Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 269781, 14 pages

http://dx.doi.org/10.1155/2015/269781

## A Novel OBDD-Based Reliability Evaluation Algorithm for Wireless Sensor Networks on the Multicast Model

^{1}Guangxi Key Laboratory of Trusted Software, School of Computer Science and Engineering, Guilin University of Electronic Technology, Guilin 541004, China^{2}Zhongxing Telecommunication Equipment Corporation (ZTE Corporation), Shenzhen 518000, China

Received 16 July 2014; Revised 4 November 2014; Accepted 5 November 2014

Academic Editor: Xue Jun Li

Copyright © 2015 Zongshuai Yan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The two-terminal reliability calculation for wireless sensor networks (WSNs) is a #P-hard problem. The reliability calculation of WSNs on the multicast model provides an even worse combinatorial explosion of node states with respect to the calculation of WSNs on the unicast model; many real WSNs require the multicast model to deliver information. This research first provides a formal definition for the WSN on the multicast model. Next, a symbolic OBDD_Multicast algorithm is proposed to evaluate the reliability of WSNs on the multicast model. Furthermore, our research on OBDD_Multicast construction avoids the problem of invalid expansion, which reduces the number of subnetworks by identifying the redundant paths of two adjacent nodes and *s-t* unconnected paths. Experiments show that the OBDD_Multicast both reduces the complexity of the WSN reliability analysis and has a lower running time than Xing’s OBDD- (ordered binary decision diagram-) based algorithm.

#### 1. Introduction

Wireless sensor networks (WSNs) have important applications in many prominent areas, such as military, medical industry, and environmental health monitoring [1]. All of these applications require a high level of reliability to operate safely and effectively [2]. If the WSN is unreliable, the tasks may not be completed wasting sensor resources. In order to minimize the cost while maximizing the reliability, first evaluating the reliability is an indispensable step before the successful deployment of WSNs [3].

WSN communication can be divided into two categories: infrastructure communication and application communication [4]. Infrastructure communication relates to transmitting the control and configuration messages from the sink to the sensors, while application communication relates to the transmission of the sensed data from the sensors to the sink. The infrastructure consists of sensors and their current deployment status; thus, infrastructure communication is needed to keep the network functional, ensuring robust operation even in hostile environments, while also optimizing the overall performance. Park and Sivakumar classify data delivery models into three models: sink to a single sensor node (unicast), sink to a group of sensors (multicast), and sink to all sensors (broadcast) [4]. In this paper, in accordance with the reliability problem experienced by WSNs on the multicast model proposed by Shrestha et al. [5], we perform in-depth research on the algorithm used to evaluate the reliability of WSNs on the multicast model, namely, the -terminal reliability of WSNs.

Computing the two-terminal reliability of WSNs is #P-hard [6]. Calculating the -terminal reliability for WSNs provides an even worse combinatorial explosion of node states with respect to the calculation of WSNs on the unicast model.

In effectively evaluating the reliability of large-scale WSNs, the process is more difficult using traditional methods, such as factoring, inclusion-exclusion algorithm, and the sum of disjoint products (SDP). Recently, an implicitly symbolic representation and manipulation technique, called a symbolic graph algorithm or symbolic algorithm [7, 8], has emerged in order to combat or ease combinatorial state explosion. Ever since Akers proposed the binary decision diagram (BDD) [9] and Bryant popularized the use of BDD by introducing decision variable ordering and simplified rules [10, 11], which make the BDD a canonical form (OBDD) for a Boolean function. Typically, symbolic algorithms based on OBDD are efficient methods in computing the network reliability. Yeh et al. [12] proposed an efficient approach to compute the reliability of an undirected -terminal network based on 2-terminal reliability functions. The approach constructs the 2-terminal reliability functions of the () terminal-pairs based on the edge expansion diagram using OBDD and then constructs the *-*terminal reliability function by combing these () 2-terminal reliability functions with OBDD’s “AND” operation. Ghasemzadeh [13] presented a method to analyze the -terminal reliability based on factoring using the “If-Then-Else(ITE)” operation of OBDD. Hardy et al. [14] studied -terminal network reliability using network decomposition based on edge deletion/contraction with OBDD and recognizes isomorphic subgraphs represented by boundary set in order to reduce redundant computations. However, Yeh, Ghasemzadeh, and Hardy’s methods assumed edges were unreliable without considering the case of node failure. This assumption is impractical for WSNs; WSN’s sensors have limited power and are usually deployed in inaccessible terrains and even hostile environments. As a result, the sensors are prone to failure as a result of energy depletion or the natural factors such as earthquakes and landslides. We cannot ignore the impact of node fault on the reliability of the WSN. So these studies cannot be directly applied to WSNs.

Shrestha et al. [5] improved upon Yeh’s algorithm [12] to analyze the reliability of WSNs on the multicast model under common cause failures, constructed the OBDD for the reliability function, and decreased the redundant computations from isomorphic subnetworks with the aid of a hash table.

In this paper, we focus on the WSN reliability analysis on the multicast model based on OBDD and node expansion. To evaluate the reliability, we have presented a symbolic algorithm named OBDD_Multicast. OBDD_Multicast finds the variable ordering of nodes using a breadth-first search (BFS), decomposes the WSN using node expansion to construct the OBDD using its “AND” and “OR” operations, and then combines these OBDDs using OBDD’s “AND” operation to find the OBDD of the reliability function. OBDD_Multicast reduces redundant computations by recognizing the redundant paths of two adjacent nodes and *-* unconnected redundant paths in addition to identifying isomorphic subgraphs. Experiments show that OBDD_Multicast reduces the complexity of WSN reliability analysis by identifying the two types of redundant paths that lead to invalid expansions and has lower running time than Xing’s OBDD-based algorithm.

#### 2. Preliminaries

##### 2.1. The Model of a WSN on the Multicast Model

The graph of a WSN should be drawn before constructing a model of the WSN. In this study, we assume all sensors belonging to a WSN were the same. We assumed if a sensor named A is in the communication range of a sensor B, then sensor B is also in the communication range of sensor A, in which case AboElFotoh’s method [6] of transforming a WSN into an undirected graph is a better choice. Figure 1 shows the basic idea behind this method, where (a) is a wireless network and (b) is the corresponding graphic model. The figure shows a bidirectional edge exists between each node if they are in range of each other. The edge A-B in Figure 1(b) indicates that there is a communication path from A to B as well as from B to A. In this study, we use a unidirectional edge to represent the bidirectional edge.