Clock synchronization is a crucial requirement for coordinated activities in distributed networks. As an effective solution, Precision Time Protocol (PTP) is tailored to provide tight synchronization which, however, suffers from packet delay variation (PDV). Therefore, it is a challenging task to mitigate the uncertainties of PDV so that clock skew and offset can be estimated with higher accuracy and robustness. Under the assumption of delay symmetry, PDV is caused by unavailable prior information about the statistical distributions of stochastic delay. This paper investigates a robust estimator that can achieve joint estimation for clock skew and offset under delay with unknown distributions by employing the competitive learning-based rival penalized expectation maximization (RPEM) algorithm to learn the unknown probability density function (pdf) of stochastic delay. Moreover, for the asymmetric delay scenario, besides the unknown delay distributions, delay asymmetry is another important influence on synchronization accuracy. Therefore, the optimal invariant robust estimator is developed to simultaneously handle the performance degradation caused by delay asymmetry and unknown delay distributions. The estimator can even cope with the scenario where the random delays in the uplink and downlink follow different distributions. The effectiveness of the estimation schemes is validated by the computer simulations.

Timestamp-free synchronization protocol is tailored to provide a global time understanding for resource-limited wireless networks as it eliminates timestamp interaction, thereby minimizing additional resource overheads. However, the existing two-hop based timestamp-free protocols are not suitable for synchronizing all nodes in multi-hop networks, as they necessitate multiple response times to establish the timestamp relationship between each pair of neighboring nodes. To this end, we introduce a novel multi-hop timestamp-free synchronization protocol. The proposed protocol allows any two nodes to be synchronized using only local timestamps and the skew estimates embedded within packets traversing the reverse path. Furthermore, considering that synchronization accuracy suffers from delay variation resulting from packet loss or retransmission in wireless networks, we derive a Pitman estimator to estimate the clock skew under arbitrary delay models, given known information. To further target unknown arbitrary delay distributions, we approximate the probability density function (pdf) of stochastic delays using a Gaussian mixture model, and then learn the pdf using the rival penalized expectation maximization algorithm. With the aid of the learned pdf, the robustness-enhanced Pitman estimator is derived, which is robust against arbitrary distributed delays without known knowledge. The effectiveness and performance enhancement of estimators are validated by simulations.

Clock synchronization is crucial requirement for coordinated activities in distributed networks. Precision Time Protocol (PTP) is a popular synchronization scheme to provide tight synchronization which, however, suffers from packet delay variation (PDV). In real networks, prior information on the statistical distribution of stochastic delays is always unavailable. Therefore, it is significant to estimate the clock skew and offset in the presence of stochastic delays with unknown statistical distributions. This paper investigates a robust estimation approach applicable to scenarios where the delay distributions are unknown, which utilizes the Gaussian mixture model to approximate the probability density function (pdf) of random delays, and then employs the rival penalized expectation maximization (RPEM) algorithm to learn mixture model parameters. The optimum invariant estimators for the clock skew and offset are derived based on the learned pdfs of random delays. The effectiveness of the estimation scheme is validated by the computer simulations.