Critical Markov Branching-Immigration Systems: Complete Classification of the State Space and Exact Rates of Convergence
5 affiliations
- 1Department of Algebra and Geometry, Karshi State University, Karshi 180119, Uzbekistan
- 2Department of Mathematical Analysis, Urgench State University Named After Abu Rayhan Biruni, Urgench 220100, Uzbekistan
- 3Department of Higher Mathematics, Tashkent State Transport University, Tashkent 100167, Uzbekistan
- 4Department of Mathematics, Kokand State University, Kokand 150700, Uzbekistan
- 5Department of Mathematics and Natural Sciences, Cyber University, Nurafshon 111500, Uzbekistan
Abstract
We study a critical homogeneous continuous-time Markov branching-immigration system whose offspring and immigration generating functions regularly vary at point 1 of orders 1+ν and δ with ν,δ∈(0,1]. The slowly varying factors are controlled through their index functions, which causes regular variation of the second order, both factors tending to finite positive limits. Two numbers govern the classification: the difference γ=δ−ν of the two orders and the limit Λ∞ of the ratio of the immigration factor to the offspring one. We classify the state space completely: it is positive recurrent for γ>0, transient for γ<0, and on the boundary γ=0, null recurrent or transient accordingly as Λ∞≤ν or Λ∞>ν; at ν=δ=1, this is the classical threshold, read under the present second-order hypotheses. In each regime, we identify the limiting generating function of the normalised transition functions, together with the leading term of the deviation from it, uniformly in the argument on compact subsets of [0,1), the transient regime under two further inequalities on the parameters. Every rate is carried by a complex-analytic argument and with no monotonicity hypothesis to the individual transition probabilities as an upper bound, which for each fixed state is the exact order when γ≠0. An exact integral identity for the survival probability of the system without immigration makes the constants explicit and yields a three-term expansion of that probability. A numerical section on explicit models confirms the constants and the rates.
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BibTeX
@article{imomov2026critical,
title = {{Critical Markov Branching-Immigration Systems: Complete Classification of the State Space and Exact Rates of Convergence}},
author = {Azam A. Imomov and Sarvar B. Iskandarov and Jakhongir B. Azimov and Hurshidjon Q. Jumaqulov and Zuhriddin A. Nazarov},
journal = {Mathematics},
year = {2026},
month = oct,
doi = {10.3390/math14203698},
url = {https://www.mdpi.com/2227-7390/14/20/3698},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Critical Markov Branching-Immigration Systems: Complete Classification of the State Space and Exact Rates of Convergence
AU - Azam A. Imomov
AU - Sarvar B. Iskandarov
AU - Jakhongir B. Azimov
AU - Hurshidjon Q. Jumaqulov
AU - Zuhriddin A. Nazarov
PY - 2026
DA - 2026/10/12
JO - Mathematics
DO - 10.3390/math14203698
UR - https://www.mdpi.com/2227-7390/14/20/3698
AB - We study a critical homogeneous continuous-time Markov branching-immigration system whose offspring and immigration generating functions regularly vary at point 1 of orders 1+ν and δ with ν,δ∈(0,1]. The slowly varying factors are controlled through their index functions, which causes regular variation of the second order, both factors tending to finite positive limits. Two numbers govern the classification: the difference γ=δ−ν of the two orders and the limit Λ∞ of the ratio of the immigration factor to the offspring one. We classify the state space completely: it is positive recurrent for γ>0, transient for γ<0, and on the boundary γ=0, null recurrent or transient accordingly as Λ∞≤ν or Λ∞>ν; at ν=δ=1, this is the classical threshold, read under the present second-order hypotheses. In each regime, we identify the limiting generating function of the normalised transition functions, together with the leading term of the deviation from it, uniformly in the argument on compact subsets of [0,1), the transient regime under two further inequalities on the parameters. Every rate is carried by a complex-analytic argument and with no monotonicity hypothesis to the individual transition probabilities as an upper bound, which for each fixed state is the exact order when γ≠0. An exact integral identity for the survival probability of the system without immigration makes the constants explicit and yields a three-term expansion of that probability. A numerical section on explicit models confirms the constants and the rates.
SN - 2227-7390
ER -