I have two independent distributions (Distr-1 and Distr-2) these distributions represent service times of two systems and sampled at run time. Basically, I have two servers in a chain, one after another, application's requests go through server 1 and then through server 2. Each server maintains its own distribution of its own service times.
My question is: By knowing the individual distributions, how can I estimate/compute a target percentile (say 90th) of the sum of these two distributions? Conceptually this is equivalent if I would record delays from the moment request enters the first server until the moment it leaves the second server (ignoring the network delay) and then I would plot the CDF of these delays for all requests.
My initial approach was to compute the 90th percentiles of both Distr-1 and Distr-2 and then sum them up together to obtain the 90th percentile of the Distr-(1+2). But soon I realized that this is not correct. I am attaching a sample plot, I know blue and red distributions and I want to obtain 90th percentile of the green one. However, I can not simply add 90th percentiles of the blue and red to obtain 90th of the green one.
Since the two input percentiles obtained at run time, I do not know if they comply to some known distributions (e.g., normal or exponential etc.) thus I have only their numerical representations (as a list of samples).
Would you have any recommendations of how I could solve this problem?