Asymptotic behavior of the covert queueing constants

Establish an analytical characterization of the limiting behavior of the covert queueing constants K_X and K_Y as the Willie-job arrival rate approaches zero or the service rate from below, including the corresponding asymptotics of the functions J(lambda_w, mu) and C(lambda_w, mu).

Background

The paper derives explicit formulas for the square-root-law scaling constants K_X and K_Y when Willie observes, respectively, the number of jobs served during each busy period and the busy-period duration. The formula for K_X is elementary, whereas K_Y depends on the function C(lambda_w, mu), which itself is defined using the integral J(lambda_w, mu) involving modified Bessel functions.

Numerical plots suggest that both constants diverge as lambda_w approaches 0 and as lambda_w approaches mu from below. The paper states that obtaining an analytical characterization of these limiting behaviors requires a detailed analysis of J(lambda_w, mu) and C(lambda_w, mu), and leaves that analysis unresolved. It is therefore an explicit problem to determine the endpoint asymptotics of both constants and the underlying functions.

References

Furthermore, both constants appear to diverge as $\lambda_w\to0$ and as $\lambda_w\to\mu$. An analytical characterization of the limiting behavior of $K_X$ and $K_Y$ requires a detailed investigation of the functions $J(\lambda_w,\mu)$ and $C(\lambda_w,\mu)$ introduced in \cref{lem:Tayor_D_KL_Stat2}. We leave this investigation for future work.

— Fundamental Limits of Covert queueing under Busy-Period Observations  (2610.13911 - Sahithi et al., 9 Oct 2026) in Section Discussion, immediately following Figure 1 and Equation K_X <= K_Y