Monte Carlo for estimating exponential convolution

TL;DR

This paper introduces a stable Monte Carlo-based method for estimating the cumulative distribution function of Hypoexponential variables, crucial for network reliability analysis, especially for very small failure probabilities.

Contribution

A new unbiased estimation algorithm for Hypoexponential distributions that remains stable and accurate for large variable counts and extremely small failure probabilities.

Findings

The estimator has bounded relative error.

It outperforms existing methods in stability and accuracy.

Effective for failure probabilities as low as 10^{-40}.

Abstract

In this note we study the numerical stability problem that may take place when calculating the cumulative distribution function of the {\it Hypoexponential} random variable. This computation is extensively used during the execution of Monte Carlo network reliability estimation algorithms. In spite of the fact that analytical formulas are available, they can be unstable in practice. This instability occurs frequently when estimating very small failure probabilities $(10^{-30}-10^{-40})$ that can happen for example while estimating the unreliability of telecommunication systems. In order to address this problem, we propose a simple unbiased estimation algorithm that is capable of handling a large number of variables. We show that the proposed estimator has a bounded relative error and that it compares favorably with other existing methods.