A modified κ-ε model is proposed for calculating transitional boundary layer flows. In order to develop the eddy viscosity model for the problem, the flow region is divided into three regions; namely, pre-transition region, transition region and fully turbulence region. In the pre-transition region, since the turbulence does not yet attain its equilibrium state in which the eddy viscosity is proportional to the distance from the wall, it is postulated that a structure like the viscous sub-layer in the turulent boundary layer prevails across the boundary layer so that the eddy viscosity is proportional to the cube of the wall distance. Further it is assumed that as the turbulent spots that have appeared at the onset of transition grow with the downstream distance in the transition region, the dependency of the eddy viscosity on the wall-distance changes gradually from that in the pre-transition region to that in the fully turbulent state. Therefore, a universal variation of the near-wall intermittency factor in the transition region is employed to represent such a transition eddy viscosity in the transition region, bridging the two different states, pre-transition and turblence eddy viscosity. In addition, the model constant $C_ε1$ in the standard κ-ε model is modified to take into account the particular characteristics of turbulence in the transition region, derived from the assumption that the mean velocity has an similarity solution. And also, the effect of pressure gradient is taken into account in the stream-wise intermittency factor because it has an universal curve regardless of pressure gradient. From the intermittecny factor in various flows, Narshima`s intermittency function, F(γ), has been found to be proportional to $χ^n$ according to the extent of pressure gradient. Three basic empirical correlations of intermittency factor being analyzed according to the results of analysis on experimental data, the best one was chosen to calculate six benchmark cases of bypass-transition flows with different free-stream turbulence intensity under zero and abitray pressure gradients. In the present test calculations, since the governing equations are integrated from a point close to the leading edge in the pre-transition region, the initial and boundary conditions of κ and ε at the onset of transition are automatically supplied. Under zero pressure gradient, it was found that the profiles of mean velocity and turbulent intensity, local maximum of velocity fluctuations, their locations as well as the streamwise variation of integral properties such as skin friction, shape factor and maximum velocity fluctuations are very satisfactorily predicted throughout the flow regions. Similarily the prediction under changing pressure gradient was also good agreement except the later part of the transition region, where better correlation is needed to improve the overall performance. Additionally, the experiment with a flat plate was conducted to verify the present modeling concept of having two diffrenent scales at least and to give available experimental data.