Colored MMFQs with Fluid Jumps

• Colored MMFQs with Fluid Jumps are an advanced fluid queue model that integrates Markov modulation with discrete jumps to capture both continuous drift and bursty arrivals.
• The model reduces state-space complexity by using colored fluid representations to efficiently track the origin and timing of fluid increments in performance analysis.
• It employs matrix-analytic techniques and solves NAREs to compute key performance metrics such as buffer occupancy, top-color probabilities, and related statistics.

Colored Markov-Modulated Fluid Queues (MMFQs) with fluid jumps constitute an advanced modeling framework for performance analysis in computer and communication systems. In this setting, a continuous fluid level—modulated by a finite-state Markov process—represents quantities such as workloads or buffer contents. The colored extension introduces a “color” attribute to the fluid, thereby providing a tractable and memory-efficient method to track when and how portions of the fluid arrived. The further generalization to include fluid jumps allows the model to capture both continuous drift and instantaneous increments, which are common in jump-like arrival or bursty service processes (Houdt, 28 Jan 2026).

1. Model Structure

Let $C \in \mathbb{N}$ denote the number of colors. The state space $S$ is partitioned into a shared background component $S_-$ and $C$ upward, color-specific subsets $S_+^{(c)}$ ($c = 1,\dots,C$). A state is specified either at the boundary as $(0,i)$, $i \in S_-$, or in the colored bulk as $(x_1,\dots,x_C,i) \in \Omega_c$, with non-negative fluid coordinates where $x_c>0$, $x_{c+1}=\dots=x_C=0$, and $i \in S_- \cup S_+^{(c)}$. The total fluid level is $\sum_{k=1}^C x_k$, with “top” color $c$ in $\Omega_c$.

• Fluid dynamics (without jumps):
  • If $i \in S_-$, the fluid level decreases at unit rate.
  • If $i \in S_+^{(c)}$, the fluid increases at unit rate, affecting only the top color $c$.
  • Reflecting boundary at level zero: if in $S_-$, the process remains at zero until an $S_+^{(c)}$ state is entered to resume upward drift.
  • When $x_c$ hits zero in downward drift, the process “pops” to the next nonzero color below (or to zero).
• Fluid jumps:

Instantaneous upward jumps occur, adding fluid with a prescribed color. Each jump type $\ell$ is governed by a phase-type (PH) distribution $(\alpha_{\ell,c'}, U_{\ell,c'})$. If in $(x_1,\dots,x_c,i)\in\Omega_c$, $i \in S_-$, a jump of type $\ell$ occurs at rate $(Q_\ell^{(c,c')})_{i,j}$, increasing the fluid at color $c'>c$. Typically, $c'=c+1$ (no color skipping).

2. Generator and Transition Structure

Transitions are classified as:

• Drift (no jump),
• Color changes (first and second kind, e.g., moving to a higher color),
• Fluid jumps.

For each $c$, transition rates are encoded in matrices:

• $T_{--}^{(c)}$: $|S_-| \times |S_-|$, background transitions in down-phase;
• $T_{-+}^{(c)}$: $|S_-| \times |S_+^{(c)}|$, transitions initiating upward drift;
• $T_{++}^{(c)}$: $|S_+^{(c)}| \times |S_+^{(c)}|$, transitions within $S_+^{(c)}$;
• $T_{+-}^{(c)}$: $|S_+^{(c)}| \times |S_-|$, returning to down-phase.

Second-kind matrices $T_{-+}^{(c,c')}$, $T_{++}^{(c,c')}$ capture transitions to higher color levels $c' > c$. Boundary transitions at $x=0$ use $T_{--}^{(0)}$, $T_{-+}^{(0,c)}$. The PH-jump matrices $Q_\ell^{(c,c')}$ and $Q_\ell^{(c)}$ govern the jump dynamics. The entire generator is a (generally infinite-state) CTMC on $\Omega$, though it is never realized explicitly; only the block structure is used in computation.

3. Kolmogorov Forward Equations

Let $f_i(x_1,\dots,x_C; t)$ denote the joint density for phase $i$ and fluid vector $x$ at time $t$. The evolution is governed by a system of PDEs:

• For each color $c$, the forward equations reduce to a one-dimensional PDE in $x_c$, coupled via boundary/matching conditions due to color changes and jumps.
• For $C=2$ colors, the interior and boundary equations explicitly describe inflow, drift, and transition phenomena:

$$\partial_t f_-(x,y) - \partial_y f_-(x,y) = f_-(x,y)T_{--}^{(2)} + f_+(x,y)T_{+-}^{(2)},$$

$$\partial_t f_+(x,y) + \partial_y f_+(x,y) = f_-(x,y)T_{-+}^{(2)} + f_+(x,y)T_{++}^{(2)},$$

plus analogous boundary and matching conditions at $y=0$ and $(0,0)$. In steady state, the system decouples along color strata, given knowledge of the first-passage “return kernels” $\Psi_c$.

4. Matrix-Analytic Solution and Key Formulas

The central objects in analysis are the return-kernel matrices $\Psi_c$, which solve nonsymmetric algebraic Riccati equations (NAREs). For $C$ colors,

• For $c=C$,

$$0 = T_{++}^{(C)}\Psi_C + \Psi_C T_{-+}^{(C)}\Psi_C + \Psi_C T_{--}^{(C)} + T_{+-}^{(C)},$$

• For $c=C-1,\dots,1$,

$$0 = T_{++}^{(c)}\Psi_c + (T_{+-}^{(c)} + \sum_{\ell>c}T_{++}^{(c,\ell)}\Psi_\ell) + \Psi_c T_{-+}^{(c)}\Psi_c + \Psi_c (T_{--}^{(c)} + \sum_{\ell>c}T_{-+}^{(c,\ell)}\Psi_\ell).$$

Defining “censored” blocks for rates, these Riccati equations yield $\Psi_c$, which together determine spectral drift matrices $K_c = T_{++}^{(c)} + \Psi_c T_{-+}^{(c)}$.

The stationary joint density in a state with positive colors $c_1<\dots<c_n$ and fluid-vector $(x_{c_1},\ldots,x_{c_n})$ is

$$[\pi_+(x), \pi_-(x)] = p_- T_{-+}^{(0,c_1)} e^{K_{c_1} x_{c_1}} \left(\prod_{i=1}^{n-1}(T_{++}^{(c_i,c_{i+1})}+\Psi_{c_i} T_{-+}^{(c_i,c_{i+1})})e^{K_{c_{i+1}} x_{c_{i+1}}}\right)[I,\Psi_{c_n}].$$

The boundary vector $p_-$ satisfies

$$p_-(T_{--}^{(0)} + \sum_{c=1}^C T_{-+}^{(0,c)}\Psi_c) = 0,$$

normalized by

$$p_-\left(e + 2[T_{-+}^{(0,1)} \ldots T_{-+}^{(0,C)}]K^{-1} e\right) = 1.$$

Closed-form expressions for buffer-occupancy CDFs, top-color probabilities, and Laplace transforms follow by integrating matrix exponentials $e^{K x}$ (Houdt, 28 Jan 2026).

5. State-Space Reduction via Coloring

Classical single-color MMFQs track only the total fluid, lacking information about the origin or timing of fluid increments. In many finite-capacity systems, distinguishing each batch or job’s contribution is essential. However, direct modeling leads to state-space explosion: an order of $O(M^N)$ (with $M$ the PH-order, $N$ the number of tracked jobs/layers).

By assigning each “job” or fluid layer a color $c$, colored MMFQs retain only minimal memory: $C$ colors suffice, reducing the state-space to $O(CM)$. For example, in the MMAP[L]/PH[L]/1/N/LCFS model, colored fluid representation with $C=N$ tracks the $N$ colored fluid levels. The computation for $\Psi_c$ and $K_c$ reduces to $O(M_a^3(\sum M_{s_\ell})^3N)$—linear in $N$, cubic in total PH-dimensions—rather than exponential, enabling tractable analysis for $N \lesssim 10^3$ or $C \lesssim 10$ on commodity hardware (Houdt, 28 Jan 2026).

6. Computation of Performance Metrics with Fluid Jumps

For MMFQs with PH-distributed fluid jumps, performance metrics are computed by embedding jumps as upward “drift” intervals, leveraging the established colored MMFQ formulas and subsequently censoring post-jump drift. Specifically:

• In the joint density formula, replace the final $[I, \Psi]$ with $\Psi$;
• In the normalization formula, omit the factor two, as up/down symmetry is broken.

The algorithmic workflow is as follows:

• Fit each jump-size distribution with a phase-type representation $(\alpha, U)$.
• Expand background states to include jump phases for construction of $T_{++}$, $T_{-+}$, etc.
• Solve the NAREs using methods such as stable doubling (SDA/ADDA) or Sylvester solvers.
• Form drift matrices $K_c$, block matrix $K$, and compute matrix exponentials $e^{K_c x}$ and $K^{-1}$.
• Assemble stationary densities and integrate for statistics such as raw moments, CDFs, and blocking probabilities (Houdt, 28 Jan 2026).

Numerical studies validate the computational advantage: scenarios with $N$ up to $10^3$ or $C$ up to $10$ are tractable in milliseconds, contrasting with infeasibility for comparable finite-state Markov chain models.

7. Applications and References to Prior Work

Colored MMFQs with fluid jumps extend the analytic arsenal for telecommunications, computer systems, and queueing models, with particular applicability to finite-buffer systems, LCFS preemption, and multi-level job cascades. Classical analysis by da Silva Soares & Latouche for finite buffers [Perform. Eval. 63(4), 2006], and approaches by Dzial et al. on jump processes [Perform. Eval. 62(1-4), 2005], provide context for these advances. Algorithms for NAREs are informed by Wang et al. (SDA/ADDA) [SIAM J. Matrix Anal. Appl. 33(1), 2012] and Guo [J. Comput. Appl. Math. 192(2), 2006]. The colored MMFQ framework, as introduced and developed by B. Van Houdt (Houdt, 28 Jan 2026), constitutes a substantial expansion in the tractability and expressive power of fluid queue models.