Energy Efficient Sensor, Relay and Base Station Placements for Coverage, Connectivity and Routing

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1 Energy Efficient Sensor, Reay and Base Station Pacements for Coverage, Connectivity and Routing Mauin Pate*, R. Chandrasekaran and S.Venkatesan Teecommunication Engineering Program Erik Jonsson Schoo of Engineering and Computer Science University of Texas at Daas, Richardson, Texas {chandra, Abstract We consider a wireess sensor network made of sensor nodes capabe of sensing and communication, reay nodes capabe of communication, and base stations responsibe for coecting data generated by sensor nodes, to be depoyed in sensor fied. We address the probem of pacing the sensor nodes, reay nodes and base stations in the sensor fied such that (i) each point of interest in the sensor fied is covered by a subset of sensors of desired cardinaity (ii) the resuting sensor network is connected and (iii) the sensor network has sufficient bandwidth. We propose severa depoyment strategies to determine optima pacements of sensor nodes, reay nodes and base stations for guaranteed coverage, connectivity, bandwidth and robustness. We study severa different objectives such as minimizing the number of sensor nodes depoyed, minimizing the tota cost, minimizing the energy consumption, maximizing the network ifetime and maximizing the network utiization. The pacement probems for reiabe as we as unreiabe/probabiistic detection modes are formuated as Integer Linear Programs (ILPs). The practicaity, effectiveness and performance of the proposed strategies are iustrated through simuations. I. INTRODUCTION Distributed sensor networks, designed to monitor and/or contro the surrounding environmenta phenomena, have the potentia to revoutionize many appications. A sensor network consists of sensor nodes and one or more base stations. Sensor nodes generate, process, and forward data (via intermediate sensor nodes) to base stations. Among the major design chaenges in the design of sensor networks is the efficient utiization of resources avaiabe to sensor nodes such as scarce bandwidth and imited energy suppy. In appications such as surveiance, target tracking and intrusion detection, the strategic pacement of sensor nodes in the sensor fied can greaty enhance the sensing quaity, reduce the cost and minimize the energy consumption, thereby increasing ifetimes of sensor nodes. Depending on properties of sensing devices, terrain and appication scenarios, the sensing capabiities of sensor nodes can be modeed as reiabe (precise) or unreiabe (imprecise). In the unreiabe sensing mode, a sensing/detection probabiity is assigned to each reading which characterizes the eve of confidence in the reading. In the reiabe detection mode readings within the sensing range are assumed to be precise. The area in which a sensor can perform its sensing, monitoring, surveiance and detection tasks with a reasonabe accuracy (i.e. the sensor readings have at east a threshod eve of sensing/detection probabiities within that area) is known as the coverage area. The union of coverage areas of individua sensors is the coverage area of a sensor network. Coverage areas can be irreguar and can be ocation dependent due to the obstructions in the terrain e.g. sensors depoyed for indoor appications, in urban and hiy areas [1]. The coverage area may aso depend on the target, e.g. a sensor might be abe to detect smaer targets from short distances and bigger targets from ong distances. The degree of the sensing coverage required depends on the appication [2]. Covering every ocation with mutipe sensors can provide robustness. Some appications may require the preferentia coverage of critica ocations based on risk or tactica importance [1]. The coverage requirements can aso change with time due to changes in environmenta conditions, e.g. the visibiity can vary due to fog or smoke. A ow degree of coverage might be sufficient in norma circumstances but, when a critica event is sensed, a high degree of coverage may be desired in that area [2]. It is desirabe to achieve the required degree of coverage and robustness with the minimum number of active sensor nodes to minimize cost, interference and information redundancy [2], [3]. However, due to the imited range of the wireess communication and obstaces such as buidings, was and trees, the minimum number of sensor nodes required for the coverage may not guarantee the connectivity of the resuting sensor network. Some technoogies require the transmitter and the receiver to be in the ine-of-sight, e.g. Infra-red, Utrasound [4]. Therefore, a sensor node pacement strategy must take connectivity and wireess channe bandwidth imitation (communication bottenecks) into consideration when deciding on where to pace sensor nodes. Since the sensor nodes have imited energy suppies, a sensor node pacement strategy, which minimizes the energy consumed in communication or maximizes the time ti the required degree of coverage is maintained, is highy desirabe. In this paper, we propose severa depoyment strategies which determine the optima pacements of sensor nodes, reay nodes and base stations for guaranteed coverage, connectivity, bandwidth and robustness. The pacement probems for reiabe as we as probabiistic detection modes are formuated as ILPs. The proposed strategies are compared using simuations.

2 II. RELATED WORK Sensor pacement probem is cosey reated to the art gaery probem [5]. Chakrabarty et a. [6] have formuated the sensor pacement probem as an ILP probem. Ray et a. [4] and Chakrabarty et a. [6] have used a framework of identifying codes to determine sensor pacements. Zou and Chakrabarty [7] have proposed a virtua force agorithm to redepoy sensors to new position after initia random pacement. Dasgupta et a. [8] have deveoped a Sensor Pacement and Roe assignment for energy-efficient INformation Gathering (SPRING) agorithm. Dhion and Chakrabarty [1] have proposed agorithms for coverage optimization under the constraints of imprecise detections and terrain properties. Gandham et a. [9] have formuated the base station ocation probem as an ILP. Cheng et a. [10] have studied the impact of pacing reays on the topoogy of biomedica sensor networks. Hou et a. [11] have formuated the probem of energy provisioning and reay pacement into a mixed-integer non-inear programming probem. III. SYSTEM MODEL The foowing assumptions are made about the system. A sensor network consists of sensors 1, reays 1 and base stations, a of these are static. Sensors are responsibe for sensing/monitoring surrounding environment as we as forwarding data they receive from other sensors and reays towards base stations. Sensors generate data packets periodicay and a packets are of the same size. Data aggregation is not performed. Reays are responsibe for forwarding data they receive (from other sensors and reays) towards the base station. Base stations are responsibe for coecting a the data generated by sensors. Base stations have sufficient hardware, sufficient software and constant power suppy. A the base stations are homogeneous. Thus, a sensor can send data packets to any base station. The critica points are ocations in the sensor fied which must be covered by the required number of sensors. The feasibe sites are the ocations where it is possibe to depoy sensors, reays or base stations. Sensors, reays and base stations can communicate with other nodes within their radio transmission range using a MAC protoco. The MAC protoco determines channe capacity i.e. the mean rate at which sensors and reays can transmit data to their neighbors over wireess channes. The predominant traffic in the network is the data traffic from sensors to the base stations. The foowing notations are used in probem formuations. Ψ Set of critica points in the sensor fied. Φ s Set of feasibe sites for sensors and reays. Φ b Set of feasibe sites for base stations. Φ Φ b Φ s. RL max Upper bound on the number of reays that can be depoyed. 1 The word node is omitted to save the space. B max Upper bound on the number of base stations that can be depoyed. 1 if critica point m Ψ is within the coverage δ im = area of a sensor paced at ocation i Φ s ; Q m Critica point m Ψ must be covered by at east Q m 1 sensors. Ω i Φ Set of those feasibe sites (neighbor set) which can be directy reached by a sensor/reay/base station A paced at ocation i Φ with a certain power eve. Set of a directed inks (i, j) where i, j Φ and j Ω i. N max Maximum number of sensors/reays that can be depoyed within the wireess communication range of a sensor/reay excuding itsef. D i E G T ij R ji The rate at which the information is generated at a sensor ocated at i Φ s. Energy suppy avaiabe at a sensor/reay. Energy consumed in generating one unit of information at a sensor. The energy required to transmit one unit of data over the ink (i, j) A. Since base stations have constant power suppy T ij =0if i Φ b. The energy required to receive one unit of data at a sensor/reay/base station ocated at j Φ. R ji =0 if j Φ b. U ij Capacity of the wireess ink (i, j) A. U max Maximum amount of data a sensor/reay can hande (transmit and/or receive) per unit time (caed the node capacity). { 1 if a sensor is paced at ocation i Φs ; x i = { 1 if a reay is paced at ocation i Φs ; y i = { 1 if a base station is paced at ocation k Φb ; z k = f ij The data fow rate over the wireess ink (i, j) A. >0 Lifetime of the network (i.e. time ti the required degree of coverage is maintained). In the foowing sections we assume that sensor readings are reiabe. In section IX we extend our ILP formuations for probabiistic sensing mode. IV. MINIMUM SENSOR PLACEMENT (MSP) Given a set of critica points to be covered, a set of feasibe sites for the sensor pacement, sensing areas of sensors to be paced at feasibe sites, parameter Q m 1, determine the minimum number of sensors and their pacements at the feasibe sites such that each critica point m Ψ is covered by at east Q m sensors. x i δ im x i Q m m Ψ, (1a) x i {0, 1},i Φ s. (1b)

3 If Q m =1 m Ψ then the above probem is an instance of the set cover probem. The set cover probem is a we known NP-compete probem for sub-sets of size 3 or more. V. MINIMUM COST PLACEMENT (MCP) The objective is to achieve desired degree of coverage at the minimum tota cost whie being abe to deiver a the data packets generated by sensors to base stations without exceeding the ink and the node capacities. Let α, β, γ denote the cost of a sensor, a reay and a base station respectivey. Formay, the probem can be stated as foows. ( f ij {k:k Φ b } {j:(k,j) A} f ij + f kj (α x i + β y i)+ {j:(j,k) A} {k:k Φ b } γ z k f ji = D i x i i Φ s, (2a) f jk )= D i x i, (2b) f ji U max x i + U max y i i Φ s, (2c) δ im x i Q m m Ψ, (2d) {k:k Φ b } z k B max, (2e) x i + y i 1 i Φ s, (2f) x i,y i,z k {0, 1},i Φ s,k Φ b, (2g) 0 f ij U ij x i + U ij y i i Φ s, (i, j) A, (2h) 0 f ij U ij x j + U ij y j j Φ s, (i, j) A, (2i) 0 f ij U ij z i i Φ b, (i, j) A, (2j) 0 f ij U ij z j j Φ b, (i, j) A. (2k) The first set of constraints (2a) ensures fow conservation at each sensor and reay. The second constraint (2b) ensures that the base stations receive a the data generated by sensor nodes. The sum of incoming and outgoing fow through a senor/reay shoud not exceed its capacity and this is ensured by the third set of constraints (2c). The fourth set of constraints (2d) ensures that each critica point m Ψ is covered by at east Q m 1 sensors. The fifth constraint (2e) imits the maximum number of base stations that can be depoyed. The sixth set of constraints (2f) ensures that a sensor and a reay, both are not depoyed at the same feasibe site. Note that this condition aso means {i:i Φ x s} i + {i:i Φ y s} i Φ s. The binary variabes (2g) determine the pacement of sensors, reays and base stations. The remaining sets of constraints (2h), (2i), (2j), (2k) 2 ensure that the fow on a wireess ink is nonnegative and the fow does not exceed ink s capacity. Note that the fow on a ink connecting feasibe sites can be positive ony when sensor(s)/reay(s)/base station(s) are depoyed at both (transmitting and receiving) ends of the ink. 2 Some of these constraints are mentioned for carity. VI. MINIMUM ENERGY PLACEMENT (MEP) Energy efficiency is a primary concern in the design of a wireess sensor network. Since a significant amount of energy is consumed in transmitting high voume of data generated by sensors, one objective is to minimize tota energy consumed in communication. (i,j) A (T ij + R ji) f ij a the constraints of the MCP probem pus the foowing constraints. x j + y i RL max, y j N max+ Φ s Φ s (x i+y i) (3a) i Φ s. (3b) The constraint in (3a) imits the number of reays that can be depoyed. The interference caused by neighboring nodes can be reduced by imiting the number of neighbors a sensor/reay can have as shown in (3b). If desired, the base stations can be counted as neighbors by adding the term {j:j Ω i} z j on the eft hand side of the set of constraints in (3b). VII. MAXIMUM LIFETIME PLACEMENT (MLP) In sensor networks the predominant traffic is the data traffic from sensors to base stations. The nodes cose to the base stations forward significanty higher number of packets. Thus nodes cose to base stations depete their energies faster than the nodes that are far away. We define the network ifetime as the time ti the required degree of coverage is maintained. It is desirabe to pace nodes such that the network ifetime is maximized. ( f ij {k:k Φ b } {j:(k,j) A} f ij + Obj = Max f kj {j:(j,k) A} f ji = D i x i i Φ s, (4a) f jk ) = D i x i, f ji U max x i+u max y i i Φ s, T ij f ij + R ji f ji +G D i x i E x i+e y i i Φ s, (4d) δ im x i Q m m Ψ, (4e) x j + (4b) (4c) y j N max + Φ s Φ s (x i + y i) i Φ s, (4f) {k:k Φ b } z k B max, y i RL max, (4g) (4h) x i + y i 1 i Φ s, (4i)

4 x i,y i,z k {0, 1},i Φ s,k Φ b, (4j) 0 f ij U ij x i + U ij y i i Φ s, (i, j) A, (4k) 0 f ij U ij x j + U ij y j j Φ s, (i, j) A, (4) 0 f ij U ij z i i Φ b, (i, j) A, (4m) 1 (1 p im x i) O m m Ψ, (6a) x i {0, 1},i Φ s. (6b) 0 f ij U ij z j j Φ b, (i, j) A, (4n) 0 <. (4o) The constraints in the optimization mode are noninear because they invove products of variabes. In order to inearize the constraints, we divide both sides of constraints invoving variabe by >0. We introduce a new variabe to repace 1/. This wi inearize a the constraints except (4d). To inearize (4d), we repace E x i + E y i with E. This transformation is vaid because at east one of the binary variabes x i and y i must be 0 (see 4i). When both x i and y i are 0 then set of constraints in (4d) are equivaent to the set of constraints in (4k) and in (4). Maximizing the ifetime () in the above probem is equivaent to minimizing in the transformed ILP probem. VIII. MAXIMUM UTILIZATION PLACEMENT (MUP) In order to improve the ifetime of the network MLP paces many nodes cose to the base station(s). This may resut in poor utiization of network resources and increased cost. It is highy desirabe to maximize the network ifetime whie depoying a reasonabe number of nodes. The network utiization (NU) is defined as (xi+yi). Obj = Max = Min (xi + yi) (xi + yi) a the constraints of MLP probem. In order to inearize constraints we divide both sides of constraints invoving variabe by > 0. We introduce a new variabe to repace 1/. Now (5) invoves product of a binary variabe (x i ) with a continuous variabe ( ). It can be inearized using the resuts of [12] by (a) repacing the product term with a new continuous variabe v i, and (b) adding additiona set of constraints (i) v i L max x i, (ii) v i and (iii) vi + L max x i L max where L max is an upper bound for. Simiary, inearize the product of binary variabe yi with continuous variabe. IX. PLACEMENT PROBLEM FOR UNRELIABLE/PROBABILISTIC DETECTION MODEL A probabiistic sensing/detection mode is usefu when the sensor readings are unreiabe. Let p im denote the probabiity that an event occurring at a critica point m Ψ is sensed/detected by a sensor paced at the feasibe site i Φ s. The goa is to pace the minimum number of sensors in the sensor fied such that an event occurring at any critica point m Ψ is detected with a probabiity of at east O m.dueto the preferentia coverage requirements, different critica points can have different threshod detection probabiities. x i (5) Ceary, the probem is non-inear because the first set of constraints in (6a) invoves product of variabes. Simpifying the constraint and taking og of both sides of the constraint wi resut into the foowing set of inequaity constraint. og(1 p im x i) og(1 O m) m Ψ Note that if x i is zero then og(1 p im x i ) is zero and if x i is one then og(1 p im x i ) is og(1 p im ). The above set of non-inear constraint can be converted into the foowing set of inear constraint. og(1 p im) x i og(1 O m) m Ψ The ILP formuations studied in the previous sections can be easiy extended for a probabiistic detection mode by transforming non-inear constraints into inear constraints as shown above. X. COMMUNICATION MODEL In adaptive transmission power mode, the transmitter is capabe of adjusting its signa power eve such that the energy consumed in transmission is minimized whie maintaining acceptabe signa to noise ratio at the receiver. In constant transmission power mode, the transmitter transmits with a constant signa power. The energy consumed in transmission incudes the energy consumed in interna processing (distance independent) and the energy consumed in ampifying the signa to achieve acceptabe signa to noise ratio at a receiver (distance dependent). In adaptive transmission power mode used in simuations, energy consumed in transmitting a bit from node i to j is given by [13] T ij = ( Dist(i, j) 3 ) 10 9 Energy unit/bit In the constant transmission power mode used in simuations, the term Dist(i, j) 3 in the above equation is repaced by a constant T ransmission Range 3 where T ransmission Range is fix (static) maximum transmission range of a node. The energy consumed in receiving/generating a bit is the same in both modes which is given by R = G = Energy unit/bit XI. SIMULATION RESULTS A sensor fied of size 100 m 100 m is divided into a uniform grid of The grid points represent the critica points to be covered and the feasibe sites for sensors as we as reays. Four feasibe sites for base stations are ocated as

5 100 m 4 m 100 m Feasibe site for base station Fig. 1. Sensor Fied TABLE I COMPARISON OF NUMBER OF SENSOR NODES REQUIRED Q min MSP MCP MEP MEP MLP MLP MUP MUP (A) (C) (A) (C) (A) (C) shown in Fig.1. We have simuated a 2D sensor fied with circuar transmission and sensing areas under reiabe sensing mode. The transmission range of each sensor as we as reay is 20.0 meters. The sensing radius of each sensor is 8.0 meters. The node capacity of each sensor as we as reay is 40.0 Kbps 3, and the ink capacity of each wireess ink is 10.0 Kbps. The rate of information generation is 512 bits/second for each sensor. The costs of sensors, reays and base stations are 10 units, 8 units and 500 units respectivey. Each sensor/reay is equipped with initia energy of Energy unit. CPLEX R optimizer [14] is used to sove ILP formuations. The character C inside the parenthesis (e.g. MEP(C)) denotes that the formuation uses constant transmission power mode and the character A inside the parenthesis (e.g. MLP(A)) denotes that the formuation uses the adaptive transmission power mode. The first set of experiments (see Tabe I) compares the number of sensors required by each scheme for various coverage requirements (Q min ). To restrict the comparison to sensors, no reays are depoyed (RL max =0and N max = ). The upper bound on number of base stations that can be depoyed (B max ) is set to 4 and Q m = Q min m Ψ. Since the cost of a base station is significanty higher than the cost of a sensor, MCP soutions require ony 1 base station. On the other hand MEP, MLP and MUP soutions require 4 base stations because depoying more base stations improves the objective function vaues. Note that the vaues of MLP and MUP soutions reported in Tabe I may not be optima 4. Resuts in Tabe. I indicate that the minimum number of sensors required for coverage (MSP) increases ineary with Q min. Note that the graph induced by the wireess connectivity of the sensors (depoyed by MSP) is connected and the resuting network has sufficient bandwidth. Therefore, the number of sensors required by MSP and MCP are the same. In MEP, MLP and MUP, pacing additiona sensors (in addition to the minimum sensors required for the coverage, 3 As per the IEEE standard, the raw data rate of a sensor can be 20, 40 or 250 Kb/s. 4 The processes were terminated after 30 hrs of computation. connectivity and bandwidth) can save the energy and improve the objective function. However, additiona sensors generate data which consume energy. In MEP, pacing additiona sensors is advantageous if the energy saved in transmission is more than energy consumed by additiona data in reaching the base station. The nodes cose to base stations are overoaded with forwarding tasks and their ifetimes are significanty shorter than the ifetimes of other nodes. MLP and MUP attempt to improve the ifetimes of these nodes by pacing many nodes cose to base stations. In MLP, additiona sensors can be paced if the energy consumed by additiona data does not decrease the network ifetime. Therefore, MLP formuations depoy more nodes than other formuations. In MUP, additiona sensors can be paced if the increase in the network ifetime due to additiona sensors is at east proportiona to the number of additiona sensors. The next set of experiments compare NU of the networks synthesized in the previous experiments. In these experiments, given the sensor positions (determined by the first set of experiments in Tabe I) and base station positions (4 base stations are ocated as shown in Fig.1), the objective is to maximize the time ti the desired coverage is maintained. Dividing the network ifetime by the number of nodes depoyed gives NU. Fig. 2 shows the NU of the networks under constant and adaptive transmission power modes. The NU of the networks synthesized by MSP and MCP are the same because the sensor positions determined by MSP and MCP in the first set of experiments happen to be identica. Since MSP and MCP do not take energy efficiency into consideration, their NU vaues are the east. Networks synthesized by MUP have the highest NU foowed by the MLP. Ceary, a judicious pacement of sensors in the sensor fied can significanty increase the network ifetime. The NU of networks in the adaptive transmission power modes are roughy doube the NU in constant transmission power modes. Ceary, equipping sensors with power contro transmitters can significanty increase the NU. To study the effect of reay pacements on objective function vaues, we iterativey soved the MEP probems (increasing the vaues of RL max by 1 in each iteration) unti the objective function vaues stop improving. MUP probems are soved by setting RL max =. The number of sensor and reay nodes required are shown in Tabe II. The saving vaues show the percentage improvements in the objective function vaues due to the pacement of reay nodes with respect to the objective function vaues when reay nodes are not depoyed. In the adaptive transmission power mode the muti-hop communication is more energy efficient than direct transmission. However, at short distances, the energy consumed in interna processing and receiving the data dominates which makes direct transmission more efficient than muti-hop transmission. Therefore, as the node density increases (average distance between nodes decreases), the number of additiona sensors and number of reays decrease (See Tabe. I and II). The saving percentage drops with the increase in node density.

6 Network Utiization (C) (A) Network Utiization (C) (A) Network Utiization (C) (A) 0 MSP MCP MEP MLP MUP Depoyment Strategy 0 MSP MCP MEP MLP MUP Depoyment Strategy 0 MSP MCP MEP MLP MUP Depoyment Strategy (a) Q min =1 (b) Q min =2 (c) Q min =3 Fig. 2. Depoyment Strategies v/s Network Utiization TABLE II COMPARISON OF NUMBER OF RELAY NODES REQUIRED Q min MEP(A) MEP(C) MUP(A) MUP(C) Sensors Reays Saving Sensors Reays Saving Sensors Reays Saving Sensors Reays Saving % % % % % % % % % % % % In the constant power mode energy can be saved by minimizing the number of hops the data packets have to traverse to reach a base station. Since the minimum hop path can be abridged at reativey few paces, the number of reays required and the percentage saving vaues are smaer in MEP(C) than in MEP(A). Resuts in Tabe. II indicate that reay pacements improve the NU of networks synthesized by MUP significanty. Overa, MUP strategy is better than other depoyment strategies. XII. CONCLUSION AND FUTURE WORK In this paper, we investigated the joint probem of determining sensor, reay and base station pacements, and finding bandwidth-constrained energy-efficient routes whie ensuring desired eve of coverage, connectivity and robustness. The practicaity, effectiveness and performance of proposed strategies have been studied through simuations. Our techniques of determining strategic pacements of nodes are usefu when a modest number of nodes are to be depoyed and reasonabe terrain information is avaiabe. For exampe, in many indoor appications (invoving factories or industria units or airports) where foor pans are avaiabe, our approaches are readiy appicabe. We recognize that ILP formuations are NP-hard to sove. Since pacement probems are off-ine probems, more computationa time and power can be devoted for their soutions. Hence, modest size of probems can be soved to optimaity and good feasibe soutions can be obtained for arge size of probems using techniques such as LP Reaxation or Lagrangian Reaxation [15]. Moreover, the soutions of ILP formuations can be usefu in benchmarking an approximation agorithm or a heuristic. It is interesting to deveop poynomia time heuristic agorithms for arge size of probems. REFERENCES [1] S. S. Dhion and K. Chakrabarty, Sensor pacement for effective coverage and surveiance in distributed sensor networks, in Proceedings of IEEE WCNC, New Oreans, LA, March 2003, pp [2] X. Wang, G. Xing, Y. Zhang, C. Lu, R. Pess, and C. Gi, Integrated coverage and connectivity configuration in wireess sensor networks, in Proceedings of ACM SenSys 2003, Los Angees, Caifornia, Nov. 2003, pp [3] H. Zhang and J. C. Hou, Maintaining sensing coverage and connectivity in arge sensor networks, University of Iinois at Urbana-Champaign, IL, Tech. Rep. UIUCDCS-R , June [4] S. Ray, R. Ungrangsi, F. D. Peegrini, A. Trachtenberg, and D. Starobinski, Robust ocation detection in emergency sensor networks, in Proceedings of IEEE INFOCOM 03, San Francisco, Apri 2003, pp [5] J.O Rourke, Art Gaery Theorems and Agorithms. Oxford University Press, New York, NY, [6] K. Chakrabarty, S. S. Iyengar, H. Qi, and E. Cho, Grid coverage for surveiance and target ocation in distributed sensor networks, IEEE Transactions on Computers, vo. 51, no. 12, pp , Dec [7] Y. Zou and K. Chakrabarty, Sensor depoyment and target ocaization based on virtua forces, in Proceedings of IEEE INFOCOM 03, San Francisco, Apr. 2003, pp [8] K. Dasgupta, M. Kukreja, and K. Kapakis, Topoogy aware pacement and roe assignment for energy-efficient information gathering in sensor networks, in Proceedings of the 8th IEEE Symposium on Computers and Communications, Kemer-Antaya, Turkey, June [9] S. R. Gandham, M. Dawande, R. Prakash, and S. Venkatesan, Energy efficient schemes for wireess sensor networks with mutipe mobie base stations, in Proceedings of IEEE GLOBECOM, San Francisco, CA, Dec [10] X. Cheng, D.-Z. Du, L. Wang, and B. Xu, Reay sensor pacement in wireess sensor networks, University of Minnesota Twin Cities, Minneapois, MN, Tech. Rep , [11] Y. Hou, Y. Shi, and H. Sherai, On energy provisioning and reaying node pacement for wireess sensor networks, The Bradey Dept. of ECE, Virginia Tech, Backsburg, VA, Tech. Rep., [12] H. P. Wiiams, Mode Buiding in Mathematica Programming. Chichester, Engand; New York: Wiey, [13] W. Heinzeman, A. Chandrakasan, and H. Baakrishnan, Energyefficient communication protoco for wireess micro-sensor networks, in Proc. of the 33rd HICSS, Maui, Hawaii, Jan. 2000, pp [14] Using the CPLEX caabe ibrary, ILOG Inc., Incine Viage, NV, [15] R. Ahuja, T. Magnanti, and J. Orin, Network Fows. New Jersey: Prentice Ha, 1993.