemma V.. Suppose λ (sn) λ (rx). If node s a nonleaf bottleneck node n an optmal DAG A, then ts rate must be greater than or equal to that of the nodes whose flow or subflow passes through node,.e., r r j for all j F. Proof: Suppose r < r j for some j F n an optmal DAG A. Ths mples that the network utlty cannot be mproved by reducng r j and ncreasng r. In other words, >, we have log r + ρ r log r j r j where ρ = λ(rx) +λ (tx). Ths mples that ρr λ (sn) +λ (tx) j r ρ. Snce can be made arbtrarly close to, we have r r j ρ, whch s a contradcton to the hypothess that r < r j. Lemma V. tells us that, n a fxed routng structure, the rate of a sensor node can not surpass that of ts ancestor bottleneck nodes. Ths mples that the rate of non-bottleneck nodes can not be ncreased unless the rates of ther ancestor bottleneck nodes are also ncreased. Ths observatons lead us to the followng heurstc DAG constructon algorthm. The algorthm has two phases. The frst phase s based on Lemma V., n whch the algorthm constructs an approxmate load-balanced spannng tree rooted at the snk such that the mnmum rate of all the bottleneck nodes n the tree s maxmzed. By dong so, the rate of the descendants of the mnmum-rate bottleneck node(s) can be maxmzed as well. The rate for each bottleneck node s estmated usng the followng Equaton (), r = π ( D + )λ (tx) + D λ (rx) + λ (sn) () u u ud U(r ) δ r * Fg.. Wth our energy management scheme, nstantaneous utlty alternates between u u and u d where π s the estmated average rechargng rate of node over a long perod of tme (such as one day), D s the number of nodes currently attached to the subtree rooted at node. Note that snce the rechargng rate of sensor nodes are constantly changng, there could be a dfferent optmal DAG at each pont n tme. Therefore, we choose to use the estmated average rechargng rate over a long perod so that the overhead of tree constructon and mantenance s low. The pseudo-code for Phase I s presented n Algorthm, whch conssts of a loop that does not termnate untl all the nodes are n the tree. In each teraton of the loop, (lnes ) t fnds a node u n the tree wth the followng propertes. Frstly, some neghbor(s) of node u s not attached to the tree yet (we refer nodes satsfyng such property the edge nodes). Secondly, the mnmum rate of ts ancestor bottleneck nodes s maxmum. Tes are broken usng hop dstance to the snk. In lnes 3, any neghbor v of node u that s not n the tree s attached to u. After expandng the tree, lne computes the rate of the newly added node v and updates the rates of all ts ancestors. Fnally, lnes update the set of bottleneck nodes and the set of edge nodes accordng to the current tree. At the end of Phase I, an approxmate load-balanced spannng tree wll be constructed. The pseudo-code of Phase II s presented n Algorthm 3. Based on Lemma V., f a node u can fnd a neghborng node v whose ancestor bottleneck node can support a hgher rate, then addng lnk (u,v) can mprove the network utlty. Therefore, n each teraton of the Foreach loop of the algorthm, t checks whether there exsts such a node v n node u s neghborhood (lnes 3 ). If so, lnes add lnk (u,v) as an alternatve outgong lnk for node u. Note that although we present the algorthm as a centralzed algorthm for clarty, t can be made a dstrbuted algorthm by utlzng load-balanced tree constructon protocols such as P []. P bulds an approxmate load-balanced data collecton tree by balancng the number of nodes n each subtree. Instead of usng the number of nodes n a subtree as the load factor, one can use the nverse of the rate computed by Equaton as the load factor for each node. Once an approxmate load-balanced tree s constructed, each node can ndependently executes the nstructons nsde the outer most For loop of Phase II algorthm and adds a lnk as an alternatve outgong lnk f t leads to a neghbor whose ancestor bottleneck nodes support a hgher rate. δ r

7 Algorthm : Phase I of the DAG constructon algorthm /* Inttalzaton. T s the set of lnks n the tree, B T s the set of bottleneck nodes on the current tree T, and C T s the set of edge nodes n the tree T. */ T B T {s} C T {s} whle C T do /* Fnd a node v C T n the tree, such that the mnmum rate of ts ancestor bottleneck nodes s maxmum among all the edge nodes. */ r max h mn v foreach z C( T do ) r mn r z, mn w A z B T r w f r max < r or (r max = r and h mn > hop dstance to snk(z)) then r max r h mn hop dstance to snk(z) v z Select any node u N v not n the tree T Add the selected node u to the tree Compute r u and update the rates for all the ancestors of node u usng () Update the set of the bottleneck nodes B T accordng to the newly computed rates Update the set of edge nodes C T Algorthm 3: Phase II of the DAG constructon algorthm Wat untl all the nodes are attached to the tree foreach u V do /* Fnd the rate that the ancestors of node u can support */ r max mn r x x A u B T h mn /* Fnd the neghbor whose ancestors can support a hgher rate. */ v foreach z N ( u do ) r mn r z, mn x A z B T r x h hop dstance to snk(z) f r max < r or (r max = r and h mn > h then r max r h mn h v z f v then Add (u,v) as an alternatve outgong lnk for node u VI. SNAPIT: A LOCALIZED ENERGY MANAGEMENT SCHEME FOR VARIABLE REPLENISHMENT RATE We ntroduce a localzed scheme called SnapIt that uses the current battery level to adapt the rate computed by QuckFx wth the objectve of mantanng the fnte-capacty battery at a target level. Ths mechansm does not requre any control sgnalng between nodes. Furthermore, we observe that by attemptng to mantan the battery at a target level, the nterval (epoch) of runnng the QuckFx algorthm can be extended, leadng to reduced control overhead. Another approach wth a smlar motvaton of utlzng energy effcently wth replenshng batteres s gven n [7]. There, each node manages energy to keep ts duty cycle perod as smooth as possble and at the same tme tres to keep the battery state close to a certan desred level. Although we do not consder dutycyclng n ths work, SnapIt enables sensor nodes to acheve perpetual operaton. If we denote the total sze of the battery of node as M, SnapIt uses the md pont,.e., M / as the target battery state. Each node takes nto account the nstantaneous energy state, B (t), of ts battery and makes slght varatons on the rate allocaton n order to keep the drft toward M /. These varatons are small enough to guarantee a total utlty close to the optmal. The optmal rate assgnment for node n epoch e s r (e) = r (π (e)), provded by QuckFx by solvng P e. Snce the soluton to P e depends on π (e), and not the state of the battery, QuckFx s the battery-state oblvous statc assgnment r (e) for all tmes t durng epoch e. QuckFx s nclned to choose the rates such that the energy conservaton constrant (3) at a node s kept actve f possble. Consequently, energy s draned at a rate, dentcal to the average replenshment rate possbly over multple successve epochs. Ths leads to a hgh rate of battery dscharge and hence a low network utlty. SnapIt chooses the rate (and hence the transmsson power), ndependently at each node based on the current state of the battery as follows: For t, (e )τ + t eτ, r SnapIt (t) = { r (e) δ, B (t) M / r (e) + δ, B (t) > M /, () for some δ >, whch we wll specfy later on. Consequently, the transmt cost (power) reduces by δ λ (sn) f the battery s less than half full and ncreases by the same amount when t s more than half full. Each node can run SnapIt based solely on the local battery state B (t) to make rate assgnments. In general, one may choose to update the rate assgnments at longer ntervals τ S >, where τ S τ. The algorthm s detaled as follows: As shown n Fg., the nstantaneous utlty assocated wth SnapIt alternates between u u (e) = log(r (e) + δ ) and u d (e) = log(r (e) δ ), dependng on the battery state. Due to concavty of the log functon, the average utlty at epoch e wll be lower (by Jensen s nequalty) than the optmal value log(r (e)), whch can only be acheved f M =. The smaller the value we select for δ, the closer the average utlty of SnapIt gets to log(r (e)). However, a small δ mples a small drft away from the complete dscharge state, and hence

8 8 Algorthm : SnapIt: Localzed Energy Management 3 foreach τ S tme unts do Check out the battery state B (t) f B (t) M / then r SnapIt (t) r (e) δ else (t) r (e) + δ r SnapIt a hgher lkelhood of complete dscharge. The mportant queston s how to choose δ such that, not only the average utlty approaches to the optmal value, but also the complete dscharge rate decays to suffcently fast. Next, we show that ths s possble under rather weak assumptons on the nstantaneous replenshment rate ρ (t). In partcular, we assume that the asymptotc sem-nvarant log moment generatng functon [ ( )] Λ ρ (s) = lm T T log E exp s ρ (t) () of ρ (t) exsts and s fnte for all s (,s max ) for some s max >. Note that ths exstence requres an exponental (or faster) decay for the tal of the sample pdf of ρ (t) and t rules out the possblty of long range dependences n the {ρ (t)} process. We also assume that the estmate, π (e) used for the average replenshment [ rate for each epoch e s unbased,.e., π (e) = τ E eτ ] t=(e )τ+ ρ (t) for all e. In presentng our result, we use the followng notaton: a n = O(b n ) f a n goes to at least as fast as b n, a n = o(b n ) f a n goes to strctly faster than b n, and a n = Θ(b n ) f a n and b n go to at the same rate. Also p SnapIt (M ) s the probablty of complete battery dscharge for node [ as a functon ] of the sze, M, of ts battery, Ū SnapIt = E U(r SnapIt (t)) and Ū = E [U(r (e))] are the tme average utltes acheved by SnapIt and the optmal rate allocaton wth an unlmted battery sze respectvely. t= Theorem VI.. If the varance, σ ρ var ( τ τ t= ρ (t) ) s bounded and the utlty functon s the log functon, U( ) = log( ), then, gven any β, SnapIt acheves p SnapIt (M ) = O(M β ) and Ū Ū SnapIt = Θ δ = βσ ρ log M. λ (sn) M Proof: See Appendx B. ( ) log M M wth the choce of Ths theorem shows that t s possble to have a quadratc decay for the probablty of complete battery dscharge, and at the same tme acheve a utlty that approaches the optmal value (that of an unlmted energy source) approxmately as /M. To understand the strength of ths SnapIt, note that there exsts no scheme that acheves (even asymptotcally) the optmal utlty wth an exponental decay for the probablty of Note that the scalng laws gven n the theorem are asymptotc n the battery sze M for fxed values of β. complete dscharge. Very brefly, the proof has the followng log M sketch. By choosng δ = κ M, we show for any choce of κ >, the desred scalng for the utlty functon s acheved. Then, by choosng κ = βσ ρ /λ (sn), we prove that we can acheve the desred quadratc decay for the probablty of complete dscharge. An extensve performance analyss of SnapIt s gven n Secton VII along wth some comparsons to the statc scheme that assgns a rate, fxed at the optmal value r (e) durng the entre epoch e. As we shall llustrate, n many scenaros, the dynamc scheme sgnfcantly reduces the battery dscharge rate, and consequently ncreases the overall utlty consderably. VII. EVALUATION In ths secton, we frst evaluate QuckFx/SnapIt and compare t wth IFRC [] usng a fxed DAG. Then, we evaluate the performance of our proposed DAG constructon algorthm. The parameters used n the smulatons are lsted n Table II. We buld the rechargng profles of the nodes usng the real solar radaton measurements collected from the Baselne Measurement System at the Natonal Renewable Energy Laboratory [8]. The data set used s Global -South Lcor, whch measures the solar resource for collectors tlted degrees from the horzontal and optmzed for year-round performance. Unless explctly specfed, we use the profle of a sunny day (Feb. st 9). The data s approprately scaled to create a rechargng profle for a solar panel wth a small dmenson (37mm 37mm). The battery capacty of sensor nodes s assumed to be mah. The epoch length τ s set to one hour and we choose to run the QuckFx algorthm one teraton every fve mnutes. Throughout the evaluaton, we focus on the performance measures durng the daytme because the energy harvestng rate s zero at nght. However, based on the applcaton s mnmum samplng rate requrement, one can determne the mnmum battery level that can support the mnmum samplng rate at nght and the SnapIt algorthm wll mantan the battery at that level to ensure the network remans actve durng the nght tme. It should also be noted that although we dd not consder the energy cost for sgnalng n our formulaton, we dd take that nto account n the smulatons. In the remanng secton we compare our algorthms wth the nstantaneous optmum computed usng MATLAB n each tme slot )wthout consderng battery state); contrast t wth a backpressure-based algorthm (IFRC); and, evaluate the senstvty of the results wth respect to the parameter δ. A. QuckFx, SnapIt and Instantaneous Optmum We frst demonstrate the operaton of SnapIt usng a small -node network whch has three levels. Node, at the frst level, s the snk. Node and node are at the second level and they are the mmedate chldren nodes of the snk. Nodes 3, and are at the thrd level. Nodes 3 and have only one parent (nodes and, respectvely) and node has two parents (both node and node ). A small network s used here because t takes a long tme for MATLAB to generate a

9 9 soluton for each epoch f the network s large. Ths actually sgnfes the mportance of our work. In ths set of smulatons, we use the same rechargng profle for all the sensor nodes. The performance metrc nclude the network utlty, the sum of data rates, the cumulatve downtme of sensor nodes, and the cumulatve battery full tme. The network utlty and the sum of data rates are computed based on the packet recepton rates at the snk. The sum of data rates s used as one of the metrcs because the utlty functon s n the logarthmc scale and has a small slope. The latter metrc makes t easer to vsualze the dfference n performance between dfferent solutons, especally at hgher data rates. Fgures 3(a), 3(b), (a) and (b) show that the network utlty and the sum rates observed at the snk are close to the optmum no matter whether SnapIt s used or not. However, f the battery level s hgh (above the target level), SnapIt wll explot the excessve energy n the battery and ncrease the rates by δ. Ths beneft s especally observable durng -8 AM n Fgure (a). In Fgure 3(c), we observe the cumulatve downtme of nodes and when the ntal battery levels of all the sensor nodes are at a very low level (.% of the full capacty). We only observe nodes and because these are the only potental bottleneck nodes n the network. Wthout usng SnapIt, the cumulatve downtme for both nodes are hgh. Ths s due to the fact that the QuckFx algorthm only runs coarsely (one teraton every mnutes), and thus ts computed rates can be naccurate and even nfeasble. The SnapIt algorthm mtgates ths problem by reducng the rates when the battery level s below the target level. Therefore, the cumulatve downtme for both nodes are zero (thus nvsble n Fgure 3(c)) when SnapIt s used. Fgure (c) shows the cumulatve battery full tme when the ntal battery levels of all the sensor nodes are at a very hgh level (99% of the full capacty). It can be observed that, wthout usng SnapIt, the batteres of nodes and spend more tme n the full state. Ths causes nodes and to mss the opportuntes to harvest more energy. In contrast, f SnapIt s used, both nodes and spend less tme n full battery state, and the addtonal harvested energy s leveraged to ncrease the network utlty. TABLE II PARAMETERS Parameter λ (sn) λ (tx) j λ (rx) α & α δ Value µw 3mW 9mW.. r B. QuckFx/SnapIt v.s. Backpressure-based Protocol Next, we compare our protocol wth a backpressure-based protocol, IFRC [], whch ams to acheve maxmn farness n WSNs [8][]. IFRC uses explct sgnalng embedded n every packet to share a node s congeston state wth the neghbors. Rate adaptaton s done by usng AIMD. Several queue thresholds are defned n IFRC. A node wll reduce ts rate more aggressvely as a hgher queue threshold s reached. Snce IFRC does not consder battery state and energy replenshment, we smlarly defned several thresholds for battery levels and energy harvestng rates so that all the nodes can mantan the battery at half of the full capacty. We use a tree nstead of a DAG when performng the comparson as IFRC assumes a tree network. The tree s constructed usng 7 nodes based on Motelab s [9] topology. We used the rechargng profles of a sunny day (Feb. st 9) as well as that of a cloudy day (Feb. nd 9) for the evaluaton. The ntal battery level for all nodes s set to % of the full capacty. Fgure clearly shows that QuckFx can acheve both hgher network utlty and sum of rates. The sum of rates s % hgher than IFRC on average. The man reason s as follows. In order to mantan the battery at %, IFRC halves the rate of a sensor and that of all ts descendants f the battery drops below %. In contrast, SnapIt only slghtly reduces the rate by a small amount, δ. It s possble to mprove the performance of IFRC, but a smarter rate control algorthm s needed. C. Effect of dfferent δ Larger δ can result n a hgher network utlty when there s extra energy n the battery. However, large δ has a negatve mpact on the battery levels as t can cause a node and ts ancestors to consume the energy at a hgher rate. We manually select three nodes and observe ther battery levels over tme. The three selected nodes A, B, and C are -hop, 3-hops, - hops away from the snk respectvely. And nodes A and B are on a path from node C to the snk,.e. both nodes A and B are ancestors of node C. Fgure shows that the performance of our soluton s not very senstve to the exact value of δ f δ s small. However, hgh values of δ (= r ) should be avoded due to the consequent hgh fluctuatons n the battery that also ncreases the chances of a node to run out of battery. D. Impact of Routng Structures on Network Utlty To evaluate the effectveness of the DAG constructon algorthm proposed n Secton V, we compute and compare the achevable network utlty and sum of the data rates over a load-balanced tree (), a load-balanced DAG (), a shortest path tree (), and a shortest path DAG (). The load-balanced tree/dag s constructed usng Algorthm wthout/wth the second phase (Algorthm 3) mprovement. We create three networks of dfferent node denstes by unformly dstrbutng nodes n regons of sze 8m 8m, m m and m m respectvely. The rest of the parameters are the same as Secton VII. However, to concentrate on the performance of dfferent routng topologes (.e. trees and DAGs), we dsable SnapIt, fx the rechargng rate, and run QuckFx for teratons to compute the nstantaneous optmal network utlty and sum of the data rates. The results are presented n Fgures 7, 8 and 9. Fgures 7(a), 8(a), and 9(a) show the nstantaneous optmal network utlty. Among the four dfferent routng topologes, can acheve the hghest network utlty. Note that, as shown n Fgures 7(b), 8(b), and 9(b), does not necessarly have the hghest sum of the data rates as our objectve s to optmze the network utlty. We also compute the proportonal

10 Network utlty - - QuckFx w/o SnapIt QUckFx w/ SnapIt Instantaneous opt Sum of data rates at snk [pkt/s] 8 QuckFx w/o SnapIt QUckFx w/ SnapIt Instantaneous opt Cumulatve down tme [s] 3 Node w/o SnapIt Node w/ SnapIt Node w/o SnapIt Node w/ SnapIt -3 : : : 8: Tme (a) Network utlty : : : 8: Tme (b) Sum of data rates : : : 8: Tme (c) Cumulatve down tme. Wth SnapIt, the downtme s zero. Fg. 3. QuckFx and SnapIt vs. Instantaneous Optmum (ntal battery =.%). Our algorthms attan smlar utlty and sum of rates as optmum whle sgnfcantly reducng the battery downtme. Network utlty - - QuckFx w/o SnapIt QuckFx w/ SnapIt Instantaneous opt -3 : : : 8: Tme (a) Network utlty Sum of data rates at snk [pkt/s] 8 QuckFx w/o SnapIt QUckFx w/ SnapIt Instantaneous opt : : : 8: Tme (b) Sum of the data rates Cumulatve battery full tme [s] 8 8 Node w/o SnapIt Node w/ SnapIt Node w/o SnapIt Node w/ SnapIt : : : 8: Tme (c) Cumulatve battery full tme Fg.. QuckFx and SnapIt vs. Instantaneous Optmum (ntal battery = 99%). Our algorthms attan smlar utlty and sum of rates as optmum whle sgnfcantly reducng the tme for whch battery s full. Network utlty QuckFx w/ SnapIt IFRC Sum of data rates at snk 3 QuckFx w/ SnapIt IFRC Network utlty QuckFx w/ SnapIt IFRC Sum of data rates at snk 3 QuckFx w/ SnapIt IFRC - : : : 8: : : : 8: Tme Tme (a) Network utlty (sunny day) (b) Sum of the data rates (sunny day) - : : : 8: : : : 8: Tme Tme (c) Network utlty (cloudy day) (d) Sum of the data rates (cloudy day) Fg.. QuckFx vs. IFRC (7-node network, ntal battery = %). Rechargng profle of nodes are obtaned by varyng a base profle by an amount randomly selected n the [-%,%] regon. Network Utlty δ = % δ = % δ = % - : : : 8: Tme Sum of data rates at snk 8 δ = % δ = % δ = % : : : 8: Tme Battery level [mah] 8 Node A Node B Node C : : : 8: Tme Battery level [mah] 8 Node A Node B Node C : : : 8: Tme (a) Network utlty usng %, % and % of nodal samplng rate as δ. (b) Sum of the data rates (c) Battery level (δ =. r ) (d) Battery level (δ = r ) Fg.. A comparson of network utlty (a), sum rates (b) and battery levels (c)(d) when δ =. r, δ =. r, and δ = r (network sze s 7 nodes). Hgh values of δ can cause rapd fluctuatons n battery levels leadng to ncreased downtme.

11 Computed Utlty 3 Computed Sum Rate [pkts] Negatve Farness Index 3 (a) Network utlty (b) Sum of the data rates (c) Negatve farness ndex Fg. 7. Node densty = nodes 8m Computed Utlty 3 Computed Sum Rate [pkts] 3 3 Negatve Farness Index 3 3 (a) Network utlty (b) Sum of the data rates (c) Negatve farness ndex Fg. 8. Node densty = nodes m Computed Utlty 8 Computed Sum Rate [pkts] 3 3 Negatve farness Index (a) Network utlty (b) Sum of the data rates (c) Negatve farness ndex Fg. 9. Node densty = nodes m farness ndex to compare the algorthms. A vector of rates r R V s sad to be proportonally far f t s feasble and for any other feasble rate vectors r R V, the aggregate of proportonal changes (respectve to r) s zero or negatve [3]. In other words, V r r r As the feasble regon of rate vectors s dfferent for dfferent routng topologes, we sort the rate vectors before computng the farness ndex. Fgure 7(c), 8(c). and 9(c) show that the negatve farness ndexes are all postve. Ths mples that the real farness ndexes are all negatve. Thus, s not only more effcent n terms of network utlty, but also proportonally farer compared to the other three routng topologes. To explore whether the DAG constructed by Algorthms and 3 can result n better performance when coarse computaton s used as n Secton VII, we apply the DAG constructon algorthm to the same network used n Secton VII-A. Snce, n the smulatons, t s assumed that all the sensor nodes have smlar varaton patterns n the rechargng rate, the DAG constructon algorthm s only run once at the daybreak. Smlar to Secton VII-A, the network utlty and sum of the data rates are computed usng the packet recepton rates perceved by the snk. As shown n Fgures and, the results are consstent wth the analytcal results. The loadbalanced DAG can acheve a hgher network utlty than the shortest path DAG. When SnapIt s enabled, the network utlty can be even mproved further. VIII. CONCLUSIONS AND FUTURE WORK Achevng proportonal farness n energy harvestng sensor networks s a challengng task as the energy replenshment rate vares over tme. In ths paper, we showed that our proposed QuckFx algorthm can be appled to track the nstantaneous optmum n such a dynamc envronment and the SnapIt algorthm successfully mantans the battery at the desred target level. Our evaluatons show that the two algorthms, when workng together, can ncrease the total data rate at the snk by % on average when compared to IFRC, whle smultaneously mprovng the network utlty. As part of the future work, we plan to explore dynamc adaptaton of our parameter δ for faster operaton, whle contnung to avod battery overflows and underflows.