# Analysis of Distributed Consensus Time Synchronization with Gaussian Delay over Wireless Sensor Networks

- Gang Xiong
^{1}and - Shalinee Kishore
^{1}Email author

**2009**:528161

https://doi.org/10.1155/2009/528161

© G. Xiong and S. Kishore. 2009

**Received: **28 June 2008

**Accepted: **6 May 2009

**Published: **11 June 2009

## Abstract

This paper presents theoretical results on the convergence of the distributed consensus timing synchronization (DCTS) algorithm for wireless sensor networks assuming general Gaussian delay between nodes. The asymptotic expectation and mean square of the global synchronization error are computed. The results lead to the definition of a *time delay balanced network* in which average timing consensus between nodes can be achieved despite random delays. Several structured network architectures are studied as examples, and their associated simulation results are used to validate analytical findings.

## 1. Introduction

Wireless sensor networks are typically comprised of inexpensive, small-sized, power-limited terminals. In a variety of applications, sensor nodes are required to maintain accurate time synchronization, for example, moving object tracking, reconnaissance and surveillance, environmental monitoring, and so forth [1]. This necessitates algorithms that achieve and maintain *global time synchronization* at all network nodes, that is, algorithms that align all nodes to a common notion of time.

Due to imperfections in low-cost hardware nodes and the decentralized nature of wireless sensor networks, global time synchronization has been recognized as a particularly challenging task. Recently, several distributed time synchronization algorithms have been proposed; one such class is *distributed consensus time synchronization* (DCTS) [2]. In the DCTS approach, a global time consensus can be sufficiently reached within a connected network by averaging pairwise local time information. In [3], Olfati-Saber et al. established a theoretical framework for the analysis of consensus synchronization algorithms. Later, a fully distributed, asynchronous DCTS algorithm was proposed in [4]; this scheme was designed to reach agreement on time offset and skew offset between network nodes using media access control (MAC) layer time-stamped packet exchanges. As an alternative, a physical layer-based DCTS algorithm was introduced in [5] by modeling sensor nodes as coupled discrete time oscillators. Based on our knowledge, the existing body of literature on the DCTS approach does not examine the effects of time delay uncertainty between network nodes. In this paper, we study the convergence of the DCTS algorithm when uncertain delays impact local pairwise time information exchange.

In [6], Xiao et al. considered distributed average consensus with additive noise and investigated the design of network link weights to minimize the mean-square deviation in steady state. In this paper, we analyze the convergence characteristics of the DCTS algorithm under Gaussian delay uncertainties. First, we determine the asymptotic expectation of the global synchronization error. Our results lead to the definition of a *time delay balanced network*, and we claim that under such network topologies average timing consensus between nodes can be achieved despite the presence of random delays. Additionally, we show that the asymptotic mean square synchronization error is lower and upper bounded by several values related to network parameters. As examples, we analyze the global synchronization error of the DCTS algorithm for several structured networks.

This paper is outlined as follows. Section 2 provides background and system model for the DCTS algorithm studied here. Section 3 presents convergence results on the synchronization error of the DCTS algorithm due to Gaussian random delays between nodes. Section 4 discusses the convergence characteristics of the global synchronization error for several structured networks. Simulation results are presented in Section 5, and we conclude our discussion in Section 6.

## 2. Background and System Model

### 2.1. Time Delay for Local Time Information Exchange

The DCTS algorithm requires local time information exchange between two or more nodes in a wireless sensor network. This exchange can occur using either MAC layer time-stamped packets or via physical layer pulse signals. In either case, the delay between two network nodes is defined as the interval between when the time information is generated by the sender node and when this information is determined by the receiver node. Furthermore, in either case, this delay can be comprised of a deterministic and a random portion. In the following, we discuss the delay sources at the two layers and argue that, in both cases, a common underlying model of Gaussian delay uncertainty can be adopted. (We have separately examined the performance of the DCTS algorithm considering alternate delay distributions, e.g., exponential delay distribution [7]. Results show similar performance bounds as those presented in this paper for the Gaussian assumption. For this reason, we constrain our discussion here to the more common Gaussian delay model.)

#### 2.1.1. Physical Layer-Based Time Delay

#### 2.1.2. MAC Layer-Based Time Delay

where is a constant equal to zero for physical layer-based schemes and for MAC layer-based schemes; and is a zero mean Gaussian random variable. The variance of , , is equal to for physical layer schemes and to for MAC layer-based schemes.

### 2.2. DCTS Algorithm With Gaussian Delay

In each iteration of the DCTS algorithm, each node processes and decodes the time-stamped message from its neighbors in the MAC layer-based approach or estimates the arrival time of its neighbors' pulse signals in the physical layer scheme. Each node then updates its local clock time using the weighted average of the time differences with its neighbor nodes. It is well known that in a connected network with nonrandom delay between nodes, this DCTS algorithm can reach *average consensus* [10]; that is, all nodes converge to the average of the initial timing differences between the nodes.

where . It should be noted that and might not be independent between nodes and since the two nodes might have identical noise coming from some potentially overlapping neighbors.

### 2.3. Network Model and Some Preliminaries

In the following, we model a wireless sensor network as an undirected graph , consisting of a set of nodes and a set of edges . (The convergence properties presented here can be easily extended for a directed graph. We omit this extension here.) Each edge is denoted as where and are two nodes connected by edge . We assume that the presence of an edge indicates that nodes and can communicate with each other reliably. We assume here a connected graph; that is, there exists a path connecting any pair of distinct nodes in the network.

where is called a Perron matrix of a graph with parameter [3]. Here, denotes the identity matrix. The eigenvalues of are and can be ordered in decreasing order: . It is worth mentioning that the constant step size which minimizes convergence time is given as [12]. (Note that the optimal is generally difficult to obtain as it involves computing the eigenvalues of the Laplacian matrix . However, in practical applications, a numerical solution can be obtained offline based on node deployment within a given wireless sensor network, and this can then be flooded to all nodes before they run the DCTS algorithm.) Let us define and , where . Then the noise vector in (8) is given as .

When there is no Gaussian delay between nodes, it can be shown that [10, 12], for a time-invariant, connected, undirected network, when
, *average consensus* can be asymptotically achieved by the DCTS algorithm, that is,
In our discussion, we also assume an undirected, connected network with a constant step size
unless otherwise stated.

In the following analysis, we use the following matrices: , and . For matrices and , it is straightforward to show that the eigenvalues of agree with those of except that is replaced by ; such that ; and and .

## 3. Convergence Analysis of DCTS Algorithm with Gaussian Delay

Let us define the average value in each iteration as . Then, mean of the average value in each iteration of the DCTS algorithm is , and the variance of the average value is It can be seen that as iteration time increases, both mean and variance increase linearly with the time index . Furthermore, the variance of increases linearly with the variance of the random Gaussian delay, .

### 3.1. Expectation and Second Central Moment of Disagreement Vector

We now define the disagreement vector as ; that is, is the difference between the updated times and the actual average times of the network nodes. Then, the disagreement vector evolves as .

Lemma 1.

The proof of this lemma is straightforward and thus omitted from the paper. Let us define the second central moment of disagreement vector as . We next note the following.

Lemma 2.

where denotes the trace of a matrix.

Proof.

Please see Appendix.

### 3.2. Asymptotic Expectation of Global Synchronization Error

Let us define ; then the eigenvalues of are . For this , we can show that.

Theorem 1.

In a network with fixed, connected topology, in (11) is a constant vector independent of the constant value of .

Proof.

Thus, for a constant step size , the steady state of expectation of disagreement vector is a constant vector regardless of . In other words, in a network with fixed topology, the expectation of global synchronization error is the same regardless of the speed of synchronization.

*tolerable synchronization error*". Let us now define the asymptotic expectation of pairwise synchronization error as

Hence, the maximum asymptotic expectation of global synchronization error between any two nodes is . It is worth mentioning that, under certain network topologies (e.g., the ring network studied in Section 4), average consensus can still be asymptotically achieved when using the DCTS approach under Gaussian delays.

Recall that . In this equation, is the disagreement vector of . When , we see that , for and . More specifically, when and , then and , implying that the DCTS algorithm achieves average consensus asymptotically. The condition above indicates that the time delay between nodes can be canceled if each node receives the same amount of time delay from all neighbors; networks that meet this condition are defined as follows.

Definition 1.

A network is called "time delay balanced network" if , for and , or equivalently, .

Otherwise we refer to the network as "*time delay unbalanced*". It is worth mentioning that a similar definition of "equal delay networks" was discussed in [13] for continuous time network synchronization. Based on the definition above, we see that time delay balance may be readily (but not exclusively) achieved in well-structured networks.

### 3.3. Asymptotic Mean Square Synchronization Error

Theorem 2.

where , , , , is the total degree in the networks, and denotes the norm of a vector.

Proof.

Please see the Appendix.

Based on this result, it can be seen that the lower and upper bounds of are determined by several values related to network parameters: eigenvalues of and , total degree of network, step size, and delay time vector.

## 4. DCTS Algorithm with Gaussian Delay in Structured Networks

In this section, we apply the DCTS algorithm under Gaussian delay for several structured networks. In particular, we study the structured networks as they are analytically tractable, provide some valuable insights, and can be used to validate our analytical findings. (Typical sensor network deployments may in fact have a random topology. We study how our results extend to such random network scenarios using simulation in Section 5.) Specifically, we analyze at the impact of Gaussian delay when using DCTS in the following networks.

Definition 2 (A Ring Network with Equal Distance ( )).

A ring network is a network that consists of a single cycle. The ring network with equal distance is a ring network that has nodes, edges, and for and .

Definition 3 (A Star Network with Equal Distance ( )).

A star network is a network that consists of edge set . The star network with equal distance is a star network that has nodes, edges, and for and .

Definition 4 (A Hypercube Network with Equal Distance Degree ( )).

A hypercube network with equal distance degree is a hypercube network that has nodes, edges and .

### 4.1. Convergence Properties for Ring Networks

where , . Since , we see that the ring network is a time delay balanced network.

### 4.2. Convergence Properties for Star Networks

The star network is time delay unbalanced. Furthermore, it should be noted that when operating the DCTS algorithm with , we get that . This is because can be simplified in this case to . As a result, we see that as becomes large, .

### 4.3. Convergence Properties for Hypercube Networks

where and . Since , the hypercube network is also time delay balanced.

## 5. Simulation Results

The simulation parameters are described as follows: initial time phase of node is , , where , and the standard deviation of delay variance is . The simulation results are based on 5000 runs. (Trends similar to the ones noted below were observed when initial time offsets between nodes were arbitrary (e.g., when they were uniformly distributed over ). We use this fixed offset assumption here for comparison purposes.)

### 5.1. Structured Networks

### 5.2. Random Networks

*average consensus achievable with tolerable synchronization error*" as described in Section 3.2.

*time delay balanced network*".

## 6. Conclusions

In this paper, we present theoretical results on the convergence of the DCTS algorithm for wireless sensor networks with general Gaussian delay between nodes. Specifically, we compute the asymptotic expectation and mean square of the global synchronization error of the DCTS algorithm. The results lead to the definition of a *time delay balanced network* in which average timing consensus between nodes can be achieved despite random delays. Furthermore, several structured network architectures are studied as examples, and their associated simulation results are used to validate analytical findings. In the future, we intend to investigate the effects of skew, link failure, and other practical conditions when utilizing the DCTS algorithm in wireless sensor networks.

## Appendices

### A. Proof of Lemma 2

Proof.

Therefore, has the exact same form as (A.3) for . Thus, (10) is valid, and we can conclude the proof.

### B. Proof of Theorem 2

Before proving the theorem, first we present some known results.

Theorem 3.

Combining (B.2) with (B.3), we have the following theorem.

Theorem 4.

We can now prove Theorem 2.

Proof.

This completes the proof.

## Authors’ Affiliations

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