## 1 Introduction
### Basic Model \(M_{0}\)  
A _smart dust (or sensor) cloud_ (a set of sensors, which may be called grain particles) is spread in an area (for a graphical presentation, see Figure 1). The deployment is random uniform. Two important network parameters (crucially affecting connectivity) are the particle density \(d\) (usually measured in numbers of \(particles/m^{2}\)), and the maximum transmission range \(\mathcal{R}\) of each grain particle. The sensors do not move. A two-dimensional framework is adopted. There is a single point in the network, which is called the \"sink\" S, that represents the fixed control center where data should be reported to. The sink is very powerful, in terms of both energy and computing power. Sensors can be aware of the direction (and position) of the sink, as well as of their distance to the sink. Such information can be e.g. obtained during a set-up phase, by having the (powerful) sink broadcast control messages to the entire network region. We assume that events, that should be reported to the control center, occur in the network region. Furthermore, we assume that these events are happening at random uniform positions. As far as energy dissipation is concerned, we assume that the energy spent at a sensor when transmitting data is proportional to the square of the transmitting distance. Note however that the energy dissipation for receiving is not always negligible with respect to the energy when sending such as in case when transmissions are too short and/or radio electronics energy is high [23]. A variation of this basic model includes multiple and/or mobile sinks. The following wall notion generalizes that of the sink and may correspond to multiple (and/or moving) sinks. A _receiving wall_\(\mathcal{W}\) is defined to be an infinite line in the smart-dust plane. Any particle transmission within range \(\mathcal{R}\) from the wall \(\mathcal{W}\) is received by \(\mathcal{W}\). \(\mathcal{W}\) is assumed to have very strong computing power, is able to collect and analyze received data and has a constant power supply and so it has no energy constraints. The wall represents in fact the authorities (the fixed control center) which the realization of a crucial event and collected data should be reported to. Note that a wall of appropriately big (finite) length may suffice. The notion of multiple sinks which may be static or moving has also been studied in [46], where [PERSON], [PERSON], [PERSON] and [PERSON] introduce \"NanoPeer Worlds\", merging notions from Peer-to-Peer Computing and Smart Dust. The \(M_{0}\) model is assumed for the LTP protocol. Figure 1: A Smart Dust Cloud. A stronger model \(M_{1}\)  
This model is derived by augmenting model \(M_{0}\) with additional (stronger) assumptions about the abilities of the sensors and the sensor deployment. In particular, the network topology can be a lattice (or grid) deployment of sensors. This grid placement of grain particles is motivated by certain applications, where it is possible to have a pre-deployed sensor network, where sensors are put (possibly by a human or a robot) in a way that they form a _2-dimensional lattice_. Note indeed that such sensor networks, deployed in a structured way, might be useful e.g. in precise agriculture, where humans or robots may want to deploy the sensors in a lattice structure to monitor in a rather homogenous and uniform way certain conditions in the spatial area of interest. Also, exact terrain monitoring in military applications may also need some sort of a grid-like shaped sensor network. Note that [PERSON] _et al._ in a state of the art survey refer to the pre-deployment possibility [1]. Also, [25] explicitly refers to the lattice case. Moreover, as the authors of [25] state in an extended version of their work, they consider, for reasons of \"analytic tractability\", a square grid topology [26]. We also assume that each grain particle has the following abilities:  
(i) It can estimate the direction of a received transmission (e.g. via the technology of direction-sensing antennae).