Abstract: We consider a system of asymmetric independent random walks on $Z^d$, denoted by $\{\eta_t,t\in R\}$, stationary under the product Poisson measure $\nu_{\rho}$ of marginal density $\rho>0$. We fix a pattern $A$, an increasing local event, and denote by $\tau$ the hitting time of $A$. By using a Loss Network representation of our system, at small density, we obtain a coupling between the laws of $\eta_t$ conditioned on $\{\tau>t\}$ for all times $t$. When $d\ge 3$, this provides bounds on the rate of convergence of the law of $\eta_t$ conditioned on $\{\tau>t\}$ towards its limiting probability measure as $t$ tends to infinity. We also treat the case where the initial measure is {\it close} to $\nu_{\rho}$ without being product.