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Colored Markov Modulated Fluid Queues

Benny Van Houdt

TL;DR

This work generalizes Markov-modulated fluid queues by introducing colored MMFQs, where multiple colors tag fluid to store a memory of the workload and event history. The authors develop a matrix-analytic framework, defining color-specific first-passage matrices $\Psi_c$ and color-wise rate matrices $K_c$, and they establish a backward-recursion approach to compute stationary distributions, including cases with fluid jumps represented by phase-type distributions. The paper also presents several practical applications (e.g., MMAP[L]/PH[L]/1/N/LCFS and multi-level cascades) that achieve tractable analysis with complexity linear in the number of colors and cubic in the PH dimensions, supported by numerical results and runtime comparisons. These colored MMFQs enable efficient analysis of queueing systems that were previously intractable due to state-space explosion, offering a flexible framework for workload and event-driven modeling in networks and computing 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.

Colored Markov Modulated Fluid Queues

TL;DR

This work generalizes Markov-modulated fluid queues by introducing colored MMFQs, where multiple colors tag fluid to store a memory of the workload and event history. The authors develop a matrix-analytic framework, defining color-specific first-passage matrices and color-wise rate matrices , and they establish a backward-recursion approach to compute stationary distributions, including cases with fluid jumps represented by phase-type distributions. The paper also presents several practical applications (e.g., MMAP[L]/PH[L]/1/N/LCFS and multi-level cascades) that achieve tractable analysis with complexity linear in the number of colors and cubic in the PH dimensions, supported by numerical results and runtime comparisons. These colored MMFQs enable efficient analysis of queueing systems that were previously intractable due to state-space explosion, offering a flexible framework for workload and event-driven modeling in networks and computing 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.
Paper Structure (26 sections, 4 theorems, 62 equations, 7 figures)

This paper contains 26 sections, 4 theorems, 62 equations, 7 figures.

Table of Contents

  1. Introduction
  2. The classic Markov-modulated fluid queue:
  3. The colored Markov-modulated fluid queue
  4. The state space
  5. The evolution of a colored MMFQ
  6. Stationary distribution of a colored Markov-modulated fluid queue
  7. Some Special Cases
  8. The matrices $T_{++}^{(c,c')}=0$ and $T_{-+}^{(c,c')}=0$ for $c' > c+1$: no color skipping
  9. The matrices $T_{-+}^{(c)}=0$ for $c < C$
  10. The Colored MMFQ with fluid jumps
  11. Applications
  12. The MMAP[L]/PH[L]/1/N/LCFS queue
  13. The MMAP[L]/PH[L]/1/N[L]/LCFS queue
  14. A finite FCFS queue with a "multi-level job" cascade
  15. Numerical and runtime results
  16. ...and 11 more sections

Key Result

Theorem 1

The stationary densities of the colored fluid queue characterized by the matrices $T_{--}^{(0)}$,$T_{-+}^{(0,c)}$, $T_{++}$, $T_{-+}$, $T_{--}^{(c)}$, $T_{+-}^{(c)}$, for $c=1,\ldots,C$ can be expressed as follows. Assume $\vec{x} \in \mathbb{R}^C$ and let $1 \leq c_1 < c_2 < \ldots < c_n \leq c$ be The stationary probability vector $p_-$ solves This vector is normalized by (where $e$ is a column

Figures (7)

  • Figure 1: Illustration of a sample path of an MMFQ with $S_+=\{1,2\}$ and $S_-=\{3,4\}$, background transitions occur at times $1.2, 2.1, 3.5, 4.2, 6$ and $7.2$.
  • Figure 2: Illustration of a sample path of a $2$-colored MMFQ with $S^{(1)}_+=\{1\}$, $S^{(2)}_+=\{2,3,4\}$ and $S_-=\{5,6\}$, background transitions occur at times $1.2, 2.1, 3.6, 4.7, 5.7, 6.4$ and $7$. There is no background transition at times $3$ and $7.9$, but the rate matrix of the background process does change from $T_{--}^{(2)}$ to $T_{--}^{(1)}$.
  • Figure 3: Illustration of a sample path of a $2$-colored MMFQ with $S^{(1)}_+=\{1\}$, $S^{(2)}_+=\{2,3,4\}$ and $S_-=\{5,6\}$. Left: original sample path, Right: censored path. Intervals $(1.2,3)$ and $(3.6,5.6)$ of the original sample path are censored out.
  • Figure 4: Illustration of the reduction of a sample path of a $3$-colored MMFQ with fluid jumps to a sample path of a $3$-colored MMFQ. $S_-=\{1,2,3\}$ and $S^{(c)}_+=\{1,2,3\} \times \{(1,1),(1,2),(2,1)\}$ if type-2 fluid jumps have an order $2$ phase-type representation and type-2 jumps are exponential. The fluid jump occurring at time $0.7$ is a type-$1$ jump of size $1.5$, the fluid jump at time $1.8$ is type-$2$ and has size $0.5$.
  • Figure 5: Loss probability of an MMAP[L]/PH[L]/1/N[L]/LCFS queue with $L=2$ job types, exponential job durations and $2$-state MMAP[L] arrivals.
  • ...and 2 more figures

Theorems & Definitions (7)

  • Theorem 1
  • proof
  • Corollary 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3