 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,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.