Verify weaker polynomial stability conditions for heavy-tailed queueing models

Establish verifiable weaker polynomial stability conditions for heavy-tailed queueing models, particularly lognormal-service models, without assuming a common geometric-drift condition, in order to support the transient-bias and ranking-stability analysis of direct sample-average learning.

Background

The paper’s theoretical verification relies primarily on a common geometric-drift condition at fixed-event-count cycle boundaries. The authors explicitly note that this convenient condition need not hold for heavy-tailed service distributions such as lognormal service, even when the queue is stable and relevant polynomial moments are finite.

The numerical lognormal-service experiment exhibits slower stabilization of objective rankings and motivates replacing geometric-drift assumptions with weaker polynomial or subgeometric stability conditions. Establishing such conditions would extend the direct sample-average and transient-bias framework beyond the light-tailed models for which the paper provides complete verification.

References

The lognormal experiment suggests that heavy-tailed service can make these finite horizons practically much more demanding: rare long services create long-lived congestion and slow stabilization of rankings. The long-horizon experiment shows that this need not prevent accurate identification, but it raises an important theoretical question about verifying weaker polynomial stability conditions without a common geometric-drift assumption.

— Direct Sample-Average Learning for Queueing Control: Gradient-Free Top-$K$ Coarse-to-Fine Optimization  (2610.26653 - Yazdi et al., 19 Oct 2026) in Section 7, Discussion, paragraph “Transient effects and heavy tails”