Cross-layer Optimization of Correlated Data Gathering in Wireless Sensor Networks

Transcription

1 IEEE TRANSACTIONS ON MOBILE COMPUTING Cross-layer Optmzaton of Correlated Data Gatherng n Wreless Sensor Networks Shbo He, Student Member, IEEE, Jmng Chen, Senor Member, IEEE, Davd K.Y. Yau, Member, IEEE, Youxan Sun Abstract We consder the problem of gatherng correlated sensor data by a sngle snk node n a wreless sensor network. We assume that the sensor nodes are energy-constraned and desgn effcent dstrbuted protocols to maxmze the network lfetme. Many exstng approaches focus on optmzng the routng layer only, but n fact the routng strategy s often coupled wth power control n the physcal layer and lnk access n the MAC layer. Ths paper represents a frst effort on network lfetme maxmzaton that jontly consders the three layers. We frst assume that lnk access probabltes are known and consder the jont optmal desgn of power control and routng. We show that the formulated optmzaton problem s convex and propose a dstrbuted algorthm, JRPA, for the soluton. We also dscuss the convergence of JRPA. When the optmal lnk access probabltes are unknown, as n many practcal networks, we generalze the problem formulaton to encompass all the three layers of routng, power control, and lnk-layer random access. In ths case, the problem cannot be converted nto a convex optmzaton problem, but there exsts a dualty gap when the Lagrangan dual method s employed. We propose an effcent heurstc algorthm, JRPRA, to solve the general problem, and show through numercal experments that t can sgnfcantly narrow the gap between the computed and optmal solutons. Moreover, even wthout a pror knowledge of the best lnk access probabltes predetermned for JRPA, JRPRA acheves extremely compettve performance wth JRPA. Beyond the metrc of network lfetme, we also dscuss how to solve the problem of correlated data gatherng under general utlty functons. Other numercal results are provded to show the convergence of the algorthms and ther advantages over exstng solutons. Index Terms Sensor networks, Correlated data gatherng, Cross-layer optmzaton INTRODUCTION In montorng and survellance applcatons n wreless sensor networks (WSNs), sensor nodes are used to montor and collect data about ther local areas, and report the data to a snk node for analyss. The sensor nodes are normally powered by energy-constraned batteres and deployed at a hgh densty. Because of the hgh densty, nformaton sensed by dfferent sensor nodes s usually spatally correlated [] [3]. Hence, for energy effcency, the sensor nodes need to decde ther source data rates to the snk n order to communcate the total network nformaton whle mnmzng redundances n ther data reports. Besdes, the communcaton strategy tself must be desgned to maxmze the network lfetme [4]. Crstescu et al. [2] propose Slepan-Wolf codng to decde the amount of data reported for each sensor node and try to fnd an optmal routng tree to mnmze the total transmsson A prelmnary verson of ths paper was presented n 200 IEEE SECON conference n Boston, MA, and appears n the conference proceedngs. Research was supported n part by Natonal Natural Scence Foundaton of Chna (NSFC) under grant numbers and , n part by Natural Scence Foundaton of Zhejang (NSFZJ) under grant number R00324, n part by Projects under grant number B0703, and n part by U.S. Natonal Scence Foundaton under grant number CNS Shbo He, Jmng Chen (correspondence author), and Youxan Sun are wth the State Key Laboratory of Industral Control Technology, Department of Control Scence and Engneerng, Zhejang Unversty, Hangzhou 30027, Chna. (emal: ferrer@zju.edu.cn; {jmchen, yxsun}@pc.zju.edu.cn). Davd K.Y. Yau s wth Department of Computer Scences, Purdue Unversty, West Lafayette, IN, USA, and Advanced Dgtal Scences Center, Sngapore (emal: yau@cs.purdue.edu). cost. Based on the Slepan-Wolf codng, they gve a closedform expresson for the source rate allocaton and devse a shortest-path tree algorthm for the optmal routng. Ther approach s amenable to a fully dstrbuted mplementaton by employng a localzed form of the Slepan-Wolf codng, whch we wll also use n our work. However, an mportant drawback of ther work s that they do not fully consder the mpact of the source rate allocaton on the underlyng communcaton layers. Specfcally, they gnore the capacty of the network n tryng to route data over a mnmum-energy network path. Hence, lnks on the selected paths wll have a hgh probablty of congeston, whch severely degrades the network performance. Ths problem s partally addressed by Yuen et al. [5], who ncorporate lnk capacty constrants n ther problem formulaton. They derve a cluster of lnks that cannot transmt data at the same tme to account for channel contenton constrants. Va congeston control, they try to avod the overloadng of each lnk. For example, when a lnk s congested, data may be routed to bypass the lnk although dong so may ncur added network overhead. However, they assume that the capacty of each lnk s fxed and cannot be adapted to optmze the transmsson. In practce, the capacty of each lnk depends on the power level of the transmsson and the MAC protocol, so that physcal-layer power control and MAC-layer random access strateges must also be ncorporated nto the overall soluton. Another mportant ssue s that, n routng sensor data to the snk node, we must truly optmze for the network lfetme. Indeed, each sensor node may behave both as a data source

2 IEEE TRANSACTIONS ON MOBILE COMPUTING 2 and a data relay. Nodes that are frequently used by other nodes as relays wll have ther energy draned quckly. When these nodes de, the network may become parttoned. A good routng strategy s thus needed to balance the traffc load and prolong the network lfetme. Ths llustrates agan the ntrcate nteractons between the communcaton layers. For example, nodes that are chosen to be on busy data paths by the routng wll need hgh data rate allocatons supported by the MAC and physcal layers, and an optmal soluton must consder ther jont effects. In addton, solutons that focus on mnmzng the total energy cost of the network [2], [5] wll not work well n practce, because f lots of data are routed through the same path to mnmze the global energy use, the nodes on ths path wll run out of energy quckly, thus parttonng the network. The above examples llustrate the mportance of a comprehensve cross-layer desgn n truly maxmzng the network lfetme, whch s the focus of ths paper. The cross-layer feature also sets our work apart from a large body of exstng work [6] [8] that uses n-network data aggregaton to gan effcency durng network-layer routng. Instead, we fnd the transmsson structure at each source node that optmzes energy consumpton across the dfferent layers; we do not need n-network computatons. To the best of our knowledge, ours s a frst attempt to solve the problem by fully recognzng the nteractons between all the routng, data lnk, and physcal layers. For example, unlke the approach n [5], we account for dynamc lnk capactes obtanable by optmal power control n the physcal layer. We also account for the energy consumpton of the ndvdual network nodes, rather than that of the total network only, n order to maxmze the network lfetme. Our proposed solutons can be mplemented n a fully dstrbuted manner. We therefore for the frst tme provde a comprehensve framework for solvng the energy-effcent data gatherng problem. We frst consder a specal case of our problem n whch the lnk-layer random access probabltes are gven a pror. We formulate ths specal case as a jont power control and routng optmzaton problem, and prove that t s convex. Through the Lagrangan dual method, the problem s decomposed nto two ndependent convex optmzaton problems, for power control n the physcal layer and routng n the network layer, respectvely, coordnated by sutable Lagrangan multplers. The proposed soluton, whch we call the jont routng and power control algorthm (JRPA), s thus realzed by two ndependent dstrbuted protocols, whch we call the power control protocol (PCP) and routng strategy protocol (RSP), for the physcal and network layers, respectvely. We prove theoretcally that the proposed protocols are convergent, and that the optmal soluton of the dual problem equals that of the prmal problem. In practce, the best lnk access probabltes are frequently unknown a pror. Hence, after solvng the specal case, we dscuss the general case, n whch all the ssues of routng, power control, and random lnk access are consdered. The general problem cannot be converted nto a convex problem, and there exsts a dualty gap between optmal solutons to the dual and prmal problems. We gve a heurstc dstrbuted algorthm, whch we call the jont routng, power control, and random access algorthm (JRPRA), to solve the general problem. We show through numercal experments that JRPRA can sgnfcantly narrow the gap between the computed and optmal solutons. Moreover, even wthout a pror knowledge of the best lnk access probabltes made avalable to JRPA, JRPRA has extremely compettve performance wth JRPA n the experments. The expermental results also show the convergence of the proposed algorthms and ther advantages over exstng approaches. Beyond network lfetme, other metrcs of network performance such as the delay of reportng data to the snk node can be mportant. We wll dscuss how dverse performance metrcs can be represented by general utlty functons, and how the correlated data gatherng can be optmzed under the general functons. The rest of the paper s organzed as follows. We dscuss related work n Secton 2. In Secton 3, we formulate the cross-layer optmzaton of network lfetme n our applcaton context. We propose a dstrbuted algorthm for the specal case problem n whch the lnk access probabltes are gven a pror. For ths case, our algorthm explots the convexty of the optmzaton to produce an effcent dstrbuted soluton that jontly optmzes routng and physcal-layer power control. In Secton 4, we consder the general optmzaton problem n whch all the three layers of routng, power control, and random lnk access are consdered jontly. The general optmzaton s non-convex, and we propose a dstrbuted heurstc algorthm for ts soluton n Secton 5. Issues about mplementng the proposed algorthms n practcal WSNs are dscussed n Secton 7. Numercal results are presented n Secton 8 to llustrate the performance of our algorthms. Secton 9 concludes. 2 RELATED WORK A common technque to explot correlated data for energy savngs n sensor networks s data aggregaton [7] [9]. For example, Luo et al. propose a dstrbuted algorthm for en-route aggregaton of data [8], and Fan et al. desgn a structure-free data aggregaton technque for wreless sensor networks [7]. In ths paper we are concerned wth how to explore the structure of data for encodng them at each sensor to reduce packet transmssons whle guaranteeng that full sensng nformaton can be reconstructed at the snk. In contrast to the many exstng data aggregaton approaches, we do not perform any n-network computatons n our soluton. Crstescu et al. use two codng frameworks the Slepan- Wolf model and a jont entropy codng model to determne the rate allocaton and routng strateges for gatherng spatally correlated data n WSNs. Ther goal s to mnmze the total cost of data transmsson, and they propose approxmaton algorthms towards the objectve [2]. Rckenbach et al. employ sngle-nput codng strateges to aggregate correlated data along communcaton paths [0]. Gven n network nodes, they propose a MEGA algorthm for a foregn codng model, whch ams to fnd a mnmum-energy data gatherng topology n O(n 3 ) tme. They also propose an approxmaton algorthm for a self-codng model wth an approxmaton rato of 2( + 2) and a runnng tme complexty of O(n + log n). Yuen et al.

3 IEEE TRANSACTIONS ON MOBILE COMPUTING 3 pont out that lnk capactes could be an mportant constrant for data gatherng n WSNs. They propose a dstrbuted framework for gatherng correlated data n these networks through the Lagrangan dual method [5]. All the above efforts focus on mnmzng the total transmsson cost. As dscussed n Secton, they may over-stress nodes on the mnmum energy paths and cause the nodes to de quckly. An alternatve approach s to maxmze the network lfetme, whch s defned as the tme untl the frst node n the WSN runs out of energy []. Chang et al. formulate maxmum lfetme routng n WSNs as a lnear programmng problem. They propose dstrbuted algorthms for constant nformaton generaton rates and arbtrary generaton processes [2]. Madan et al. adopt a Lagrangan dual approach for the maxmum lfetme routng, and propose partally and fully dstrbuted algorthms as solutons []. Hua et al. use data aggregaton n an optmal routng strategy to maxmze the network lfetme [3]. They present an approxmaton functon for the ntegrated data aggregaton and routng, based on whch a dstrbuted gradent search algorthm s desgned. Lang et al. address the onlne data gatherng problem [4]. Gven a sequence of data gatherng queres, they am to maxmze the number of queres answered untl the frst falure of a node n the network. Kalpaks gves novel algorthms to solve the optmal routng problem usng n-network data aggregaton [5]. Ths body of work all focuses on the network layer only, and does not consder the mpact of other layers. Cross-layer desgn has been appled n network protocol desgn. Madan et al. use the approach n optmzng routng and lnk schedulng (but not the physcal layer) for the lfetme of WSNs [6]. Ther problem deals wth multple ndependent traffc flows and does not explot the data correlatons between these flows. Long et al. use jont congeston control, random access, and power control to maxmze the utlty (e.g., throughput) of a wreless mesh network [7]. Chang et al. provde an extensve survey on cross-layer network utlty maxmzaton (NUM) problems [8]. A set of mathematcal tools for solvng varous NUM problems are presented n [9]. Ths paper uses cross-layer desgn but, dfferent from the pror work, focuses on the problem of gatherng spatally correlated data n a WSN. Focusng on the network layer only, Cu et al. [20] approxmate the lfetme of a wreless network wth an objectve functon that network nodes can compute usng local nformaton only. We leverage ther technque n our dstrbuted mplementaton. In summary, none of the exstng efforts optmze across all the routng, data lnk, and physcal layers for the network lfetme maxmzaton problem, and our work shows the clear advantages of such a comprehensve cross-layer approach. 3 PROBLEM STATEMENT Assume that there are N sensng nodes and one snk node n the regon of nterest. We can model the WSN as a drected graph G = (V, L), where V ncludes N sensng nodes and one snk node, and L denotes the drected lnk set. (, j) L means that sensor node can transmt data to sensor node j and d j s the dstance between and j. A sensor node montors a local regon (of arbtrary shape) determned by ts sensng range for nformaton about the regon. For sensor node, denote by y the random varable of the nformaton montored by. Gven a subset X of N, let y X = (y ) X denote the vector formed by the rates of data flows generated by the sensor nodes n X. A summary of the notaton used n ths paper s gven n Table. Note that we denote sets by captal letters, varables by lowercase letters, vectors by bold lowercase letters, and matrces by bold captal letters. Where there s no confuson, we may abuse notatons and use captal letters to denote both sets and ther cardnaltes. TABLE Notaton defntons Symbol Defnton r rate of data flow generated by sensor node ; p j transmsson power allocated to lnk (, j) p j maxmum transmsson power of lnk (, j) x j total rate of data flows over lnk (, j) q j random access probablty assocated wth lnk (, j) h j gan of lnk from transmtter to recever j S j set of sensor nodes whose transmssons may nterfere wth the recever of lnk (, j), excludng L set of lnks whose transmssons are nterfered by transmssons from node, excludng the outgong lnk from γ j sgnal to nterference and nose rato H(y) entropy of random varable y H(y y 2 ) entropy of y condtoned on y 2 σ j background nose assocated wth lnk (, j) 3. Slepan-Wolf codng for correlated data A physcal quantty (e.g., temperature) n the sensng regon montored by sensor node s gven as a random varable (RV) denoted by y. If we want to gan full nformaton about the physcal quantty montored by, we have to transmt at least H(y ) amount of data f s the only node communcatng ts data to the snk node. However, f the physcal quanttes assocated wth dfferent sensng regons are correlated and all the sensor nodes send ther data, then t s possble for each node to transmt less than H(y ) amount of data wthout causng any nformaton loss. Slepan and Wolf [2] showed that, for two correlated sensor nodes and j, ther full nformaton can be encoded wth a total rate equal to ther jont entropy H(y, y j ), f the ndvdual data rates of the sensors are at least equal to the condtonal entropes H(y y j ) and H(y j y ) for the two nodes, respectvely. In our problem, allocatng a data rate r, N, for each sensor node can communcate full nformaton about the regon of nterest f and only f these rates satsfy [2] r H(y X y X c), () X where X s an arbtrary subset of N, X c = N X, and y X c = (y ) X c. We assume that each sensor generates raw data about the regon contnuously at rate y, whch can be reduced to r by the Slepan-Wolf codng.

4 IEEE TRANSACTIONS ON MOBILE COMPUTING Power control and random access In ths secton, we present an nterference-based model for random access at the lnk layer. A lnk (, j ) s under the nterference from another lnk ( 2, j 2 ) f the dstance between the nodes j and 2 s less than some threshold [22]. For communcatng data (both ts own data and data relayed for other nodes) to the snk, each sensor node has a transmsson probablty of q. When determnes to transmt data, t chooses one of ts outgong lnks, say (, j) L, wth a probablty q j, such that q j = q. We assume j: that each sensor s equpped wth one half-duplex transcever and cannot transmt and receve packets smultaneously. For lnk (, j) to be used for successful data transmsson, t s necessary that () chooses lnk (, j) to transmt the data, () none of s neghbors choose as the recever of data, and () j does not choose to send data on any of ts outgong lnks. The probablty that () () are satsfed s gven by ρ j, and we have ρ j = q j ( q j ) ( q j ). j j:(j,) L :(j, ) L Further to () () above, for the transmsson on lnk (, j) to be successful, t s necessary that the receved sgnal at j s not garbled by another concurrent transmsson not nvolvng and j. We use the sgnal to nterference and nose rato (SINR) model n [7] to characterze the condton of non-nterference, n whch the average SINR of lnk (, j) s gven by γ j = σj 2 + θ h j p j S j (,j ) L h jp j q j, where θ s the orthogonalty factor. Then, the transmsson over lnk (, j) s successful f the SINR, γ j, s above a threshold γ (0) j. The capacty of lnk (, j) can n turn be determned by the nterference-based communcaton model as { ρj log γ c j = j, γ j γ (0) j 0, otherwse. 3.3 Flow conservaton constrants We adopt mult-path routng to forward the sensed data to the snk node. For each sensor node, assocate a routng varable x j wth each lnk (, j) L. x j > 0 means that the lnk (, j) s selected by sensor node to forward messages to sensor node j, and x j = 0 means that the lnk (, j) s not selected. Hence x s the routng decson varables as well as varables of flow rates. We assume lossless transmssons n ths paper. For each sensor node, N, t has to satsfy the flow conservaton constrant that the total data transmsson rate by s equal to the receved data rate by plus the rate of data generated by tself. Formally, we have x j x j = r, N, (2) j: j:(j,) L where r s the rate of data generated by after Slepan-Wolf codng of the data. 3.4 Network lfetme model We now gve the energy model used n ths paper. We only consder the energy consumed for transmttng and recevng sensory data. We assume that the sensory data are transmtted over small dstances because the sensors are close together. In ths case, power control wll not affect the total power consumpton of a sensor sgnfcantly [23], [24], and hence we do not consder ts effects explctly n our model. Let e t j and e r j denote the energy consumpton per second for transmttng and recevng one unt of data over lnk (, j), respectvely. The power consumpton rate at sensor node, denoted by w, s equal to w = x j e t j + x j e r j. (3) j: j:(j,) L We assume that the sensor node has a gven lmted amount of ntal energy e. Hence, ts energy lfetme t s gven by t = e w. (4) The lfetme of the network s defned as the tme from the ntal deployment of the network to the tme that the frst sensor runs out of energy n the network. The maxmum network lfetme problem s thus: max mn t. (5) 3.5 Cross-Layer optmzaton model We need to maxmze the network lfetme whle guaranteeng that full nformaton about the network can be communcated to the snk node. The goal can be expressed as the followng cross-layer desgn problem: ( max mn e x j e t j + x j ej) r (6) s.t. j: j:(j,) L γ j γ (0) j, (, j) L x j c j, (, j) L 0 p j p j, (, j) L q 0, N constrants (), (2), (3), (4) There are two challenges n solvng the above problem n a dstrbuted manner. Frst, we wll need to dssemnate nformaton about the remanng energes of the sensor nodes throughout the network, whch results n hgh communcaton overhead. Exstng work has used approxmaton algorthms for solvng the maxmum network lfetme problem. Followng [20], we can convert the objectve functon of max-mn farness (6) nto one that maxmzes the aggregate utlty z α α, as α, where z = /t. As dscussed n [20], the computaton of z only needs local nformaton exchange between the sensor nodes. We wll use ths approxmaton approach to desgn effectve dstrbuted algorthms wth much less communcaton overhead than a brute-force approach. Second, global network nformaton s needed when usng Slepan-Wolf codng to allocate data rates to the sensor nodes.

5 IEEE TRANSACTIONS ON MOBILE COMPUTING 5 Addressng the problem, Crstescu et al. show that a localzed form of the Slepan-Wolf codng can be employed to gve a dstrbuted approxmaton algorthm for the rate allocaton wth a known approxmaton rato [2]. Specfcally, let N denote the subset of sensor node s neghbors that are closer to the snk node than, where node j s sad to be s neghbor f the dstance between them s less than a certan threshold. The localzed Slepan-Wolf codng specfes that each sensor node should encode ts data at a rate r equal to the entropy of ts assocated RV condtoned on the RVs of N,.e., r = H(y y N ). Then the optmzaton of (6) can be approxmated as the problem: PP : mn z α α (7) s.t. γ j γ (0) j, (, j) L (8) 0 x j c j, (, j) L (9) 0 p j p j, (, j) L (0) x j x j = r, N () j: q 0, N j:(j,) L constrants (3), (4). To solve (PP), we have to decde a rate allocaton for each sensor node, the power allocaton and random access probablty for each lnk, and the routng of data from the sensors to the snk. Usng Slepan-Wolf codng, we can use a shortest path algorthm such as Bellman-Ford to compute the dstance between each sensor node and the snk, and then assgn a rate to each sensor node accordngly. The remanng problem s then to jontly optmze the power control, lnk random access, and routng to maxmze the network lfetme. 4 JOINT POWER CONTROL AND ROUTING The cross-layer optmzaton problem above s non-convex and non-separable. We now consder a restrcted specal case of the problem n whch the lnk random access probabltes are known a pror. Wth the random access fxed, we only focus on the jont power control and routng problem. We wll show that after sutable transformatons of the varables and nequalty constrants, the problem (PP) can be expressed as an equvalent convex problem. We wll consder the general cross-layer optmzaton problem nvolvng all the routng, lnk access, and physcal layers n Secton 5. Let p j = log p j. Takng log of (8), we change the nequalty to log γ j γ (0) j, (2) where γ (0) j = log γ (0) j and γ j s gven by γ j = σj 2 + θ h j e p j S j (,j ) L. (3) h jq j e p j We have the followng theorem. Theorem : Assume that q s fxed. After a log change of p n nequalty (8), (PP) s a convex optmzaton problem. Proof: When α 2, functon zα α s an ncreasng and convex functon of z, N. Obvously, z s a lnear thus convex functon of x. Accordng to the composton rules that preserve convexty [25], zα α s a convex functon of x, N. z Hence, the sum of N convex functons,.e., α α, s a convex functon of x. The constrants (0) and () are clearly convex. The constrants (9) and (2) are convex provded that log γ j s a concave functon. In fact, log γ j = log h j + p j log(σ 2 j + θ S j (,j ) L h je p j qj ). The functon log( a e x ) s convex f a 0 [6]. In that case, log γ j s a concave functon. Hence (PP) s convex. 4. Lagrangan dual decomposton The power allocaton varables p and routng varables x are coupled by the nequalty (9). We use the Lagrangan dual method to decouple them n the dual problem. Snce the orgnal problem s convex (Theorem ), there s no dualty gap between the optmal prmal and dual solutons. Defne the Lagrangan as [26, Sec.6] L(x, p, µ) = z α α + µ j (x j ρ j log γ j ) = L (x, µ) L 2 (p, µ), where L (x, µ) = z α α + µ j x j, L 2 (p, µ) = µ j ρ j log γ j. Let D (µ) be the soluton of the followng mnmzaton: DP : L (x, µ) x j x j = H(y y N ), N j: j:(j,) L s.t. x j e t j + x j e r j = e z, N j: j:(j,) L x j 0 and D 2 (µ) be the maxmum of the problem DP 2 : L 2 (p, µ) { log γ s.t. j γ (0) j, p j p j where p j = log p j. The dual problem (DP) to the prmal problem (PP) s max D(µ), µ 0 where the Lagrangan dual objectve functon D(µ) = D (µ) D 2 (µ). Because of the convexty of (PP), the dual,

6 IEEE TRANSACTIONS ON MOBILE COMPUTING 6 problem (DP) can be solved by a gradent projecton method,.e., µ j can be updated accordng to µ j (t + ) = [µ j (t) + δ(x j ρ j log γ j )] +, (4) where δ s the step sze and [a] + = max(0, a). The dual objectve functon D(µ) s decomposed nto two ndependent subproblems: DP, for routng n the network layer, and DP 2, for power control n the physcal layer. The decomposton corresponds to a vertcal decomposton across the protocol stack. The two subproblems are coordnated by the Lagrangan multplers µ. In the followng, we wll desgn two dstrbuted algorthms for the power control and routng, respectvely, whch corresponds to a horzontal decomposton across the protocol stack. 4.2 Routng desgn We proceed to solve DP,.e., determne routng of the data from the sensor nodes to the snk. The network flows have to follow the flow conservaton constrant and be routed such that the energy consumpton can be balanced among the sensors. Assocatng Lagrangan multpler λ wth each flow conservaton constrant, we get the relaxed functon: L (x, λ, µ) = z α + λ ( = µ j x j x j α + z α α + j: j:(j,) L x j r ) (µ j + λ λ j )x j λ r. (5) Takng dervatve of L (x, λ, µ), we obtan L = z α 2 x j L = λ e t j e j: + z α 2 e r j j x j + µ j + λ λ j, e j x j r. j:(j,) L As DP s a convex problem, t can be solved by λ (t + ) = [λ (t) + δ L λ (t)] +, (6) x j (t + ) = [x j (t) δ L x j (t)] +. (7) Algorthm summarzes the Routng Strategy Protocol (RSP). Algorthm Routng Strategy Protocol (RSP) ) At each teraton of protocol, for each lnk (, j): a) Collect z, z j, λ, λ j, e, e j from nodes and j; b) Update the routng varable x j accordng to (7); c) Communcate the new x j to the nodes and j. 2) At each teraton of protocol, for each sensor node : a) Get x j, (, j) L, x j, (j, ) L, from network; b) Update λ accordng to (6); c) Communcate the new λ to the lnks for whch s ether the sender or recever. Remark: The problem DP s convex but not strctly convex. Hence, ts optmal solutons are not unque, whch means that the optmal power allocaton s not unque. Ths s because we only need a power allocaton that can guarantee a suffcent capacty for each lnk. Any power allocaton that can satsfy the requrement s feasble. As dscussed n Secton 8, the levels of lnk power obtaned by PCP are typcally not too hgh. 4.3 Power control In ths secton, we gve a dstrbuted protocol for optmal power allocaton n the physcal layer. The frst objectve for power control s to make sure that the SINR of lnk (, j), γ j, s larger than the threshold γ (0) j. The second objectve s to determne sutable lnk capactes that can attan the requred data rates whle maxmzng the network lfetme. The two objectves can be acheved by solvng DP 2. Note that DP 2 s also a convex optmzaton problem, and we can use further Lagrangan multplers β to relax the problem. Let L 2 ( p, µ, β) = µ j ρ j log γ j It can be easly obtaned that L 2 = log γ j γ (0) j β, j L 2 = µ j ρ j + β j p j γ j q j h j h j β j ( γ (0) j log γ j ). (8) (,j ) L (µ j ρ j + β j ) e pj e p j The prmal-dual method [9] can be used to solve the problem (8),.e., the power allocaton varables and Lagrangan multplers can be updated accordng to. β j (t + ) = [β j (t) δ L 2 β j (t)] +, (9) p j (t + ) = [ p j (t) + δ L 2 p j (t)] p j, (20) where [a] p j = mn(a, p j ). We ntegrate the above desgn choces n Algorthm 2: Algorthm 2 Power Control Protocol (PCP) ) At each teraton of protocol, for each lnk (, j): a) Collect p j and h j from the nterference set S j; b) Update β j accordng to (9); c) Communcate the updated β j to the sensors n L. 2) At each teraton of protocol, for each sensor node : a) Collect β j, h j, h j, and p j from lnk (, j ), (, j ) L ; b) Update p j accordng to (20); c) Communcate the updated power allocaton to S j.

7 IEEE TRANSACTIONS ON MOBILE COMPUTING Jont routng and power control algorthm (JR- PA) We are now ready to establsh the Jont Routng and Power control Algorthm (JRPA) for the correlated data gatherng problem. In the physcal and network layers, respectvely, PCP and RSP operate ndependently to update ther power allocaton and routng strateges. The nterface varables µ are used to coordnate the two strateges. That s, JRPA utlzes µ to control the performance of the two sub-algorthms and regulate the power allocaton and routng strateges towards the optmal soluton. The detaled algorthm s gven n Algorthm 3. Algorthm 3 Jont Routng and Power Control Algorthm (JRPA) ) Intalzaton: a) For each lnk (, j), choose ntal routng varable x j and Lagrange multpler µ j. b) For each sensor node, choose ntal power level p j for each lnk (, j); use shortest path method to decde the dstance from the snk node and communcate wth ts neghbors to decde ts rate allocaton r. 2) At each teraton, use PCP to solve the power allocaton; 3) Perform RSP to solve the routng; 4) Update each µ j accordng to (4). stoppng condton s satsfed. We have the followng theorem for the convergence of the proposed JRPA algorthm. Theorem 2: Choosng a small enough δ and startng from an arbtrary ntal pont, 0 p p, x 0, the algorthm JRPA converges statstcally to the optmal soluton of (DP). Proof: Let j (t) = x j (t) ρ j log γ j (t), (, j) L, then (t) denotes the subgradent vector of the dual problem (DP). Defne the Lyapunov functon as V(µ(t)) = 2δ From (4), we have (µ j µ j (t)) 2. V(µ(t + )) V(µ(t) + δ (t)) (µ j µ j(t) δ j (t)) 2 = 2δ = 2δ { (µj (t) µ j )2 2δ j (t)(µ j µ j(t)) +δ 2 2 j (t)} = V(µ(t)) + ( j (t)(µ j (t) µ j ) + 2 δ 2 j (t)) Accordng to the defnton of subgradent, j (t)(µ j (t) µ j ) D(µ(t)) D(µ ), hence we obtan V(µ(t + )) V(µ(t)) (D(µ ) D(µ(t))) + V(µ()) t τ= { D(µ ) D(µ(t)) + 2 δ 2 j (t) 2 δ 2 j (τ) Due to the convexty of the prmary problem (PP), the gradent j, (, j) L are bounded. Let B be a constant satsfyng (t) B, t > 0, and notce that 2 j V(µ(t + )) 0, we get } t (D(µ ) D(µ(τ))) V(µ()) + δbt 2. (2) τ= Fg.. The framework of the algorthm JRPA. An llustraton of JRPA s gven by Fg.. By adoptng the Lagrangan multplers, the maxmum network lfetme problem s transformed nto a prmal-dual problem, where the sub-prmal problem can be dvded nto two subproblems: power control and routng desgn. At each teraton, the algorthm wll check f the stoppng condton, gven as the well-known saddle pont optmalty crteron, s satsfed. The two subproblems are solved ndependently, coordnated by the Lagrangan multplers µ. The teraton contnues untl the From (2), By utlzng the concavty of the dual functon D(µ) and Jenson s nequalty, we get where µ(t) = t D(µ ) D( µ(t)) V(µ()) t t µ(τ). Hence we have τ= + δb 2, lm sup (D(µ ) D( µ(t))) δb t 2. Accordng to the defnton of statstcal convergence n [27], gven a small enough δ, µ(t) converges statstcally to the optmal value µ.

8 IEEE TRANSACTIONS ON MOBILE COMPUTING Complexty dscusson A key ssue about JRPA s ts communcaton complexty n terms of the number of message exchanges requred by the algorthm. The analyss s as follows. For PCP, at each teraton, nformaton about each lnk (, j) only has to be communcated to sensor nodes n the nterference set S j to obtan p j and h j, and each sensor node only has to communcate wth lnks n the set L to obtan β j, h j, h j, and p j. Ths nformaton s local-dependent only. For RSP, although the functon (5) s coupled wth x, and updatng x j and λ needs nformaton from the network, the requred nformaton s all local-dependent snce t concerns the sensor nodes and j and the lnk (, j) only. Hence, at each teraton of the protocols, each node only has to exchange a (small) constant amount of nformaton wth each of ts neghbors. 5 GENERAL CASE: JOINT ROUTING, POWER CONTROL, AND RANDOM ACCESS JRPA works well when the best lnk access probabltes are known a pror. Such knowledge s frequently unavalable n practcal networks. In ths secton, we consder the general case of problem (PP). That s, we do not assume that the lnk access probabltes are gven, but wll jontly optmze the routng, power control, and random access. Wth the change, the lnk capacty constrant (9) cannot be transformed nto a convex functon f(x, p, q) 0, and the problem (PP) s no longer convex. Thus there exsts a dualty gap between the prmal and dual problems. In ths secton, we wll develop a heurstc algorthm for solvng the general optmzaton problem. In the next secton, we wll show by numercal results that the algorthm s effectve n approxmatng the optmal solutons. Let q j = log q j, q = e q j, and ρ j = e q j ( j j:(j,) L j: Makng log of Eq. (9), we get ( e q j ))( :(j, ) L e q j ). log x j log ρ j + log(log γ j ). (22) We now gve the Jont Routng, Power control, and Random access Algorthm (JRPRA). The man procedure s smlar to that of JRPA (see Algorthm 3) n the prevous secton, and we wll just state the dfferences between the algorthms, n terms of updates for the varables β j, p j, q j, x j and µ j. In JRPRA, the PCP component of JRPA s replaced by a correspondng PCRAP protocol, where the lnk power β j, p j, and q j are updated accordng to β j (t + ) = [β j (t) δ(log γ j γ (0) j )]+, (23) p j (t + ) = [ p j (t) + δ L 2 p j (t)] p j, (24) q j (t + ) = [ q j (t) + δ L 2 q j (t)]. (25) Here, [a] = mn(0, a) and L 2 p j, L 2 p j = β j + µ j log γ j L 2 q j = µ j θ γ j e qj h j L 2 q j are gven by (,j ) L (β j + µ j log γ j ) e pj h e p, j j :(j, ) L j :(j,) L µ j e q j e qj µ j e q j e q :(, ) L (,j ) L (β j + µ j log γ j )θ γ j e qj h j h j We summarze the PCRAP protocol n Algorthm 4 e pj e p j. Algorthm 4 Power Control & Random Access Protocol (PCRAP) ) At each teraton of protocol, for each lnk (, j): a) Collect p j, q j and h j from the nterference set S j from sensor node ; b) Update β j and q j accordng to (23) and (25), respectvely; c) Send the new β j, q j to the sensor nodes n L. 2) At each teraton of protocol, for each sensor node : a) Collect β j, h j, h j, and p j from lnk (, j ), (, j ) L ; b) Update p j by (24); c) Communcate the new power allocaton to S j. In the network layer, RSP remans the same as n JRPA, except that the routng varables x j are updated accordng to where L x j x j (t + ) = [x j (t) + δ L x j (t)] +, s gven by L = z α 2 e t j x j e and µ j s updated accordng to µ j (t + ) + z α 2 e r j j + µ j + λ λ j, e j x j = [µ j (t) + δ(log x j log(log γ j ) log ρ j )] +. (26) We summarze the JRPRA algorthm n Algorthm 5. Algorthm 5 Jont Routng, Power Control, and Random Access Algorthm (JRPRA) ) Intalzaton: Smlar to JRPA. 2) At each teraton, perform PCRAP to decde the power allocaton and lnk access probabltes; 3) Use modfed RSP to solve the routng; 4) Update each µ j accordng to (26).

9 IEEE TRANSACTIONS ON MOBILE COMPUTING 9 6 DISCUSSIONS: CROSS-LAYER OPTIMIZA- TION UNDER GENERAL UTILITY FUNCTIONS So far, motvated by energy constrants n WSNs, we have focused on network lfetme exclusvely as the goal of algorthm desgn. In ths secton, we dscuss how our approach can be generalzed to other forms of network performance dependng on the applcaton context. General forms of network performance can be represented by general utlty functons [28], [29]. Let U (x, c) denote the utlty functon assocated wth sensor node. U (x, c) gves the gan that can obtan from the whole network. Accordng to the network utlty maxmzaton (NUM) framework n [30], [3], the functon U (x, c) n actual applcatons usually satsfes the followng two propertes: ) U (x, c) s a concave functon of (x, c); 2) U (x, c) s twce contnuously dfferentable. Under a general utlty functon, our problem formulaton can be stated as follows. GPP : mn U (x, c) (27) s.t. γ j γ (0) j, (, j) L 0 x j c j, (, j) L 0 p j p j, (, j) L j: x j j:(j,) L q 0, N constrants (3), (4). x j = r, N (28) Note that for consstency wth the rest of the paper, we mnmze U (x, c) nstead of maxmze U (x, c). The specfc choce of utlty functons depends on the applcaton context. For nstance, when both long network lfetme and low transmsson delay are desrable, the utlty functon could be chosen as follows: U (x, c) = zα α ω where ω s a weght constant. j: zα α x j c j x j (29) s used to approxmate the network lfetme (as n the prevous sectons), and x j c j x j gves the average number of packets n the queue of lnk (, j) under an M/M/ queung model [3]. As such, the utlty functon (29) specfes a weghted tradeoff between the network lfetme and queueng delay metrcs. Other forms of utlty functons may be used for other performance metrcs, such as log( ) for network throughput [30]. We proceed to dscuss how to solve the general formulaton (27). If x and c n the utlty functon U (x, c) are not coupled,.e., U (x, c) = U () (x) + U (2) (c), then smlar to Secton 4, the Lagrange dual method can be employed to decompose the problem: L(x, p, q, µ) = U () (x) + + U (2) (c) µ j log x j µ j (log(log γ j ) + ρ j ) (30) If Eq. (30) s separable n terms of x, p, and q (c conssts of p and q), t can be dvded nto two subproblems: () routng desgn, and () power control and random access control. We can then use a smlar algorthm to Algorthm 5 to solve the problem (27). If the utlty functon U (x, c) s not separable, e.g., the functon n Eq. (29), exstng decouplng methods [9] cannot be used to decouple the utlty functon (.e., the objectve functon) drectly, but auxlary varables wll have to be ntroduced to ensure separablty. Take the utlty functon n Eq. (29) as an example. As c j s a functon of the power allocaton to lnk (, j ) wthn the nterference range of lnk (, j), the objectve functon (29) s hard to separate. The problem (27) can be transformed nto DGPP: mn zα α ω j: s.t. the constrants gven by (28) x j y j x j (3) y j = c j, (, j) L, (32) where y are auxlary varables ntroduced to decouple the objectve functon. The key observaton about y j s that t can be taken as a local varable of lnk (, j). Hence, the objectve functon (3) becomes separable, and the DGPP problem can be solved smlarly to the problem (27). Hence, by usng the approach developed n the prevous sectons and leveragng exstng decouplng methods n the lterature, we can solve the cross-layer optmzaton problem under general utlty functons. 7 IMPLEMENTATION ISSUES We dscuss how the JRPA and JRPRA algorthms can be mplemented n a practcal WSN for gatherng spatally correlated data. 7. Modelng the data correlatons We assume knowledge of the dstrbuton y, and that correlatons n ts components can be modeled as condtonal entropes. In practce, jont dstrbutons of nformaton about a montored regon may not be exactly known. However, a wdely accepted model found to provde excellent approxmatons n many real applcaton scenaros s the Gaussan process [32]. For example, successful applcatons of the Gaussan process to model temperature dstrbutons n real lfe can be found n [33]. In our problem context, snce there s a fnte number of sensor nodes n the RoI, we can adopt the multvarate normal dstrbuton, a specal case of the Gaussan process, to estmate the dstrbuton of the correlated data. Specfcally, the jont N- dmensonal multvarate normal dstrbuton, G N (ν, K), for the spatal data Y N measured at the N nodes, s gven by f(y N ) = 2π det (K) /2 e (0.5(Y N ν) T K (Y N ν)), (33) where K s the covarance matrx of Y N, and ν s the mean vector. Obvously, the multvarate normal dstrbuton

10 IEEE TRANSACTIONS ON MOBILE COMPUTING 0 s decded by the parameters K and ν. One advantage of ths dstrbuton s that, for y, the condtonal dstrbuton f(y y c) s also a normal dstrbuton, whose varance K y y c and mean vector ν y y c are gven by ν y y c = ν y + K y,y c Ky c,y c (y c ν c), (34) ky 2 y c = ky 2,y K y,y c Ky c,y c K y c,y. (35) In a feld deployment, sensors can measure ther dstances from each other usng a specalzed protocol [2] or a more general localzaton protocol [5]. Correlatons of data between pars of (close-by) sensors can then be estmated based on the measured dstances and doman knowledge, e.g., gradual varatons of temperature over space [33]. Alternatvely, the correlatons can be calbrated. Durng a startup phase, all the sensor nodes transmt what they sense to the snk node. After collectng enough data samples, the snk node computes the condtonal probablty dstrbutons of the sensed data (for pars of reasonably close-by nodes) and sends the results to the sensors. The sensor nodes then use the condtonal dstrbutons to mplement the proposed protocols and algorthms. To account for dynamc changes n the external envronment, f and when some of the sensors observe sgnfcant changes n ther measurements, re-calbratons of the correlatons may be performed for these sensors. 7.2 Implementaton of Lagrangan multplers We now address the mplementaton of Lagrangan multplers for JRPA and JRPRA. We wll focus on the case of JRPA only, as JRPRA can be handled n a smlar way. The man ssue s how to mplement the Lagrangan multplers µ, β, and λ. Lagrangan multplers, n general, can be nterpreted as the prces for the supply/demand of a goods [34]. µ j, (, j) L, corresponds to the congeston of lnk (, j) and can be nterpreted as the congeston prce. From (4), we can see that when the rate of lnk (, j) exceeds the capacty of (, j), µ j wll ncrease, and vce versa. Random Exponental Markng (REM) [35] s known to be a hghly effectve algorthm for evaluatng the path congeston prce. In our problem, each lnk (, j) adopts a smlar REM process to measure ts congeston level and update the prce µ j accordngly. For β, t can be nterpreted as the prce of not reachng a specfed sgnal-to-nterference-and-nose rato (SINR). Hence, each sensor node can measure ts SINR at each teraton, and update the SINR prce accordng to (9). For λ, t can be nterpreted as the prce of nconsstent coordnaton of the ncomng and outgong flows [36]. It can be mplemented smlarly as µ and β. 8 NUMERICAL RESULTS In ths secton, we report numercal experments to evaluate the performance of JRPA and JRPRA proposed n Sectons 4 and 5, respectvely. We also dscuss the advantages of these algorthms over the proposed soluton n [5]. We consder the WSN topology shown n Fg. 2, whch conssts of k + 4 sensor nodes and one snk node. We denote the snk node by k + 5. The dstances between (, 2), (2, 3), Fg. 2. Topology of the smulated WSN., (k+, k+2), (k+, k+3) and (k+2, k+4) are all equal to 5, and the dstances between (k+3, k+5) and (k+4, k+5) are equal to 0. We only consder the lnks through whch sensory data can be forwarded closer to snk node. A transmsson on one lnk nterferes wth a transmsson on another f the dstance between the recever of the frst lnk and the sender of the second lnk s less than 5 m. For the SINR nterference model, we set σ 2 j = 5 0 3, h j = 0.097/d j, γ (0) j = 50 and θ = /256. The correlaton model f(y N ) can be decded n applcatons as dscussed n secton 7.. For smplcty, we set k j = e d2 j. Accordng to [2], the entropy of y condtoned on y N, denoted by H(y y N ), s gven by H(y y N ) = ( 2 log 2πe det K[y ), y N ], (36) det K[y N ] where K[y] denotes the covarance matrx formed by the random varables y. When the network starts, each node uses Bellman-Ford to get ts shortest path from the snk node. Then a node, say, forms the set N by exchangng local messages wth ts neghbors. By usng (36), the data rate orgnatng from node, r, can be obtaned. Note that by usng Slepan- Wolf codng, the sensor nodes whch are closer to the snk node wll have hgher orgnal data rates, thus reducng the communcaton cost. Followng the communcaton model n [28], we set e r j = 50n J/bt and e t j = ϱ + ςdm, where ϱ = 50nJ/bt, ς = 0.003p J/b/m 4, and m = 4. The ntal energes of the sensors k + 4 are set to be 2500 J. The snk node k + 5 s assumed to have enough energy, and we do not account for ts energy use. We wll frst set k =,.e., there are fve sensor nodes and one snk node n the network. We use the baselne topology to show the convergence of our proposed algorthms. Snce we can not show all the results f we adopt a large sze of networks due to space lmtaton. Then we vary k to be 2, 3, 4, 5, and show the advantages of our proposed algorthms over exstng approaches, as the sze of the network ncreases. 8. Algorthm performance evaluaton JRPA algorthm: In ths specal case problem, we assume that the best random access probabltes (separately determned) are known, and use them as fxed nput to the JRPA algorthm. (More generally, any consstent set of the probabltes could be used, although the performance would vary.) Settng the step sze α = 0., we collect the values of p and x at each teraton, and the results are shown n Fg.3. From Fg. 3(a) and Fg. 3(b), we can see that p and x all

11 IEEE TRANSACTIONS ON MOBILE COMPUTING 0.9 p x 46 x q 46 q 56 Power of each lnk (W) Iteraton number (n) (a) The convergence of PCP n JRPA. p 35 p 23 p 46 p 2 p 56 Rate of each lnk (Kpbs) Iteraton number (n) x 24 x 35 (b) The convergence of RSP n JRPA. x 2 x23 Access probablty of each lnk (W) q q q 23 q Iteraton number (n) (c) The convergence of PCRAP n JRPRA. Fg. 3. The convergence of JRPA and JRPRA. TABLE 2 Soluton comparson between JRPA and JRPRA. Routng varables (kb/s) x 2 x 23 x 24 x 35 x 46 x 56 JRPA JRPRA Power varables (W) p 2 p 23 p 24 p 35 p 46 p 56 JRPA JRPRA Access probabltes q 2 q 23 q 24 q 35 q 46 q 56 JRPA JRPRA TABLE 3 Optmalty Descrpton. Routng varables x 2 x 23 x 24 x 35 x 46 x 56 Lfetme JRPRA h Optmal soluton of (7) h Optmal soluton of (6) h converge to the optmal solutons. As dscussed, the optmal dual soluton s also prmarly optmal JRPRA algorthm: We evaluate the JRPRA algorthm n Secton 5. In ths case, the random access probabltes q are no longer fxed. Instead, they are decded n the teratve process of the algorthm. Fg. 3(c) gves the results. The updates of p and x are smlar to those n JRPA. So we plot the updates for q only and lst the fnal acheved soluton n Table 2. From Fg. 3(c), note the convergence of JRPRA. Table 2 shows the solutons obtaned by JRPA and JRPRA. The power allocaton p and routng strategy x are very smlar n the two algorthms. These results ndcate that JRPA s a TABLE 4 Soluton comparson between our and exstng approaches. Routng varables x 2 x 23 x 24 x 35 x 46 x 56 Lfetme MnE h MnE-NC h JRPRA h Bandwdth (JRPA) specal case of JRPRA (wth q fxed). More mportantly, JR- PRA acheves extremely compettve performance wth JRPA, but does so wthout a pror knowledge of the best lnk access probabltes. The general case of problem (7) may not be a convex optmzaton problem, n whch case there wll exst a gap between the optmal soluton of (7) and that obtaned by JRPRA. In a number of test cases, we compare the optmal solutons of (7) obtaned by a centralzed optmzaton algorthm n MATLAB and those obtaned by JRPRA n Table 3. We can see that JRPRA n fact attans the optmal solutons of (7) n the tests, whch valdates the effectveness of JRPRA. In our problem formulaton, we use problem (7) to approxmate problem (6). Table 3 shows n addton that we can acheve good approxmaton ratos, where the problem (7) s solved n a dstrbuted way wth neglgble reductons ( %) n the network lfetme. 8.2 Algorthm comparson We proceed to dscuss the advantages of the proposed algorthms over exstng solutons. Smlar to [5], much exstng work focuses on the mnmum energy consumpton (MnE) problem for correlated data gatherng n WSNs. Also, many exstng efforts on the optmal routng problem do not account for the spatal correlaton of nformaton between the sensors. We denote the use of MnE n ths case by MnE-NC. These metrcs do not optmze the network lfetme as we seek, snce mnmzng the total energy consumpton may unduly stress certan sensor nodes and cause them to de quckly, and not explotng the spatal correlaton of nformaton may lead to the communcaton of unnecessary data and waste energy. We can see these effects n Table 4. The solutons obtaned by MnE [5], MnE-NC, and JRPRA are lsted n the table. As also shown n Table 2, the network lfetmes acheved by JRPRA s about 4420 h. The network lfetme obtaned by MnE s 46 h, whch s sgnfcantly shorter than that obtaned by JRPA. MnE-NC obtans a stll shorter network lfetme of 957 h. We conclude that the lmtatons of MnE and MnE-NC n prolongng the network lfetme are apparent relatve to JRPRA. Lastly, Yuen et al. [5] assume that the capacty of each lnk s fxed. If the transmsson data rate over a shortest-path

12 IEEE TRANSACTIONS ON MOBILE COMPUTING 2 lnk exceeds the lnk s capacty, the extra traffc has to be routed to bypass the lnk, causng extra energy consumpton. For example, n Fg. 2, f node k + 3 has to transmt 4 kb of data to node k + 5 and the capacty of lnk (k + 3, k + 5) s 3 kb, then node k + 3 s forced to transmt kb of data to node k+4. In contrast to ther approach, we use power control to adaptvely provson suffcent bandwdth for each lnk. By comparng the actual lnk rates on the 5th row of Table 4 and the provsoned lnk bandwdth on the 7th row, we can see that the power control s effectve. TABLE 5 Solutons obtaned by the four algorthms when n = 7. Routng varables x 2 x 23 x 34 x 35 x 46 x 57 x 67 MnE MnE-NC JRPRA TABLE 6 Lfetme of the networks wth ncreasng number of sensor nodes. Sensor nodes (n) MnE 46 h 3598 h 369 h 283 h 2559 h MnE-NC 957 h 398 h 087 h 900 h 753 h JRPRA 4420 h 4068 h 3806 h 3458 h 3328 h 8.3 Network performance under varyng number of nodes In ths secton, we vary k to be 2, 3, 4, 5, and report results for MnE, MnE-NC, and JRPRA. These results are shown n Table 5, 9, 0 and (See Tables 9, 0 and n the Appendx. A of the supplemental fle). The correspondng network lfetmes acheved by the varous algorthms are lsted n Table 6. We can see that as the number of sensor nodes ncreases, the network lfetme decreases sharply for MnE and MnE-NC, whereas the decrease s much slower for JRPRA. The performance gan s acheved by optmzng for the network lfetme globally, rather than for the total energy consumpton. We conclude that the proposed algorthms are sgnfcantly more energy effcent than MnE and MnE-NC. Importantly, the energy savngs become more pronounced as the network sze ncreases. 8.4 Network performance under varyng correlaton models In ths secton, we proceed to nvestgate the mpact of the data correlaton model on the network lfetme, n a network of large node degree as shown n Fg. 4. In the fgure, sensor node s the snk node, and nodes 2 7 are sensng nodes. The vertcal and horzontal dstances between two adjacent sensng nodes are both 5 m, and the vertcal and horzontal dstances from the snk node to the sensng nodes 2, 6, 0, and 4 are all 7.5 m. The other network parameters reman the same as before. Fg. 4. A large node degree extended network topology. We employ a popular data correlaton model, the ratonal quadratc (RQ) co-varance functon, to show ts mpact on the performance of the proposed JRPRA algorthm [37]: K RQ (x, x ) = φ 2 3( + d2 (x, x ) 2φ 2 2 φ ) φ, (37) where φ, φ 2, and φ 3 are parameters of the RQ functon. As φ s closely related to the degree of correlaton, we set φ 2 = 2, φ 3 = 0.4, and smulate JRPRA under varyng φ,.e., φ = 0.07, 0.06, 0.05, 0.04, and We frst study the mpact of the data correlaton on r, = 2,, 7, whch are the allocated rates at the sensng nodes by usng local Slepan-Wolf codng n JRPRA. The results are shown n Table 7. We can see that the data correlaton model has a sgnfcant mpact on r, = 2,, 7. Most of the sensng nodes can reduce the rate wthout mparng performance usng Slepan-Wolf codng. The smaller the φ, the smaller the r at each sensng node. Especally, when φ = 0.03 or 0.04, some nodes (e.g., nodes 2 and 3) even have a rate of 0. Ths s because accordng to the RQ functon, a smaller φ mples a larger correlaton n the functon. We conclude that the proposed algorthm can acheve better performance n a dense sensor network whose nodes have a hgh data correlaton. We now nvestgate the optmalty of JRPRA. We denote the optmal algorthm for (7) as OPT, and compare the results between JRPRA and OPT when φ = 0.07, as shown n Table 8 and Fg. 5. Fg. 5 shows the fast convergence of JRPRA. Table 8 ndcates that the solutons obtaned by JRPRA and OPT are very close to each other. We summarze the results for φ = 0.06, 0.05, 0.04, 0.03 n Tables 2, 3, 4 and 5, whch are shown n the Appendx. B of the supplemental fle due to space lmtaton. All the results demonstrate that the solutons obtaned by JRPRA are close to the optmal ones. 9 CONCLUSIONS We solved the problem of optmal gatherng of correlated data n WSNs for maxmum network lfetme. We adopted a comprehensve cross-layer approach, whch has advantages over exstng solutons. We frst consdered a specal case of the problem and showed that t s convex. The JRPA soluton decomposed the problem nto 2 ndependent convex sub-problems of power control and routng. We presented two dstrbuted protocols, PCP and RSP, for the 2 layers to fnd the optmal power allocaton and routng, respectvely. We

13 IEEE TRANSACTIONS ON MOBILE COMPUTING 3 TABLE 7 Rate allocaton for each sensor node under dfferent φ. φ \ Sensor TABLE 8 Solutons obtaned by JRPRA and OPT when φ = Routng varables (kb/s) x 2 x 32 x 42 x 53 x 6 x 76 x 84 x 86 x 98 x 0, JRPRA OPT Routng varables (kb/s) x,7 x,0 x 2,0 x 3, x 4, x 5,3 x 5,4 x 6,2 x 6,4 x 7,6 JRPRA OPT x 0, x 6 x x 4, Rate of each lnk (Kpbs) x 86 x 32 x 96 x 42 x 76 x 53 x 84 Rate of each lnk (Kpbs) 0.5 x 6,4 x,0x3, x 7,6 x 2,0 x 6,2 x,7 x 5,4 x 5, Iteraton number (n) Iteraton number (n) (a) (b) Fg. 5. Routng varable updates durng teratons. then solved the general case of the optmzaton nvolvng all 3 layers of routng, power control, and lnk access, usng the heurstc dstrbuted algorthm JRPRA. Numercal results valdated our analyss and confrmed the effectveness of the solutons. They demonstrated the advantages of the proposed algorthms over exstng approaches. In the comparson between JRPRA and JRPA, we show that JRPRA acheves extremely compettve performance wth JRPA, but does so wthout a pror knowledge of the best lnk access probabltes. REFERENCES [] H. Luo, H. Tao, H. Ma, and S.K. Das. Data fuson wth desred relablty n wreless sensor networks. IEEE Transactons on Parallel and Dstrbuted Systems, 22(3):50 53, 20. [2] R. Crstescu, B. Beferull-Lozano, and M. Vetterl. On network correlated data gatherngs. In IEEE Conference on Compute Communcatons (INFOCOM), [3] H. Luo, Y. Lu, and S.K. Das. Routng correlated data wth fuson cost n wreless sensor networks. IEEE Transactons on Moble Computng, 5(): , [4] H. Nshyama, A. E. Abdulla, N. Ansar, Y. Nemoto, and N. Kato. Hymn to mprove the longevty of wreless sensor networks. In Proceedngs of IEEE global communcatons conference (Globecom), 200. [5] K. Yuen, B. Lang, and B. L. A dstrbuted framework for correlated data gatherng n sensor networks. IEEE Transactons on Vehcular Technology, 57(): , [6] H. Luo, Y. Lu, and S.K. Das. Routng correlated data n wreless sensor networks: A survey. IEEE Network, 2(6):40 47, [7] K. Fan, S. Lu, and P. Snha. Structure-free data aggregaton n sensor networks. IEEE Transactons on Moble Computng, 6(8): , [8] H. Luo, Y. Lu, and S.K. Das. Dstrbuted algorthm for en route aggregaton decson n wreless sensor networks. IEEE Transactons on Moble Computng, 8(): 3, [9] H. Luo, J. Luo, Y. Lu, and S.K. Das. Adaptve data fuson for energy effcent routng n wreless sensor networks. IEEE Transactons on Computers, 55(0): , [0] P. Rckenbach and R. Wattenhofer. Gatherng correlated data n sensor networks. In ACM Workshop on Foundatons of Moble Computng (DIALM-POMC), [] R. Madan and S. Lall. Dstrbuted algorthms for maxmum lfetme routng n wreless sensor networks. IEEE Transactons on Wreless Communcatons, 5(8): , [2] J. Chang and L. Tassulas. Maxmum lfetme routng n wreless sensor networks. IEEE/ACM Transactons on Networkng, 2(4):609 69, [3] C. Hua C. and T. Yum. Optmal routng and data aggregaton for maxmzng lfetme of wreless sensor networks. IEEE/ACM Transactons on Networkng, 6(4): , [4] W. Lang and Y. Lu. Onlne data gatherng for maxmzng network lfetme n sensor networks. IEEE Transactons on Moble Computng, 6():2, [5] K. Kalpaks, K. Dasgupta, and P. Namjosh. Effcent algorthms for maxmum lfetme data gatherng and aggregaton n wreless sensor networks. Computer Networks, 42:697 76, [6] R. Madan, S. Cu, S. Lall, and A. Goldsmth. Cross-layer desgn for lfetme maxmzaton n nterference-lmted wreless sensor networks. IEEE Transactons on Wreless Communcatons, 5(): , [7] C. Long, B. L, Q. Zhang, B. Zhao, B. Yang, and X. Guan. The

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Optmal flow control for utlty-lfetme tradeoff n wreless sensor networks. Computer networks, 53: , C. Rasmussen and C. Wllams. Gaussan Processes for Machne Learnng. Massachusetts Insttute of Technology Press, Shbo He s currently a Ph.D canddate n Control Scence and Engneerng at Zhejang Unversty, Hangzhou, Chna. He s a member of the Group of Networked Sensng and Control (IIPCnesC) n the State Key Laboratory of Industral Control Technology at Zhejang Unversty. Hs research nterests nclude coverage, cross-layer optmzaton, and dstrbuted algorthm desgn problems n wreless sensor networks. 4 Jmng Chen (M 08 SM ) receved B.Sc degree and Ph.D degree both n Control Scence and Engneerng from Zhejang Unversty n 2000 and 2005, respectvely. He was a vstng researcher at INRIA n 2006, Natonal Unversty of Sngapore n 2007, and Unversty of Waterloo from 2008 to 200. Currently, he s a full professor wth Department of control scence and engneerng, and the coordnator of group of Networked Sensng and Control n the State Key laboratory of Industral Control Technology at Zhejang Unversty, Chna. Hs research nterests are estmaton and control over sensor network, sensor and actuator network, coverage and optmzaton n sensor network. He currently serves assocate edtors for several nternatonal Journals ncludng IEEE Transactons on Industral Electroncs. He s a guest edtor of IEEE Transactons on Automatc Control, Computer Communcaton (Elsever), Wreless Communcaton and Moble Computer (Wley) and Journal of Network and Computer Applcatons (Elsever). He also serves as a Co-char for Ad hoc and Sensor Network Symposum, IEEE Globecom 20, general symposa Co-Char of ACM IWCMC 2009 and ACM IWCMC 200, WCON 200 MAC track Co-Char, IEEE MASS 20 Publcty Co-Char, IEEE DCOSS 20 Publcty Co-Char, IEEE ICDCS 202 Publcty Co-Char and TPC member for IEEE ICDCS 200, IEEE MASS 200, IEEE SECON 20, IEEE INFOCOM 20, IEEE INFOCOM 202, IEEE ICDCS 202 etc. Davd K. Y. Yau receved the B.Sc. (frst class honors) from the Chnese Unversty of Hong Kong, and the M.S. and Ph.D from the Unversty of Texas at Austn, all n computer scence. He s currently Dstngushed Scentst at the Advanced Dgtal Scences Center, Sngapore, and Assocate Professor of Computer Scence at Purdue Unversty, West Lafayette, IN, USA. Dr. Yau was the recpent of an NSF CAREER award for research n qualty of servce provsonng. Hs other areas of research nterest are protocol desgn and mplementaton, wreless and sensor networks, network securty, network ncentves, and smart grds. Dr. Yau served as Assocate Edtor of IEEE/ACM Trans. Networkng ( ); Vce General Char (2006), TPC co-char (2007), and TPC Area Char (20) of IEEE Int l Conf. Network Protocols (ICNP); TPC co-char (2006) and Steerng Commttee member ( ) of IEEE Int l Workshop Qualty of Servce (IWQoS); and TPC Track co-char (Network/Web/P2P Protocols and Applcatons) of IEEE Int l Conf. Dstrbuted Computng Systems (ICDCS) 202. He s a member of the IEEE. Youxan Sun receved the Dploma from the Department of Chemcal Engneerng, Zhejang Unversty, Chna, n 964. He joned the Department of Chemcal Engneerng, Zhejang Unversty, n 964. From 984 to987, he was an Alexander Von Humboldt Research Fellow and Vstng Assocate Professor at Unversty of Stuttgart, Germany. He has been a full professor at Zhejang Unversty snce 988. In 995, he was elevated to an Academcan of the Chnese Academy of Engneerng. Hs current research nterests nclude modelng, control, and optmzaton of complex systems, and robust control desgn and ts applcaton. He s author/co-author of 450 journal and conference papers. He s currently the drector of the Insttute of Industral Process Control and the Natonal Engneerng Research Center of Industral Automaton, Zhejang Unversty. He s Presdent of the Chnese Assocaton of Automaton, also has served as Vce-Charman of IFAC Pulp and Paper Commttee and Vce-Presdent of Chna Instrument and Control Socety.