PDF of sum of independent exponential r.v.s of different parameter values If we have N independent exponential random variables of different parameter values: 
$x_1$ ~ exp($m_1$), $x_2$ ~ exp($m_2$), ..., $x_N$ ~ exp($m_N$)
Is there a closed form (and SIMPLE) answer for the pdf of the sum of those random variables, i.e., the pdf of $\sum_{i=1}^{N} {x_i}$? If not, is there an approximation (with a reference if available) where this pdf is approximated using another known pdf?
Thanks

Of course, if the rates $\lambda_1 = \lambda_2 = \cdots = \lambda$ are equal, then
$S_n = \sum_{i=1}^n X_i \sim Gamma(n, rate = \lambda).$ For $n=2,\;$ $\lambda_1 = 1/4,$ and $\lambda_2 = 1/5,$ here is a simulation of
the distribution of the sum that approximates the density function.
 lam1 = 1/4;  lam2 = 1/5;  m = 10^6;  
 x1 = rexp(m, lam1);  x2 = rexp(m, lam2);  s = x1 + x2
 mean(s);  sd(s)
 hist(s, br=50, prob=T, col="wheat")
   c1 = lam2/(lam2 - lam1);  c2 = lam1/(lam1 - lam2)
   curve(c1*dexp(x,lam1)+c2*dexp(x,lam2), 0,max(s), 
      n = 1001, lwd=2, col="blue", add=T)


Notes: (1) In R, the curve statement requires the
use of the parameter x.
(2) Our factors c1 and c2 are not exactly the same as the $\alpha_i$
in the Comment.
(3) Because of the subtractions in the denominators, one cannot simply set $\lambda_1 = \lambda_2 = \lambda$
in the expression for the PDF of $S_2$ to get the PDF of
$Gamma(2, \lambda).$
