Analyze Speed Up in A Dispatcher-Woker Model
Here is how you can compute its distribution.  Let $F(t) = \Pr[X_i \le t]$ (the cumulative distribution function of $X_i$), and $G(t) = \Pr[Y \le t]$ (the cumulative distribution function of $Y$).  Then

$$G(t) = \Pr[Y \le t] = \Pr[X_1 \le t, \dots, X_N \le t] = \Pr[X_1 \le t] \times \dots \times \Pr[X_N \le t] = F(t)^N.$$

Now the probability density function of $Y$, $g(t)$, is the derivative of the cumulative distribution function, so

$$g(t) = G'(t) = N F(t)^{N-1} f(t),$$

where $f(t)$ is the probability density function of $X_i$.  This directly gives you the distribution of $Y$: namely, $g(\cdot)$ is the distribution of $Y$.

This calculation also gives you a good way to compute tail bounds.  For instance, if you want to compute the probability that the whole process needs more than $t$ time, i.e., that $Y &gt; t$, you can compute this as $1-G(t) = 1 - F(t)^N$.

See also https://math.stackexchange.com/q/89030/14578 for other useful facts about the distribution of $Y$.