All Labs

Queueing & Bottlenecks Lab

Explore how arrival rates, service rates, and the number of servers affect queue behavior. Compare M/M/1, M/M/c, and M/M/1/K models to understand utilization thresholds, bottleneck formation, and capacity planning.

Guided Experiment: The Utilization Threshold

What happens to queue length and wait times as utilization approaches 1? Is the relationship linear or non-linear?

Write your hypothesis in the Lab Report panel, then click Next.

Visualization (M/M/1)

Queue Diagram
Queue (Lq ≈ 3.2)BusyServerDone
Utilization Gauge
00.51.0
ρ = 0.8
Probability Distribution P(n)
020%116%213%310%48%57%65%74%83%93%102%P(n)n (customers in system)

Controls

4 / minutes
5 / minutes

Performance Metrics (M/M/1)

Utilization
ρ = 0.8(80.0%)
System is stable
P0P_0Probability system empty
0.2
LLAvg customers in system
4
LqL_qAvg customers in queue
3.2
WWAvg time in system
1minutes
WqW_qAvg wait in queue
0.8minutes
lambdatexteff\\lambda_{\\text{eff}}Effective throughput
4/minutes
Model Parameters
Model: M/M/1Servers: 1Stable (ρ < 1)

Data Table

(0 rows)
#TrialModelλμcρLLqWWq
0 / 500
0 / 500
0 / 500

Reference Guide

M/M/1 Single Server

The simplest queueing model. One server with Poisson arrivals and exponential service times. Stable only when the arrival rate is less than the service rate.

ρ=λμ,L=ρ1−ρ,W=1μ−λ\rho = \frac{\lambda}{\mu}, \quad L = \frac{\rho}{1 - \rho}, \quad W = \frac{1}{\mu - \lambda}

As utilization approaches 1, queue length grows without bound (hyperbolic growth).

M/M/c Multi-Server

Multiple identical servers share a single queue. Customers are served by the first available server. More efficient at spreading load.

ρ=λcμ,L=Lq+λμ,W=Lλ\rho = \frac{\lambda}{c\mu}, \quad L = L_q + \frac{\lambda}{\mu}, \quad W = \frac{L}{\lambda}

Adding servers reduces queue length but with diminishing returns as each additional server contributes less.

Little's Law

The most fundamental result in queueing theory. Relates the average number in the system to the arrival rate and average time in the system.

L=λeff⋅W,Lq=λeff⋅WqL = \lambda_{\text{eff}} \cdot W, \quad L_q = \lambda_{\text{eff}} \cdot W_q

This law holds for any queueing discipline and requires no assumptions about arrival or service distributions.

Stability & Bottlenecks

For infinite-capacity queues, stability requires utilization below 1. Finite-capacity systems (M/M/1/K) are always stable but reject excess arrivals.

ρ<1 (stable),λeff=λ(1−PK) (M/M/1/K)\rho < 1 \text{ (stable)}, \quad \lambda_{\text{eff}} = \lambda(1 - P_K) \text{ (M/M/1/K)}

Bottlenecks occur when utilization exceeds 0.8-0.9, where small increases in load cause large jumps in wait times.