Queueing Theory
Queueing theory models waiting lines. The exam focuses on the M/M/1 system: Poisson arrivals, exponential service times, one server, infinite capacity, and first-come-first-served.
M/M/1 formulas
Let λ = arrival rate and μ = service rate in the same time unit. Utilisation ρ = λ/μ; the system is stable only if ρ < 1.
- Probability server is idle: P0 = 1 − ρ.
- Average number in system: Ls = λ/(μ − λ) = ρ/(1 − ρ).
- Average number waiting: Lq = λ²/[μ(μ − λ)] = ρ²/(1 − ρ).
- Average time in system: Ws = 1/(μ − λ).
- Average waiting time: Wq = λ/[μ(μ − λ)]. Little's law links them: Ls = λWs and Lq = λWq.
Worked example: customers arrive at 6 per hour and are served at 10 per hour. ρ = 0.6. Ls = 6/(10−6) = 1.5 customers; Lq = 36/[10(4)] = 0.9 customers. Ws = 1/4 hour = 15 minutes; Wq = 6/(10×4) = 0.15 hour = 9 minutes.
Traps: using λ in place of ρ in the formulas; applying M/M/1 when λ ≥ μ; mixing per-hour and per-minute rates; and confusing Ls with Lq or Ws with Wq.