Let $X_1,\ldots,X_N$ be independent Poisson distributed random variables with unequal parameters $\lambda_1,\ldots,\lambda_N$.

Is there any closed form expression or at least a good approximation for the distribution (I am most interested in the CCDF) of their minimum $Y = \min\limits_{1\leqslant i\leqslant N}(X_{i})$?