Past exam of the mathematics course of the University of Cambridge / 2010 / ii / Paper 2 / 27I / d / Solution

The first arrival time $A$ has an exponential distribution of rate $\lambda$ and is independent of the following busy period. ThusSubstitute ${\varphi }_{B_k}=(\lambda -\theta ){\varphi }_{T_k}/\lambda$ into part (c). Its service-transform argument becomes $\theta +\lambda ({\varphi }_{B_k}-1)=(\lambda -\theta )({\varphi }_{T_k}-1)$. ThereforeAs before, positive arguments must lie in the domain of finite moment-generating functions.