TY - JOUR

T1 - Tails in a Fixed-Point Problem for a Branching Process with State-Independent Immigration

AU - Foss, Sergey

AU - Miyazawa, Masakiyo

PY - 2020

Y1 - 2020

N2 - 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.

AB - 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.

KW - heavy tail asymptotics

KW - branching process

KW - state-independent immigration

KW - fixed-point equation

KW - single-server feedback queue

KW - long tail

KW - dominantly varying tail

KW - (intermediate) regularly varying tail

KW - RANDOM TIME-INTERVAL

KW - HEAVY

M3 - Article

VL - 26

SP - 613

EP - 635

JO - Markov Processes And Related Fields

JF - Markov Processes And Related Fields

SN - 1024-2953

IS - 4

ER -