 Solution

At a common large threshold b, the header tail has exponent $-\log \rho$, whereas the payload tail has exponent $1-\rho$. Since $-\log \rho >1-\rho$ for $0<\rho <1$, the payload queue has the heavier tail and dominates overflow on the large-buffer exponential scale at fixed load. The inequality follows from the strict convexity of minus the logarithm, or from integrating $1/u>1$ between rho and one.

The explanation is that a header always consumes one slot, while a payload can be unusually large. A rare accumulation of work can therefore overflow the byte buffer without accumulating as many as 1000 packets. Replacing every payload by its mean would lose precisely this source of variability.

For the actual capacity 1000, retaining the exact continuous-time prefactors gives a qualification that a tail-exponent comparison alone cannot detect. The two infinite-buffer proxies are $p_H={\rho }^{1000}$ and $p_P=\rho e^{-1000(1-\rho )}$, andThus payload overflow is more likely under this proxy exactly when the displayed expression is positive. For ordinary loads it is positive; for example it is positive at $\rho =0.9$. At $\rho =0.999$ it is approximately $-0.00050017$, so the header proxy is slightly larger. Consequently, without a numerical load, there is no universal exact ordering for the fixed capacity, although the large-buffer exponential ordering is unambiguous. Very near full utilization both 1000-unit buffers have substantial tail probabilities, so equal numerical capacities alone do not establish that the design is sensible. These are the requested infinite-space estimates, rather than an exact loss model for the interacting finite buffers.