Achieving Relative Time Synchronization in Wireless Sensor Networks

Abstract

Time synchronization is required for a variety of distributed control applications in wireless sensor networks (WSNs). In this paper, a novel distributed time synchronization technique for WSNs is presented. The proposed technique does not require all the sensor nodes to have the same time periods or starting times. Furthermore, the communication topology of the WSN is not assumed to be static; it can change with time. The WSN is modeled by a time-varying discrete-time system. The objective is to design local control inputs to achieve relative time synchronization by using the neighbor sensor nodes information. By employing tools from nonnegative matrix and graph theories, the convergence analysis is presented. Numerical examples are presented to demonstrate the effectiveness of the proposed protocol.

1. Introduction

Distributed time synchronization in wireless sensor networks (WSNs) has been an important topic of research in the control community [1–3]. In most distributed control applications, coordination among sensor nodes requires precise time synchronization [4, 5]. Time synchronization in a WSN is also required for many other tasks such as data fusion, transmission scheduling, and nodes sleep and wake-up scheduling. Therefore, it is important to design distributed protocols to achieve time synchronization in a WSN.

A common approach to design time synchronization protocols is to create a hierarchical structure within the WSN. The most prominent examples are the reference broadcast Synchronization (RBS) [6], timing-sync protocol for sensor networks (TPSN) [7], and flooding time synchronization protocol (FTSP) [8]. The implementation of RBS requires that the WSN should be divided into distinct clusters and each cluster should have a leader or reference node, all the leader nodes synchronize themselves with respect to each other and sensor nodes within the same cluster synchronize themselves with respect to their leader node. The implementation of TPSN and FTSP requires that a spanning tree rooted at a reference node should be built in the WSN and the time difference of any sensor node can be obtained with respect to the reference node. Although the hierarchy-based protocols have been experimentally tested in [8] and their performance is remarkable, they require substantial overhead to rebuild the spanning tree or clusters if a sensor node dies or a new sensor node is added.

Distributed time synchronization protocols which do not require a hierarchal structure have been proposed in [3, 9, 10], where the authors have considered a continuous time system to represent the clock of a sensor node. It is assumed that the clock of a sensor node has the same continuity property as the physical time and it can measure in any time interval. In actual implementation, the clock of a sensor node has some frequency or time period. A clock cannot measure a time interval which is shorter than its time period. Hence, it is more reasonable to use a discrete-time system to model a clock.

Discrete-time systems to model clocks in a WSN have been used in [1, 11, 12]. For the design of time synchronization protocol, the authors have assumed that all the sensor nodes should periodically exchange and update their clock states at the same time instants. This assumption is unrealistic since it requires synchronization of all clocks, which is the goal of the protocol. In this paper, we propose a time synchronization protocol which relaxes the three main assumptions used in the literature. First, the proposed protocol does not require all the sensor nodes to have the same time periods or starting times. Second, all the sensor nodes are not restricted to exchange and update their clock states at the same time instants. Third, the communication topology of the WSN is not assumed to be static. We use the discrete-time clock model presented in [1]. We assume that each sensor node knows its time period but does not know its exact starting time. Each sensor node exchanges its clock readings with its neighbors. The objective of the time synchronization protocol is to synchronize the clocks in a WSN with respect to a virtual clock, which is not physically present or accessible to any sensor node. The virtual clock is only used to demonstrate the concept of time synchronization in a WSN as the time periods and starting times of sensor nodes can be different.

The remainder of this paper is structured as follows: Section 2 presents some basic definitions and results from graph and matrix theories. The problem is formulated in Section 3. The main results and analysis are presented in Section 4. Numerical examples to demonstrate the effectiveness of the proposed protocol are presented in Section 5. Finally, concluding remarks are given in Section 6.

2. Preliminaries

In this section, we present some definitions and results from graph and matrix theories which will be used in this paper.

Let 𝟏=[1,…,1]^𝑇∈ℝ^𝑛, 𝐼_𝑛∈ℝ^𝑛×𝑛 denote an identity matrix, and the symbol ℕ denote the set of nonnegative integers. A matrix 𝑆 is nonnegative if all its elements are nonnegative [13]. A non-negative, finite, and square matrix 𝑆 is called stochastic if the sum of each row is 1. A stochastic matrix 𝑆 is called indecomposable and aperiodic (SIA) if there exists a column vector 𝑤 such that lim_𝑘→∞𝑆^𝑘=𝟏𝑤^𝑇 [14]. Let ∏^𝑘_𝑖=1𝑆_𝑖=𝑆_𝑘,𝑆_𝑘−1,…,𝑆₁ denote the left product of the matrices 𝑆_𝑘,𝑆_𝑘−1,…,𝑆₁. For any matrix 𝑆, we write 𝑠_𝑝𝑞 to represent the element which lies in the 𝑝th row and 𝑞th column of 𝑆.

Consider an undirected (directed) graph 𝒢 which consists of a vertex set, 𝒱(𝒢)={𝑣₁,…,𝑣_𝑛}, and an edge set, ℰ(𝒢)⊂{(𝑣_𝑖,𝑣_𝑗)∶𝑣_𝑖,𝑣_𝑗∈𝒱(𝒢)} where an edge is an unordered (ordered) pair of vertices [15]. The index set of neighbors of vertex 𝑣_𝑖 is denoted by 𝒩_𝑖={𝑗∶(𝑣_𝑗,𝑣_𝑖)∈ℰ(𝒢),𝑗≠𝑖} and its cardinality by |𝒩_𝑖|. A path from 𝑣_𝑖1 to 𝑣_𝑖𝑘 is a sequence, 𝑣_𝑖1,…,𝑣_𝑖𝑘 of distinct vertices such that (𝑣_𝑖𝑗,𝑣_𝑖𝑗+1)∈ℰ(𝒢) for any 𝑗=1,…,𝑘−1. An undirected (directed) graph 𝒢 is connected (strongly connected) if there is a path between any two distinct vertices. A directed tree is a directed graph where every vertex except the root vertex has exactly one parent vertex and the root vertex can be connected to any other vertices via paths. A spanning tree of directed graph 𝒢 is a directed tree whose edge set belong to ℰ(𝒢) and vertex set is the same as 𝒱(𝒢). The Laplacian matrix of 𝒢 is denoted by 𝐿=[𝑙_𝑝𝑞]∈ℝ^𝑛×𝑛 and is defined as follows: 𝑙_𝑝𝑞=−1if𝑝≠𝑞,𝑞∈𝒩_𝑝,𝒩_𝑝if𝑝=𝑞,0otherwise. (1) Let 𝑆=[𝑠_𝑝𝑞]∈ℝ^𝑛×𝑛 be a nonnegative matrix. Then, 𝒢(𝑆) is defined as a directed graph having vertex set 𝒱(𝒢(𝑆))={𝑣₁,…,𝑣_𝑛}, and edges from 𝑣_𝑞 to 𝑣_𝑝 if and only if 𝑠_𝑝𝑞>0. We present some lemmas which will be used later.

Lemma 1 (see [16]). Let 𝑆 be a stochastic matrix. If 𝒢(𝑆) contains at least one spanning tree such that the root vertex of that spanning tree has a self-loop in 𝒢(𝑆), then 𝑆 is SIA.

Lemma 2 (see [14]). Let Γ={𝑆₁,𝑆₂,…,𝑆_𝑘} be a finite set of SIA matrices such that for each sequence 𝑆_𝑖1,𝑆_𝑖2,…,𝑆_𝑖𝑗 of positive length, the matrix product 𝑆_𝑖𝑗,𝑆_𝑖𝑗−1,…,𝑆_𝑖1 is SIA. Then, for each infinite sequence 𝑆_𝑖1,𝑆_𝑖2,…, there exists a column vector 𝑤 such that the following holds: lim_𝑗→∞𝑆_𝑖𝑗,𝑆_𝑖𝑗−1,…,𝑆_𝑖1=𝟏𝑤^𝑇. (2)

3. Problem Formulation

We consider a WSN having 𝑛 sensor nodes. Each sensor node has a local clock which consists of a software program and a hardware oscillator. The oscillator periodically generates time interrupts using its time period and the software computes the local time of the sensor node at those time instants when the interrupts are generated [1]. Let 𝜏_𝑖, 𝑥_𝑖 and 𝑡^𝑖₀ denote the time period, clock value, and starting time of sensor node 𝑖, respectively, where 𝑖=1,…,𝑛. Each sensor node knows its time period but does not exactly know its starting time. Rather, its estimate, denoted by 𝑡^𝑖₀̂ for sensor node 𝑖, is known. Let 𝑇_𝑖={𝑡^𝑖_𝑘̃∶𝑘=0,1,….,} represent the sequence of time instants when the clock of sensor node 𝑖 updates its state, where 𝑡^𝑖_𝑘̃∶=𝑘𝜏_𝑖+𝑡^𝑖₀. The time computed locally by sensor node 𝑖 can be expressed as follows: 𝑥_𝑖𝑡̃^𝑖_𝑘+1=𝑥_𝑖𝑡^𝑖_𝑘̃+𝜏_𝑖,𝑥_𝑖𝑡^𝑖₀̃=𝑡^𝑖₀̂,𝑖=1,…,𝑛. (3) If each sensor node does not exactly know its starting time, then the clocks in the WSN would not be synchronized. To achieve time synchronization, we apply a control input to adjust the clock readings of sensor nodes. The control input is applied periodically at a certain known integer multiple of the local clock time period. Let 𝑡^𝑖_𝑠∶=𝑡̃^𝑖_{0+𝑠𝑧𝑖} be the time instants when the control input is applied to the clock of sensor node 𝑖, where 𝑧_𝑖 is a known integer, 𝑠∈ℕ and 𝑖=1,…,𝑛. We can write the following: 𝑥_𝑖𝑡^𝑖_𝑠+1=𝑥_𝑖𝑡^𝑖_𝑠+𝑧_𝑖𝜏_𝑖+𝑢_𝑖𝑡^𝑖_𝑠,𝑥_𝑖𝑡^𝑖₀=𝑥_𝑖𝑡^𝑖₀̃, (4) where 𝑢_𝑖(𝑡^𝑖_𝑠) denotes the control input or protocol which is needed to be designed. A virtual continuous time clock can be expressed as follows: 𝑣(𝑡)=𝑎𝑡+𝑏, (5) where 𝑡 represents the time of WSN, 𝑣 represents the clock readings, and 𝑎 and 𝑏 represent the time period and initial offset of the virtual clock, respectively. The virtual clock is not available to any sensor node. It is only used to demonstrate the concept of relative time synchronization. The clocks in a WSN are said to be synchronized if there exists a common virtual clock in the form of (5) such that the following holds: 𝑡̃lim_{𝑖𝑘→∞}𝑥_𝑖𝑡^𝑖_𝑘̃−𝑡^𝑖_𝑘̃+𝑏=0,∀𝑖=1,…,𝑛. (6) As each clock knows its own time period, we use 𝑎=1 for the common virtual clock. The sensor nodes communicate with their neighbors and exchange clock readings. The communication topology of the WSN can be modeled by a time-varying undirected graph, 𝒢(𝑡^𝑖_𝑠), where 𝑖=1,…,𝑛 and 𝑠∈ℕ. The neighbors of sensor node 𝑗 at 𝑡^𝑖_𝑠 consist of those sensor nodes which are able to communicate with node 𝑗 at 𝑡^𝑖_𝑠, where 𝑖,𝑗=1,…,𝑛 and 𝑠∈ℕ. Let 𝒩_𝑖(𝑡^𝑖_𝑠) denote the index set of sensor node 𝑖 neighbors and let 𝐿(𝑡^𝑖_𝑠) represents the Laplacian matrix associated with 𝒢(𝑡^𝑖_𝑠). The following assumptions are required to ensure that the clocks can be synchronized.(A1) There exists a known integer 𝜅 such that each sensor node 𝑖 can obtain information at least once from any possible neighbor during [𝑡^𝑖_𝑠,𝑡^𝑖_𝑠+𝜅], where 𝑖=1,…,𝑛 and 𝑠∈ℕ. (A2) There exists an integer 𝑝≥0 such that the union of graphs 𝒢(𝑡^𝑖_𝑠),𝒢(𝑡^𝑖_𝑠+1),…,𝒢(𝑡^𝑖_𝑠+𝑝), is connected, for all 𝑖=1,…,𝑛, and 𝑠∈ℕ. (A3) For any two sensor nodes 𝑖 and 𝑗, the following holds: 𝑡^𝑖_𝑠∈𝑇_𝑗,∀𝑠. (7)It should be noted that (A1) is made to ensure that each senor node can receive information from its possible neighbors in a finite interval of time. Assumption (A2) is made to ensure that union of communication topologies during a specific time interval remains connected, although at some particular time instants the communication topology may not be connected. Assumption (A3) is made so that by using the proposed protocol the relative time synchronization error in the WSN can be driven to zero. In actual implementation, if (A3) can not be ensured to hold for all sensor nodes, then it can be relaxed to consider neighboring nodes only. If (A3) is not ensured for neighboring nodes, then a small approximation error may exist due to the fact that the starting times and time periods of the sensor nodes may be different. In the next section, we present the main results to achieve relative time synchronization.

4. Main Results

We present the following protocol for achieving time synchronization: 𝑢_𝑖𝑡^𝑖_𝑠=𝛼_𝑖𝑡^𝑖_{𝑠𝑗∈𝒩𝑖𝑡𝑖𝑠}𝑥_𝑗𝑡^𝑖_𝑠−𝑥_𝑖𝑡^𝑖_𝑠,𝑖=1,…,𝑛, (8) where 𝛼_𝑖(𝑡^𝑖_𝑠) denotes a gain which needs to be designed. For synchronized distributed control systems, a protocol similar to (8) has been presented in [17]. In this paper, we study protocol (8) for asynchronous distributed control systems. The following theorem presents the main results.

Theorem 3. Consider a WSN which satisfies assumptions (A2) and (A3). Then, the clocks of the sensor nodes achieve relative time synchronization using protocol (8) if 𝛼_𝑖(𝑡^𝑖_𝑠)∈(0,1/|𝒩_𝑖(𝑡^𝑖_𝑠)|), ∀𝑖=1,…,𝑛 and 𝑠∈ℕ.

Remark 4. If the communication topology of the WSN is static, then we can allow 𝛼_𝑖(𝑡^𝑖_𝑠)∈(0,1/𝑑), ∀𝑖=1,…,𝑛 and 𝑠∈ℕ, where 𝑑 denotes the maximum of the diagonal entries of 𝐿.

The proof of Theorem 3 consists of three main steps. In the first step, we combine the local clock models and define a new time sequence to represent the local clock models by a single time-varying system. In the second step, we use the new time sequence to compute an upper bound on the time interval in which each sensor node obtains information at least once from all its neighbors. Finally, we complete the proof by utilizing the properties of connected graphs and product of SIA matrices.

4.1. Representation by a Single System

Before grouping the local clock models to represent them by a single system, we define a new state as follows: 𝑦_𝑖𝑡^𝑗_𝑠=𝑥_𝑖𝑡^𝑗_𝑠−𝑡^𝑗_𝑠−𝑡₀,𝑖,𝑗=1,…,𝑛. (9) We rewrite the model expressed by (4) and control protocol (8) as follows: 𝑦_𝑖𝑡^𝑖_𝑠+1=𝑦_𝑖𝑡^𝑖_𝑠+𝑢_𝑖𝑡^𝑖_𝑠,𝑦_𝑖𝑡^𝑖₀=𝑥_𝑖𝑡^𝑖₀,𝑢_𝑖𝑡^𝑖_𝑠=𝛼_𝑖𝑡^𝑖_{𝑠𝑗∈𝒩𝑖(𝑡𝑖𝑠)}𝑦_𝑗𝑡^𝑖_𝑠−𝑦_𝑖𝑡^𝑖_𝑠,𝑖=1,…,𝑛. (10) Let 𝑡_ℎ=max{𝑡^𝑖₀̃,𝑖=1,…,𝑛}, and we define a new time sequence 𝑇∶={𝑡_𝑟}, where 𝑟∈ℕ, which is the union of the distinct values of the sequences {𝑡^𝑖_𝑠≥𝑡_ℎ,𝑖=1,…,𝑛,𝑠∈ℕ}. We order the values of 𝑇 such that 𝑡_𝑟<𝑡_𝑟+1. We mention here that the distances between consecutive elements of 𝑇 are nonuniform. The time-synchronization objective can be rewritten as follows: lim_{𝑡𝑟→∞}𝑦_𝑖𝑡_𝑟−𝑏=0,𝑖=1,…,𝑛, (11) for some 𝑏. Let 𝑦(𝑡_𝑟)=[𝑦₁(𝑡_𝑟),…,𝑦_𝑛(𝑡_𝑟)]^𝑇 denote the group state vector. The dynamics of the overall system can be written as follows: 𝑦𝑡_𝑟+1=𝐼_𝑛−𝐴𝑡_𝑟𝐿𝑡_𝑟𝑦𝑡_𝑟, (12) where 𝐴(𝑡_𝑟)=[𝑎_𝑝𝑞], 𝐿(𝑡_𝑟)=[𝑙_𝑝𝑞], 𝑎_𝑝𝑞=𝛼_𝑝𝑡_𝑟if𝑝=𝑞,∃𝑠s.t.𝑡_𝑟=𝑡^𝑝_𝑠,𝑙0otherwise,_𝑝𝑞=−1if𝑞∈𝒩_𝑝𝑡_𝑟,∃𝑠s.t.𝑡_𝑟=𝑡^𝑝_𝑠,𝒩_𝑝𝑡_𝑟if𝑝=𝑞,∃𝑠s.t.𝑡_𝑟=𝑡^𝑝_𝑠,0otherwise. (13) It should be noted that 𝐿(𝑡_𝑟) is a time-varying Laplacian matrix which can be unsymmetric and the graph associated with 𝐿(𝑡_𝑟) may not be strongly connected. Let Φ(𝑡_𝑟)=𝐼_𝑛−𝐴(𝑡_𝑟)𝐿(𝑡_𝑟), the solution of (12) can be written as follows: 𝑦𝑡_𝑟=Φ𝑡_𝑟Φ𝑡_𝑟−1⋯Φ𝑡₀𝑦𝑡₀,𝑟=0,1,…. (14) If 𝛼_𝑖(𝑡^𝑖_𝑠)∈(0,1/|𝒩_𝑖(𝑡^𝑖_𝑠)|)∀𝑖,𝑠, then Φ(𝑡_𝑟) are always stochastic matrices with positive diagonal entries. The product Φ(𝑡_𝑟)Φ(𝑡_𝑟−1)Φ(𝑡_𝑟−2)⋯Φ(𝑡₀) is also a stochastic matrix with positive diagonal entries. With this preparation, we proceed to the next step.

4.2. Computation of an Upper Bound on the Time Interval

The following lemma is quite useful in the computation of an upper bound on the time interval in which each sensor node can obtain information at least once from its neighbors.

Lemma 5. Let 𝑔=max{𝑧_𝑖𝜏_𝑖}/min{𝑧_𝑖𝜏_𝑖} where 𝑖=1,…,𝑛. For any sensor node 𝑖=1,…,𝑛 and 𝑠=1,2,…, the number of elements in the set {𝑡_𝑝∶𝑡_𝑝∈[𝑡^𝑖_𝑠,𝑡^𝑖_𝑠+1)} is not greater than ℎ, where ℎ=(⌈𝑔⌉)(𝑛−1)+1 and ⌈𝑔⌉ denotes the smallest integer not less than 𝑔.

Proof. Let 𝑗 be the sensor node such that 𝑧_𝑗𝜏_𝑗≥𝑧_𝑖𝜏_𝑖 where 𝑖=1,…,𝑛,𝑖≠𝑗. When sensor node 𝑗 updates its control input at 𝑡^𝑗_𝑠, the maximum number of times any sensor node 𝑖 can update its control input before the time instant 𝑡^𝑗_𝑠+1 cannot exceed ⌈𝑔⌉, where 𝑖=1,…,𝑛,𝑖≠𝑗. Since there are 𝑛−1 possible values of sensor node 𝑖 (excluding 𝑖=𝑗), we conclude that the number of elements in {𝑡_𝑝∶𝑡_𝑝∈[𝑡^𝑗_𝑠,𝑡^𝑗_𝑠+1)} is not greater than ℎ.

Using Lemma 5 and assumption (A1), we can state that each sensor node obtains information from all its neighbors at least once during the time interval [𝑡_{𝑚−𝜅ℎ},𝑡_𝑚], where 𝑚=𝜅ℎ,𝜅ℎ+1,….

4.3. Technical Analysis

Let 𝒢(𝑡_𝑚) be the graph obtained by the union of 𝒢(Φ(𝑡_𝑚)),…,𝒢(Φ(𝑡_{𝑚−𝜅ℎ})). Using (A2), we conclude that 𝒢(𝑡_𝑚) is strongly connected. If 𝒢(𝑡_𝑚) is strongly connected, then, by using Lemma 1 together with [18, Lemma 3.9], we establish that the product of Φ(𝑡_𝑚)⋯Φ(𝑡_𝑚−𝑞) is SIA.

Let 𝐹₀=∏^ℎ−1_𝑙=0Φ(𝑡_𝑙), 𝐹₁=∏^2ℎ−1_𝑙=ℎΦ(𝑡_𝑙), 𝐹₂=∏^3ℎ−1_𝑙=2ℎΦ(𝑡_𝑙) and so on. Then, it can be easily shown that all possible unique 𝐹_𝑗’s are finite, where 𝑗=0,1,…. Any product which involves a finite combination of the unique 𝐹_𝑗’s is SIA. So, we can apply Lemma 2 to get the following conclusion: lim^𝑘_{𝑘→∞𝑗=0}𝐹_𝑗=𝟏𝑤^𝑇. (15) The vector 𝑤 depends on the order in which the matrices Φ(𝑡_𝑙) are multiplied, where 𝑙=0,1,…, and it cannot be computed in advance. We can further write the following: lim_{𝑡𝑟→∞}𝑦𝑡_𝑟=𝟏𝑤^𝑇𝑦𝑡₀=𝑤^𝑇𝑦𝑡₀⋮𝑤^𝑇𝑦𝑡₀. (16) Hence, we complete the proof of time synchronization protocol by taking 𝑏=𝑤^𝑇𝑦(𝑡₀).

5. Numerical Examples

In this section, we present two examples to demonstrate the effectiveness of the proposed time synchronization protocol.

Example 1 (static topology, small-size WSN). We consider a WSN having 10 sensor nodes and label them as 1,…,10. We assume that the communication topology of the WSN is not changing with respect to time (𝜅=0) and is shown in Figure 1. Using Theorem 3, we choose 𝛼_𝑖=0.3,∀𝑖. The values for different parameters satisfying (A2) and (A3) are listed in Table 1. By applying protocol (8), the WSN was simulated for 50 seconds. Figure 2 shows a plot of the clock readings for 50 seconds. Figure 3 illustrates a plot of the local control inputs applied to the sensor nodes. Based on the plots, it can be concluded that this WSN achieves distributed time synchronization using the proposed protocol.

Table 1 

Parameter values used to simulate the WSN as shown in Figure 1.

Figure 1 

Communication topology of a WSN considered in Example 1 having 10 sensor nodes. The lines indicate the presence of communication channels for exchanging clock readings.

Figure 2 

Plot of the local clock readings of a WSN considered in Example 1. The topology of the WSN is shown in Figure 1.

Figure 3 

Plot of the local control inputs applied to the WSN considered in Example 1. The topology of the WSN is shown in Figure 1.

Example 2 (switching topology, medium-size WSN). We consider a WSN having 45 sensor nodes and label them with 1,…,45. We consider that the communication links in the WSN are changing randomly with respect to time. Figure 4 shows the union of possible communication topologies of the WSN. We randomly select the parameter values for different sensor nodes from Table 1 and choose 𝜅=3. For simulation purposes, we compute the relative time synchronization error of sensor node 𝑖 with respect to its neighbor nodes, denoted by 𝑒_𝑖(𝑡_𝑟), as follows: 𝑒_𝑖𝑡_𝑟=_{𝒢𝑗∈𝒩𝑖𝑡𝑟}𝑥_𝑖𝑡_𝑟−𝑥_𝑗𝑡_𝑟,𝑖=1,…,45. (17) Applying Theorem 3, we simulate the WSN for 80 seconds and select the values for 𝛼_𝑖 at run time based on the communication topology. Figure 5 shows a plot of the sum of time synchronization errors. Based on the plots, it can be concluded that, by using protocol (8) and applying Theorem 3, this WSN also achieves distributed time synchronization.

Figure 4 

The graph of union of communication topologies of a WSN considered in Example 2 having 45 sensor nodes labeled as 1,…,45. The lines indicate the presence of communication channels only and do not indicate if at a particular time instant any information is exchanged on the communication link.

Figure 5 

Plot of the sum of the relative time synchronization errors in the WSN considered in Example 2. The graph of the union of communication topologies of the WSN is shown in Figure 4.

6. Conclusions and Future Research Directions

In this paper, we presented a distributed time synchronization protocol for WSNs. The time periods and starting times of the sensor nodes were not restricted to be the same. By employing tools from stochastic matrix and algebraic graph theories, we presented the convergence analysis. In this paper, we considered that a sensor node could access the clock readings of its available neighbors without any delays. In actual implementation, time delay occurs between transmission and reception of information in a WSN. In the future work, we can consider this by modifying the protocol to handle time-delayed measurements from neighbors.

Acknowledgment

This research work was supported by the Natural Sciences and Engineering Research Council of Canada. Feng Xiao’s work was supported in part by NSFC (61273030).