Queueing models

Objective. Analyse arrival, service and congestion.

1. Learn
2. Worked example
3. Practice
4. Feedback
5. Continue

Learn: the key idea

Queueing models describe how customers arrive, wait and are served, and analyse congestion using arrival rate λ, service rate μ and traffic intensity ρ = λ/μ. For the M/M/1 queue, with Poisson arrivals and exponential service, the system is stable only if ρ < 1, and the expected number in the system is L = ρ/(1 − ρ). For example, if λ = 3 customers per hour and μ = 4 per hour, ρ = 0.75 and L = 0.75/0.25 = 3 customers on average. Little's law L = λW then converts this into expected waiting time W.

Worked example

In a queueing model of a single-server bank counter, customers arrive at λ = 6 per hour and are served at μ = 8 per hour. Assuming an M/M/1 queue, find the expected number of customers in the system.

1 $Compute the traffic intensity ρ = λ/μ = 6/8 = 0.75, which is less than 1, so the queue is stable.$
2 $Use the M/M/1 formula for expected number in the system, L = ρ/(1 - ρ).$
3 $Substitute ρ = 0.75 to get L = 0.75/0.25.$
4 $Simplify 0.75/0.25 = 3, giving the expected number in the system.$

Answer: 3 customers

Concept mastery

Based on your past attempts at this lesson — the question types to practise again come first.

Complete a practice set to see your concept breakdown here.

Practice

Practice happens on its own screen, one question at a time. Answers stay hidden until you submit yours, and you can stop and pick up at the same question later.

Start practising

Common mistake. A common mistake is computing L = ρ/(1−ρ) when ρ ≥ 1, ignoring that the queue is unstable and grows without bound; stability requires checking ρ = λ/μ < 1 before applying steady-state formulas.

Your progress

Tick each step as you finish it. Your exact reading position is saved automatically, so Continue drops you back on the same line.

0 of 5 steps complete

Your place saves automatically as you read.

My dashboard