How to Derive the Core Formulas of a Single-Server Queueing System

Single-Server Queueing System Theory Derivation

The M/M/1 model is the simplest queueing model: arrivals follow a Poisson process with rate λ, service times are exponentially distributed with rate μ, there is a single service station, an infinite source of customers, unlimited system capacity, and arrivals are independent of service times. The service discipline is first‑come‑first‑served.

In the exponential distribution, the probability of one arrival in a small interval Δt is λΔt, the probability of one departure is μΔt, and the probability of more than one arrival or departure is negligible (order (Δt)²).

At time t, the state transition probabilities lead to the following balance equations (after simplification):

... (after algebra we obtain the steady‑state probabilities).

For the whole system in steady state, the state balance equations are:

P_i : probability that the system has i customers

λ : arrival rate of customers

μ : service rate

Since the sum of all state probabilities equals 1, we have Σ_{i=0}^∞ P_i = 1.

Using a power‑series expansion, we obtain:

... (the resulting expression P_i = (1‑ρ)ρ^i, where ρ = λ/μ is the traffic intensity, reflecting the utilization of the service facility.)

The average number of customers in the system (including those waiting and being served) is the mean queue length L, and the average number waiting in line is L_q. The average time a customer spends in the system, also called the average residence time W, is derived from Little’s law.

The average waiting time before service begins is W_q, and the average service time per customer is 1/μ . Consequently, the following relationships hold:

L = ρ / (1‑ρ)

L_q = ρ² / (1‑ρ)

W = 1 / (μ‑λ)

W_q = λ / (μ(μ‑λ))

These metrics provide a complete performance analysis of the M/M/1 queue.

Reference

ThomsonRen GitHub: https://github.com/ThomsonRen/mathmodels