- Open Access
Source positioning in a large-scale tiny-sensor network of arbitrary topology
© Penkin et al.; licensee Springer. 2014
- Received: 21 June 2013
- Accepted: 9 April 2014
- Published: 28 April 2014
To effectively localise a source node in a dense wireless tiny-sensor network with an arbitrary 2D/3D node distribution, a novel approach suitable to describe the hop progress of source-to-sink path in such a system is proposed. In this approach, the network topology is described as a regular lattice and relates the statistical parameters of the hop count in the source-to-sink propagation path to the fractal properties of a percolating cluster. Based on this approach, a mathematical model is developed to estimate the probability P(r,t) of successful reception of the message by a sink node spaced at distance r from a source in a given time t. The accuracy analysis of the positioning method demonstrates its high performance for the very broad spectrum of values of the occupation probability pocc.
- Random network
- Lattice model
- Percolation theory
- Multihop channel
- Hop count
- Source-to-sink path
- Localization algorithm
At present, environmental sensing using electromagnetic waves can be performed using two different approaches. The first approach implies the use of devices that sense an environment remotely (e.g. radars or lidars). Another approach requires a number of sensor nodes, which are distributed in an application area, perform sensing directly inside the phenomenon and wirelessly communicate with each other and with the external reference nodes (sinks). The collected local data are then transmitted through such a wireless sensor network towards an end user in a multihop fashion. The ability to operate and monitor in harsh, complex and inaccessible environments makes the latter approach more promising and ensures a wide range of applications for wireless sensor networks. In particular, there already exist low-end sensor-network-based solutions aiming to detect a sniper location, monitor volcano intensity, protect forest against fire, control water waste in homes and the like. All these actual sensor networks consist of a small number of macro-scale nodes with their positions carefully engineered or pre-determined and capable of coarse-grained sensing. Further development of such sensor networks towards a wireless tiny-sensor network (WTSN) with superior fine-grained sensing capabilities would enable a wide spectrum of appealing applications in civil engineering and the health-care sector. Similar to its macro-scale counterpart, each node in a WTSN integrates a sensing, processing and communication unit and a power source. Due to its size restrictions, the tiny node imposes severe constraints on its power component. Since the communication coverage of such a node is thus very limited, the WTSN is foreseen as a swarm of densely deployed nodes within an application area: due to the high node density, the network can still support a long-distance transmission using multihop communication. The simple functionality of tiny nodes implies that the WTSN topology cannot be retrieved (a node has no energy to reconnoitre its surroundings). A large number of these nodes in turn entails that a random scattering over the area to be monitored is the most convenient or perhaps the only option to deploy them. All these features make the WTSN very distinguishable from the current sensor systems. Therefore, novel solutions at the networking level are needed to transfer and extract information gathered by the WTSN.
In particular, an efficient method capable of localising a transmitting node in the WTSN is required. The use of GPS on all nodes is unacceptable in terms of energy resources, node dimensions and its inability to operate indoor, so other techniques have to be considered. At present, there are two such major GPS-free techniques for positioning in multihop environments: centralised and distributed localisation. In the first approach, all ranges between all nodes in the network are measured, marked by unique node numbers and are transmitted to a global central unit, which processes and computes the location of each node in the network[9–11]. Evidently, the centralised approach is not a good candidate for the WTSN since it requires a very large communication overhead. Distributed localisation algorithms[12–15] require communication and ranging only with direct neighbouring nodes and are in general more robust and energy efficient than centralised schemes. With distributed localisation, each node in the network is capable of processing and computing to remove a lack of reliance on a single point of possible failure and provide with an optimum load balancing. Both approaches mentioned above require a large amount of energy to perform the prerequisite self-organization stage since the nodes must reconnoitre their surroundings to form a network topology. Due to modest processing capabilities and very limited energy resources of the nodes, the distributed localisation however becomes unaffordable and impracticable for the WTSN. In view of this, we propose a pioneering approach to estimate the location of a source node in the WTSN with arbitrary positions of nodes without consuming extra energy on the self-organization step. This approach employs an analysis of hop progress in such a network and uses time differences of last-arriving signals to its borders to position a source (initially transmitting) node.
Due to the complexity and the unknown locations of sensor nodes, determination of the hop count in the source-to-sink path of the WTSN is intractable with existing solutions, which are based on statistical laws (e.g. the Poisson distribution). Models available in the literature on the probabilistic analysis of the hop count in a large-scale and decentralised system are currently limited to two-dimensional (2D) networks. In particular, the probability of establishing a connection of a given length through a planar network with uniformly distributed nodes with a given number of hops has been investigated in. There it was shown that node density does not greatly affect the hop count but has a huge effect on whether a connection can be made at all. In turn, the distribution of Euclidean distances to the n th node in a Poisson process is studied in. For homogeneous sensor networks, the connectivity probability in one or two hops is derived and the connectivity in multiple hops is examined with analytical bounds in. In, the distribution of the hop distance and its expected value in uniformly distributed networks are analysed by means of numerical simulations. The results indicate that directional nodes might substantially decrease the hop count compared to omnidirectional ones, although the connectivity of a large-scale sensor system will degrade. Hence, as there are merely solutions suitable to determine the hop count in specific 2D WTSN structure, which require heavy computational resources to treat large-scale networks, a different model capable of analysing the source-to-sink multihop path in a 2D/3D WTSN and obtaining its hop count in a low-cost manner is developed here. It is notable that the model capable of analysing the hop progress can also be used to estimate the time required to pass a message through the WTSN and reach a sink node as well as determine other relevant parameters, such as degradation of network throughput and raised source localization cost[20–22].
In the proposed model, the WTSN topology is mapped to a regular lattice, where each lattice site is occupied by a sensor node with certain occupation probability. By relating statistical parameters (such as mean and variance) of the source-to-sink hop count to fractal properties of the percolating cluster in the corresponding lattice, the probability P(r,t) of successful reception of sensed data by a sink node spaced at Euclidean distance r from a source in a given time interval t can in particular be estimated. This probability quantifies the source-to-sink channel in the WTSN and may be regarded as a measure of the network reliability. Based on the hop progress model, the new positioning algorithm that requires no energy on the self-organization step is eventually proposed.
The remainder of the paper is organized as follows. Concepts of percolation theory and the system modelling from the percolation perspective are discussed in Section 2. In Section 3, a technique to determine the hop count of the source-to-sink channel in a WTSN and the probability P(r,t) is developed, whereas simulation and numerical results are provided in Section 4. Finally, the source-positioning method for the WTSN is introduced and discussed in Section 5, whereas the paper concludes in Section 6.
In the WTSN, the sensed data spreads through the flooding protocol (i.e. a node transmits a signal either when it is triggered by its sensor or when a signal is received from neighbouring nodes) as it is the most suitable way for a network with unknown node topology. To minimize interference overhead, a node is regarded not to broadcast the same message twice if it comes from different transmitting nodes with a delay in time. Each node is assumed to radiate omnidirectionally with a coverage distance b d , meanwhile the region covered by the WTSN is represented by a regular lattice structure with a given lattice spacing a d (in the following, the suffix d indicates the dimension).
Due to the regularity, the probability that a site of this lattice is occupied by a node is uniform and can be obtained as pocc = ρ · (a d ) d , where ρ is the given node density [m-d], and (a d ) d indicates the d-dimensional space occupied by a single site. Two occupied sites are interconnected only if either there exists a one-hop link or both of them share such links with another occupied site. The one-hop link exists solely when the sites are spaced at a distance less than b d , while occupied sites separated by a large range (i.e. ≫b d ) can still establish a connection through a multihop channel. The occupied sites form clusters in the lattice. When the occupation probability pocc is small, there is a sparse population of occupied sites, and clusters composed of small numbers of these sites predominate. However, by increasing pocc, more occupied sites become interconnected and thus become part of the same cluster. Eventually, for pocc to be large enough, the lattice experiences a critical phase transition: i.e., once pocc reaches the percolation threshold p c , an infinitely large percolating cluster of interconnected sensor sites emerges for the first time. For a finite-sized lattice, this cluster is bounded by the lattice edges, thus also called the spanning cluster. The percolation threshold value p c depends merely on the lattice geometry and the ‘lattice’ node coverage: the number and organization of sites are interconnected through a one-hop link. Note that due to an exceedingly complex structure of the percolating cluster in higher dimensions, it is cumbersome to determine the percolation threshold analytically except for the 1D and a few 2D lattices. In this respect, the value p c is typically estimated via a numerical experiment.
where pcl is the probability a site belongs to the percolating cluster (the power of two is due to the fact that both source and sink nodes must be part of this cluster), whereas p(r,t) is the probability a signal conveyed through such a cluster is received within the time interval t by the sink node at the distance r from the source. The analysis of pcl is carried out in Section 4, while the probability p(r,t) is modelled in the next section.
where μ and σ are the mean and variance corresponding to the hop count in the ‘hull’ multihop path, respectively. The parameter h is the actual number of hops made in time t, while the minimum number of hops needed to cover the source-to-sink distance r is equal to n. Hence, the probability p(r,t) increases monotonically from zero (h < n) to 1 (h ≫ μ), where μ formally corresponds to the average number of hops to be made to reach the sink node through a ‘hull’ multihop path.
As can be seen in Equation 4, the probability p(r,t) can be determined once the effective amplitudes c μ ,c σ and the fractal dimensions d μ ,d σ are known. Such fractal parameters merely depend on the lattice geometry and are estimated by means of numerical experiments (the modelling is simple and shown in Section 4). It is also notable that once these parameters are known, the probability p(r,t) can be assessed for any distance r without extra numerical simulations as the fractal parameters are independent of the source-to-sink separation. This is an essential advantage of the proposed model since; unlike other simulation solutions, it can treat a very large-scale sensor network in a simple way.
The percolation threshold p c and the size of a percolating cluster are numerically explored as both coefficients are of particular interest: the value p c distinguishes percolating from fragmented systems, while its size is directly related to the probability pcl that an occupied site belongs to the percolating cluster. To determine the fractal parameters related to the cluster hull, the hop count of a multihop path only composed of sites belonging to the hull is investigated as well.
To estimate these parameters, a set of Monte Carlo simulations are carried out. Due to computer limitations, the largest source-to-sink distance of a 2D lattice being analysed is 2,000 sites, whereas that of a 3D system is bounded by 500 sites. To get accurate results, each set of numerical experiments requires Q = 3,000 iterations. Per simulation, a signal is only transmitted to the adjacent occupied sites, which are found based on the left-hand maze rule in order to keep its propagation along the cluster hull (by keeping the left hand in contact with one wall of a maze, the player is guaranteed not to get lost and will reach a different exit if there is one). Once there exists a multihop path between the source site and sink spaced at distance n, the outcome is assumed to be successful and the hop count h of this path is stored. After applying this procedure Q times, the value p c is determined as the ratio between the number of successful outcomes and the total number of iterations, while the mean value μ and the standard deviation σ are eventually determined through using their common representations.
The percolation threshold of a 2D structure is found to be equal to pc(2) = 0.4073 ± 0.0007. This result is in agreement with the data presented in. The percolation threshold of a 3D lattice is significantly less than pc(2) and determined to be pc(3) = 0.0977 ± 0.0008. The further estimations are performed under the assumption that the occupation probability pocc exceeds the percolation threshold to ensure that a percolating cluster arises.
Fractal characteristics of 2D and 3D lattice structures
Eventually, once the parameters r and t are specified, the lowest density of nodes (i.e. the minimum value of pocc) needed to keep the probability P(r,t) above a given threshold can be determined by using the proposed model. Alternatively, for a given node density and time interval t, the distance r in the network can be obtained so that the probability P(r,t) still exceeds some specified level. Since the probability P(r,t) can basically be considered as a good indicator for the performance of a large-scale sensor network, the proposed model is relevant for handling such a system. The source-to-sink path with the maximum number of hops, which can be rigorously defined through the fractal characteristics of the hull of the percolating cluster, is also important from a communication perspective as its modelling helps to avoid inter-message interference in the WTSN (i.e. the source node must send more than one message to complete its report). However, the proposed model is not limited to such predictions on communication properties of a WTSN but also can be used to effectively position a source node in this network.
Once a source node in the WTSN triggers, its position is assumed to be detectable only when this node is located at one of the network borders (i.e. an end user can only get access and treat such ‘outward’ sensor nodes). If a message originates from one of the ‘inward’ nodes, there is no direct way to localise this node as it is inaccessible from an end user point of view. Nonetheless, if such an ‘inward’ source belongs to the percolating cluster, the originating message spreads throughout the WTSN and is only terminated by the lattice borders. By using the data collected at these borders and applying the developed model of hop progress in the WTSN, the position of the inward source node can reliably be estimated.
Evidently, the accuracy of the positioning method for the 2D WTSN can be treated in the same way as above. The ratio Δ(2) is thus reduced from Equation 9 and has the squared Z p and no term containing N k due to the two dimensionality.
In this paper, we propose a novel method to obtain in a simple manner the worst case number of hops of the source-to-sink path in a very large-scale network with an arbitrary network complexity and the unknown locations of the wirelessly connected nodes. The model assumes that the network topology can be represented by a regular lattice, where each lattice site is occupied by a sensor node with occupation probability pocc. The value pocc should exceed the percolation threshold p c for the emergence of a percolating cluster. Then, by relating statistical parameters (such as mean and variance) of the hop count of the source-to-sink path to the fractal parameters of the percolating cluster, the probability P(r,t) indicating a successful arrival of the sensed signal to a sink node spaced at distance r from a source within a specified time t has been mathematically expressed. The simple approach to estimate fractal parameters of the percolating cluster based on the left-hand maze rule has been used.
The numerical analysis has been performed for 2D and 3D conceptual lattices to better understand the impact of randomness on a large-scale network. It has been shown that the network with pocc being close to p c is an unreliable system from an application perspective. Meanwhile, it has also been demonstrated that the occupational probability pocc greatly affects the level of randomness in the network, i.e. the value P(r,t) increases pretty fast by slightly increasing pocc beyond p c . From this, the practical conclusion can be drawn that it is more effective to increase P(r,t) through adding extra sensors rather than by expanding the time t. The raise of P(r,t) with increasing number of sensors is much faster in 3D network than that in 2D due to having an extra degree of freedom. The inverse problem, i.e. the determination of the occupational probability needed to maintain the required P(r,t), can also be solved using the developed model. Note that although the model’s applicability is currently limited by nodes radiating omnidirectionally, the approach shown in the paper implies that the model can be expanded to nodes with directional coverage once the fractal parameters in the respective lattice structure are estimated.
An effective source-positioning method, which can localise a source node in a large-scale network with arbitrary positions of nodes and without retrieving the network topology, has been proposed based on the developed connectivity model. This localization method, which can be applied in very large scale networks, exploits the fractal nature of the percolating cluster. The accuracy analysis of the method has demonstrated its high performance for the very broad spectrum of values of the occupation probability pocc. As the counterpart approaches are impartially restricted to position in very large-scale networks, this method is the first of its kind to localise a source in these networks and thus is useful for designing such networks in the near future.
The authors thank the anonymous reviewers for their constructive and helpful comments.
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