The origin of the water masses exported from the Arctic to the North Atlantic along both sides of Greenland is investigated using an original numerical method. A quantitative Lagrangian analysis is applied to the monthly climatological 3-D output of a global ocean/sea ice high-resolution model. It allows quantification of the different branches of the export to the North Atlantic, as well as related timescales and water mass transformations. In the model, the outflow through Davis Strait consists in equal parts of Pacific and Atlantic water, whilst the export through Fram Strait consists almost fully of Atlantic water (contrary to observations). Pacific water is transferred quickly (O(10 years)) to the North Atlantic through the Beaufort Gyre, where gradual warming and salinification occur. Atlantic water exiting in the surface layer along both sides of Greenland remains about 10 years in the Arctic Basin and undergoes cooling and significant freshening. Below the surface water, Atlantic water exiting through the intermediate and deep layers in Fram Strait follows different pathways in the Arctic, with trajectories being subject to topography constraints. The travel time depends strongly on the pathway (from 1 to 1000 years). The intermediate outflow consists mainly of water entering the Arctic at Fram Strait, while half the deep outflow is composed of water from the Barents Sea. We find that the Barents Sea Branch, which contributes to both the outflows at Fram and Davis straits, is almost fully transformed after a year due to heat exchanges with the very cold atmosphere (in the Barents Sea).