Numerical instability in B&D models

TL;DR

This paper addresses numerical instability issues in Birth and Death models when calculating extinction times, proposing a solution and deriving new formulas to improve computational accuracy.

Contribution

It introduces a novel approach and derivation to mitigate numerical ill-conditioning in B&D models for extinction time calculations.

Findings

Proposed a method to reduce numerical ill-conditioning.

Derived new formulas for more stable calculations.

Improved accuracy in computing extinction times.

Abstract

When computing the expected value of time till extinction of a Birth and Death process, the usual textbook approach results in an extreme case of numerical ill-conditioning, which prevents us from getting accurate answers beyond the first few low-lying states; in this brief note we present a potential solution, together with a novel derivation of related formulas.