Distributed Detection of Randomly Located Targets in Mobility-Assisted Sensor Networks with Node Mobility Management
© T.Wimalajeewa and S. K. Jayaweera. 2010
Received: 13 July 2009
Accepted: 13 April 2010
Published: 20 May 2010
Performance gain achieved by adding mobile nodes to a stationary sensor network for target detection depends on factors such as the number of mobile nodes deployed, mobility patterns, speed and energy constraints of mobile nodes, and the nature of the target locations (deterministic or random). In this paper, we address the problem of distributed detection of a randomly located target by a hybrid sensor network. Specifically, we develop two decision-fusion architectures for detection where in the first one, impact of node mobility is taken into account for decisions updating at the fusion center, while in the second model the impact of node mobility is taken at the node level decision updating. The cost of deploying mobile nodes is analyzed in terms of the minimum fraction of mobile nodes required to achieve the desired performance level within a desired delay constraint. Moreover, we consider managing node mobility under given constraints.
The problem of distributed detection and decision fusion in stationary wireless sensor networks has been extensively studied by many authors in different contexts [1–6]. However, stationary sensor networks may not suit for some applications, for example, in situations where it might be necessary to deploy a huge number of static nodes with limited coverage to monitor a large region within a desired performance level. In such situations, if relatively a small number of nodes are allowed to move, the system performance can be improved over time due to improvement in sensing coverage . Deploying mobile nodes in a sensor network, however, may not be as cost-effective as deploying static nodes. Also, nodes will have to spend node energy for mobility in addition to sensing and communication. Thus, it is desirable to allow only a fraction of the nodes of the network to be mobile according to the requirement.
In this paper, we consider the problem of detecting a randomly located stationary target in a hybrid sensor network made of both mobile and static nodes. At the initial deployment stage, static and mobile nodes may scatter in the region of interest in random fashion, if the network does not have prior information about Phenomenon of Interest (PoI). Mobile nodes may be required to perform on-demand for different applications after the initial deployment. Due to energy constraints, we assume that the mobile nodes are kept stationary until a target is detected with certain confidence level, or useful statistics regarding the target locations are available. Note that since mobile nodes are required to perform on-demand for different functionalities, it is not possible to locate them in a certain area for a specific task. We assume that, at each time step, a mobile node can move to a limited number of locations from its current position, where these candidate locations are determined by physical factors related to mobile sensors and the environment. At each time step, mobile nodes move in a direction chosen based on the proposed mobility management schedule to maximize the detection probability during a desired delay constraint. At each time step, each node makes a local binary decision based on its observations and transmits it to the fusion center. The fusion center combines local decisions from all static and mobile nodes to reach at a final decision at the corresponding time instance. Specifically, we develop two decision fusion models to make the final decision where in the first model, the impact of the node mobility is taken into account to update the decision at the fusion center, while in the second model, the impact of node mobility is taken at the node-level decision updating. Since allowing more nodes to be mobile increases the cost, we characterize analytically the required minimum fraction of mobile nodes to be directed to move in order to achieve a desired performance level within a desired delay constraint. We investigate the performance gain achieved by the hybrid sensor network when the network parameters are changing and discuss the scenarios where the node mobility is essentially improves the network performance.
The organization of the paper is as follows. Important related work is discussed in Section 2. Section 3 explains the sensor network and the observation models, and presents the problem formulation. In Section 4, we develop a decision fusion model in which the fusion center updates the decisions over time while nodes make binary decisions based on the observations collected during one time step when the target location is random. Also, mobility management schedule is proposed to maximize the detection probability at the fusion center within a desired delay constraint. In this discussion, the effect of the node mobility is taken into account at the fusion center decision updating. In Section 5, a decision fusion model is developed in which the effect of the node mobility is taken into account at the node-level decisions. In Section 6, we develop an analytical procedure to find the minimum number of mobile nodes that should be incorporated with static nodes to achieve a desired performance level within a desired delay constraint. Performance results are given in Section 7, and the concluding remarks are given in Section 8.
2. Related Work
Distributed detection and decision fusion are analyzed by many authors in different contexts, for example, [1–6, 8–10], to name a few. However, many of these existing analysis on target detection have considered stationary sensor networks, where sensor nodes are deployed with fixed positions or in a random fashion. Since the performance of such a stationary sensor network is limited by network size, sensing ranges, and so forth, recently, mobile sensor nodes have been suggested to enhance the system performance in wireless sensor network applications .
Use of node mobility in mobile sensor networks for relocation after initial random placement was previously suggested in [11, 12]. However, in their models, nodes only make a one-time movement to achieve a better (uniform) coverage. Using mobile nodes as data collection points (sinks) in sensor networks was studied by [13–15]. Liu et al. in  showed that the coverage can be improved by a mobile sensor network with continuous mobility over the time, compared to that with a static network. Surveillance coverage of mobile sensor networks under Brownian motion random node mobility model was addressed in . Managing mobile node mobility in target tracking applications in mobile sensor networks is addressed in .
Since deploying mobile nodes for continuous performance (coverage, detection, and tracking) improvement might not be as cost-effective as deploying static nodes, it is useful to consider networks consisting of both static and mobile nodes where the mobile nodes are allowed to move only if necessary. The target tracking performance of an integrated mobile-static sensor network was addressed in . In , the mobile nodes are used to aid the data propagation when the communication ranges of static nodes are limited. The target detection in a hybrid sensor network is addressed by [19, 20] where they have proposed a two-phase detection model for target detection assuming known target locations. Although we address a similar problem, our work is different from [19, 20] in several contexts. (i) In this paper we explicitly present two decision-fusion models for target detection when the target location is random. (ii) We consider constrained mobility for mobile nodes where each node can move only in a predetermined set of candidate directions from their current locations. (iii) We evaluate the cost of deploying mobile nodes in terms of the minimum fraction of mobile nodes that should be directed to move to achieve a desired performance level within a desired delay constraint, analytically. Moreover, [19, 20] did not allow for the possibility of imperfect communication links between nodes and the fusion center.
3. Problem Formulation and System Model
We consider a hybrid sensor network made of number of total sensors. We assume that there are number of static nodes and a maximum of number of mobile nodes initially deployed in a square region with dimensions . Note that when mobile nodes are not in the mobile configuration, they make measurements at their stationary configuration. Let and be the fractions of mobile and static nodes, respectively. Let to be the location of the th static node which is assumed to be fixed after initial deployment. Let be the set of all node indices in the network, and let and to be the sets containing mobile and static node indices, respectively.
3.1. Problem Formulation
In this paper, we assume that the network is kept stationary until a target is detected at a certain confidence level. We also assume that the network does not have any information regarding sensing field at the time of deployment. Information regarding possible target locations may be available to the network after initial deployment and the target can be shown in a particular target location during a certain period of time. Because of these factors, it is not possible to deploy mobile sensors to cover possible target locations at the time of deployment. On the other hand, mobile nodes may be required to perform on-demand for different purposes. The key contributions in this paper are threefold.
Develop decision fusion architectures for the target detection by hybrid sensor network when the target location is random. Specifically, we propose two decision fusion architectures where in the first one, the effect of the node mobility is taken into account for the decision updating at the fusion center, and nodes make binary decisions based on the observations during one step movement. In the second model, nodes take the effect of the node mobility into account for node-level decision updating.
Manage node mobility to improve (maximize) the system performance within a desired delay constraint after a target is initially detected by the stationary configuration at certain confidence level.
The cost of mobile nodes is evaluated in terms of the minimum number of mobile nodes required to achieve a desired performance level within a desired delay constraint.
3.2. Node Mobility Model
3.3. Observation Model
for , where is the signal strength received from the target at time , is the measurement noise process at the th node which is assumed to be white Gaussian with mean zero and the autocovariance function where denotes the Dirac delta function.
4. Detection Performance with Decision Fusion Architecture 1: Fusion Center Updating Decisions Over Time
In this section we consider the performance dynamics of the hybrid sensor network when the exact target location is unknown. At the stationary configuration, we assume that the network monitors the Field of Interest (FoI) continuously, and mobile nodes are directed to move when a possible target is detected with relatively lower confidence level by the stationary configuration. More precisely, let and be the overall (system) detection and false alarm probabilities at time . If and , in particular and , we say that a target is detected at time-zero with a low confidence level. The target location coordinates and are assumed to be random variables with known statistics. Moreover, in this paper we assume that once appeared, the target remains active for a known period of time.
Let denote . Note that according to the signal model (3) assumed in the paper, the signal strength received by a sensor node is decreasing as the distance between the node location and the target location is increasing. If a simple constant threshold testing is performed on the received signal strength  (or on energy ) at a sensor node to determine the target is present/absent, it can be seen that more false alarms will occur at the nodes located relatively far away from the target location if the threshold is chosen too small, or miss probability will be higher at sensors located closer to the target location, if the threshold is too large. Reference  has provided an approach to select an optimal threshold such that the performance at the fusion center is maximized for a static sensor network. However, in this paper since mobile nodes are directed to move when required, maintaining a constant threshold test on signal strength (or energy) to determine the presence/absence of a target would not essentially reflect the performance gain achieved by node mobility. Thus it is required to have a dynamically varying threshold at sensor nodes to exploit the impact of node mobility in an effective way. Thus, in this paper, we consider that th mobile node to perform likelihood ratio testing on its observations. Explicitly, we assume that each node performs -level Neyman-Pearson (N-P) test to detect the presence/absence of the target at each time .
4.3. Performance Evaluation at Fusion Center with Noisy Communication
4.4. Mobility Management for Mobile Nodes
If the fusion center were to compute the movement plan beforehand for each mobile node, in general, the optimization has to search over as many as variables leading to a search space of size where is the cardinality of the set . Although this brute-force approach will result in the optimal solution, it is computationally expensive. Thus, in the following we propose a near-optimal approach for each mobile node to select its best movement direction at each time step based on its own performance measure; that is, each node moves in a direction at each time step which would lead to maximum individual performance at time .
where denotes the second and higher order terms in the Taylor series expansion. It is seen from (21) that if and the sum were to be independent of each other, then will be monotonically decreasing with increasing . It was shown in  that with high probability, is indeed decreasing when the sum is increasing. Thus, with high probability, maximizing the detection probability at the fusion center at time is equivalent to maximizing the sum . Since each mobile and static node performs their detection problems independent of each other, maximizing over all possible movement plans for will maximize the sum at time .
Now the optimization problem is equivalent to finding the set which maximizes the sum of detection probabilities up to time at the th mobile node as given in (11). Let be the sum of detection probabilities at th mobile node up to time where as given by (11) is the detection probability related to the decision made by th mobile node based on observations during time interval . In the following, we convert the required problem into an time expansion graph, so that the required problem becomes a shortest path problem and the solution for the optimization problem can be obtained, for example, via forward dynamic programming.
Let be the state space at time (stage) for the th mobile node which represents the set of directions that the th mobile node can move at time . We assume that each mobile node has the same candidate set of directions that it can move at a given time step (however, this assumption can be generalized to have different candidate sets for different mobile nodes).
In solving the shortest path algorithm via dynamic programming for the original optimization problem in (19), the movement plan for each mobile node needs to be computed beforehand at time which also requires the knowledge of the candidate set of locations at each time. In the following we show that a sequential approach where the th mobile node determines its movement direction at time based on only its current information and expected information at time yields closer performance compared to that with dynamic programming approach under certain conditions.
where is the average detection probability at the th mobile node at time step if the direction is selected at time , is the step index at which for the first time. The average detection probability at th mobile node at time is as given by (11). From the simulation results, we see that, when the candidate set of directions that any mobile node can move at a given time is the same, and a node moves at the same speed in all directions, the performance of this scheme coincides with the near-optimal scheme which is computed based on shortest path algorithm.
5. Detection Performance with Decision Fusion Architecture 2: Nodes Updating Decisions Over Time
5.3. Decision-Fusion Performance with Noisy Communication
5.4. Mobility Management for Mobile Nodes
Similar to the scenario in Section 4, we need to find the best movement schedule for each mobile node in order to maximize the detection probability at the fusion center within a desired delay constraint or before the target disappears. The idea is to find the optimal movement schedule for each mobile node such that the detection probability at the fusion center within a desired delay constraint is maximized. As in Section 4.4, let be the desired delay constraint and be the optimal set of movement directions at each time step for node . Following a similar approach as in Section 4.4, it can be shown that with high probability, maximizing the detection probability at the fusion center at time (35) is equivalent to maximizing the sum . Since mobile and static nodes perform their detection problems independent of each other, maximizing each for over all possible movement plans will maximize the sum at time . Similar to Section 4.4, it can be shown that maximizing at th mobile node is equivalent to maximizing at the th mobile node, given by (28).
Note that if the exact target location is known, then maximizing (28) at the th mobile node is equivalent to maximizing the total energy collected during the interval , as given in (31). Then, the approach given in Section 4.4 can be directly used to find the optimal movement directions at each time step, where now the metrics of branches of the trellis in Figure 4 are replaced by which represents the energy collected during transition from state to .
6. Minimum Set of Mobile Nodes
Since allowing nodes to be mobile is expensive in terms of energy, it is important to determine the minimum number of mobile nodes (from the set ) that should be directed to move to achieve a certain detection probability within a given delay constraint or before the target disappears. In the following, we consider the problem of finding the smallest set of mobile nodes in order to maintain the maximum detection probability achieved by time is greater than some threshold value. For the discussion given below, we assume the case where exact target location is known with the decision-fusion model as given by Section 5, in which nodes are updating decisions over time.
See Appendix C.
7. Performance Results
As can be seen in Figure 6(b), with decision-fusion model 2, in which the nodes update decisions over time, the performance is improved significantly by adding a relatively large number of nodes compared to that with the model 1 under same network conditions. According to the decision-fusion model 2, static nodes also collect energy over time, and decisions are getting more accurate as the time goes. For moderate and higher nominal SNR values, a static node may collect sufficient energy at its stationary locations compared to that collected by a mobile node while moving towards possible target locations, since for large and moderate , even sensors located far away from the target location will receive signals with considerable strength. However, with the decision-fusion model 2, when the fraction of mobile nodes, is increasing the performance gain over a stationary network becomes significant.
In Figure 10, we compare the detection performance when the node mobility management is performed via dynamic programming approach and the sequential approach. In Figure 10 we assume that the target is located at the origin and results correspond to decision fusion model 2. It can be seen from Figure 10 that when each mobile node uses same speed and same set of direction at each time step, the detection performance with sequential approach fairly matches that of the dynamic programming approach. Figure 10 also depicts that when the desired system false alarm probability is small, adding mobility greatly improves the detection performance.
In this paper, we proposed two decision-fusion models for target detection using a hybrid sensor network in which the node mobility is taken into account at node-level and at the fusion center and analyzed the impact of node mobility to the overall performance under both schemes. The mobile nodes in the network are kept stationary until a target is detected with a low confidence level or statistical information on target locations are available and are directed to move to maximize the detection probability during a desired delay constraint once a target is detected within a certain confidence level. We proposed a node mobility management scheme in order to maximize the detection probability within a desired delay constraint. Since deploying mobile nodes in a sensor network is not as cost-effective as deploying static nodes, we evaluate the cost of allowing nodes to be mobile in terms of the minimum number of mobile nodes required to achieve a desired performance level within desired delay constraint.
This research was performed while the first author was with the Department of Electrical and Computer Engineering at the University of New Mexico. The paper was supported in part by the U.S. National Science Foundation (NSF) under the Grant CCF-0830545.
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