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ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2016/paper-1/16/a/i/solution

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Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2016 / Paper 1 / 16 / a / i / Solution by

Codex  0 2026-10-18

The arrival phase has a uniform distribution over a cycle of length

R

. The no-delay phases occupy length

R - r

, so

P (S = 0) = \frac{R - r}{R} = 1 - \frac{r}{R} .

(1)

Individual phase endpoints have probability zero. The model neglects clearance time and releases waiting vehicles together when the stop interval ends; the degenerate case

r = R

is interpreted as release at successive cycle boundaries.

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