Finite-state counterexamples to Ross's second queueing conjecture
Abstract
We construct finite-state Markov arrival-rate processes for which faster modulation increases the mean stationary workload of a stable single-server queue, disproving the universal ordering in Leskelä's formulation of Ross's second conjecture. A 64-state example has positive transition rates between every pair of states, independent service times uniform on $[0.99,1.01]$, and an explicit positive workload gap. A 128-state example additionally has strictly positive arrival-rate autocovariance at every lag. More generally, finite-state examples can exhibit any prescribed finite number of separated workload rises and falls. The proofs use a second-order expansion with a nonnegative, speed-uniform remainder under only a finite second service moment. We characterize the service distributions for which the second-order workload coefficient is nonincreasing for every bounded stationary environment: the function s↦𝔼(1−cos(sB))/s2 must be nonincreasing. Mixtures of exponential service distributions satisfy this criterion. An exact Palm identity transfers the workload reversals to customer waiting times.
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BibTeX
@misc{shmalo2026finitestate,
title = {{Finite-state counterexamples to Ross's second queueing conjecture}},
author = {Yitzchak Shmalo},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.25084},
eprint = {2609.25084},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25084},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Finite-state counterexamples to Ross's second queueing conjecture
AU - Yitzchak Shmalo
PY - 2026
DA - 2026/09/19
PB - arXiv
DO - 10.48550/arxiv.2609.25084
UR - https://arxiv.org/abs/2609.25084
AB - We construct finite-state Markov arrival-rate processes for which faster modulation increases the mean stationary workload of a stable single-server queue, disproving the universal ordering in Leskelä's formulation of Ross's second conjecture. A 64-state example has positive transition rates between every pair of states, independent service times uniform on $[0.99,1.01]$, and an explicit positive workload gap. A 128-state example additionally has strictly positive arrival-rate autocovariance at every lag. More generally, finite-state examples can exhibit any prescribed finite number of separated workload rises and falls. The proofs use a second-order expansion with a nonnegative, speed-uniform remainder under only a finite second service moment. We characterize the service distributions for which the second-order workload coefficient is nonincreasing for every bounded stationary environment: the function s↦𝔼(1−cos(sB))/s^2 must be nonincreasing. Mixtures of exponential service distributions satisfy this criterion. An exact Palm identity transfers the workload reversals to customer waiting times.
ER -