Three-dimensional (3D) extrusion bioprinting typically requires an ad hoc trial-and-error optimization of the ink composition toward enhanced resolution. The ink solutions are solidified after leaving cone-shaped or cylindrical nozzles. The presence of ink instability not only hampers the extrusion resolution but also affects the behavior of embedded cellular components. This is a key factor in selecting (bio)inks and bioprinting design parameters for well-established desktop and handheld bioprinters. In this work, we developed an analytical solution for the process of ink deposition and compared its predictions against numerical simulations of the deposition. We estimated the onset of ink instability as a function of ink rheological properties and nozzle geometry. Our analytical results suggest that enhancing the shear-thinning behavior of the ink shortens the toe region of the deposition. Such an extrusion process is often desired, as it leads to faster depositions. However, we demonstrated that such conditions increase the possibility of lateral buckling of the strand once touching the substrate defined as instability in this study. The present study serves as a benchmark for detailed simulations of the extrusion process for optimal bioprinting.