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A disk wind can cause perturbations that propagate throughout the disk via diffusive processes. On reaching the inner disk, these perturbations can change the disk luminosity, which in turn, can change the wind mass loss rate, $\dot{M}_w$. It has been argued that this so-called "wind driven relaxation cycle" might explain the observed variability in some disk accreting objects. Here, we study the response of the innermost mass accretion rate $\dot{M}_a$ to the loss of matter at different rates and radii. We allow the wind launching radius, $R_L$, to scale with $\dot{M}_a$. We computed a grid of time-dependent models for various $\dot{M}_w$-$\dot{M}_a$ and $R_{L}$-$\dot{M}_a$ dependencies. We find that the disk behaviour significantly differs for the 'variable $R_L$' case compared to the 'fixed $R_L$' case. In particular, much stronger winds are required to destabilize the disk in the former than the latter case. However, the $\dot{M}_a$ amplitude does not grow significantly even for unstable cases because the oscillations saturate at a low level either due to disk depletion or due to the wind being launched at very small radii, or both. This result implies that disk winds are unlikely to be responsible for state transitions as those require large changes in the inner disk. Despite modest changes at the inner disk regions, the disk surface density at large radii can vary with a large amplitude, i.e., from 0 to a few factors of the steady state value. This dramatic variation of the outer disk could have observable consequences.