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# Queuing theory

**[Queuing theory](https://en.wikipedia.org/wiki/Queueing_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.

## Example

This example shows how queue length grows nonlinearly with utilization.

```c
// 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;
}
```