  We estimate the rate of inspiral for a population of stellar mass BHs in the
star cluster around the super massive black hole at the center of Milky Way
mass galaxies. Our approach is based on an orbit averaged Fokker Planck
approach. This is then followed by a post-processing approach, which
incorporates the impact of the angular momentum diffusion and the GW
dissipation in the evolution of system. We make a sample of 10000 BHs with
different initial semi-major and eccentricities with the distribution of
$f_c(a)/a$ and $e$, respectively. Where $f_c(a)$ refers to the phase-space
distribution function for cth species. Angular momentum diffusion leads to an
enhancement in the eccentricity of every system in the above sample and so
increases the rate of inspiral. We compute the fraction of time that every
system spends in the LISA band with the signal to noise ratio $\rm{SNR} \geq
8$. Every system eventually approaches the loss-cone with a replenishment rate
given by the diffusion rate of the cluster, $\mu/ \rm{Gyr}^{-1} \lesssim 1 $.
This small rate reduces the total rate of the inspiral for individual MW mass
galaxies with an estimate $R_{obs} \lesssim 10^{-5} yr^{-1}$. It is expected
though that a collection of $N_{gal} \simeq 10^4$ MW mass galaxies lead to an
observable GW signal in the LISA band.
