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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/ii/paper-3/27k/a/solution

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Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 27K / a / Solution by

Codex  0 Created 2026-09-24 Updated 2026-10-03

Let regeneration cycles have lengths

C_{i}

, contain

A_{i}

arrivals, and accrue queue-length reward

R_{i} = \int_{cycle i} Q (t) d t .

(1)

Fubini's geometric identity writes this area as the sum, over customers in the cycle, of their time in the system, up to boundary terms whose contribution vanishes over many cycles. The renewal-reward theorem therefore gives

L = \frac{E R _{1}}{E C _{1}}, λ = \frac{E A _{1}}{E C _{1}}, W = \frac{E R _{1}}{E A _{1}} .

(2)

Hence Little law is

L = λW .

(3)

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