Queue Utilisation
Fraction of a server's capacity being used by arriving traffic.
Formula
Variables
Example
λ=80, μ=100: ρ = 0.8
How it works
Queue utilisation measures how heavily a single server is loaded by comparing the rate of jobs arriving to the rate it can complete jobs. If arrivals happen at rate λ and service can be performed at rate μ, then the ratio λ/μ tells you what fraction of the server’s capacity is being consumed on average. A value below 1 means the server can keep up in the long run; values at or above 1 indicate the queue will tend to grow because work arrives as fast as, or faster than, it can be cleared.
When to use it
Use this in queueing models for routers, call centres, web servers, or any system with one processing resource handling incoming tasks. It is especially useful when checking whether a network node is stable and how close it is to saturation.
Worked example, step by step
Suppose packets arrive at a switch at λ = 45 packets/s and the switch can process μ = 60 packets/s. ho = λ / μ ho = 45 / 60 ho = 0.75 So the queue utilisation is 0.75, meaning the server is busy 75% of the time on average.
Common mistakes to avoid
- Mixing up arrival rate and service rate, which flips the utilisation result
- Using inconsistent units, such as arrivals per second with service rate per minute
- Assuming ρ can exceed 1 in a stable single-server queue without causing the backlog to grow
Did You Know?
As utilisation approaches 1, queuing delay heads toward infinity — why networks never run fully loaded.
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