Hypoexponential Distribution - General Case

General Case

In the general case where there are distinct sums of exponential distributions with rates and a number of terms in each sum equals to respectively. The cumulative distribution function for is given by

F(t)=1-\left(\underset{j=1}{\overset{a}{\prod}}\lambda_j^{r_j}\right)\underset{k=1}{\overset{a}{\sum}}\underset{l=1}{\overset{r_k}{\sum}} \frac{\Psi_{k,l}(-\lambda_k)t^{r_k-1}\exp(-\lambda_k t)}{(r_k-l)!(l-1)!},

with

\Psi_{k,l}(x)=-\frac{\partial^{l-1}}{\partial x^{l-1}}\left(\underset{j=1,j\neq k}{\overset{a}{\prod}}\left(\lambda_j+x\right)^{-r_j}\right) .

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