Yaglom limit for Stochastic Fluid Models

TL;DR

This paper analyzes the limiting conditional distribution (Yaglom limit) for stochastic fluid models (SFMs), a key class in matrix-analytic methods, by studying the long-run behavior conditioned on non-absorption. It develops a Laplace-Stieltjes transform framework, introduces a Riccati equation for ${\bf \Psi}(s)$, and applies the Heaviside principle to extract square-root type singularities at a critical point $s^*$, with $s^*=oldsymbol{\delta}^*$ indicating the dominant spectral interaction. The main contributions are proving the existence and uniqueness of the Yaglom limit for SFMs and providing explicit constructions of the limiting measures ${\boldsymbol{\mu}}(dy)^{(x)}$ and ${\boldsymbol{\mu}}(dy)^{(0)}$ in terms of transforms and auxiliary matrices, along with a clear procedure to compute them in both scalar and matrix-parameter cases. The paper includes simple scalar and matrix examples to illustrate the theory, and demonstrates that the Yaglom limit can depend on the initial level $X(0)$, offering practical insight for matrix-analytic investigations of conditional distributions in SFMs.

Abstract

In this paper we provide the analysis of the limiting conditional distribution (Yaglom limit) for stochastic fluid models (SFMs), a key class of models in the theory of matrix-analytic methods. So far, transient and stationary analyses of the SFMs have been only considered in the literature. The limiting conditional distribution gives useful insights into what happens when the process has been evolving for a long time, given its busy period has not ended yet. We derive expressions for the Yaglom limit in terms of the singularity $s^*$ such that the key matrix of the SFM, ${\bfΨ}(s)$, is finite (exists) for all $s\geq s^*$ and infinite for $s<s^*$. We show the uniqueness of the Yaglom limit and illustrate the application of the theory with simple examples.

Figures (3)

• Figure 1: The values of $\hbox{\boldmath$\mu$}(dy)^{(0)}_{11}/dy$ and $\hbox{\boldmath$\mu$}(dy)^{(0)}_{12}/dy$ in Example \ref{['ex1']} for $b=1$, $a=4,\ 3,\ 2$ (dotted, solid, dashed line, respectively).
• Figure 1: The plot of \ref{['z1']}-\ref{['z']} for $s=0$ (left) and $s=-2$ (right), when $\lambda=2.5$.
• Figure 2: The plot of \ref{['eq:cubic']} for $s=0$ (top left) and $s=-2$ (top right) and $s=-1.1178$, when $\lambda=2.5$.

Theorems & Definitions (24)

• Definition 1.1
• Remark 1.2
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Remark 2.4
• Theorem 3.1: Heaviside principle
• Definition 3.2: Semiexponentiality
• Proposition 3.3
• Lemma 4.1