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MRF), Potts Model, Wireless Sensor Networks (Wsns American Journal of Intelligent Systems 2012, 2(7): 157-162 DOI: 10.5923/j.ajis.20120207.01 Adaptive Scheduling in Wireless Sensor Networks Based on Potts Model Afsoon Assa, Majid Vafaei Jahan* Department of Computer Engineering, Islamic Azad University, Mashhad Branch, Mashhad, Iran Abstract Recent advances in wireless sensor network's technology have developed using this technology in various fields. For the importance and intense application of these networks, many efforts and researchers have been done to confront challenges. In this paper, to conquer one of the most important challenges of these networks, that is the limitation in energy resources, an adaptive scheduling algorithm based in Potts model was applied. According to this model, for every element within a system, q different states are considered. Each element converts its state to a new state or remains in that state due to its current state and its adjacent neighbours and with the effect of environment. In this project, this model was used in wireless sensor networks such that each sensor node is considered as an element of sensor network system and three active, inactive and standby states are defined for that node, which it selects one of these three states according to its current state and neighbours and environment affect and adapts its activity on the environment in a way that the least energy to be consumed. By comparing this algorithm and similar algorithm (with two states), it is observed that in identical conditions, Potts model with three states represent better results and more lifetimes for network. Keywords Ising Model, Markov Random Field (MRF), Potts Model, Wireless Sensor Networks (Wsns) and unpredictable. So if operational cycles of Nodes are 1. Introduction reduced, when nodes are not active, an event may occur, which its incorrect detection and tracking led to serious Wireless Sensor Networks (WSNs)[1] consist of the consequences and irreparable losses. Many researches were limited resources constrained devices, called sensor node, done due to wireless sensor Networks's challenge and energy which communicate wirelessly to one another. These nodes saving of these Networks, particularly in application of are able to recognize changes of states occur in their impact target tracking and various algorithms has been designed. domain and to report processed obtained data to respective One of these algorithms is an algorithm of adaptive sensor centres. This property causes usage of this technology activity scheduling (A-SAS) algorithm[2] in distributed various applications. One of them are to track a target which sensor networks, where the sensor network is modeled as a is used in different applications in benign environments and Markov random field on a graph. The sleep and wake modes also harsh ones. As mentioned previously, these networks of a sensor node are modeled as spins with ferromagnetic were comprised of nodes with limited energy resources; neighborhood interactions in the Statistical Mechanics therefore, one of the most important challenges of these setting. According to this algorithm, node's activities are networks is energy limitation and attempt to increase the scheduled such that network nodes could be able to detect operational lifetime of network. To use effectively of sensor targets all the time and determine the target's path correctly. networks, which have been different spins.requires The main goal of this paper is to develop the algorithm such resource-aware operation and takes care of a resource as that besides detect objective correctly, nodes to be scheduled much as possible; this is because sensor nodes are distributed in a way that to consume the least energy. In the proposed only once, and resources of sensor nodes are hardly ever algorithm, Potts model was used to develop the algorithm. rechargeable. To decrease energy consumption of network, it The paper is organized in four sections, including the present is usually tried to reduce node's duty cycles, but in such one. In section 2, Ising model and Potts model will be applications of Sensor networks for event detection and introduced. The proposed algorithm will be studied in tracking, it is impossible to use a fixed rule to decrease section 3. In section 4, after evaluating algorithm efficiency, node's duty cycles because events of interest are often rare the algorithm will be compared with Ising model with the same conditions, and in the last section, conclusions will be * Corresponding author: given. [email protected] (Majid Vafaei Jahan) Published online at http://journal.sapub.org/ajis Copyright © 2012 Scientific & Academic Publishing. All Rights Reserved 2. Introducing to Ising and Potts Model 158 Afsoon Assa et al.: Adaptive Scheduling in Wireless Sensor Networks Based on Potts Model The Potts model studies long term behaviour of complex This equation computes the probability of finding a 1 systems. The model is able to investigate how the internal particular state ω in possible states set Ω, where, β = ,T 퐾푇 element of the system reacts with each other based on certain represents the temperature of the system, and k = 1.38 × characteristics that each element has. As these reactions take 10-23 (joules/Kelvin) is the Boltzmann constant. However, place macroscopic properties of the system will evolve. The computing the partition function is only tractable for small Potts model has proven to be a very useful tool for simulating lattices and small values of q. In general, this function is real systems. This model is used in the area of mathematical NP-hard to compute. modelling so called statistical mechanics. In this model, the Mathematicians explore properties of the Potts model system is mapped to a graph such that graph nodes are the partition function in a variety of ways. One way is to comprising elements of the system, and its edges are the interpret it as an evaluation of the Tutte polynomial[7]. connections among these elements. For many applications, it Another is to approximate the function using a simulation is expedient to presume that the graph has a regular structure, technique such as the Metropolis Algorithm[8]. This such as a lattice. On the vertices of these graphs, anything calculation is not exact; however, it allows researchers to use can be considered, like atom, human, liquids and cells and the Potts model to investigate complex applications. even sensor node. Potts model models how the nearest elements with different spins are related to other elements of the lattice[3]. 3. The Proposed Method This model is the extended of Ising model[4,5]. In Ising model, two states are considered for each vertex, but in Potts The idea of using Potts model in wireless sensor networks model, q different states can be considered for every vertex. was expressed in an algorithm called A-SAS[2]. This Since the elements are assigned different spins and reacts algorithm is a distributed algorithm for wireless sensor with one another depending on their position on the lattice networks to be able to detect and track rare and random and their specific spins, there will be some measure of events. However, in this algorithm, Ising model was used, overall energy of the system. The function which measures that is for sensor node, just two active and inactive states the overall energy of a complex is the Hamiltonian[6]. were considered. The algorithm which is represented here is extended of (1) A-SAS algorithm using Potts model. Therefore, instead of two states for nodes, three states are considered and the third Where, J is the interaction energy between adjacent state as standby is added. The sensor nodes are presumed to elements of the system, and σi is the spin value assigned to be equipped with relevant sensing transducers and data vertex i in the state ω, and a 1 is placed on edges between processing algorithms as needed for detection and tracking neighbors with like spins and a 0 on edges with elements of events under consideration. Node's task is that until which have been different spins. resources are available, they detect and track rare events. Problem assumptions: (1) the network only deployed once. (2) Sensor nodes are static. The neighbours are fixed and . predetermined. (3) Sensor node connected to its nearest neighbours through a short single-hop wireless 2.1. The Potts Model Partition Function communication. For a graph with n vertices which its vertex can have q Sensor network is considered as a weighted graph. If G ≜1 n different spins, there are q states. In fact, Potts model studies (S, E, W) is supposed a weighted graph where S = {s1, s2, ... these states. The Potts model probability function is the sN} and N ∈ℕ is the set of all nodes of the sensor network function which calculates the probability of finding the and E is the set of all edges of graph G which each edge lattice in a particular state. This function depends on the specifies the distance between two nodes of S that are Boltzmann distribution in statistical mechanics. Equation 2 connected together with single-hop and is determined by an shows above function with exponential distribution. edge (si, sj) ∈ E. Function W expresses the weight of the graph's edge and has the value W((si, sj)) = wij. The strength (2) of interaction this function is dictated by factors such as physical distance between the sensor nodes. Where, To describe the problem, some definitions have to be stated: (1) Markov Random Field: suppose ∂ ≜ {∂i} si ∈ S be (3) the neighbour set for a node si. Then, with respect to ∂, a random field is called a Markov Random Field (MRF) if and Equation (3) is called Partition function of this model. only if hold the Markov properties (I.e.
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