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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2011/ii/paper-4/26j/iii/solution

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Past exam of the mathematics course of the University of Cambridge / 2011 / ii / Paper 4 / 26J / iii / Solution by

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

The service-period arrival count has probability generating function

g (s) = j \geq 0 \sum a_{j} s^{j} = \int e^{- λ (1 - s) t} d G (t), ∣ s ∣ \leq 1,

with

g (1) = 1

and

g^{'} (1) = ρ

. Let

Π (s) = \sum_{j \geq 0} π_{j} s^{j}

. In stationarity the recurrence and independence give

Π (s) = g (s) [π_{0} + \frac{Π ( s ) - π _{0}}{s}],

initially for

0 < s < 1

. Rearrangement gives

Π (s) = π_{0} (s - 1) g (s) / (s - g (s))

. Letting

s ↑ 1

and using

Π (1) = 1

shows

π_{0} = 1 - ρ

. Hence

Π (s) = \frac{( 1 - ρ ) ( s - 1 ) g ( s )}{s - g ( s )} .

Analytic continuation gives the identity inside the unit disc; boundary values follow by convergence of the probability series, with removable limiting values wherever the displayed quotient is indeterminate, particularly at

s = 1

.

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