A scenario for a periodic filling and emptying of the accretion disc of the microquasar GRS 1915+105 is proposed, by estimating the mass transfer rate from the donor and comparing it with the observed accretion rate onto the primary black hole. The mass of the Roche-lobe-filling donor (1.2±0.2M
☉), the primary black hole mass (14±4M
☉) and the binary orbital period of 33.5d (Greiner et al.,
2001Natur.414..522G) predict for the donor spectral type and K-magnitude around K6 III and -2.6, respectively. The He-core of 0.28M
☉ of such a giant leads to evolutionary expansion along the giant branch with a conservative mass transfer rate of {dot}(M)
d=(1.5±0.5)x10
–8M
☉/year. On the other hand, the average observed accretion rate onto the primary is ten times larger: {dot}(M)
obs=2.0x(η/0.1)
–1(d/12.5 kpc)
2x10
–7M
☉/y, where η is the efficiency of converting accretion into radiation. We propose a duty cycle with (5-10)(η/0.1) per cent active ON-state. The timescale of the (recurrent) OFF-state is identified as the viscosity time scale at the circularization radius (14R
☉) and equals t
visc=370(α/0.001)
–4/5 years, where α is the viscosity parameter in the α-prescription of a classical disc. If the viscosity at the outer edge of the disc is small and η is close to the maximum available potential energy (per rest mass energy) at the innermost stable orbit, the present activity phase may still last another 10-20 years. We also discuss other solutions allowing a broader range of donor masses (0.6-2.4M
☉).