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Adding Percentiles of Erlangian Distributions

In networking, enterprise computing and many other areas the issue of adding percentiles of a performance metric, such as the response time, arises regularly. Percentiles cannot be added using the arithmetic sum, and surprisingly there are no known formulae that permit us to do so correctly. In this paper, we obtain an exact analytical expression for adding percentiles of random variables which can be represented by a series of generalized exponential stages (e.g., Erlang, hypoexponential and two-stage Coxian). We demonstrate the applicability of our results by an example in which we use our expressions in the Dijkstra's algorithm to calculate the shortest 'percentile delay' path.