We study a heterogeneous wireless sensor network (WSN) where$N$heterogeneous access points (APs) gather data from densely deployed sensors and transmit their sensed information to$M$heterogeneous fusion centers (FCs) via multi-hop wireless communication. The heterogeneous optimal deployment of APs and FCs is modeled as an optimization problem with total wireless communication power consumption of the network as its objective function. We consider both static WSNs, where APs and FCs retain their deployed position, and mobile WSNs where APs and FCs can move from their initial deployment to their optimal locations. Based on the derived necessary conditions for the optimal deployment in static WSNs, we propose an iterative algorithm to deploy APs and FCs. In addition, we study the necessary conditions of the optimal movement-efficient deployment in mobile WSNs with constrained movement energy and present iterative algorithms to find such deployments, accordingly. Simulation results show that our proposed deployment algorithms outperform the existing methods in the literature, and achieve a lower total wireless communication power in both static and mobile WSNs.

We study a heterogeneous two-tier wireless sensor network in which N heterogeneous access points (APs) collect sensing data from densely distributed sensors and then forward the data to M heterogeneous fusion centers (FCs). This heterogeneous node deployment problem is modeled as an optimization problem with the total power consumption of the network as its cost function. The necessary conditions of the optimal AP and FC node deployment are explored in this paper. We provide a variation of Voronoi Diagram as the optimal cell partition for this network and show that each AP should be placed between its connected FC and the geometric center of its cell partition. In addition, we propose a heterogeneous two-tier Lloyd algorithm to optimize the node deployment. Furthermore, we study the sensor deployment when the communication range is limited for sensors and APs. Simulation results show that our proposed algorithms outperform the existing clustering methods like Minimum Energy Routing, Agglomerative Clustering, Divisive Clustering, Particle Swarm Optimization, Relay Node placement in Double-tiered Wireless Sensor Networks, and Improved Relay Node Placement, on average.

We study a wireless ad-hoc sensor network (WASN) where N sensors gather data from the surrounding environment and transmit their sensed information to M fusion centers (FCs) via multi-hop wireless communications. This node deployment problem is formulated as an optimization problem to make a trade-off between the sensing uncertainty and energy consumption of the network. Our primary goal is to find an optimal deployment of sensors and FCs that minimizes a Lagrangian combination of sensing uncertainty and energy consumption. To support arbitrary routing protocols in WASNs, the routing-dependent necessary conditions for the optimal deployment are explored. Based on these necessary conditions, we propose a routing-aware Lloyd-like algorithm to optimize node deployment. Simulation results show that our proposed algorithm outperforms the existing deployment algorithms, on average.

To provide a reliable wireless uplink for users in a given ground area, one can deploy Unmanned Aerial Vehicles (UAVs) as base stations (BSs). In another application, one can use UAVs to collect data from sensors on the ground. For a power-efficient and scalable deployment of such flying BSs, directional antennas can be utilized to efficiently cover arbitrary 2-D ground areas. We consider a large-scale wireless path-loss model with a realistic angle-dependent radiation pattern for the directional antennas. Based on such a model, we determine the optimal 3-D deployment of N UAVs to minimize the average transmit-power consumption of the users in a given target area. The users are assumed to have identical transmitters with ideal omnidirectional antennas and the UAVs have identical directional antennas with given half-power beamwidth (HPBW) and symmetric radiation pattern along the vertical axis. For uniformly distributed ground users, we show that the UAVs have to share a common flight height in an optimal power-efficient deployment, by simulations. We also derive in closed-form the asymptotic optimal common flight height of N UAVs in terms of the area size, data-rate, bandwidth, HPBW, and path-loss exponent.

In many quantization problems, the distortion function is given by the Euclidean metric to measure the distance of a source sample to any given reproduction point of the quantizer. We will in this work regard distortion functions, which are additively and multiplicatively weighted for each reproduction point resulting in a heterogeneous quantization problem, as used for example in deployment problems of sensor networks. Whereas, normally in such problems, the average distortion is minimized for given weights (parameters), we will optimize the quantization problem over all weights, i.e., we tune or control the distortion functions in our favor. For a uniform source distribution in one-dimension, we derive the unique minimizer, given as the uniform scalar quantizer with an optimal common weight. By numerical simulations, we demonstrate that this result extends to two-dimensions where asymptotically the parameter optimized quantizer is the hexagonal lattice with common weights. As an application, we will determine the optimal deployment of unmanned aerial vehicles (UAVs) to provide a wireless communication to ground terminals under a minimal communication power cost. Here, the optimal weights relate to the optimal flight heights of the UAVs.

We study a heterogeneous two-tier wireless sensor network in which N heterogeneous access points (APs) collect sensing data from densely distributed sensors and then forward the data to M heterogeneous fusion centers (FCs). This heterogeneous node deployment problem is modeled as a quantization problem with distortion defined as the total power consumption of the network. The necessary conditions of the optimal AP and FC node deployment are explored in this paper. We provide a variation of Voronoi diagrams as the optimal cell partition for this network, and show that each AP should be placed between its connected FC and the geometric center of its cell partition. In addition, we propose a heterogeneous two-tier Lloyd-like algorithm to optimize the node deployment. Simulation results show that our proposed algorithm outperforms the existing methods like Minimum Energy Routing, Agglomerative Clustering, and Divisive Clustering, on average.

We study a mobile wireless sensor network (MWSN) consisting of multiple mobile sensors or robots. Three key factors in MWSNs, sensing quality, energy consumption, and connectivity, have attracted plenty of attention, but the interaction of these factors is not well studied. To take all the three factors into consideration, we model the sensor deployment problem as a constrained source coding problem. %, which can be applied to different coverage tasks, such as area coverage, target coverage, and barrier coverage. Our goal is to find an optimal sensor deployment (or relocation) to optimize the sensing quality with a limited communication range and a specific network lifetime constraint. We derive necessary conditions for the optimal sensor deployment in both homogeneous and heterogeneous MWSNs. According to our derivation, some sensors are idle in the optimal deployment of heterogeneous MWSNs. Using these necessary conditions, we design both centralized and distributed algorithms to provide a flexible and explicit trade-off between sensing uncertainty and network lifetime. The proposed algorithms are successfully extended to more applications, such as area coverage and target coverage, via properly selected density functions. Simulation results show that our algorithms outperform the existing relocation algorithms.

We study a mobile wireless sensor network (MWSN) consisting of multiple mobile sensors or robots. Two key issues in MWSNs - energy consumption, which is dominated by sensor movement, and sensing coverage - have attracted plenty of attention, but the interaction of these issues is not well studied. To take both sensing coverage and movement energy consumption into consideration, we model the sensor deployment problem as a constrained source coding problem. Our goal is to find an optimal sensor deployment to maximize the sensing coverage with specific energy constraints. We derive necessary conditions to the optimal sensor deployment with (i) total energy constraint and (ii) network lifetime constraint. Using these necessary conditions, we design Lloyd-like algorithms to provide a trade-off between sensing coverage and energy consumption. Simulation results show that our algorithms outperform the existing relocation algorithms.

Multi-tier networks have many applications in different fields. We define a novel two-tier quantizer that can be applied to different node deployment problems including the energy conservation in two-tier wireless sensor networks consisting of N access points (APs) and M fusion centers (FCs). We aim at finding an optimal deployment of APs and FCs to minimize the average weighted total, or Lagrangian, of sensor and AP powers. For one FC, M = 1, we show that the optimal deployment of APs is simply a linear transformation of the optimal N-level quantizer for density f, and the sole FC should be located at the geometric centroid of the sensing field. We also provide the exact expression of the AP-Sensor power function and prove its convexity. For more than one FC, M > 1, we provide a necessary condition for the optimal deployment. Furthermore, to numerically optimize the AP and FC deployment, we propose three Lloyd-like algorithms and analyze their convergence. Simulation results show that our algorithms outperform the existing algorithms.

We study a two-tiered wireless sensor network (WSN) consisting of N access points (APs) and M base stations (BSs). The sensing data, which is distributed on the sensing field according to a density function f, is first transmitted to the APs and then forwarded to the BSs. Our goal is to find an optimal deployment of APs and BSs to minimize the average weighted total, or Lagrangian, of sensor and AP powers. For M = 1, we show that the optimal deployment of APs is simply a linear transformation of the optimal N-level quantizer for density f, and the sole BS should be located at the geometric centroid of the sensing field. Also, for a one-dimensional network and uniform f, we determine the optimal deployment of APs and BSs for any N and M. Moreover, to numerically optimize node deployment for general scenarios, we propose one-and two-tiered Lloyd algorithms and analyze their convergence properties. Simulation results show that, when compared to random deployment, our algorithms can save up to 79% of the power on average.

We study the heterogeneous wireless sensor networks (WSNs) and propose the necessary condition of the optimal sensor deployment. Similar to that in homogeneous WSNs, the necessary condition implies that every sensor node location should coincide with the centroid of its own optimal sensing region. Moreover, we discuss the dynamic sensor deployment in both homogeneous and heterogeneous WSNs with limited communication range for the sensor nodes. The purpose of sensor deployment is to improve sensing performance, reflected by distortion and coverage. We model the sensor deployment problem as a source coding problem with distortion reflecting sensing accuracy. Traditionally, coverage is the area covered by the sensor nodes. However, when the communication range is limited, a WSN may be divided into several disconnected sub-graphs. Under such a scenario, neither the conventional distortion nor the coverage represents the sensing performance as the collected data in disconnected sub-graphs cannot be communicated with the access point. By defining an appropriate distortion measure, we propose a Restrained Lloyd (RL) algorithm and a Deterministic Annealing (DA) algorithm to optimize sensor deployment in both homogeneous and heterogeneous WSNs. Our simulation results show that both DA and RL algorithms outperform the existing Lloyd algorithm when communication range is limited.

We study the heterogeneous wireless sensor networks (WSNs) and propose the necessary condition of the optimal sensor deployment. Similar to that in homogeneous WSNs, the necessary condition implies that every sensor node location should coincide with the centroid of its own optimal sensing region. Moreover, we discuss the dynamic sensor deployment in both the homogeneous and the heterogeneous WSNs with limited communication range for the sensor nodes. The purpose of sensor deployment is to improve sensing performance, reflected by distortion and coverage. We model the sensor deployment problem as a source coding problem with distortion reflecting sensing accuracy. However, when the communication range is limited, a WSN is divided into several disconnected sub-graphs under certain conditions as we will discuss in this paper. In such a scenario, neither the conventional distortion nor the coverage represents the sensing performance as the collected data in disconnected sub-graphs cannot be communicated with the access point. By defining an appropriate sensing performance measure, we propose a Restrained Lloyd (RL) algorithm and a deterministic annealing (DA) algorithm to optimize sensor deployment in both the homogeneous and heterogeneous WSNs. Our simulation results show that both the DA and the RL algorithms outperform the existing algorithms when communication range is limited.