On the expected time a branching process has K individuals alive

TL;DR

This paper derives a formula for the expected total time a homogeneous branching process spends with exactly K individuals alive, independent of the specific life length distribution, given a constant birth rate and normalized mean life length.

Contribution

It provides a universal formula for the expected time with K individuals alive in a branching process, regardless of the life length distribution, under specified conditions.

Findings

Expected time formula: rac{elta^{K-1}}{k(1elta)^K}

Result holds for any life length distribution with mean 1

Applicable to homogeneous continuous-time branching processes

Abstract

Consider a homogeneous time-continuous branching process where individuals have constant birth rate $\delta$, and life length distribution $Q$ having mean $E(Q)=1$. Let $X(u)$ denote the number of individuals alive at time $u$, and assume that $X(0)=1$. Let $K$ be a positive integer and define $A_K:=\int_0^\infty 1_{\{X(u)=K\}}du$, the accumulated time that the branching process has exactly $K$ individuals alive. In this paper we prove that $E(A_K)=\delta^{K-1}/\left(k(1\vee\delta)^K\right)$, irrespective of the life length distribution $Q$, subject to the normalizing condition $E(Q)=1$.