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.

Particle swarm optimization (PSO) is a swarm intelligence technique that optimizes a problem by iterative exploration and exploitation in the search space. However, PSO cannot achieve the preservation of population diversity on solving multimodal optimization problems, and once the swarm falls into local convergence, it cannot jump out of the local trap. In order to solve this problem, this paper presents a fast restarting particle swarm optimization (FRPSO), which uses a novel restarting strategy based on a discrete finite-time particle swarm optimization (DFPSO). Taking advantage of frequently speeding up the swarm to converge along with a greater exploitation capability and then jumping out of the trap, this algorithm can preserve population diversity and provide a superior solution. The experiment performs on twenty-five benchmark functions which consists of single-model, multimodal and hybrid composition problems, the experimental result demonstrates that the performance of the proposed FRPSO algorithm is better than the other three representatives of the advanced PSO algorithm on most of these functions.