Queuing theory is the mathematical study of waiting lines, predicting queue length, wait time, and system behavior given arrival rate $\lambda$ and service rate $\mu$. The utilization ratio $\rho = \lambda/\mu$ must stay below 1 for stability, but queue length $L = \rho/(1-\rho)$ diverges nonlinearly—at 90% utilization, the queue is roughly 9 times longer than at 50%.

This explains why systems can suddenly develop enormous latency under modest load increases and why capacity planning requires real headroom rather than just meeting average load.

This example shows how queue length grows nonlinearly with utilization.

// compile: gcc -o queuing queuing.c -lm
// run: ./queuing
// description: demonstrate nonlinear queue growth as utilization approaches 1
 
#include <stdio.h>
#include <math.h>
 
int main() {
    printf("Utilization (rho) | Avg Queue Length (L)\n");
    printf("%-17s | %-20s\n", "---", "---");
 
    double rhos[] = {0.5, 0.7, 0.8, 0.9, 0.95, 0.99};
 
    for (int i = 0; i < 6; i++) {
        double rho = rhos[i];
        double L = rho / (1.0 - rho);
        printf("%.2f             | %.1f\n", rho, L);
    }
 
    printf("\nKey insight: queue length grows much faster than linearly\n");
    printf("at high utilization, causing latency cliffs.\n");
 
    return 0;
}