⏳ Industrial Engineering Dashboard: M/M/c Queueing Simulator

A modern, web-based interactive simulation engine designed to visualize and analyze the M/M/c Queueing Model in real-time. This dashboard bridges the gap between theoretical Operations Research formulas and practical system behavior by simulating random arrivals (Poisson process) and service times (Exponential distribution) right in the browser.

🚀 Click Here for Live Demo

✨ Key Features

• Real-Time Stochastic Simulation: Watch customers arrive and get served dynamically based on probabilistic models, not pre-scripted animations.
• Live Chart.js Integration: Tracks and graphs the number of entities in the system ($L$) and in the queue ($L_q$) instantaneously as the simulation runs.
• Theoretical vs. Empirical: Automatically calculates steady-state expectations (Probability of zero entities, utilization rate, etc.) and allows you to compare them against the live running simulation.
• Time-Warp Engine: Includes a safe, anti-crash speed multiplier that allows you to simulate hours of queueing behavior in mere seconds.
• Glassmorphism UI: A highly polished, dark-themed interface inspired by modern analytics tools.

🧮 Mathematical Background (M/M/c Model)

The underlying engine relies on the classic Markovian queueing theory:

• Arrivals: Follow a Poisson process with rate $\lambda$ (Lambda).
• Service Times: Follow an Exponential distribution with rate $\mu$ (Mu).
• Servers: $c$ identical servers working in parallel.

The Steady-State formulas calculated natively in the dashboard:

System Utilization ($\rho$): $$\rho = \frac{\lambda}{c \mu}$$

Probability of Zero Entities ($P_0$): $$P_0 = \left[ \sum_{n=0}^{c-1} \frac{(\lambda/\mu)^n}{n!} + \frac{(\lambda/\mu)^c}{c!} \left( \frac{1}{1-\rho} \right) \right]^{-1}$$

Average Length of Queue ($L_q$): $$L_q = \frac{P_0 (\lambda/\mu)^c \rho}{c! (1-\rho)^2}$$

Note: The system mathematically warns the user and prevents theoretical calculation if $\rho \ge 1$ (unstable system capacity).

🛠️ Installation & Usage

This project is fully client-side. No complex dependencies, server setups, or databases required.

1. Clone the repository:

   git clone [https://github.com/onurfgg/mmc-queueing-simulator.git](https://github.com/onurfgg/mmc-queueing-simulator.git)
   

Onur Furkan Gök

Linkedin GitHub