Queueing theory

Enables estimation of system latency given arrival and service rates.

Web server: high request arrival rate approaching service capacity; Call center: predicting average wait time for callers.

Performance engineering, operations research, network traffic modeling

measure λ and μ → compute R = 1/(μ - λ) → assess if response time meets SLA

R = 1/(μ - λ)

• arrival rate λ
• service rate μ
• utilization ρ

• Using utilization ρ = λ/μ
• the formula can be written as R = (1/μ) * 1/(1-ρ)
• which highlights the impact of load.

Using λ ≥ μ leads to division by zero or negative response time → predicts impossible performance; Ignoring the exponential service assumption and applying the formula to deterministic services → inaccurate latency estimates; Substituting μ with average throughput instead of service rate → underestimates response time.

Average number in system L = λ/(μ - λ) – measures queue length, not time; Little's Law L = λ * R – relates the two but does not provide R directly.

Coming from deterministic queuing models: assuming constant service time → misuse of the exponential assumption in M/M/1, which can underestimate variability.

• Utilization M/M/1
• Average number in system M/M/1
• Little's Law
• M/M/c response time

Do not use when λ ≥ μ because the system becomes unstable; As λ approaches μ, response time grows sharply, indicating potential overload; The formula assumes Poisson arrivals and exponential service times, so deviations affect accuracy.

Provides a quick analytical check for capacity planning and SLA verification, preventing under-provisioning of resources.

Imagine a single checkout lane: the waiting time equals one divided by how much faster the cashier works than customers arrive.

Assumes a single server, infinite buffer, and first‑come‑first‑served discipline.

After mastering R = 1/(μ - λ), progress to the M/M/c response time formula: R = (1/μ) * (C(ρ) / (c(1-ρ))) where C(ρ) accounts for multiple servers.