= Solution
Let $\rho=\lambda/C<1$ and measure payload in kilobytes. A unit-mean exponential payload requires an exponential <service time> of rate C. Headers remain until service is complete, so Q counts all unfinished packets, including the packet in service. Its process is an <M M 1 queue> with birth rate lambda and death rate C whenever nonempty.
The <detailed balance equations> give $\pi_{k+1}=\rho\pi_k$, and normalization gives $\pi_k=(1-\rho)\rho^k$. In the stationary infinite-space approximation,
$$
\boxed{\mathbb P(Q\geq q)=\rho^q,\qquad q=0,1,2,\ldots.}
$$
For a noninteger positive threshold replace q by its ceiling. At capacity 1000 the requested tail estimate is $\rho^{1000}$; literal exceedance of that capacity uses the adjacent integer threshold 1001. We compare the stated capacity-tail proxies consistently below.
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