Maximizing Lifetime of Sensor-Target Surveillance in Wireless Sensor Networks

Transcription

1 Maxmzng Lfetme of Sensor-Target Survellance n Wreless Sensor Networks Ha Lu, Xaowen Chu, Yu-Wng Leung Computer Scence, Hong Kong Baptst Unversty Xaohua Ja, Peng-Jun Wan Computer Scence, Cty Unversty of Hong Kong Computer Scence, Illnos Insttute of Technology Abstract The paper addresses the maxmal lfetme problem n sensor-target survellance networks. Gven a set of sensors and targets n an Eucldean plane, each sensor can watch all targets wthn ts survellance range and each target should be watched by at least one sensor at any tme. The problem s to schedule the sensors to watch the targets and forward the sensed data to the base staton, such that the lfetme of the survellance network s maxmzed, where the lfetme s the duraton that all targets are watched and all actve sensors are connected to the base staton. We propose an optmal soluton to acheve the maxmal lfetme. Our soluton conssts of three steps: ) compute the maxmal lfetme of the survellance network and fnd a workload matrx and data flows by usng the lnear programmng technque; ) decompose the workload matrx nto a sequence of schedule matrces by usng the perfect matchng technque; ) determne the sensor-target survellance trees based on the above obtaned schedule matrces and data flows, whch specfy the actve sensors and the routes to pass sensed data to the base staton. The proposed optmal soluton s llustrated by a numerc example. Keywords-maxmum lfetme; sensor networks; schedulng; I. INTRODUCTION A sensor-target survellance network normally conssts of a set of sensor nodes (sensors for short), a set of targets and the base staton (BS). Sensors are utlzed to watch targets and collect sensed data to the BS. In manufacture and transportaton, sensors are usually used to montor temperature, humdty, or bomedcal value of some hot spots of a regon or a buldng/contaner. For example, sensors are employed to montor the cargo contaners whch carry dangerous gas/lqud durng the long ourney of shpment or durng the storage at a port. The sensors self-organze nto a mult-hop network whch s connected to the BS. The sensors perodcally sample the ar and forward the sensed data to the BS by mult-hop transmssons. Alarm messages wll be sent once value of the reported data exceeds a predetermned threshold whch mples leakage of the gas/lqud. In these applcatons, locatons of targets are usually statc and sensors are utlzed to montor the targets. Another example s to use noncontact nfrared sensors to montor temperature of rotatng machnes n the manufactory. Overheatng of the machnes wll be detected and reported to the BS for further analyss and subsequence actons. Snce sensors are powered by batteres and have strngent power budget, they are requred to coordnate wth each other to montor the targets n turn, and fnd energy-effcent routes to forward the sensed data to the BS. In the paper, we study the maxmal lfetme problem n sensor-target survellance networks. We assume that each sensor has an ntal energy reserve, a fxed survellance range and an adustable transmsson range whch s lmted by the maxmum transmsson range of the sensor. A sensor, e.g., acoustc, magnet and temperature sensor, can watch all the targets wthn the survellance range, and each target should be watched by at least one sensor at any tme. The problem s to schedule a subset of sensors to be actve at a tme to watch all the targets and fnd the routes for the actve sensors to forward data to the BS, such that the lfetme of the entre survellance network s maxmzed. The lfetme s the duraton up to the tme when there exsts one target that can no longer be watched by any sensors or data cannot be forwarded to the BS any more due to the depleton of energy of the sensor nodes. The rest of the paper s organzed as follows. Secton II s related work whch states the dfference between our work and exstng solutons. The formal defnton of the problem s gven n secton III. In secton IV, we present our optmal soluton whch conssts of three steps. A numerc example s presented n secton V. We conclude our work n secton VI. II. RELATED WORK There are three maor technques for maxmzng the lfetme of wreless sensor networks: the use of energy effcent routng, the ntroducton of on/off modes for sensors, and the ntegrated solutons of the above two technques. Energy effcent routng: Extensve research has been done on energy effcent data gatherng and nformaton dssemnaton n sensor networks. Some well-known energy effcent protocols were developed, such as Drected Dffuson [7] and LEACH [5]. Recently, a clusterng archtecture to mprove the lfetme of two-tered sensor networks was studed n []. Both sngle-hop and mult-hop routng were consdered and the problem was formulated as nteger lnear programmng (ILP). An optmal flow routng algorthm for upper-ter aggregaton and forwardng nodes n two-ter sensor networks was proposed n [6]. The maxmum lfetme can be acheved n both cases that the flows are splttable and non-splttable. Constructon of a data gatherng tree to maxmze the lfetme of sensor networks was proved to be NP-complete n [5]. The proposed approxmaton algorthm starts wth an arbtrary tree and teratvely reduces the load on overloaded nodes. Constructng data gatherng trees n both grd and general graphs was studed n [6]. Authors proposed a Mnmal Stener Tree based algorthm whch provdes a constant approxmaton rato for grd graphs, and a randomzed algorthm whch guarantees a polylogarthmc performance bound for general graphs. On/off schedulng: Another mportant technque used to prolong the lfetme of sensor networks s the ntroducton of /09/$ Ths full text paper was peer revewed at the drecton of IEEE Communcatons Socety subect matter experts for publcaton n the IEEE "GLOBECOM" 009 proceedngs.

2 swtch on/off modes for sensors. J. Carle et al dd a good survey on energy effcent area montorng of sensor networks [], and ponted out that the best method for conservng energy s to turn off as many sensors as possble, whle stll keepng the system functonng. An analytcal model was proposed n [] to analyse network capacty and data delvery delay, aganst the sensor dynamcs n on/off modes. Work n [4] s to determne on/off modes of sensors, such that the requred amount of data s delvered to the BS and lfetme of the network s maxmzed. The proposed algorthm acheves 0.7 of the maxmal lfetme. The on/off schedulng was studed n target (pont) coverage n wreless sensor networks [4]. The problem s to fnd the maxmum number of subsets of sensors (a sensor can appear n several subsets), such that each subset can suffcently cover all targets n the regon. However, the network aspect,.e., how to forward the sensed data to the BS, was not consdered n [4]. Moreover, the proposed heurstc n [4] s applcable to only homogeneous networks where sensors have unform transmsson/survellance range and ntal energy reserves, whle the optmal soluton proposed n ths work can be appled to heterogeneous networks. Integrated solutons: Most of exstng studes addressed only one aspect,.e., ether energy effcent routng or on/off schedulng. We have proposed the ntegrated solutons whch combne the both technques to maxmze the lfetme of sensor-target survellance networks. The proposed ntegrated solutons are optmal n the scenaros where a target s requred to be watched by one sensor [9] or k sensors [0]. Our contrbuton: All our pror work [9] [0] assumes that a sensor s able to watch only one target at a tme, whch greatly lmts applcatons of the optmal solutons. In fact, most of current sensors, e.g., acoustc, magnet and temperature sensors, can watch/montor all the targets as long as the dstance between the sensor and the target s less than the sensor s survellance range. Ths paper studes a general maxmal lfetme problem where each sensor can watch all the targets wthn the survellance range whch matches real applcatons of sensor-target survellance networks. We propose an optmal soluton to the problem. III. PROBLEM SPECIFICATION We frst ntroduce the followng notatons. Note that S() may overlap wth S() for, and T() may overlap wth T() for. Tab.. Notatons. B, S, T base staton, set of sensors (n= S ), set of targets (m= T ). S() set of sensors that are able to watch target, =,,m. T() set of targets that are wthn the survellance range of sensor, =,, n. N() set of neghbors of sensor, =,, n. E ntal energy reserve of sensor, =,,n. d dstance between sensor and,, =,,,n, B. R data rate generated from sensors whle watchng e S, e T, e R targets. energy requred for sensng, transmttng, recevng one unt data. We assume that a sensor s able to watch all targets wthn ts survellance range and each target should be watched by at least one sensor at any tme. Each sensor has an ntal energy reserve, a fxed survellance range and an adustable transmsson range. In a transmsson from s to s, s adusts ts transmsson range to exactly reach s to save energy. Transmttng one unt data from s to s costs energy e T (d ) α, where α s the sgnal declne factor. We assume the postons of targets, sensors and the BS are statc and are gven n pror. The Maxmal Lfetme problem of Sensor-Target Survellance (MLSTS for short) s, for gven S and T, to schedule the sensors to watch the targets and route the sensed data to the BS, such that the lfetme of the survellance network s maxmzed. The lfetme of the network s the length of tme untl there exsts a target, say, such that all sensors n S() run out of ther energy or the sensed data cannot be forwarded back to the BS due to the dsconnecton of the network. Each sensor n the network s ether n the actve mode for sensng/forwardng data or n the sleep mode. The lfetme of the survellance network can be dvded nto a sequence of sessons, such that each sensor stcks to the same mode wthn a sesson. In each sesson, a set of sensors are scheduled to watch the targets and forward sensed data to the BS. Other sensors that have no sensng/forwardng tasks n ths sesson go to sleep to save energy. Some actve sensors can sense and forward data smultaneously. In the next sesson, another set of sensors are scheduled to work n smlar way. Some sensors may work contnuously for multple sessons. The soluton of MLSTS s to determne these sessons, each of whch specfes whch sensor s scheduled to watch whch target and how the sensed data s forwarded to the BS. IV. OUR OPTIMAL SOLUTION The MLSTS problem can be solved n three steps. Frst, we compute the upper bound on the maxmal lfetme of the network, a workload matrx and data flows of sensors. Second, we completely decompose the workload matrx nto a sequence of schedule matrces to acheve the upper bound. Fnally, we determne a sensor-target survellance tree of each sesson whch specfes the actve sensors and the routes to forward data to the BS. We present our optmal soluton step by step. A. Fnd Maxmal Lfetme We use lnear programmng (LP) technque to fnd the maxmal lfetme of the survellance network. Let L denote the lfetme of the survellance network. We ntroduce two varables: x : total tme sensor watchng target, S, T. f : amount of data transmtted from sensor to sensor (the recever could be the BS). The problem can be formulated as the followng: Obectve: Max L s.t. x = L T; () S( ) er x + e ( d) f + e f E S T α R T() N() { B} N() ; R x + f = f x T() N() N() { B} S; () S () 0, f 0. (4) /09/$ Ths full text paper was peer revewed at the drecton of IEEE Communcatons Socety subect matter experts for publcaton n the IEEE "GLOBECOM" 009 proceedngs.

3 Note that topology nformaton that ndcates whch sensor s connected to whch target/sensor s contaned n S(), T() and N(), =,, n, =,,m. Eq. () specfes that for each target n T, the total tme that sensors watch t s equal to the lfetme of the network. That s, each target should be watched throughout the lfetme of the survellance network. Ineq. () mples that the total energy cost of a sensor node shall not exceed ts ntal energy reserve. There are three components of energy cost of a sensor node: the cost for sensng data, the cost for transmttng data (whch s dependent on the transmsson dstance), and the cost for recevng data. Eq. () s for flow conservaton. It mples that for each sensor n S, the total amount of data sensed and data receved should be equal to the amount of data transmtted. Snce all data flows are orgnated from targets and do not return to the targets, t wll not lead to cycles n our soluton. All data flows wll eventually reach the BS. The above formulaton s a typcal LP formulaton, where x, n and m, and f,,=,,,n,b, are real number varables and the obectve s to maxmze L. So the optmal results of x, f, and L can be computed n polynomal tme. However, L, obtaned from the LP formulaton ()~(4), s the upper bound on the lfetme, and each x specfes only the total tme that sensor should watch target n order to acheve ths upper bound L. Each f specfes only the total amount of data transmtted from sensor to sensor or the BS. Our task s to fnd a schedule that specfes from what tme up to what tme whch sensor watches whch targets and through whch route to pass the sensed data to the BS n each sesson. In the next two steps we wll fnd the schedule and routes that wll fnally acheve the optmal lfetme L. The values of x, n and m, obtaned from the LP, can be represented as a matrx: xx... xm X n m = x x... x m.... xn xn... x nm n m We call matrx X n m workload matrx, for t specfes the total length of tme that a sensor watch a target. In the next step, we fnd the detaled schedule n each sesson for sensors watchng targets based on the workload matrx. B. Compute Schedule of Each Sesson In each sesson of the survellance lfetme, a set of sensors are scheduled to watch the targets. Snce each sensor s able to watch all targets wthn ts survellance range, t s dffcult to determne how many sensors are requred n each sesson. Wthout loss of generalty, we assume that each sensor can watch at most k, k, targets wthn ts survellance range at a tme. Note that ths assumpton s more general than that each sensor can watch all targets wthn ts survellance range whch can be handled by settng k, k m, to the maxmum node degree for watchng targets. Suppose there s no swtchng of watchng targets n each sesson. That s, each target s contnuously watched by the same sensor n the sesson. Schedule of each sesson can be represented as a matrx, where there are exactly one postve number n each column, representng each target should be watched by one sensor; and at most k postve numbers n each row, representng each sensor can watch at most k target at a tme. The rest elements n the matrx are zeros. All the nonzero elements n the matrx have the same value, whch s the tme duraton of ths sesson. Now, our task becomes to decompose the workload matrx nto a sequence of schedule matrces of sessons: xx... x m 0c ct xx... x m c c 00ct... c t xx... x m = c + 0c xn xn... x nm n m 00 c...0 c0 c...0 0ct =P + P + P q, where c, =,,,q, s the length of tme of sesson, and q the total number of sessons. We call ths sequence of sesson schedule matrces P, =,,,q, the schedule matrces. In schedule matrx P, all elements are ether 0 or c, each column has exactly one non-zero element, and each row has at most k non-zero elements (t could be all 0, ndcatng the sensor has no sensng task n ths sesson). Snce each sensor can watch at most k targets and all targets should be watched at any tme, we have kn m (otherwse no feasble solutons). We frst consder a smple case of kn=m,.e., the number of targets s exactly k tmes of the number of sensors n the network, whch mples that all sensors should work untl the survellance operaton of the network termnates. Then, we extend the results to the general cases of kn>m. ) A Smple Case kn=m Let R, =,,,n, and C, =,,,m, denote the sum of elements n row and the column n the workload matrx, respectvely. Accordng to eq. (), we have: C = L, =,,,m. (5) Snce each sensor can watch at most k targets, we have: R kl, =,,,n. (6) Note that the sum of R equals to the sum of C. That s, n m R = C = m L. Snce kn=m, we have: = = n = R = nkl. (7) Combnng (6) and (7), we have: R = kl, =,,,n. (8) (5) and (8) gve an mportant feature of the workload matrx when kn=m that the sum of elements n each row s equal to kl and the sum of elements n each column s equal to L. Ths feature wll guarantee the possblty of decomposng the workload matrx nto schedule matrces n Theorem. The basc dea of decomposng the workload matrx, X n m, s to represent t as a bpartte graph G(S T, E), where one sde are sensors S=(s, s,, s n ) and the other are targets T=(t,,, t m ). For each non-zero element x n X n m, there s an edge from s to t and the weght of the edge s x. Consderng each schedule matrx of sesson, each column has exactly one non-zero element that specfes each target should be watched by one sensor, and each row has exactly k non-zero elements whch mples that each sensor should watch k targets to guarantee all targets are under survellance. That s, k dstnct /09/$ Ths full text paper was peer revewed at the drecton of IEEE Communcatons Socety subect matter experts for publcaton n the IEEE "GLOBECOM" 009 proceedngs.

4 targets are watched by one sensor n each sesson, whch can be represented as one sensor matchng k targets n the bpartte graph G. Thus, the problem of fndng a schedule matrx s equvalent to fndng a k-matchngs n G,.e., one sensor matchng k targets. The k-matchng algorthm proposed n [0] s appled to fnd a sequence of schedule matrces from the workload matrx. For completeness, we brefly ntroduce basc dea of the k- matchng algorthm. It replaces each sensor node n G by k duplcate nodes. The lnks adacent to the orgnal sensor are adacent to each of the duplcate nodes. In the new bpartte graph, denoted by G k, one sensor (duplcate) matches exactly one target. Thus, the problem of fndng a k-matchng n G s transformed to the problem of fndng a perfect matchng n G k. The perfect matchng algorthm n [] could be adopted and ts tme complexty s O(log V ), where V s the number of nodes n the bpartte graph. The k-matchng algorthm works as follows. Each tme G s converted to G k, we fnd a perfect matchng on G k and merge the duplcate sensors to obtan a k- matchng where one sensor matches exactly k targets. Each k- matchng s correspondng to a schedule matrx. Let c be the smallest weght of edges n the k-matchng. We update G by deductng c from the weght of the m edges n the k-matchng and remove the edges whose weght becomes zero. Ths operaton s repeated untl there s no matchng n G k. s s (a) G t t s s t s t s s s t s t s s t s t (b) G (c) perfect matchng n G (d) -matchng n G Fg. Decomposton of the workload matrx. For example, suppose a workload matrx s 0 and 0 k=. The matrx s represented as a bpartte graph G(S T, E), where S={s, s } and T={t,, t, } (Fg. (a)). Then, targets s and s are replaced by duplcates s, s and s, s, respectvely, resultng a new graph G (Fg. (b)). A perfect matchng s found n G (Fg. (c)) and a -matchng s obtaned by mergng the duplcates sensors (Fg. (d)). We obtan the schedule matrx 0 0 based on the -matchng 0 0 n Fg. (d). The follow theorems state that there exsts a k-matchng n every round of the decomposton and the number of decomposton rounds s bounded. Theorem. In every round of decomposton, there exsts a k-matchng whch s correspondng to the workload matrx computed from LP formulaton ()~(4). Theorem. The number of decomposton rounds s bounded by the number of non-zero elements n X n m. Thus, the sensor-target watchng schedule of each sesson can be computed when kn=m. In the next secton, we wll dscuss the general cases of kn>m. ) General Cases kn>m To solve general cases kn>m, our basc dea s to transform the case to kn=m by ntroducng some dummy targets nto the network. That s to fll the workload matrx X n m wth some dummy columns, such that the sum of elements n each column s equal to L and the sum of elements n each row s kl. Let Z n (kn m) be the dummy matrx, whch has (kn m) columns. By appendng the columns of the dummy matrx to the rght hand sde of X n m, the resultng matrx, denoted by W n kn, s n the form as: xx... x m zz... z( kn m) W n kn =. xx... xm zz... z( kn m) xn xn... xnm zn zn... zn( kn m) n kn To make matrx W n kn have the features of (5) and (8),.e., the sum of elements n each column s equal to L and the sum of elements n each row s kl, the dummy matrx Z n (kn-m) should satsfy the condtons: kn m z = kl R, for =,,, n. (9) = n z = L, for =,,, kn m. (0) = The FllMatrx algorthm n [9] s appled to determne elements of Z n (kn m). The algorthm s to greedly assgn value to each element n Z n (kn m) from top-left corner to bottom-rght corner. Each tme, t assgns the sum of the remanng undetermned elements of the row (or column), as much as possble, to the current element wthout volatng condtons of (9) and (0). Let R and C record the sum of the remanng undetermned elements of row and column, respectvely, for =,,..,n and =,,,kn m. Suppose we are gong to determne z,.e., elements of z kl, for k=,, and l=,,, are already determned so far. If C > R, set z = R and the undetermned elements of row are assgned to 0. If R > C, set z = C and the undetermned elements of column are assgned to 0. If R = C, set z = R and all undetermned elements of row and column are assgned to 0. We can see that condtons (9) and (0) hold n the assgnment process. Thus, general case kn>m can be smoothly transformed to the case kn=m. The complete algorthm to decompose the workload matrx s as follows. DecomposeMatrx Algorthm { f kn>m fll the matrx X n m to obtan W n kn ; construct a bpartte graph G from W n kn ; whle there exst edges n G do fnd k-matchng and correspondng schedule matrx P ; deduct P from W n kn and remove correspondng edges n G; endwhle Output W n kn =P +P + +P q ; } /09/$ Ths full text paper was peer revewed at the drecton of IEEE Communcatons Socety subect matter experts for publcaton n the IEEE "GLOBECOM" 009 proceedngs.

5 Theorem. The tme complexty of the DecomposeMatrx algorthm s O(n m log kn ). Gven a workload matrx X n m, usng the proposed algorthm, we can fll the matrx as W n kn and decompose W n kn nto a sequence of schedule matrces: W n kn = P +P + +P q. () ' Let P denote the matrx whch contans the frst m columns n P (.e., the nformaton for the m vald targets by droppng the kn m dummy columns), =,,, q. By removng the dummy columns n P, we have: X n m = P ' ' ' + P Pq. () The above dscussons conclude that a workload matrx s decomposable to a sequence of schedule matrces such that each value of x, n, m, can be actually met. In the next secton, we wll determne a sensor-target survellance tree for each sesson, such that the maxmal lfetme L can be fnally acheved. C. Determne Sensor-Target Survellance Trees We have obtaned a sequence of schedule matrces. Each schedule matrx specfes the actve sensors watchng targets n the sesson. That s, the number of sessons s the number of schedule matrces. To allow the actve sensors send ther sensed data to the BS, we need to construct a sensor-target survellance tree n whch the root s the BS and all leaf nodes are the actve sensors that are watchng targets. Intermedal nodes of the tree are the sensors whch forward data for others. The sensed data flow from actve sensors to the BS along the tree. From LP formulaton ()~(4) n secton IV-A, we have obtaned a data flow f from any sensor to sensor, ncludng the BS. To forward data to the BS, each sensor, say, needs to follow ts outgong flow f n order to acheve the maxmal lfetme L. Suppose sensor has l downstream nodes, denoted by s, s,, s l, to forward ts data to the BS (.e., f, f,, f l have non-zero values). Snce there s no orderng of data flow f, f,, f l, we smply let sensor pass ts outgong data frst to s untl flow f s saturated, then swtch to s untl the value of f s met,, and fnally t pass the last flow f l to s l. The outgong data of sensor nclude ts own sensed data and the data t helps others to forward to the BS, as shown n the left hand sde of eq. (). By followng the data flow obtaned from the LP formulaton, the optmal routes, n terms of energy effcency, can be determned and thus the maxmal lfetme L s acheved. The process wll be llustrated by a concrete example n secton V. Theorem 4. The tme complexty and space complexty of the proposed optmal soluton are O(n m).5 and O(m n 4 ), respectvely. Note that computaton of the optmal schedule s an one-off operaton at the system ntalzaton stage. Once the BS dssemnates the schedule to all sensors, each sensor operates accordng to the schedule and no addtonal computaton on the schedule s requred. When the network starts operaton, each sensor wll watch targets, turn off to sleep, receve and forward data accordng to ts schedule and the correspondng flows. Note that some sensors may work contnuously for multple sessons. There s no need to synchronze sensors to swtch target watchng at the end of sesson. Each sensor operates accordng to ts own schedule ndependently from the others. A sensor can watch a set of targets contnuously untl t s tme to swtch to another set of targets or go to sleep. The sensors perform ther own schedule based on ther local clocks whch may drft away from each other from tme to tme. To ensure a target wll be watched by another sensor contnuously before the current one swtches to other targets or goes to sleep, clocks of the sensors need to be synchronzed. There are some clock synchronzaton protocols, e.g., [8] and [], avalable for wreless sensor networks. When schedulng sensors to watch targets, the system can add a small bufferperod (n the order of mllseconds dependng on the clock error) n the front and at the end of a workng sesson to ensure that a target wll be watched contnuously at sensor swtchng. Note that compared wth the duraton of a workng sesson the buffer-perod s several orders of magntude smaller. BS V. A NUMERIC EXAMPLE s Fg. A sensor-target survellance network. Tab. Intal energy reserves of sensors. s s s We randomly place the BS, 6 sensors (crcles) and 6 targets (trangles) n a 0 0 two-dmensonal regon as shown n Fg.. The survellance range of sensors s set to 0.4 0, and the maxmum transmsson range s set to There s a dashed edge between a sensor and a target f the target s wthn the survellance range of the sensor. There s an arc from sensor s to s f s s wthn the maxmum transmsson range of s (n ths example, the maxmum transmsson ranges for all sensors are unform and thus arcs are replaced by sold edges). The ntal energy reserves of sensors are random numbers generated n [0, 00] wth the mean value 50, as shown n Tab.. To smulate the energy consumed on dfferent tasks, we set e T = 0., e R = 0.. These values are n proportonal to the actual power consumpton for transmttng and recevng data, respectvely, as ponted out n []. Experments n [] further showed that energy cost of sensng data, such as montorng temperature and humdty, s comparable to the energy cost of recevng data. Thus, we set e S = e R = 0. and the sensng data rate R =. The sgnal declne factor α s set to. s t s t /09/$ Ths full text paper was peer revewed at the drecton of IEEE Communcatons Socety subect matter experts for publcaton n the IEEE "GLOBECOM" 009 proceedngs.

6 Frst, the lnear programmng, descrbed n secton IV-A, s utlzed to compute the maxmal lfetme L, n terms of hours, workload matrx X 6 6 and data flows f that acheve L: L=4.6044hrs, X 6 6=, ( f ) = Second, we run the DecomposeMatrx algorthm n secton IV-B to decompose the workload matrx nto 4 schedule matrces as below. k equals to 5 snce s can watch fve targets n Fg.. That s, each sensor can watch at mos targets n each sesson. It s equvalent to that each sensor can watch all targets wthn ts survellance range. We can see that the whole lfetme of the network can be dvded nto 4 sessons, each of whch lasts hr,.69hrs,.67hrs and hr, respectvely. X 6 6 =P +P +P +P 4 = s t s t s s t (a) sesson (b) sesson s s t t s s s t (c) sesson (d) sesson 4 Fg. Sensor-target survellance trees of 4 sessons. s s s t t Fnally, the sensor-target survellance trees are determned based on the above schedule matrces and the data flows matrx (f ) 6 7. The survellance trees of 4 sessons are shown n Fg.. We can see that the sensor-target survellance trees satsfy the data flow constrants and the maxmal lfetme L s acheved. VI. CONCLUSION We have proposed an optmal soluton to maxmze the lfetme of sensor-target survellance networks. The proposed soluton was llustrated by a numerc example. ACKNOWLEDGMENT Ths work s supported n part by grants from Research Grants Councl of Hong Kong [Proect No. CtyU407 and HKBU009], CtyU and FRG/08-09/II-4. P.-J. Wan s supported n part by NSF under grant CNS-088. REFERENCES [] A. Bar, A. Jaekel, S. Bandyopadhyay, Clusterng Strateges for Improvng the Lfetme of Two-tered Sensor Networks, Computer Communcatons, vol., no. 4, pp , 008. [] C.F. Chassern, M Garetto, Modelng the Performance of Wreless Sensor Networks, INFOCOM 04, 004. [] J. Carle and D. Smplot-Ryl, Energy-Effcent Area Montorng for Sensor Networks, IEEE Computer, vol. 7, no., pp , 004. [4] M. Carde, M. Tha, Y. L, W. Wu, Energy-effcent Target Coverage n Wreless Sensor Networks, INFOCOM 05, 005. [5] W. Henzelman, A. Chandrakasan, and H. Balakrshna, Energyeffcent Communcaton Protocol for Wreless Mcrosensor Networks, HICSS 00, Jan 000. [6] Y.T. Hou, Y. Sh, J. Pan, S.F. Mdkff, Maxmzng the Lfetme of Wreless Sensor Networks through Optmal Sngle-Sesson Flow Routng, IEEE Transactons on Moble Computng, vol. 5, no. 9, pp , 006. [7] C. Intanagonwwat, R. Govndan, and D. Estrn, Drected Dffuson: a scalable and robust communcaton paradgm for sensor networks, MOBICOM 00, 000. [8] A. Krohn, M. Beql, C. Decker, T. Redel, Syncob: Collaboratve Tme Synchronzaton n Wreless Sensor Networks, INSS 07, 007. [9] H. Lu, X. Ja, P. Wan, C.-W. Y, S. Makk, and N. Pssnou, Maxmzng Lfetme of Sensor Survellance Systems, IEEE/ACM Transactons on Networkng, vol. 5, no., pp. 4-45, 007. [0] H. Lu, P. Wan, and X. Ja, Maxmal Lfetme Schedulng for Sensor Survellance Systems wth K Sensors to Target, IEEE Transactons on Parallel & Dstrbuted Systems, vol. 7, no., 006. [] A. Savvdes, C.C. Han, M. Srvastava, Dynamc Fne-graned Localzaton n Ad-Hoc Networks of Sensors, MOBICOM 0, 00. [] O. Smeone, U. Spagnoln, Y. Bar-Ness, S.H. Strogatz, Dstrbuted Synchronzaton n Wreless Networks, IEEE Sgnal Processng Magazne, pp. 8-97, Sep008. [] T. Uno, A Fast Algorthm for Enumeratng Bpartte Perfect Matchngs, LNCS, pp , 00. [4] Y. Wu, S. Fahmy, N.B. Shroff, Energy Effcent Sleep/Wake Schedulng for Mult-hop Sensor Networks: Non-convexty and Approxmaton Algorthm, INFOCOM 07, pp , 007. [5] Y. Wu, S. Fahmy, N.B. Shroff, On the Constructon of a Maxmum-Lfetme Data Gatherng Tree n Sensor Networks: NP- Completeness and Approxmaton Algorthm, INFOCOM 08, 008. [6] Y. Yu, B. Krshnamachar, V.K. Prasanna, Data Gatherng wth Tunable Compresson n Sensor Networks, IEEE Transactons on Parallel & Dstrbuted Systems, vol. 9, no., pp , /09/$ Ths full text paper was peer revewed at the drecton of IEEE Communcatons Socety subect matter experts for publcaton n the IEEE "GLOBECOM" 009 proceedngs.