## 3 Pairwise Clock Synchronization under Gaussian Delays  
In general, probability density function (PDF) models have been employed for modeling the random portion of delays in WSNs. Examples of PDFs that have been adopted in the literature include exponential, Gaussian, Weibull and Gamma [18; 19; 20; 21]. In this section, we use the Gaussian distribution to characterize the random delay in WSNs, _i.e._, \(X_{i}\) and \(Y_{i}\) are Gaussian random variables with parameters \((\mu_{1},\sigma_{1})\) and \((\mu_{2},\sigma_{2})\), respectively. The reasons for selecting the Gaussian distribution for modeling the random delays are as follows:  
* The central limit theorem (CLT) states that the PDF of the sum of a large number of independent and identically-distributed (i.i.d.) random variables is approximately normally distributed. Therefore, the Gaussian model is appropriate if the random delays are assumed to be the summation of multiple independent random variables. * Experimental results based on two Texas Instruments ez430-RF2500 evaluation boards were recorded in [22] to demonstrate the fitness of the Gaussian distribution in modeling the random portion of delays in WSNs. The most representative pairwise clock synchronization algorithms in the literature focused on developing maximum likelihood estimators (MLEs) for the clock offset and skew. The MLE is one  
Figure 3: Two-way message exchange mechanism (with skew). of the most widely-used estimators in the area of parameter estimation, since it achieves asymptotically the Cramer-Rao lower bound (CRLB). Based on the two-way message exchange model in Equation (4), where only the clock offset is considered, [18] derived the MLE for clock offset by assuming a known fixed delay \(d\) and symmetric random delays, _i.e._, \(\mu_{1}=\mu_{2}=\mu\) and \(\sigma_{1}=\sigma_{2}=\sigma\). In this framework, the likelihood function is expressed as:  
\[p(U_{i},V_{i}|\theta)=(2\pi\sigma^{2})^{-N}e^{-\frac{1}{2\sigma^{2}}\left[ \sum\limits_{i=1}^{N}(U_{i}-d-\theta-\mu)^{2}+\sum\limits_{i=1}^{N}(V_{i}-d+ \theta-\mu)^{2}\right]}\]  
The MLE of clock offset is derived by differentiating the log-likelihood function with respect to \(\theta\) and setting the corresponding derivative to zero, and it assumes the expression:  
\[\theta_{\text{ML}}=\frac{\overline{U}-\overline{V}}{2} \tag{5}\]  
where \(\overline{U}\) and \(\overline{V}\) stand for the sample means of observations \(\{U_{i}\}_{i=1}^{N}\) and \(\{V_{i}\}_{i=1}^{N}\), respectively. It can be verified that the estimator Equation (5) is unbiased and achieves the CRLB. The MLE estimator Equation (5) represents also the minimum variance unbiased estimator (MVUE). Following the same assumptions, _i.e._, a known fixed delay and symmetric random delays, the joint maximum likelihood estimates of the clock offset and skew were also proposed in [18] based on the two-way message exchange model in Equation (2). Specifically, the likelihood function is first formulated as:  
\[p(U_{i},V_{i}|\theta,f^{\prime})=(2\pi\sigma^{2})^{-N}e^{-\frac{1}{2\sigma^{2} }\sum\limits_{i=1}^{N}\{[f^{\prime}(\theta-T_{i}^{b})+(T_{i}^{a}+d)]^{2}+[f^{ \prime}(\theta-T_{i}^{c})+(T_{i}^{d}-d)]^{2}\}}\]  
where \(f^{\prime}=1/(1+f)\). Differentiating the log-likelihood function with respect to \(\theta\) leads to an equation of \(\theta\) in terms of \(f^{\prime}\). In addition, differentiating the log-likelihood function with respect to \(f^{\prime}\) results in an equation of \(f^{\prime}\) in terms of \(\theta\). The joint estimates of \(\theta\) and \(f^{\prime}\), denoted as \(\hat{\theta}_{\text{ML}}\) and \(\hat{f}^{\prime}_{\text{ML}}\), can be obtained by combining those two equations, and the MLE of \(f\) is obtained as \(\hat{f}_{\text{ML}}=1/(1+\hat{f}^{\prime}_{\text{ML}})\), due to the invariance principle [23]. However, in some practical cases, the value of the fixed delay \(d\) is unknown and sometimes even represents a parameter that needs to be estimated, as is the case with the node localization application [24].