  Let $P$ be an irreducible and reversible transition matrix on a finite state
space $V$ with invariant distribution $\pi$. We let $k$ chains start by
choosing independent locations distributed according to $\pi$ and then they
evolve independently according to $P$. Let $\tau_{\mathrm{cov}}(k)$ be the
first time that every vertex of $V$ has been visited at least once by at least
one chain and let $t_{\rm{cov}}(k)=\mathbb{E}[\tau_{\mathrm{cov}}(k)]$ with
$t_{\rm{cov}}=t_{\rm{cov}}(1)$. We prove that $t_{\rm{cov}}(k)\lesssim
t_{\rm{cov}}/k$. When $k\leq t_{\mathrm{cov}}/t_{\rm{rel}}$, where
$t_{\rm{rel}}$ is the inverse of the spectral gap, we show that this bound is
sharp. For $k\leq t_{\mathrm{cov}}/t_{\rm{mix}}$ with $t_{\rm{mix}}$ the total
variation mixing time of $(P+I)/2$ we prove that $k \cdot
\max_{x_1,\ldots,x_k}\mathbb{E}_{x_1,\ldots,x_k}[\tau_{\rm{cov}}(k)] \asymp
t_{\rm{cov}}$.
