Past exam of the mathematics course of the University of Cambridge / 2008 / ii / Paper 4 / 26I / d / Solution

Only arrivals at times $u\le t$ remain after the window closes. Such an arrival survives until $t+v$ exactly when its service time exceeds $t+v-u$. The same independent Poisson thinning givesAt $v=0$ this agrees with part (b), while for large $v$ the parameter tends to zero.