 Research Article
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Distributed Detection of Randomly Located Targets in MobilityAssisted Sensor Networks with Node Mobility Management
EURASIP Journal on Advances in Signal Processing volumeÂ 2010, ArticleÂ number:Â 695026 (2010)
Abstract
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 decisionfusion 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.
1. Introduction
The problem of distributed detection and decision fusion in stationary wireless sensor networks has been extensively studied by many authors in different contexts [1â€“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 [7]. Deploying mobile nodes in a sensor network, however, may not be as costeffective 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 ondemand 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 ondemand 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 nodelevel 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 nodelevel 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, [1â€“6, 8â€“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 [7].
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 onetime movement to achieve a better (uniform) coverage. Using mobile nodes as data collection points (sinks) in sensor networks was studied by [13â€“15]. Liu et al. in [7] 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 [16]. Managing mobile node mobility in target tracking applications in mobile sensor networks is addressed in [17].
Since deploying mobile nodes for continuous performance (coverage, detection, and tracking) improvement might not be as costeffective 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 mobilestatic sensor network was addressed in [18]. In [18], 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 twophase 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 decisionfusion 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 ondemand for different purposes. The key contributions in this paper are threefold.

(1)
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 nodelevel decision updating.

(2)
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.

(3)
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
We assume limited mobility of mobile nodes where at each timestep each mobile node can only move in one of the predetermined set of locations (or directions) as shown in Figure 1 for example, and the maximum total distance it can move in any direction is bounded. This mobility model is justifiable in cases where a node can move to a limited number of locations from its current position due to terrain constraints. Let the velocity of mobile node at time be where is the speed of the node that is assumed to be constant and is the direction of node at time . Denote is the maximum distance that the th mobile node can move with the available resources. At each time step , mobile node moves with an average speed of in a direction selected from a set . Selection of at each time step is considered in later sections. Let be the location of the th mobile node at time . Under this mobility model, the location of the th mobile node at time is given by
for , and , where is the selected direction at time and and are and coordinates of the initial location of the th mobile node.
3.3. Observation Model
At each timestep, both mobile and static nodes make observations on the presence/absence of the target and make a binary decision on whether the target is present or absent. We consider the observation models for mobile and static nodes as given below at time under hypotheses (target present) and (target absent)
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.
The received sensing signal represents the attenuated (over distance) signal emitted by the target. Depending on the sensing modalities, (such as acoustic, seismic, IR, etc.), different models for received signal strength can be used. For this discussion, we assume the following model for the signal , which assumes that the signal emitted by the target decays as the distance from the target to the sensing node increases [2, 21]
where is the signal strength emitted by the target, is the distance between the th mobile node and the target at time for , is the location of the stationary target and is the path loss index that is assumed to be 2 throughout. Note that for static nodes (3) reduces to
where . However, the results presented in this paper can be generalized to other sensing modalities as well.
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 timezero 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.
After initially detected by a lower confidence level, we assume that mobile and static nodes make binary decisions at each time instant for based on the observations collected during the current time interval. Formally, the th node performs the following hypothesis testing problem at time
Each node transmits its local decision to the fusion center over a noisy communication channel. The fusion center combines these local decisions from mobile and static nodes and the previous information at the fusion center to make a final decision. The corresponding decisionfusion architecture is shown in Figure 2, where the symbols used in Figure 2 are defined in Sections 4.1 and 4.3. As shown in Figure 2, at each timeinstant , each node performs a local decision based on the observations collected at the current time interval . The corresponding local decisions are transmitted to the fusion center over a noisy communication channel at each time step . The fusion center updates the final decision based on the noisy corrupted decisions received from both static and mobile nodes at time and the previous information at the fusion center at time to reach a final decision. In this architecture, the impact of the mobility of mobile nodes is taken into account at the fusion center while the mobile nodes make a local decision based on observations collected at onestep movement.
4.1. Detection Performance at th Mobile Node
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 [2] (or on energy [20]) 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 [2] 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 NeymanPearson (NP) test to detect the presence/absence of the target at each time .
According to the detection problem at the th mobile node as given by (5), the log likelihood ratio based on the observations collected during time interval , , conditioned on the target location , at the th mobile node can be shown to be [22]
for where and for and as defined in (3). Computation of for a given target location is given in Section 5.1. Then the log likelihood ratio is given by
where and . Computation of is associated with the specific probabilistic model for the target location distribution. For the evaluation used in this paper, the closedform expression for with assumed target location distribution model is given in Section 7. Assuming no point masses in the pdf of , the optimal decision rule at the th mobile node at time for the hypothesis problem (5) is given by (according to the NPcriteria [22])
where is uniquely determined such that, the false alarm probability at the th node at time , for . Note that we assume each node performs the same level NP test at each time . The decision rule (8) can be further simplified to
where is the new decision statistic and is the new threshold, at the th mobile node for .
Proposition 1.
For level NP test, the threshold and the detection probability at the th mobile node at time are given by
respectively, where = and = . The function is defined as .
The proof of Proposition 1 is given in Appendix A. The block diagram of the detector at the th mobile node is shown in Figure 3.
4.2. Detection Performance at th Static Node
Proposition 2.
For static nodes, the optimal threshold and the detection probability for the level NP test are given by
See Appendix B for the proof of Proposition 2. Note that the detection threshold for a static node is a constant over time.
4.3. Performance Evaluation at Fusion Center with Noisy Communication
To make the final decision, let us assume that the nodes send their local decisions to the fusion center over binary symmetric channels (BSC) which can be used to model noisy channels [2, 23]. Let be the received signal at the fusion center from the th node at time for Note that at , the decision center has agreed that a target is detected at a lower confidence level if and . Thus, at time , the fusion center has the decision, under , with probability and with probability . Similarly, under , with probability and with probability . For , we assume that the th node transmits its local decision over a BSC with a crossover probability , and that the channels of nodes are independent of each other. The received signals at the fusion center under the two hypotheses at time for are given by
under and
under where , , and for
For the fusion center to perform optimal fusion rule, it should have the knowledge of detection and false alarm probabilities of local nodes at each time step. Although local false alarm probabilities can be easily made available at the fusion center since they are the same at each node and do not change with time, the analogy is not convenient with local detection probabilities. Thus, in this paper, we assume that the fusion center makes a final decision at time using the counting rule based on the received signals from all nodes at time and the previous available at the fusion center at time . Denote the decision statistic at the fusion center to be , where
where for , and for which can be updated recursively over time. The final decision at the fusion center is then given by
where is the threshold of the counting decision rule at the fusion center at time . The decision threshold is selected so that the overall system false alarm probability at the fusion center is less than a desired level, say . Note that for general nonidentical BSCs, is a sum of independent but nonidentical binary random variables. Using the LindebergFeller central limit theorem for independent random variables under certain regularity conditions (see Appendix D) [24], it can be shown that is distributed under two hypotheses as
where , = + + , = + + and = + + . Then the detection probability at the fusion center can be shown to be
where .
4.4. Mobility Management for Mobile Nodes
In this section, we 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 disappeared). We assume that each mobile node moves with the same speed such that for . Note that each mobile node can move a distance of during each time period of in a direction selected from the set . Let be the candidate locations of mobile node at time . Note that if there are no terrain constraints such that nodes can move heading to the possible target locations on a straight line, a certain number of steps can be made along a straight line as time goes, and there is only one direction. The following discussion is applicable, if mobile nodes are not in a position to direct towards the possible target locations on a straight line from their original locations due environmental and terrain constraints. Let be the desired delay constraint which is equal or less than the average time that the target remains active after appearing. The requirement is to maximize the detection probability at the fusion center, , where , with the best movement plan for each mobile node. Equivalently, we need to find the best direction that th mobile node should move at time , for and , to obtain the maximum (over all possible movements) detection probability at time . Let be the best set of movement directions at each time step for node . Now the problem can be formulated as follows:
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 bruteforce approach will result in the optimal solution, it is computationally expensive. Thus, in the following we propose a nearoptimal 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 .
Note that the detection probability at the fusion center at time is given by (18). The required optimization problem is to find the best movement plan for each mobile node for each such that is maximized. Maximizing in (18) is equivalent to minimizing the argument in the function. Note that in the following analysis we assume for simplicity but the similar results will hold for general values of and . Let be the argument of the function in(18)
where is the length vector containing all ones, ,,, ,,, is the length vector containing all the elements in the sum so that . Then as given by [20], since for and , using the first order Taylor series expansion around central point, can be approximated as,
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 [20] 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 .
Note that for the th mobile node is given by
Hence maximizing at th mobile node is equivalent to maximizing where is given by (11).
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).
For clarity, let us write the sum of detection probabilities as
where is the average detection probability corresponding to the decision made based on the observations during the interval which is given by (11). Now, we construct a trellis as shown in Figure 4, where the states of the trellis at time (stage) represent the directions (states) from the finite set . In Figure 4, the trellis diagram is preceded by and followed by which are two dummy nodes. We take that represents the detection probability for the decision based on observations collected during transition from state to . This represents the detection probability for the decision based on the observations collected during the time interval when the th mobile node selects the direction at time instant given that the direction selected at time is . Now, branch from to is assigned the metric where represents the average detection probability for the decision based on observations collected by the th mobile node if it selects the direction from its original location. Branch from state to is assigned the metric for . Then finding maximum value of the sum from time to time over all possible directions is equivalent to finding the shortest path between the node to as in the graph shown in Figure 4, and the optimal is the set of states in the shortest path. Note that this shortest path can be computed by forward dynamic programming with average complexity of order per mobile node.
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.
We consider the following approach where mobile nodes select best direction to move at time sequentially. The idea is to select the best location for the th mobile node at time step such that the observations collected during time interval would lead to best detection performance over all possible directions. According to the signal model (3), when a mobile node is getting closer to the target, the SNR at the node is increased, subsequently increasing the detection probability at the th mobile node. Hence, the direction at time is chosen as in the following:
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 nearoptimal scheme which is computed based on shortest path algorithm.
5. Detection Performance with Decision Fusion Architecture 2: Nodes Updating Decisions Over Time
In this section, develop an alternate formulation for decision fusion in the hybrid sensor network when the nodes are updating decisions over time, where the impact of the node mobility is taken into account at the node level decisions. As in Section 4, let and be the overall (system) detection and false alarm probabilities at time . If and , for particular and , we say that a target is detected at timezero with a lower confidence level. After a target is detected with a lower confidence level, mobile and static nodes perform the following hypothesis testing problem, at time , based on the observations collected until time :
Note that in this section, we consider that each node performs hypothesis testing (25) based on the observations collected during the interval , in contrast to Section 4. The decision fusion architecture in this case is shown in Figure 5. As shown in Figure 5, at each time instant , each distributed node performs a local detection based on the observations collected at the current time interval and previous observations up to time which can be computed recursively for These local decisions are transmitted to the fusion center over a noisy communication channel at each time step . The fusion center combines these noise corrupted decisions and the initial decision at time to reach at a final decision on whether the target is present or absent.
5.1. Detection Performance at th Mobile Node
Similar to Section 4, we assume that each node performs level NP detector to decide whether the target is present or absent based on the observations collected during time interval . The decision statistic and the threshold for the NP detector corresponding to (9) now can be shown in
where is the decision statistic which can be computed recursively at each time step and is the corresponding threshold, for where as defined in Section 4.1.
Proposition 3.
With decisionfusion model 2, for level NP test, the threshold and the detection probability at the th mobile node, at time are given by,
Note that at the th mobile node is which is essentially the instant total signal energy received during the period , for a given target location. Then we have
where , , , and . Note that (29) holds only if which is shown to be true in the following.
Then is given by
5.2. Detection Performance at the th Static Node
Similarly, for the th static node, the level NP threshold and the detection probability at time are given by the following proposition.
Proposition 4.
For static nodes, the optimal threshold and the detection probability for the level NP test are given by
5.3. DecisionFusion Performance with Noisy Communication
Similar to Section 4.3, we evaluate the decision fusion performance at the fusion center at time with BSC channels. Now, since the effect of the mobility is taken at the node level, the decision statistic at the fusion center is taken as, ,
where with probability and with probability under and with probability and with probability under as in Section 4.3. for is same as given by (13) and (14) under two hypotheses where now , , and with and are given by (28) and (33), respectively. The detection probability corresponding to the decision rule based on majority rule is given by (following a similar approach as in Section 4.3)
where = + + and = + + and = is the threshold of the detector which ensures that the false alarm probability is less than or equal to , and = + + and = + + .
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 decisionfusion model as given by Section 5, in which nodes are updating decisions over time.
Let the required detection probability threshold at time be . The problem is to find the minimum set of mobile nodes, that should be used in the network to reach a desired performance level within a desired delay constraint. Formally, we can write the optimization problem as
where as before, . Assuming that , the inequality (36) can be further simplified as given (for simplicity, we assume perfect communication channels such that for all )
where + and = + and . Note that here and are given by (28) and (33) without the outer expectation with respect to target locations (since we assume exact target locations for this analysis). Note that hereafter, we use and interchangeably when there is no ambiguity. Since mobile nodes which are not moving also make observations at their stationary configuration, we will have the set of static nodes as when the set of mobile nodes is . Note that (39) is obtained from (38) since . If we constrain which is a reasonable assumption, the argument of the function in (37) should be negative. Also, since for , to satisfy , the following condition should hold . Note that
To make sure the condition is satisfied, from (41), it can be seen that at the worst case. This says that if for some and , the required performance cannot be achieved during the required time delay even if all nodes are mobile such that . In the following, we consider only the case where ensuring . Thus, (40) is obtained from (39). The inequality (40) can be further simplified as follows:
The problem is to find the minimum size set such that, inequality (42) is satisfied. To find this, in general we need to search over a maximum of possibilities.
In the following we will show how to obtain the solution with reduced complexity under certain conditions. Note that as discussed in Section 5.4, the maximum for each at time can be computed. Without loss of generality, let us arrange s in descending order for such that . Then, the set denoted by consists of the indices of first mobile nodes. Now, define two functions and so that,
for with .
Let be the set containing all static node indices and the mobile node indices from to , for . Clearly, and . Define to be
Theorem 1.
If s are arranged in descending order and , , and are defined as in (43), (44) and (45), respectively, then we can find a unique so that
Then, is the minimum number of mobile nodes that should be used to meet the desired criteria where the minimum set .
Proof.
See Appendix C.
7. Performance Results
In this section, we evaluate the performance of the proposed target detection schemes using a hybrid sensor network. We assume that there is a total of sensors deployed in a square region of area where the center is at . We assume that mobile node speed is constant for all directions and the same for all nodes. The timestep â€‰s and each mobile node's speed is â€‰m/s. We define the nominal SNR at each node to be . We also assume that the communication between nodes and the fusion center is over i.i.d. BSCs so that for all . At each time step, we assume that a mobile node can move a distance of in directions corresponding to dueeast, northeast, duenorth, northwest, duewest, southwest, duesouth, and southeast or remain at the current location. and coordinates of the target location are assumed to be binary with the following distribution:
This type of target location model is justifiable in situations when there is a finite number of possible surveillance locations that the target can appear probabilistically, based on the prior knowledge or new information received after initial deployment. Note that with this target location model, is given by
In Figure 6, the time varying detection performance is shown with the node mobility algorithms presented in the paper. In the following figures, we assume that mobile nodes perform sequential node mobility management as discussed in Sections 4.4 and 5.4, since it can be seen that the performance with mobility algorithm based on sequential approach is much closer to that with evaluated based on forward dynamic programming algorithm (see Figure 10 for comparison). In Figure 6, we have let â€‰dB, false alarm probability at local nodes , false alarm probability at the fusion center , initial detection and false alarm probabilities , , , and , , , . The detection performance is shown when the fraction of mobile nodes is varying for . Figure 6(a) corresponds to the decisionfusion architecture 1 (Section 4) while Figure 6(b) corresponds to decision fusion model 2 in Section 5. It can be seen from Figure 6 that with scheme 1, adding a small number of nodes boosts the detection performance significantly compared to the performance with all static network. With the scheme 1, as time goes, since mobile nodes getting closer to possible target locations, according to the given probability distribution for target locations, a mobile node can make a binary decision based on the information collected at current time interval with a higher confidence level, when compared to a stationary sensor. On the other hand, stationary sensors make binary decisions based on its observations collected during current time interval, and the quality of these decisions remains the same over time since nodes are stationary. Then, according to the decision fusion model 1 described in Section 4, fusion center receives binary decisions over the time with higher confidence level when there are mobile sensors, resulting in an improved performance compared to an allstationary network.
As can be seen in Figure 6(b), with decisionfusion 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 decisionfusion 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 decisionfusion model 2, when the fraction of mobile nodes, is increasing the performance gain over a stationary network becomes significant.
Figure 7 shows the variation of overall detection probability with the system false alarm probability (ROC: receiver operating characteristics) with two decision fusion models for with different delay constraints. Other relevant parameters have the same values as in Figure 6. As can be seen from Figure 7, the fusion model 1 outperforms the model 2 especially when the delay constraint is relatively large. It also can be seen that for a given delay constraint and for moderate values of system false alarm probability , the model 1 outperforms model 2. However, both schemes show similar performance when is getting large which intuitively makes sense.
Figure 8 shows the deviation of the detection performance due to the averaging strategy used in developing local decisions in this paper. The curves in Figure 8 correspond to the performance deviation by applying the decision fusion architecture 1 if the target happens to appear at when and are varying. From Figure 8, it can be seen that for relatively large the deviation in detection performance is almost negligible. For moderate values of , it is seen that the deviation in detection performance is not very small, but not too large as well. Note that when the local nodes perform noncoherent detection (unknown target locations), a higher SNR is required to achieve the same performance level compared to that with a coherent detector (known target location) [22]. From the simulation results, it is seen that this deviation is mostly acceptable for a wide range of network parameters.
It is noted that the initial detection performance may affect the distance that the mobile nodes should move in order to achieve a certain desired performance. Figure 9 shows the dependence of the initial detection performance on average moving distance of a mobile node in order to achieve a desired performance. In Figure 9, we have let the system false alarm probability , , , local node false alarm probability . As can be seen from Figure 9, the initial detection helps to reduce average moving distance of a mobile node, especially when the desired detection probability takes moderate values.
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 Figure 11, the minimum fraction of mobile nodes required to achieve a given desired probability at the fusion center is shown when the desired delay constraint â€‰s and and are varying. Clearly Figure 11 shows the trade off between the required number of mobile nodes and the local decision qualities when achieving a desired overall system performance level. Thus, Theorem 1 in Section 6 is important in determining the required fraction of nodes to be mobile depending on the requirement.
8. Conclusions
In this paper, we proposed two decisionfusion models for target detection using a hybrid sensor network in which the node mobility is taken into account at nodelevel 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 costeffective 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.
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Acknowledgments
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 CCF0830545.
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Appendices
A. Proof of Proposition 1
Average false alarm probability corresponding to the decision rule (9) at the th mobile node is given by
where . Under we can show that is distributed as
where . Then the false alarm probability (A.1) becomes
If the allowable false alarm probability at the th mobile node is , the threshold is given by
The detection probability at the th mobile node at time based on the decision rule (9) is given by
Under , conditioned on , is distributed as
where . Thus the pdf of under is given by
Note that we use for simplicity, when there is no ambiguity.
Then, the detection probability (A.5) at the th mobile node is given by
B. Proof of Proposition 2
When nodes are static, we have for . Then, from Section 4.1, we have where , , and . Then, the decision statistic in the decision rule (9) reduces to
and the threshold is given by where is found such that . The false alarm probability associated with the decision rule (9) for the th static node reduces to
To achieve , the threshold can be computed as, . Then, the corresponding detection probability (11) for the th static node reduces to
for , where we use the relations , and for static nodes in obtaining (B.3).
C. Proof of Theorem 1
When , all mobile nodes are in stationary mode. Then, we have
If , we have for since from the definition of and . Also, from the Claim 1 (given below), we can see that then for all . Then we have , where no need for any node to be mobile to achieve the desired performance level within a desired delay constraint. Now, if , in the following we prove that, we can find a unique such that for and for . The uniqueness of such is followed from Claim 1. If for all , it implies that the required performance level can not be achieved within the desired delay constraint even if all mobile nodes are directed to move.
To prove the uniqueness of , we prove the following.
Claim 1.
If then for . Also, if we have
Prf
First we prove that if then . Note that when necessary, we use and to denote the detection probability at the th mobile node at time at its stationary configuration and mobile configuration, respectively. We have
Now adding and subtracting to the right hand side of (C.3), we will get
where we dropped argument so that and for simplicity. Substituting for from (45) and using the fact that , after simplification (C.4) reduces to (C.2).
Note that we use to denote the detection probability at the th mobile node at time at its stationary configuration, as mentioned before. In (C.2), since mobility towards the target improves the detection probability at the th mobile node, we have . Using this fact, and with the assumption that for (which holds true in practice for sufficient ) the second term of the right hand side of the inequality (C.2) is positive. Then we have
as required. Following a similar approach, we can prove that if .
D. Regularity Conditions to Apply LF Central Limit Theorem in Sections 4.3 and 5.3
LindebergFeller Central Limit Theorem (LF CLT): suppose is a sum of independent random variables with and . The LF CLT states that under certain regularity conditions the sum converges in distribution to a Gaussian random variable with mean and the variance as [24]. For the applicability of LFCLT, it was shown in [25] two sufficient conditions should be satisfied:
 (i)
 (ii)
for where and are positive values.
To apply the LFCLT in Section 4.3, first we prove that the sufficient conditions are satisfied under (in the following we assume for simplicity). Note that we can write as
where for and for . Under , it can be seen from (13) that is a Binary random variable with mean and variance . Similarly, is a Binary random variable with mean and the variance . Then assuming perfect communication channels (such that ) we have
where last inequality results because . Note that from (11), if the local false alarm probability is set such that , is positive and finite for any . Let and . Then we have and for , . Similarly, we can show that we can find two positive values and such that and for , . Following a similar procedure, it can be shown that the two regularity conditions are satisfied under as well.
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Wimalajeewa, T., Jayaweera, S.K. Distributed Detection of Randomly Located Targets in MobilityAssisted Sensor Networks with Node Mobility Management. EURASIP J. Adv. Signal Process. 2010, 695026 (2010). https://doi.org/10.1155/2010/695026
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DOI: https://doi.org/10.1155/2010/695026