The Galactic massive black hole (MBH), with a mass of M•= 3.6 × 106 M⊙, is the closest known MBH, at a distance of only 8 kpc. The proximity of this MBH makes it possible to observe gravitational waves (GWs) from stars with periapse in the observational frequency window of the Laser Interferometer Space Antenna (LISA). This is possible even if the orbit of the star is very eccentric, so that the orbital frequency is many orders of magnitude below the LISA frequency window, as suggested by Rubbo, Holley-Bockelmann & Finn (2006). Here we give an analytical estimate of the detection rate of such GW bursts. The burst rate is critically sensitive to the inner cut-off of the stellar density profile. Our model accounts for mass segregation and for the physics determining the inner radius of the cusp, such as stellar collisions, energy dissipation by GW emission and consequences of the finite number of stars. We find that stellar BHs have a burst rate of the order of 1 yr−1, while the rate is of the order of ≲0.1 yr−1 for main-sequence stars and white dwarfs. These analytical estimates are supported by a series of Monte Carlo samplings of the expected distribution of stars around the Galactic MBH, which yield the full probability distribution for the rates. We estimate that no burst will be observable from the Virgo cluster.