Arrival rate: arrivals per minute
Service times
When a minute ends, its arrival count (the colored box in the strip) flies into the left chart: that chart is the arrival-rate distribution, and for Poisson arrivals it matches the Poisson(λ) dots. Each finished service sends its service time into the right chart. The bright bar is where the newest value landed; the blue marks are what the chosen distribution predicts. Data flies in at speeds up to 8 min/s; faster runs add it directly.
Performance: simulated vs theory
Big number: measured in the simulation since the last setting change. Blue: the formula value. They converge as more customers pass through; press “Run 2,000 customers” to see it.
More distributions
Time between arrivals
Line length over time
Number in system, P(n)
Effect of the number of channels: M/M/1 vs M/M/s
“One fast channel” is a single server working s times faster (rate sμ). It has the same total capacity as s channels; comparing it with M/M/s shows the trade-off between pooling and speed.
How the theory numbers are computed
If the count of arrivals in a minute is Poisson(λ), the gaps between arrivals are Exponential with mean 1/λ. They describe the same Markovian process (letter M).
P(N=k) = e^(−λ) λ^k / k! f(t) = λ e^(−λt)One channel, ρ = λ/μ must be below 1.
L = ρ/(1−ρ) · Lq = ρ²/(1−ρ) · W = 1/(μ−λ) · Wq = ρ/(μ−λ)a = λ/μ, ρ = λ/(sμ) < 1.
P₀ = [ Σ_{n=0}^{s−1} aⁿ/n! + aˢ/(s!(1−ρ)) ]⁻¹
P(wait) = aˢ P₀ / (s!(1−ρ)) · Lq = P(wait)·ρ/(1−ρ)Holds for every queue here. With a limited line, λ is replaced by the rate of people let in, λ(1 − P(turned away)).
L = λW · Lq = λWq · W = Wq + 1/μSolved as a birth–death chain with pₙ ∝ Πλ/(min(k,s)μ) up to s + waiting room. Works for ρ ≥ 1 too.
P(turned away) = p_{s+K}No exact formula. The page uses the Allen–Cunneen approximation with squared coefficients of variation c² (Exponential 1, Normal CV², Constant 0).
Wq(G/G/s) ≈ Wq(M/M/s) · (c²ₐ + c²ₛ)/2