= Stationary workload of an exponential-payload queue
{title2=$\mathbb P(W\geq r)=\rho e^{-(1-\rho)r},\quad r>0$}
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.
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