Formulas for the distribution of sums of independent exponential random variables

This paper derives a new type of formula for the probability that, among a collection of items with s-independent exponential times to failure, a certain subset of them fails in a given order before a certain time, and all the remaining items survive beyond that time. This formula is in the form of a power series that satisfies a certain constant coefficient linear differential equation with specified initial conditions. This provides an alternative to existing closed-form formulas of the "exponomial" variety, viz., a nonlinear combination of exponential terms, where the coefficients of the exponential terms are polynomials in the mission time. Some results are given which quantify the computation effort required to achieve a specified accuracy using partial sums of the infinite series; a simple example illustrates these results. This approach can be very efficient for system reliability analysis where the product of the mission time and the sum of the failure rates down any path leading to system failure is small. Further work is needed to expand the practical applicability of this approach to cases where some rates are large and/or the mission time is long.