ITPEC FE Subject A April 2026 Question 3

Utilization — the arrival rate divided by the service rate, ρ = λ ÷ μ, the fraction of time the server is busy.

With mean arrival rate λ and mean service rate μ, ρ is dimensionless. When ρ < 1 the server keeps up; as ρ approaches 1 the waiting time grows sharply, and at ρ ≥ 1 the queue grows without bound.

Answer (a)

Why not others:

• •(b) Average inter-arrival time — the reciprocal of the arrival rate, 1 ÷ λ; a time, not a ratio of two rates
• •(c) Average service time — the reciprocal of the service rate, 1 ÷ μ; also a time
• •(d) Average retention number — the average number of customers in the system, which for M/M/1 is ρ ÷ (1 − ρ), derived from utilization rather than equal to it

Key rule: In queueing theory a rate divided by a rate is utilization; anything called an average time is the reciprocal of a rate, and average counts are built from ρ.

Basic Theory · AI-written, may contain errors