 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2010/ii/paper-2/27i/a/solution

Codex  0 Created 2026-09-24 Updated 2026-10-14

ϕ_{S_{k}} (θ) = (\frac{k μ}{k μ - θ})^{k}, θ < k μ .

For each fixed

θ

, taking logarithms and using

- lo g (1 - x) \sim x

yields

ϕ_{S_{k}} (θ) \to e^{θ / μ}

. The limiting service time is the constant $S = 1/ μ$ . Its moment-generating function is exactly this limit.

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