Scheduling in Queueing Systems with Uncertain and Evolving Holding Costs

arXiv:2505.21331v2 Announce Type: replace Abstract: In content moderation for social media platforms, the cost of delaying the review of a content is proportional to its view trajectory, which fluctuates and is apriori unknown. Motivated by such uncertain and evolving holding costs, we consider a queueing model where job states evolve based on a Markov chain with state-dependent instantaneous holding costs. We demonstrate that in the presence of such uncertain and evolving holding costs, the two canonical algorithmic principles, instantaneous-cost ($c\mu$-rule) and expected-remaining-cost ($c\mu/\theta$-rule), are suboptimal. By viewing each job as a Markovian ski-rental problem, we develop a new index-based algorithm, Opportunity-adjusted Remaining Cost (OaRC), that adjusts to the opportunity of serving jobs in the future when uncertainty partly resolves. We show that the suboptimality gap of OaRC scales as $\tilde{O}(\sqrt{N})$, where $N$ is the system size. This bound shows that OaRC achieves asymptotic optimality for overloaded systems when the system size $N$ scales to infinity. Moreover, the bound is independent of the state-space size, which is a desirable property when job states contain contextual information. We corroborate our results with an extensive simulation study based on two holding cost patterns (online ads and user-generated content) that arise in content moderation for social media platforms. Our simulations based on synthetic and real datasets demonstrate that OaRC consistently outperforms existing practice, which is based on the two canonical algorithmic principles.