Interactive Case Study Presets
Business Service Counter
Moderate queueing delay. Balanced operational state.
λ=20, μ=25High Congestion ATM
Slightly bottlenecked system. Slow queues, high delays.
λ=15, μ=16Drive-Thru Lane
High throughput. Rapid serving, optimized flow.
λ=45, μ=60CPU Task Scheduler
Very fast service vs rare tasks. Minimal delay state.
λ=8, μ=25Queuing Parameters
customers / hour
customers / hour
Stability Status:
Satisfied
For an M/M/1 system to operate in steady state, service rate must exceed arrival rate ($\mu > \lambda$).
Currently: 25 > 20
Queue Unstable!
The arrival rate ($\lambda$) equals or exceeds the service rate ($\mu$). The queue length will grow infinitely.
Sim Controls
Sim Time: 0.00h
Live Utilization Gauge
0.0%
Traffic Intensity (ρ)
Queue Animator (Stochastic Flow)
Queue: 0
Server: 0/1
Served: 0
* Each element represents an individual customer drawn with random exponential properties.
Scale: Exponential Inter-arrival & Service Rates
Real-Time Lab Report: Analytical vs. Empirical
Sample Size: 0 pts| Performance Metric | Formula Notation | Analytical (Ideal) | Empirical (Simulated) | Relative Error |
|---|---|---|---|---|
| Server Utilization | $\rho = \lambda / \mu$ | - | - | - |
| Avg. Number in System | $L = \frac{\lambda}{\mu - \lambda}$ | - | - | - |
| Avg. Number Waiting (Queue) | $L_q = \frac{\lambda^2}{\mu(\mu - \lambda)}$ | - | - | - |
| Avg. Time in System | $W = \frac{1}{\mu - \lambda}$ | - | - | - |
| Avg. Waiting Time in Queue | $W_q = \frac{\lambda}{\mu(\mu - \lambda)}$ | - | - | - |