An applied mathematics research paper and interactive dashboard exploring how continuous-time dynamical systems can model server task distribution and queue stability. Developed coupled differential equation models to evaluate classical balancing strategies under continuous task arrival and processing rates, backed by a numerical simulation dashboard.

Why?

Traditional queueing theory often analyzes server workloads discretely, which can obscure macroscopic fluid-like behaviors during traffic shifts. Modeling server load continuously via differential equations enables closed-form asymptotic stability analysis, phase portraits, and direct evaluation of how routing algorithms behave under saturated compute capacities.

How?

• Mathematical Modeling & System Dynamics

  • Formulated autonomous continuous-time systems for decoupled, directly coupled, and $n$-server clusters.
  • Modeled proportional and saturated resource allocation modes using bounding activation multipliers $S(x_i) = \frac{x_i}{x_i+1}$.
  • Derived trace-determinant stability criteria and verified negative eigenvalue asymptotic convergence for coupled clusters.

• Algorithm Analysis

  • Modeled continuous formulations of Round Robin, Weighted Round Robin, Least Connections, and Weighted Least Connections.
  • Compared equilibrium convergence, proving how weighting arrival rates by expected wait time $W_i = \frac{x_i}{\mu_i}$ optimizes cluster-wide load distribution.

• Interactive Simulation & Tooling

  • Developed a standalone Streamlit visualizer utilizing the Runge-Kutta 4th Order (RK4) numerical integration method for high numerical stability.
  • Rendered real-time phase portraits and time-series load trajectory curves under customizable arrival and coupling gain factors.