Continuous-time Markov chain | M/M/1 queue | stationary distribution π_n = ρⁿ(1−ρ)
ρ = λ/μ = 0.60
Stable? ρ < 1: YES
E[N] = ρ/(1−ρ) = 1.50
P(extinction) = 1.00 (ρ<1)
Current state: 0
Time: 0.00
M/M/1 queue: arrivals Poisson(λ), service Exp(μ).
π_n = (1−ρ)ρⁿ for ρ = λ/μ < 1
Generator: q_{n,n+1}=λ, q_{n+1,n}=μ (n≥1), q_{0,1}=λ
Balance: λπ_n = μπ_{n+1} → π_n = ρⁿπ_0
Extinction: starting at X(0)=1, P(reach 0) = min(1,(μ/λ)¹)