 Stationary workload of an exponential-payload queue

With packet arrival rate lambda, independent exponential payloads of mean one and byte-service rate C, let $\rho =\lambda /C<1$. The number N of unfinished packets has law $(1-\rho ){\rho }^n$. Conditional on N=n, the remaining payloads are n independent unit-mean exponentials, including the in-service residual by memorylessness. Summing their Laplace transforms gives $\mathbb{E}{e}^{-sW}=(1-\rho )(1+s)/(1+s-\rho )$. Thus W has mass $1-\rho$ at zero and, conditional on positivity, is exponential with rate $1-\rho$. The probability of at least r headers is instead ${\rho }^r$ for integer r.