Birth-Death Process / M/M/1 Queue

Poisson Arrivals · Exponential Service · Little's Law · Utilization

Queue Parameters

0.8
1.0
15

Live Metrics

P(n) Distribution

Queue Length Over Time

Theory (M/M/1)

Birth-death: λ_n = λ, μ_n = μ (state-independent)
Balance: λ·P(n) = μ·P(n+1)
Stationary: P(n) = (1−ρ)·ρⁿ, ρ = λ/μ
Little's Law: L = λ·W
L = ρ/(1−ρ), Lq = ρ²/(1−ρ)
W = 1/(μ−λ), Wq = ρ/(μ−λ)
ρ < 1: stable (ergodic)
ρ ≥ 1: queue grows without bound!
Poisson arrivals: memoryless, bursty