Tails in a Fixed-Point Problem for a Branching Process with State-Independent Immigration

#### S.G. Foss, M. Miyazawa

2020, v.26, Issue 4, 613-636

ABSTRACT

We consider a fixed-point equation for a non-negative integer-valued random variable, that appears in branching processes with state-independent immigration. A similar equation appears in the analysis of a single-server queue with a homogeneous Poisson input, feedback and permanent customer(s).

It is known that the solution to this equation uniquely exists under mild first and logarithmic moments conditions. We find further the tail asymptotics of the distribution of the solution when the immigration size and branch size distributions are heavy-tailed. We assume that the distributions of interest are dominantly varying and have a long tail. This class includes, in particular, (intermediate, extended) regularly varying distributions.

We consider also a number of generalisations of the model.

Keywords: heavy tail asymptotics, branching process, state-independent immigration, fixed-point equation, single-server feedback queue, long tail, dominantly varying tail, (intermediate) regularly varying tail