// multi-utility computation suite · offline · instant · precise
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sci.MM1-queue-waiting-time Calculator
Calculates M/M/1 queue metrics from arrival rate λ and service rate µ: utilisation ρ = λ/µ, mean queue length Lq = ρ²/(1−ρ), mean wait Wq = ρ/(µ−λ). At ρ = 0.9, mean queue length = 8.1 — doubling service capacity from ρ=0.9 to ρ=0.45 reduces queue length from 8.1 to 0.37.
Inputs
Lambda Per S
Amount per unit of time or per unit quantity. Check the denominator before interpreting.
Mu Per S
Amount per unit of time or per unit quantity. Check the denominator before interpreting.
Results
mean wait in queue Wq (s)
Arithmetic average — sum divided by count. Simple and familiar, but pulled by outliers. If your data contains extreme values, the median may be more representative.
mean time in system W (s)
Arithmetic average — sum divided by count. Simple and familiar, but pulled by outliers. If your data contains extreme values, the median may be more representative.
mean queue length Lq
The computed linear length.
server utilisation ρ
The value at the specified point or condition.
M/M/1: Wq = ρ/[μ(1−ρ)]
Reference formula or conversion factor shown for context.
system stability
Measure of how robust the system, structure, or process is to perturbations. Low stability means small changes can cause large deviations.