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Approximating fluid queues with quasi birth-and-death processes with rational arrival process components

Nigel Bean, Angus Lewis

TL;DR

This work develops a quasi birth-and-death process with rational arrival process components ($\text{QBD-RAP}$) to approximate a fluid queue and proves weak convergence via generator convergence. It combines a grid-based discretization of the level domain with concentrated matrix-exponential distributions (CMEs) to model deterministic-like timing while preserving probabilistic consistency. The authors demonstrate that the QBD-RAP's phase process matches the fluid queue's phase process and establish convergence of the associated generators, enabling distributional convergence through standard semigroup results. The approach yields a numerically accurate, probability-consistent method suitable for fluid-fluid queues and FRAP-type systems, with explicit constructions for boundaries and intra-level densities via a closing operator.

Abstract

A fluid queue is a stochastic process which moves linearly with a rate that is determined by the state of a continuous-time Markov chain (CTMC). In this paper we construct an approximation to a fluid queue using a quasi birth-and-death process with rational arrival process components (QBD-RAP) and prove weak convergence via convergence of generators. The primary motivation for constructing the new approximation is to achieve a better approximation accuracy than existing methods while also ensuring that all approximations to probabilities have all probabilistic properties, such as non-negativity and bounded above by 1.

Approximating fluid queues with quasi birth-and-death processes with rational arrival process components

TL;DR

This work develops a quasi birth-and-death process with rational arrival process components () to approximate a fluid queue and proves weak convergence via generator convergence. It combines a grid-based discretization of the level domain with concentrated matrix-exponential distributions (CMEs) to model deterministic-like timing while preserving probabilistic consistency. The authors demonstrate that the QBD-RAP's phase process matches the fluid queue's phase process and establish convergence of the associated generators, enabling distributional convergence through standard semigroup results. The approach yields a numerically accurate, probability-consistent method suitable for fluid-fluid queues and FRAP-type systems, with explicit constructions for boundaries and intra-level densities via a closing operator.

Abstract

A fluid queue is a stochastic process which moves linearly with a rate that is determined by the state of a continuous-time Markov chain (CTMC). In this paper we construct an approximation to a fluid queue using a quasi birth-and-death process with rational arrival process components (QBD-RAP) and prove weak convergence via convergence of generators. The primary motivation for constructing the new approximation is to achieve a better approximation accuracy than existing methods while also ensuring that all approximations to probabilities have all probabilistic properties, such as non-negativity and bounded above by 1.

Paper Structure

This paper contains 19 sections, 8 theorems, 104 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Fluid Queues
  4. Matrix exponential distributions
  5. QBD-RAPs
  6. QBD-RAP Approximation
  7. Set up
  8. Construction
  9. Up to the first event after entering a level
  10. On a change of phase
  11. Boundary conditions
  12. QBD-RAP generator
  13. Initial conditions
  14. The distribution of the fluid level at time $t$
  15. Distributional results
  16. ...and 4 more sections

Key Result

Theorem 2.3

A process $N$ is a Marked RAP if there exist matrices $\boldsymbol S$, $\boldsymbol D_i,\,i\in\mathscr K$, and a row vector $\boldsymbol \alpha$ such that $dev(\boldsymbol S)<0$, $dev(\boldsymbol S+\boldsymbol D)=0$, $(\boldsymbol S+\boldsymbol D)\boldsymbol e = 0$, $\boldsymbol D = \sum_{i\in\maths Conversely, if a point process has the property (eqn: brap) then it is a Marked RAP.

Figures (1)

  • Figure 1: The density function for a concentrated matrix exponential of order 21 from hht2020 (blue) and corresponding conditional density functions of the residual lives, $R_i(0.3)$ given $Z>0.3$ (red), and $R_i(0.6)$ given $Z>0.6$ (green). Observe how the density function of the $Z/|c_i|$ (blue) approximates a point mass at $\Delta=1$, while the density functions of $R_i({0.3})$ given $Z>0.3$ (red) and $R_i({0.6})$ given $Z>0.6$ (green) approximate point masses at $0.7$ and $0.4$, respectively.

Theorems & Definitions (14)

  • Definition 2.1: ab1999
  • Definition 2.2: bn2010
  • Theorem 2.3
  • Lemma 3.1
  • Corollary 3.1.1
  • Corollary 3.1.2
  • Lemma 3.2
  • Theorem 4.1
  • proof
  • Lemma 4.2: The generators are close to each other at a collection of points
  • ...and 4 more