# How to estimate a percentile of a sum of two independent distributions

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?

• Just to clarify. You have two random variables that are independent from one another and what you want to know is the distribution of their sum. The reason that you want to know this is that you want to compute the 90th percentile of their sum?
– user31790
May 20, 2017 at 10:06
• Yes, that is correct, however, I would like to be able to compute any percentiles, for example, 95th or 99th etc. May 20, 2017 at 11:28
• Why would the service times be independent? wouldn't they depend on the characteristics of whatever they were serving and hence be dependent? May 20, 2017 at 16:38