Continuous k-similarity trajectories search over a data stream is an important problem in the domain of spatio-temporal databases. Given a set of trajectories T and a query trajectory Tq over road network G, the system monitors trajectories within T, reporting k trajectories that are the most similar to Tq whenever one time unit is passed. Some existing works study k-similarity trajectories search over trajectory data, but they cannot work in a road network environment, especially when the trajectory set scale is large. In this paper, we propose a novel framework named RNDLP (Road Network-based Distance Lower-bound-based Prediction) to support CKTRN over trajectory data. It is a distributed framework based on the following observation. That is, given a trajectory Ti and the query trajectory Tq, when we have knowledge of D(Ti), we can compute the lower-bound and upper-bound distances between Tq and Ti, which enables us to predict the scores of trajectories in T and employ these predictions to assess the significance of trajectories within T. Accordingly, we can form a mathematical model to evaluate the excepted running cost of each trajectory we should spend. Based on the model, we propose a partition algorithm to partition trajectories into a group of servers so as to guarantee that the workload of each server is as the same as possible. In each server, we propose a pair-based algorithm to predict the earliest time Ti could become a query result, and use the predicted result to organize these trajectories. Our proposed algorithm helps us support query processing via accessing a few points of a small number of trajectories whenever trajectories are updated. Finally, we conduct extensive performance studies on large, real, and synthetic datasets, which demonstrate that our new framework could efficiently support CKST over a data stream.