Operations research 5 things

4 Things About Queues Every OR Student Needs

4 things about queuing theory that explain every waiting line…

Lesson notes

Key points

λ: arrival rate: Lambda is the average number of customers arriving per unit of time.
μ: service rate: Mu is the average number of customers one server can finish per unit time.
ρ = λ/μ: Utilisation rho is lambda divided by mu, the fraction of time the server is busy.
ρ must stay below 1: If arrivals equal or exceed service capacity, the queue grows without limit.
M/M/1 notation: M/M/1 means random arrivals, random service times, and a single server.

Takeaway: Utilisation ρ = λ/μ must stay below 1 for a stable queue.

Quick check: Customers arrive at 8 per hour; the server handles 10 per hour. What is ρ?

1.25
0.8
2

ρ = λ/μ = 8/10 = 0.8, so the server is busy 80% of the time.

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