Data Gathering in Wireless Sensor Networks with Multiple Mobile Collectors and SDMA Technique Sensor Networks

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1 This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 0 proceedings. Data Gathering in Wireless Sensor Networs with Multiple Mobile Collectors and SDMA Technique Sensor Networs Miao Zhao and Yuanyuan Yang Department of Electrical and Computer Engineering, Stony Broo University, Stony Broo, NY, USA Abstract In this paper, we consider data gathering in wireless sensor networs (WSNs) by utilizing multiple mobile collectors and spatialdivision multiple access (SDMA) technique. In particular, multiple mobile collectors, for convenience, called SenCars in this paper, are deployed in a WSN and wor independently and simultaneously to collect data. The sensing field is divided into several non-overlapping regions, each having a SenCar. Each SenCar gathers data from sensors in the region while traversing their transmission ranges. Sensors directly send data to their associated SenCars without relay in order to achieve uniform energy consumption. We also consider exploiting SDMA technique by equipping each SenCar with two antennas. With the support of SDMA, two distinct compatible sensors in the same region can successfully mae concurrent data uploading to their associated SenCar. Intuitively, if each SenCar can always simultaneously communicate with two compatible sensors, the data uploading time in each region can be cut into half in the ideal case. We focus on the problem of minimizing maximum data gathering time among different regions, which consists of two parts: the data uploading time of the sensors in this region and the moving time of the associated SenCar on a tour. We refer to this problem as data gathering with multiple mobile collectors and SDMA, or DG-MS for short, and formalize it into an integer linear program. We then propose a region-division and tour-planning algorithm to provide a practically good solution to the problem. Simulation results demonstrate that the proposed scheme significantly outperforms other non-sdma or single mobile collector schemes by efficiently shortening and balancing the data gathering time among different regions. I. INTRODUCTION Wireless sensor networs (WSN) have recently emerged as a new information-gathering paradigm with a diversity of applications. A WSN is typically composed of low-cost, low-power, densely-deployed and randomly distributed sensors []. Besides monitoring the environment by taing spatial or temporal measurements, sensors are also responsible for routing sensing data bac to an inside or outside sin [], []. This may lead to nonuniform energy consumption among sensors due to the fact that sensors near the sin need to relay more pacets from sensors far away from the sin. As a result, their energy will be depleted much faster than others. Thus, how to efficiently aggregate the information from the scattered sensors, generally referred to as data gathering problem, is an important and challenging issue as it largely determines networ lifetime. In recent studies, mobility has received much attention as an effective solution to alleviate the aforementioned nonuniformity problem, see, for example, []-[]. Different from traditional routing, schemes involving mobility use a special type of mobile nodes for facilitating connectivity among static sensors. These mobile nodes can move arbitrarily close to sensors and collect data from them via short-range communications. In these schemes, routing burden has been taen over by mobile nodes and energy can be greatly saved at sensors. A recent wor in [] employed a single mobile collector, called SenCar, in WSNs and focused on optimizing the data gathering tour. However, in a large-scale WSN, utilizing only a single SenCar may lead to a long data gathering cycle and data buffer Research supported by NSF grant numbers ECCS-00 and ECS- 0 and ARO grant number WNF overflow at sensors. To deal with these problems, in this paper, we consider deploying multiple SenCars, which wor simultaneously in a sensing field to collect data from sensors. In our scheme, the sensing field is divided into several nonoverlapping regions, and each SenCar is responsible for gathering sensing data in a region. The mobility control on the Sen- Cars maes it possible for each sensor to directly upload data to the SenCar in its region via single-hop transmission when the SenCar traverses its transmission range. Besides deploying multiple SenCars, we also consider utilizing the spatial-division multiple access (SDMA) technique in data gathering by mounting multiple antennas on each Sen- Car. SDMA is a multiuser multiple-input and multiple-output (MIMO) technique, specifically with multiple receive antennas []. It enables multiple senders to simultaneously transmit data to a receiver. In WSNs, sensors act as source nodes that generate data and data pacets are converged to a data collector from multiple sensors. Such many-to-one traffic pattern perfectly meets the requirements of applying SDMA technique. To elaborate, if each SenCar is equipped with two antennas and each sensor has a single antenna, two compatible sensors in the same region can mae concurrent data uploading to the associated SenCar by utilizing SDMA technique. Thus, data uploading time in each region would be cut into half in the ideal case and the data gathering cycle can be dramatically shortened. Hence, applying SDMA is well suitable to data gathering in WSNs, achieving spatial reuse to further leverage the effect of employing multiple SenCars. In this paper, we focus on data gathering with multiple mobile collectors and SDMA technique (DG-MS), which involves a joint design of mobility control on SenCars and the utilization of SDMA in sensor data transmissions. For each SenCar in a region, the data gathering time mainly consists of two parts: data uploading time of sensors in this region and moving time of the SenCar on the tour. The objective of our study is to minimize maximum data gathering time among different regions. In particular, we are interested in following questions: how to divide the sensing field into a certain number of regions to balance data gathering time among different regions, how to better enjoy the benefit of SDMA to shorten data uploading time in each region, and how to move each SenCar on a short tour in each region. We will examine these issues and find an effective solution to the DG-MS problem. The main contributions of our wor in this paper can be summarized as follows. () Introduce a joint design of multiple mobile collectors and SDMA technique for data gathering in WSNs and formalize it into the DG-MS problem. () Formulate the DG-MS problem into an integer linear program (ILP) and propose a heuristic region-division and tourplanning algorithm to provide a practically good solution. () The proposed scheme can radically resolve the problem of non- ----//$.00 0 IEEE

2 This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 0 proceedings. uniform energy consumption among sensors by applying mobility control to achieve single-hop transmission between each sensor and its associated SenCar. () DG-MS wors well not only in a connected WSN but also in a disconnected networ since SenCars play as virtual bridges to connect separated regions. () The proposed scheme achieves much shorter data gathering time with respect to other mobile data gathering schemes due to the concurrent use of multiple SenCars and simultaneous data uploading among sensors by utilizing SDMA. () A commercially appealing feature of the scheme is that sensors can be ept as simple as before and all intelligent operations are performed by SenCars. II. DATA GATHERING WITH MULTIPLE MOBILE COLLECTORS AND SDMA TECHNIQUE In this section, we describe the DG-MS scheme in detail. For the sae of clarity, we first define some terms that will be used in the rest of this paper. The sensing field is divided into several non-overlapping regions, each having a SenCar. We assume that each SenCar can forward the gathered data to one of the nearby SenCars when they move close enough, such that data can be eventually forwarded to the SenCar that will visit the static data sin. While SenCars are moving through the sensing field, they can only stop at certain positions to poll nearby sensors to gather data pacets. We define the positions that SenCars can stop for polling as polling points. When a SenCar moves to a polling point, it polls nearby sensors with the same transmission power as sensors, such that sensors that receive the polling messages can upload pacets to their associated SenCar within a single hop. The dis-lie shaped area centered at a polling point with the radius equal to the transmission range of a sensor is defined as the coverage area of a polling point. All the sensors in the coverage area of a polling point form the neighbor set of this polling point. Though a sensor may be located in the coverage areas of multiple polling points, it is associated with only one polling point for data uploading. In other words, the associated sensors of a polling point are not necessarily all the sensors in its neighbor set. Each SenCar is equipped with two antennas, which means that at each time slot, up to two sensors can send data simultaneously to a SenCar by utilizing SDMA technique. We say any two sensors in the coverage area of the same polling point to be compatible if a SenCar arriving at this polling point can successfully decode the multiplexing signals concurrently transmitted from these two sensors. Note that due to limited transmit power of sensors, not any two sensors in the same neighbor set of a polling point are compatible. For detailed discussions at physical layer on utilizing SDMA for concurrent data uploading, interested readers may refer to []. Different association patterns of sensors with polling points correspond to different compatibility relationship among them because the channel state information varies with the association pattern. Two compatible sensors associated with the same polling point are qualified to be a compatible pair to upload data simultaneously when the SenCar arrives at this polling point. There are two main operations in a data gathering tour: the SenCar s movement which specifies SenCar s moving tour from one polling point to another, and the operation of data uploading which specifies how sensors interact with the SenCar. Thus, Sin SenCar Sensing Field SenCar a g b f d c e a g Time slots a c f Scheduler e b d g Sensors Polling points Selected polling points SenCar moving path Compatible Compatible pair Coverage area of centered polling point Fig.. Two SenCars gather data simultaneously and compatible sensors can concurrently upload data to SenCars with the support of SDMA technique. the data gathering time in a region is the sum of the moving time of the SenCar and the data uploading time of sensors in this region. A SenCar arriving at a polling point in its region collects data from associated sensors and then moves straightly to the next polling point in the tour. Thus, the moving tour of a SenCar consists of a number of polling points in its region and the straight line segments connecting them. Note that a SenCar does not need to visit every polling point in its region. However, the polling points on the tour should cover all the sensors in this region. We call these polling points selected polling points. Fig. gives an example of such data gathering, in which there are two non-overlapping regions and polling points over the field. Two SenCars wor simultaneously and each of them moves along the selected polling points in its region. When a SenCar arrives at a selected polling point, sensors associated with this polling point are scheduled to communicate with the SenCar. Two sensors in a compatible pair can upload data simultaneously in a time slot, while an isolated sensor (i.e., a sensor not in any compatible pair) will send data to the SenCar separately. The compatibility relationship among sensors is denoted by the lins between them. Sensors a to g are associated with the same polling point. In a possible schedule, a and b, c and d, and f and g are three compatible pairs such that only time slots are needed in total for every sensor to upload a pacet. We assume that sensors can turn to sleep mode for power saving after data uploading. Thus, finding optimal data gathering that shortens data gathering latency is equivalent to minimizing maximum data gathering time among different regions. We refer to this problem as DG-MS problem. It consists of several tightly coupled subproblems: determining selected polling points and sensor association patterns, finding compatible pairs among sensors, assigning selected polling points and their associated sensors to different regions, and determining the order for each SenCar to visit selected polling points in its region. III. DG-MS PROBLEM AND HEURISTIC ALGORITHM We now formally describe the DG-MS problem. Given a set of sensors S = {,,...,N s }, a set of polling points P = {,,...,N p }, and a set of SenCars K = {,,...,N }, find: () a set of subsets of P, denoted by P, P,...,P N, which represent the selected polling points in different regions that satisfy P P... P =Φand P P... P = P P, () a set of subsets of S, denoted by S, S,...,S N, b f c d e

3 This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 0 proceedings. 0 (a) Step : Find the maximum compatible pairs among sensors (b) Step : Determine the selected polling points and association pattern of sensors (c) Step : Find the minimum spanning tree among the selected polling points (d) Step : Divide the selected polling points into two parts based on their weights Region..0 Region Original neighbor set of each polling point: : {} : {,,} : {} : {,} : {,} : {,} : {,,,,} : {,,,} : {,,} : {,} : {,,} : {,,,} : {,,} : {,,,} : {,} : {,,0} : {,,,0} : {,} : {,,,} 0: {,} : {0} : {0} : { } : {} : {} Updated neighbor set of each polling point: : {} : {} : { } : {,} : {,} : { } : {,,,} : {,} : { } : { } : { } : {,} : {,,} : { } : {,} : {,0 } : { } : {} : {,} 0: { } (e) Original neighbor set of each polling point (f) Updated neighbor sets based on maximum compatible pairs. Fig.. Illustration of the region-division and tour-planning (RDTP) algorithm. which represent the sensors assigned in different regions that satisfy S S... S =Φand S S... S = S, () the compatible pairs among the sensors in S i, i =,,,N, () the sequence by which each SenCar visits the selected polling points in P i, i =,,,N, such that maximum data gathering time among a total of N regions can be minimized. We can formulate the DG-MS problem as an ILP as shown in the Appendix. However, the complexity of the ILP solution is generally high, which may not be suitable for large-scale sensor networs. Therefore, we propose a heuristic region-division and tour-planning algorithm (RDTP) to give a practically good solution to the problem. For clarity, we describe algorithm details in four steps as follows. To better understand the algorithm, we also provide an example in Fig., where the sensing field will be divided into two regions and there are 0 sensors in S (plotted as labeled dots) and polling points in P (plotted as small numbered circles). The first step of our algorithm deals with how to fully extract the benefits of SDMA. To collect data as fast as possible, sensors should be organized into the maximum number of compatible pairs for data uploading. This can be formalized as a matching problem in a compatibility graph, where each vertex represents a sensor, and two vertices are adjacent to each other if the two sensors are compatible. In order to eep the graph simple, there is only one edge between two vertices even if two corresponding sensors are compatible in the coverage area of multiple polling points. A group of compatible pairs corresponds to a set of vertex-disjoint edges in the graph. A set of vertex-disjoint edges is defined as a matching in graph theory []. Therefore, finding a maximum number of compatible pairs among all the sensors, which is the first step of the RDTP algorithm, is equivalent to finding a maximum matching in the compatibility graph. Maximum matching can be found by some existing algorithms, such as Edmonds Blossom Algorithm []. Fig. (a) shows the compatibility graph of the example, where there is an edge between any two sensors if they are compatible in the coverage area of some polling points. We can see that the number of maximum compatible pairs among 0 sensors is, which are plotted by the bold lines as the maximum matching in the corresponding compatibility graph. : { } : { } : { } : { } : { } Selected polling points and their associated sensors: : {,} : {} : {,,} : {,} : {,} : {,,,} : {,0 } : {,} : {,} (g) Selected polling points. The second step of the RDTP algorithm is to determine selected polling points and association patterns of sensors based on maximum compatible pairs. Since we want the selected polling points to result in short moving tours for SenCars, it is preferred to find a minimum number of polling points that can achieve maximum number of compatible pairs. Since it is required that any two sensors in a compatible pair are associated with the same polling point, we first update the neighbor set of each polling point based on maximum compatible pairs obtained in the first step of the algorithm. In particular, in the case that the two sensors of a compatible pair are not compatible in a particular neighbor set, or one sensor of a compatible pair is in a neighbor set while the other is not, the neighbor sets need to be updated by deleting such sensors from them, in order for any two sensors of a compatible pair to be treated as a single element in all the neighbor sets. Fig. (e)-(f) list the original and updated neighbor sets in the example. Now, finding the minimum number of selected polling points is reduced to finding the minimum number of updated neighbor sets of polling points in P such that the selected neighbor sets contain all the sensors, which is the well-nown Minimum Set Cover (MSC) problem. We can utilize the greedy algorithm for MSC problem [] to find the selected neighbor sets, whose corresponding polling points are the selected polling points. We use P to denote the set of all selected polling points defined in the DS- MS problem. Since one of the SenCars needs to visit the static data sin, P should include the sin. Without loss of generality, we assume that the data sin is located at the position of polling point in our example. Fig. (b) shows that polling points,,,,,,, and are selected into P by the greedy algorithm. Also, the updated neighbor sets of these selected polling points explicitly indicate the association pattern of sensors. Fig. (g) lists the finally selected polling points and their associated sensors in the example. So far we have found compatible pairs among sensors and selected polling points. In the next two steps, we determine the region that each SenCar wors on by dividing the selected polling points into N parts. In the third step of the algorithm, we organize the selected polling points into a tree structure and assign each of them a

4 This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 0 proceedings. TABLE PROCEDURE OF DIVIDING SELECTED POLLING POINTS AND THEIR ASSOCIATED SENSORS INTO N PARTS. Procedure Division (T, N ) For all v on T do Calculate w(v) according to Eq. (); end for m N ; While m> W T w(r T ); v the farthest leaf vertex on T with minimum w(v); While w(v) <W T /m v PA(v); end while Build subtree t rooted at v; Add all vertices on t to P m; Add corresponding associated sensors to S m ; Remove subtree t from T ; Update w(v) for each v on the remaining T ; m m ; end while Assign remaining selected polling points on T to P ; Assign corresponding associated sensors to S ; Find approximate shortest tours that visit selected polling points in P, P,...,P N, respectively. weight, based on which the operation of region division can be done in the next step. Specifically, we find the minimum spanning tree T (V,E) among the selected polling points in P and treat the data sin as the root of the tree, denoted by r T. For example, in Fig. (c), the minimum spanning tree among selected polling points is shown by bold lines among them with polling point as the root. Let w(v) represent the weight for selected polling point v. Next, we calculate the weight for each selected polling point in P according to the following criteria. w(v) = α( f u M u )+ βl e, v P () u V (subt(v)) e E(subT(v)) where α and β are constant coefficients, which represent the time cost for a sensor to upload its data and for a SenCar to move a unit distance, respectively, subt(v) denotes the subtree of T rooted at v, V ( ) and E( ) represent the vertices and edges on the tree, f u denotes the set of associated sensors to selected polling point u, M u denotes the compatible pairs associated with u, and L e is the length of edge e. The first part of w(v) represents the sum of data uploading time at selected polling points on the subtree rooted at v, while the second part indicates the sum of moving time along edges on the subtree. Therefore, the value of w(v) implicitly reveals the expected data gathering time if a SenCar carries out a tour for data collection by visiting selected polling points on the subtree rooted at v. Clearly, the root has the largest weight among all the vertices on T and the weight is also considered as the total weight of T, denoted by W T. In the example, the weight for each selected polling point is labeled as shown in Fig. (c) with α and β set to. and., respectively. W T is equal to., which is also the weight of the root on T (i.e., polling point ). Now, the remaining issue is how to divide selected polling points and their associated sensors into a certain number of parts, each for a region, so as to balance data gathering time among these regions. Suppose there are a total of N available SenCars, which means that the selected polling points will be divided into N parts. In the fourth step of the algorithm, TABLE REGION-DIVISION AND TOUR-PLANNING (RDTP) ALGORITHM Inputs: Set S containing N s sensors Set P containing N p polling points Set K containing N SenCars Neighbor family set F = {f i i P},wheref i is the neighbor set of polling point i Distance matrix D = {d i,j } P P,whered i,j is the segment length between polling points i and j Compatibility relationship matrix C(P) ={c m,n,i } S S P Outputs: AsetP of selected polling points with P = P P... P A set of subsets of P, P, P,...,P N, each representing the selected polling points in a region A set of subsets of S, S, S,...,S N, each representing the sensors associated in a region Moving tour of each SenCar RDTP algorithm: Step : Find maximum compatible pairs M based on C(P); Step : Update F according to M; Find minimum set cover of F by the greedy algorithm; Add corresponding polling points of selected neighbor sets to P ; Step : Find minimum spanning tree T among selected polling points in P ; Calculate the weight of each vertex on T ; Step : Divide P into P, P,...,P N and divide S into S, S,..., S N by iteratively finding a subtree of T ; Find approximate shortest tours that visit selected polling points in P, P,...,P N, respectively. we focus on solving this problem. The basic idea is to partition minimum spanning tree T into N parts, by iteratively finding a subtree t based on the weight of each vertex on T and pruning t from T. In order to balance data gathering tass in different regions, it is desirable for these subtrees to have a similar total weight. We use a variable m to represent the number of currently available SenCars in each stage of the algorithm and initialize it to N. To build a subtree in each iteration, first find a farthest leaf vertex v on T with the minimum weight. If w(v) < W T /m, find its parent vertex on T, denoted by PA(v), and let v = PA(v). Chec its weight and repeat this up-tracing process until w(v) W T /m. Record this vertex v and consider it as the root of subtree t. All the vertices on t will belong to the same part and are removed from T, which means that the corresponding selected polling points on t will be assigned to the same region (or a SenCar). After that, update T, W T, m and w(v) for each vertex on the updated T and repeat this operation. When there is only one available SenCar left, all the remaining selected polling points and their associated sensors are assigned to this SenCar and the procedure terminates. To better understand it, let us tae a loo at the example in Fig (c). Polling point, as a farthest leaf vertex on T, has the minimum weight equal to. in the first iteration. Thus, v =. Since m = N =and W T =., w() <W T /m =.. Consequently, chec the weight of polling point, which is the parent vertex of polling point on T. Since w() is still less than W T /m, this up-tracing procedure continues until polling point is found with the weight larger than W T /m. Thus, polling point is considered as the root of subtree t. All the vertices on t, which are polling points,,, and, are assigned to P and their associated sensors are correspondingly assigned to S. After removing these selected polling points from T, update w(v) for each vertex on the remaining T, which are listed in Fig. (d). Also, W T is

5 This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 0 proceedings. The average data gathering time T (s) N 000 p =, D=0m, r=0m, v m =0.m/s, v d =0bps SDMA+two SenCars (RDTP) non SDMA+two SenCars 00 SDMA+single SenCar non SDMA+single SenCar Number of sensors in the field N s The average data gathering time T (s) N p =, N =, D=0m, r=0m Case I: v m =m/s, v d =0bps Case II: v m =0.m/s, v d =bps Number of sensors in the field N s The average data gathering time T (s) N 00 s =0, N p =, r=0m, v m =0.m/s, v d =0bps 00 N = N = 00 N = Side length of sensing field D (m) Fig.. (a) Data gathering time as a function of N s. (b) Data gathering time of RDTP under different settings of v m and v d. (c) Data gathering time of RDTP under different settings of N. recalculated with the result equal to 0. and m is updated to. Since only one SenCar is left (i.e., m =), all the remaining selected polling points and their associated sensors are simply assigned to P and S, respectively. Finally, since selected polling points in each region are determined, the moving tour of a SenCar is to visit each selected polling point in its region exactly once. Then finding the shortest tour of a SenCar is equivalent to the Traveling Salesman Problem (TSP) []. We can use the approximate or heuristic algorithms for the TSP problem to determine the moving tour for each SenCar. As a result, the moving tours of the two SenCars in the example are as given in Fig. (d): SenCar : and SenCar :. The details of the procedure in step are shown in Table. Finally, we summarize the RDTP algorithm in Table. IV. PERFORMANCE EVALUATIONS In this section, we present the simulation results to evaluate the performance of RDTP algorithm for the DG-MS problem and compare it with other three mobile data gathering schemes. In the simulations, we consider a D D square sensing field, where a total of N s sensors are randomly distributed, and N p polling points are located at the intersections of grids with each polling point having the same distance to its adjacent polling points in horizontal and vertical directions. The radius of the coverage area of each polling point r =0m, which is also the transmission range of each sensor. A total of N SenCars are available. We assume that the size of sensing data q in each sensor is Mb, the effective data uploading rate of each sensor v d =0bps, and the moving velocity of each SenCar v m = 0.m/s, if not stated otherwise. Fig. (a) plots the data gathering time by different schemes when N s varies from to 0, where N p = and D = 0m. We compare four mobile schemes: without SDMA and with a single SenCar (non-sdma+single-sencar), with SDMA and with a single SenCar (SDMA+single-SenCar), without SDMA and with two SenCars (non-sdma+two-sencars), and with SDMA and with two SenCars (SDMA+two-SenCars, which is also the RDTP scheme). When multiple SenCars are used, data gathering time refers to the maximum time among different regions. It can be seen that data gathering time of all the schemes increases as N s increases. However, RDTP always outperforms other schemes due to the concurrent use of multiple collectors and simultaneous data uploading among sensors with the support of SDMA technique. For instance, it achieves % time saving compared with non-sdma+single-sencar scheme when N s = 0. Shorter data gathering time leads to longer networ lifetime since sensors can turn to power-saving mode once the data gathering in their region is done. We also notice that the advantages of RDTP over other schemes become more evident when the networ becomes denser with more sensors. This is reasonable because more sensors would provide more opportunities to utilize SDMA for concurrent data uploading. Fig. (b) shows that data gathering time of RDTP varies with N s under different settings of v m and v d, where D = 0m. There are polling points and two available SenCars. We consider two configurations of (v m,v d ), which are (v m =m/s, v d =0bps) and (v m =0.m/s, v d = bps), to represent two different cases. It is noticed that when N s is small, the moving velocity of SenCar v m has a greater impact on data gathering time than v d. Higher moving velocity, such as v m =m/s in Case I, results in shorter data gathering time even with a smaller v d than the other case. It is reasonable since the moving time of each SenCar is dominant when sensors are sparsely scattered. On the contrary, when N s is large, the impact of v d on data gathering time overwhelms that of v m. For example, when N s increases to more than 0, the data gathering time for Case II, which has higher effective data uploading rate at v d = bps, is smaller than that of Case I. This is because that more sensors mae data uploading time dominant in each region and they provide more opportunities to extract the benefit of SDMA technique to the maximum extent. Fig. (c) plots the data gathering time of RDTP varying with D under different settings of N, where N s =0and N p =. We can see that as D increases, data gathering time increases. The reason for the increase is two fold. First, maximum compatible pairs among sensors decrease as the sensing area becomes larger since the sparse distribution maes it less possible for any two sensors to be compatible. Second, as the number of polling points is given, the distance among the selected polling points becomes larger with the increase of D. Apparently, the moving tour of each SenCar may also be longer than that with a smaller D. It is also noticed that data gathering time is shortened with more available SenCars as the load of data gathering is shared and balanced among different SenCars. V. CONCLUSIONS In this paper, we have considered data gathering in WSNs by applying multiple mobile collectors and SDMA technique. We formalized this problem as DG-MS problem and formulated it into an integer linear program. Correspondingly, we proposed a heuristic region-division and tour-planning (RDTP) algorithm

6 This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 0 proceedings. to provide a practically good solution to the problem. Simulation results demonstrate that RDTP algorithm can effectively shorten the data gathering time compared to other non-smda or single mobile collector schemes. APPENDIX: DG-MS PROBLEM FORMULATION In this appendix, we formulate the DG-MS problem as the following integer linear program. We summarize the notations used in Table. Minimize max T () K ( ) d T = q x v n,i, i,j e i,j, i,j P u d m,n,i, + v m n Si P m,n Si P subject to x n,i, f n,i I i,, n S, i P, K () x n,i, = n S () i P K x n,i, I i, i P, K () n S I i, = i P () K I i, K () i P ( xm,i, ) +x u m,n,i, n,i, c m,n,i m, n S, i P, K () u m,n,i, n S () m S\{n} i P K u m,n,i, m S () n S\{m} i P K u m,n,i, = u n,m,i, m, n S, i P, K () e i,j, = I j, j P, K () i P\{j} e i,j, = I i, i P, K () j P\{i} y i,j, P e i,j, i, j P, K () y i,, = I i, i P () i P i P\{} y j,i, y i,j, = I j, j P, K () i P\{j} i P\{j} In the above formulation, objective function () minimizes the maximum data gathering time among different regions over the sensing field, which aims to shorten the data gathering latency of different regions. Constraints ()-() ensure that a sensor should be associated with one and only one selected polling point within the coverage area the sensor is located, so that its sensing data can be collected by a SenCar visiting this polling point. Constraints ()-() indicate that a selected polling point can only belong to one region and each region consists of at least one selected polling point. Constraint () reveals that any two sensors that are qualified to be a compatible pair must be both associated with the same selected polling point in a region and be compatible in the coverage area of this polling point. Constraints ()-() enforce that each sensor belongs to at most one compatible pair. Constraints ()-() ensure the fact that each selected polling point on the moving tour of a SenCar should have one arc pointing toward it and another arc pointing away from it. Constraint () restricts that flow can tae place only when the arc is on the moving tour of a SenCar. Constraint () TABLE NOTATIONS USEDINTHEFORMULATION OF DG-MS PROBLEM Indices: S = {,,...,N s} A set of sensors. P = {,,...,N p} A set of polling points. K = {,,...,N } A set of mobile collectors or a set of regions. Constants: f n,i = {0, } Location indicator. If sensor n is in coverage n S, i P area of polling point i, f n,i =, otherwise, f n,i =0. c m,n,i = {0, } Indicator of compatibility relationship. If sensors m, n S, i P m and n are compatible when they are both in coverage area of polling point i, c m,n,i =, otherwise, c m,n,i =0. d i,j 0 Length of arc a i,j, i.e., distance between polling i, j P point i and polling point j. q>0 Size of the sensing data of each sensor. v d > 0 Effective data uploading rate of each sensor. v m > 0 Moving velocity of the SenCar. Variables: I i, = {0, } Indicator of selected polling point. If polling i P, K point i is selected and assign to region, I i, =, otherwise, I i, =0. x n,i, = {0, } Indicator of sensor association. If sensor n is n S, i P, associated with polling point i in region, K x n,i, =, otherwise, x n,i, =0. u m,n,i, = {0, } Indicator of compatible pair. If sensors m and n m, n S, i P, are selected as a compatible pair when they are K both associated with polling point i in region, u m,n,i, =, otherwise, u m,n,i, =0. e i,j, = {0, } Indicator of selected line segment in moving i, j P, K tour of the associated SenCar in region. If moving tour contains arc a i,j, e i,j, =, otherwise, e i,j, =0. y i,j, 0 Flow value from polling point i to polling point i, j P, K j in region on arc a ij. specifies that the units of flow entering polling point are equal to the number of selected polling points on the tour for SenCar since polling point is the position of the static data sin, which is assumed to be visited by SenCar as the starting and ending points of its tour. Constraint () specifies that for each selected polling point on the moving tour of a SenCar, the units of outgoing flow are one unit more than that of the incoming flow, which excludes the solution of the moving tour of a SenCar with loops [][]. REFERENCES [] I. F. Ayildiz, W. Su, Y. 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