The asymptotic behavior of the optimal TSP tour length is well known from the classical Beardwood--Halton--Hammersley theorem. We extend this result to the Traveling Salesman Problem with Drone (TSPD), a cooperative routing problem in which a truck and a drone jointly serve customers. Using a nonmonotone subadditive Euclidean functional framework, we establish the existence of an almost sure limit for the optimal TSPD makespan scaled by the square root of the problem size. We derive explicit upper and lower bounds for the speed-scaled Euclidean TSPD model: upper bounds are obtained via structured ring-based tour constructions and Monte Carlo evaluation, while lower bounds are derived using nearest-neighbor distance distributions and the $k$-traveling salesman problem. Computational results illustrate how tight the bounds are. We also extend the analysis to the Rectilinear--Euclidean mixed TSPD model, in which truck travel is measured by the rectilinear distance and drone travel by the Euclidean distance.

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