In this paper, the correlation functions in the CA (cellular automaton) traffic models incorporating the `slow-to-start' rule, such as VDR model, BJH model and T2 model, are systematically studied at different traffic densities. The results show that there are anti-correlations and correlations between cars at low density. When anti-correlation disappears with the increases of density, it means the transition from free flow to jamming. In order to study the characteristics of phase transition, we study the order parameters of these models with the delay probability and slow-to-start probability. We found that the CA traffic model with ‘slow-to-start’ rule will change the characteristics of phase transition. Independently of the delay probability and in the case of less than maximal velocity, the transition from free flow to jamming in the CA model with slow-to-start probability not exceeding 0.5 is the second phase transition, which has an analogy to one in the deterministic NaSch model. Otherwise, the first phase transition will appear. Under conditions of stochastic delay, the crossover phenomena will occur. When the limiting velocity has larger values, it will show the first phase transition in spite of the delay probability.