Colored Markov Modulated Fluid Queues

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.

💡 Research Summary

The paper introduces a novel extension of the classic Markov‑modulated fluid queue (MMFQ) by adding a “color” attribute to the fluid, resulting in the colored MMFQ and its further generalization colored MMFQ with fluid jumps. In a traditional MMFQ the fluid is indistinguishable; only the total fluid level matters. By assigning a color to each infinitesimal fluid packet when it arrives, the model can keep track of the contribution of different job classes, energy sources, or packet types. The colors are ordered (color 1 at the bottom, color C at the top) and the queue always stores fluid in this order. Consequently, the background Markov process has color‑dependent transition matrices: for each possible top‑color c there are four matrices (T^{(c)}{–}, T^{(c)}{-+}, T^{(c)}{++}, T^{(c)}{+-}) governing transitions while the fluid level is decreasing, increasing, or staying constant. When the top color changes (because the fluid of that color is depleted), the corresponding matrices switch instantly, providing a built‑in memory of the most recent events.

Two families of background transitions are defined. The first family stays within the same color set Ω_c and uses the four matrices above. The second family, called “second‑kind” transitions, moves the system from a state with top‑color c to a state with a higher color c′ ( c′>c ). These transitions introduce a new color of fluid and are described by matrices (T^{(c,c’)}{-+}) and (T^{(c,c’)}{++}). The same structure is replicated at the boundary (0,i) to allow the system to start increasing fluid using any chosen color.

To compute the stationary distribution, the authors derive color‑specific nonsymmetric algebraic Riccati equations (NAREs). For the highest color C the NARE coincides with the classic MMFQ equation: \