A new traffic flow cellular automaton (CA) model situated between Nagel-Schrecke nberg (NS) type and Fukui-Ishibashi (FI) type is defined and studied. This new m odel adopes the gradual acceleration scenario for all cars as NS model and the s tochastic delay scenario for only the car with speed limit as FI model. It is pr oved that the fundamental diagram of the average traffic flow speed as the funct ion of the car density for the new model is exactly the same as for FI model. Th is means that if the acceleration scenario of the FI traffic flow model in which a car may increase in velocity rapidly from zero to the velocity limit M or the maximum velocity permitted by empty spacing ahead, is changed to a gradual way in which the velocity of a car may increase by only one unit at most in one time step, the asymptotic steady state behavior of traffic flow will not be changed by interaction between cars and the self-organization evolution of the traffic flow.