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

Sergey Foss, Masakiyo Miyazawa

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Pages (from-to)613-635
Number of pages23
JournalMarkov Processes And Related Fields
Issue number4
Publication statusPublished - 2020


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

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