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Heavy‑tailed strong tail optimality beyond regular variation

Prove that the strong tail optimality results established for regularly varying job size distributions in the M/G/1 extend to intermediate regularly varying distributions; in particular, generalize the necessary condition used in Wierman–Zwart (2012, Proposition 1) so that the conclusion inf_π C_π = 1 holds under intermediate regular variation.

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Background

Appendix A shows that for regularly varying job size distributions, several policies (e.g., SRPT, LAS) are in fact strongly tail‑optimal, by leveraging a necessary condition from Wierman and Zwart (2012).

The authors explicitly conjecture that the regular variation assumption can be relaxed to intermediate regular variation, noting that the only proof step that relies on full regular variation is the cited necessary condition; a suitable generalization would carry the result through.

References

We further conjecture that the regularly varying requirement on $S$ can be relaxed to requiring $S$ be intermediate regularly varying, namely

\limsup_{\delta \to 0} \limsup_{t \to \infty} \frac{\P{S > t}{\P{S > (1 - \delta) t} = 1.

This property suffices for the computation in our proof above, and it suffices for many of the prior works showing $C_\pi = 1$ for various policies~\citep{nunez-queija_queues_2002, scully_characterizing_2020, scully_when_2024}. The only step of the proof that requires (non-intermediate) regular variation is \cref{eq:heavy-tailed_necessary_condition}, the result of \citet[Proposition~1]{wierman_tailoptimal_2012}. If their result could be generalized to give the same necessary condition for $C_\pi < \infty$ even when $S$ is intermediate regularly varying, it would imply $\inf_\pi C_\pi = 1$ in that setting, too.

— Strongly Tail-Optimal Scheduling in the Light-Tailed M/G/1 (2404.08826 - Yu et al., 12 Apr 2024) in Appendix A (Strong tail optimality for heavy‑tailed job size distributions)