Questions: Queueing Theory: The M/M/1 Queue and Little's Law

Question 1

Customers arrive at a single-server M/M/1 queue at rate λ = 8 per hour, and the service rate is μ = 10 per hour. What is the mean time a customer spends in the system?

• A. 6 minutes
• B. 24 minutes
• C. 30 minutes
• D. 48 minutes

Answer: (C)

W = 1/(μ - λ) = 1/(10 - 8) = 1/2 hour = 30 minutes. The value 24 minutes is the mean waiting time before service begins, Wq = ρ/(μ - λ) = 0.8/2 hour, and 6 minutes is the mean service time 1/μ.

Question 2

In the same queue (λ = 8, μ = 10), what fraction of time is the server idle, and what is the mean number in the system?

Model answer: Idle fraction π₀ = 1 - ρ = 0.2; mean number L = ρ/(1 - ρ) = 0.8/0.2 = 4.

With ρ = λ/μ = 0.8, the stationary distribution is πₙ = 0.2 · 0.8ⁿ. As a check with Little's law, L = λW = 8 × (1/2) = 4.

Question 3

An M/M/1 queue with arrival rate equal to service rate (λ = μ) settles into a steady state with a finite mean queue length.

• T. True
• F. False

Answer: False

When ρ = 1 the balance equations force all πₙ to be equal, which cannot sum to 1, so no stationary distribution exists. The queue length behaves like a null-recurrent random walk and does not settle down.

Question 4

A coffee shop has on average 12 customers inside, and customers arrive at 1 per minute. Assuming a stable system, how long does a customer stay on average?

Model answer: 12 minutes

Little's law gives W = L/λ = 12/1 = 12 minutes. The law needs only long-run averages, so it applies without knowing the arrival or service distributions.