[论文解读] Colored Markov Modulated Fluid Queues

Benny Van Houdt|arXiv (Cornell University)|Jan 28, 2026

Advanced Queuing Theory Analysis被引用 0

一句话总结

The paper introduces colored Markov-modulated fluid queues (MMFQs) and MMFQs with fluid jumps, adding color-based memory to track fluid contributions and enable tractable analysis of higher-dimensional queueing systems.

ABSTRACT

Markov-modulated fluid queues (MMFQs) are a powerful modeling framework for analyzing the performance of computer and communication systems. Their distinguishing feature is that the underlying Markov process evolves on a continuous state space, making them well suited to capture the dynamics of workloads, energy levels, and other performance-related quantities. Although classical MMFQs do not permit jumps in the fluid level, they can still be applied to analyze a wide range of jump processes. In this paper, we generalize the MMFQ framework in a new direction by introducing {\bf colored MMFQs} and {\bf colored MMFQs with fluid jumps}. This enriched framework provides an additional form of memory: the color of incoming fluid can be used to keep track of the fluid level when certain events took place. This capability greatly enhances modeling flexibility and enables the analysis of queueing systems that would otherwise be intractable due to the curse of dimensionality or state-space explosion.

研究动机与目标

• Motivate and extend Markov-modulated fluid queues (MMFQs) to incorporate multiple colors of fluid for enhanced memory and modeling flexibility.
• Develop a mathematical framework for colored MMFQs that generalizes classic MMFQs and supports fluid jumps.
• Enable tractable analysis of queueing systems that suffer from state-space explosion by leveraging color-based structure.

提出的方法

• Define the colored MMFQ state space with ordered colors and color-dependent background transitions.
• Derive the stationary distribution using a set of first passage probability matrices Ψ1,…,ΨC and a backward-recursive construction.
• Establish matrices Kc and a block-structured generator K to express the stationary densities and solve a system of nonsymmetric algebraic Riccati equations (NAREs).
• Provide conditions for positive Harris recurrence and a verification method via invariant vectors ξ(c) e.
• Extend the framework to colored MMFQs with fluid jumps by censoring jump intervals and using phase-type distributions to model jump sizes.
• Present computational simplifications for special cases (no color skipping, and certain transitions) to reduce complexity.

实验结果

研究问题

• RQ1 How can MMFQs be extended to incorporate multiple colors of fluid to capture richer workload histories?
• RQ2 How can the stationary distribution of colored MMFQs be computed and what conditions ensure existence of a stationary distribution?
• RQ3 How can fluid jumps be integrated into colored MMFQs without sacrificing tractability?
• RQ4 What are practical simplifications or special cases that reduce computational complexity for colored MMFQs?
• RQ5 How can colored MMFQs be applied to complex queueing models that are intractable under classical MMFQs?

主要发现

• A colored MMFQ framework is developed, where the total fluid is partitioned into colored layers and the top color determines the active background-rate matrices.
• The stationary distribution is characterized through Ψ matrices solving backward NAREs and a product-form-like expression involving exponentials of Kc and Ψc, culminating in a closed-form for π+(x), π−(x).
• A set of conditions for positive Harris recurrence is provided, via invariant vectors ξ(c) and their relation to sub-generator properties of KC matrices.
• The approach extends to MMFQs with fluid jumps by censoring jump intervals and using phase-type (PH) distributions to model jump sizes, preserving tractability.
• Special cases enable linear or Sylvester-equation-based solutions, reducing computational complexity in practical settings.
• Applications are demonstrated to queueing models (e.g., MMAP[L]/PH[L]/1/N/LCFS) where traditional methods fail due to state-space explosion