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On a finite quasi birth-death process with catastrophes and its diffusion approximation

Giulia Di Nunno, Barbara Martinucci, Serena Spina

TL;DR

The paper develops and analyzes a finite quasi birth-death process with catastrophes, modeling a multi-type Ehrenfest-like system that resets to zero at random times. It employs generating-function and matrix-analytic methods to obtain transient distributions, moments, and asymptotic (stationary) behavior, including explicit forms in the λ=μ special case. A diffusion approximation is derived, showing convergence to a family of jump Ornstein–Uhlenbeck processes on a star graph, with explicit stationary density and moments. The results provide a tractable framework for systems that combine gradual, level-structured evolution with abrupt catastrophic resets, with applications to population dynamics, queueing, and networked systems.

Abstract

We study a multi-type Ehrenfest process modeled as a finite quasi-birth-death (QBD) process. We assume that the transitions are allowed only to the two adjacent levels of the same phase and are characterized by linear rates. The crucial element lies in the phase switching mechanism at the origin, which is governed by an irreducible stochastic matrix. The process evolution is interrupted by catastrophic events, whose occurrences are controlled by a Poisson process. Each catastrophe resets the system state to zero, initiating a new cycle of evolution until the next resetting event. We conduct a comprehensive analysis, addressing both the transient and long-term behavior of this process. Furthermore, we derive a diffusive approximation, by proving its convergence to a reflected Ornstein-Uhlenbeck jump diffusion process.

On a finite quasi birth-death process with catastrophes and its diffusion approximation

TL;DR

The paper develops and analyzes a finite quasi birth-death process with catastrophes, modeling a multi-type Ehrenfest-like system that resets to zero at random times. It employs generating-function and matrix-analytic methods to obtain transient distributions, moments, and asymptotic (stationary) behavior, including explicit forms in the λ=μ special case. A diffusion approximation is derived, showing convergence to a family of jump Ornstein–Uhlenbeck processes on a star graph, with explicit stationary density and moments. The results provide a tractable framework for systems that combine gradual, level-structured evolution with abrupt catastrophic resets, with applications to population dynamics, queueing, and networked systems.

Abstract

We study a multi-type Ehrenfest process modeled as a finite quasi-birth-death (QBD) process. We assume that the transitions are allowed only to the two adjacent levels of the same phase and are characterized by linear rates. The crucial element lies in the phase switching mechanism at the origin, which is governed by an irreducible stochastic matrix. The process evolution is interrupted by catastrophic events, whose occurrences are controlled by a Poisson process. Each catastrophe resets the system state to zero, initiating a new cycle of evolution until the next resetting event. We conduct a comprehensive analysis, addressing both the transient and long-term behavior of this process. Furthermore, we derive a diffusive approximation, by proving its convergence to a reflected Ornstein-Uhlenbeck jump diffusion process.
Paper Structure (11 sections, 19 theorems, 124 equations, 10 figures, 1 table)

This paper contains 11 sections, 19 theorems, 124 equations, 10 figures, 1 table.

Table of Contents

  1. Introduction
  2. The model
  3. Transient regime
  4. The special case: $\lambda=\mu$
  5. Asymptotic regime
  6. The diffusion approximation
  7. Conclusions
  8. Proof of Proposition \ref{['asymptotic']}
  9. Proof of Proposition \ref{['prob_asympt_l_equal_m']}
  10. Approximation of $g(\lambda,\mu,\xi,N)$
  11. Asymptotic variance

Key Result

Proposition 3.1

The pgf (FPgrande) satisfies the following partial differential equation, for $z\in [0,1]$ and $t>0$: with initial condition and boundary conditions

Figures (10)

  • Figure 1: State transition diagram in the case $d=3$ and $N=2$.
  • Figure 2: The marginal probabilities $p(k,t)$ as function of $t$ for $N=3$, $\lambda=\mu=1$ and $\xi=0$ (full line), $\xi=0.5$ (dashed line), $\xi=1$ (dot-dashed line), $\xi=2$ (dotted line).
  • Figure 3: The expected value $M(t)$ as function of $t$ for $N=2$ on the left and $N=3$ on the right, $\lambda=\mu=1$ and $\xi=0$ (full line), $\xi=0.5$ (dashed line), $\xi=1$ (dot-dashed line), $\xi=2$ (dotted line).
  • Figure 4: The variance $Var(t)=M^2(t)-[M(t)]^2$ is plotted as a function of $t$ for $N=2$ on the left and $N=3$ on the right, $\lambda=\mu=1$ and $\xi=0$ (full line), $\xi=0.5$ (dashed line), $\xi=1$ (dot-dashed line), $\xi=2$ (dotted line).
  • Figure 5: The probabilities $\rho(k)$ given in (\ref{['prob_asympt']}) are plotted for $N=10$, $\lambda=1,\mu=3$, $\lambda=\mu=1$, $\lambda=3,\mu=1$ from left to right, with $\xi=2$.
  • ...and 5 more figures

Theorems & Definitions (42)

  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • proof
  • Proposition 4.1
  • Proposition 4.2
  • ...and 32 more