In practice, the cost of delaying a job can grow as the job waits. Such behavior is modeled by the Time-Varying Holding Cost (TVHC) problem, where each job's instantaneous holding cost increases with its current age (a job's age is the time since it arrived). The goal of the TVHC problem is to find a scheduling policy that minimizes the time-average total holding cost across all jobs. However, no optimality results are known for the TVHC problem outside of the asymptotic regime. In this paper, we study a simple yet still challenging special case: A two-class M/M/1 queue in which class 1 jobs incur a non-decreasing, time-varying holding cost and class 2 jobs incur a constant holding cost. Our main contribution is deriving the first optimal (non-decreasing) index policy for this special case of the TVHC problem. Our optimal policy, called LookAhead, stems from the following idea: Rather than considering each job's current holding cost when making scheduling decisions, we should look at their cost some $X$ time into the future, where this $X$ is intuitively called the ``lookahead amount." This paper derives that optimal lookahead amount.

翻译：在实践中，延迟处理作业的成本可能随着作业等待时间的增加而增长。此类行为可通过时变持有成本（TVHC）问题进行建模，其中每个作业的瞬时持有成本随其当前“年龄”（即自作业到达后经过的时间）增加而增长。TVHC问题的目标是寻找一种调度策略，以最小化所有作业的时间平均总持有成本。然而，在渐近范围之外，TVHC问题尚未有最优性结果被证实。本文研究一个简单但仍具挑战性的特例：一个两类M/M/1队列系统，其中第1类作业具有非递减的时变持有成本，而第2类作业具有恒定持有成本。我们的主要贡献是为TVHC问题的这一特例推导出首个最优（非递减）索引策略。我们提出的最优策略称为“前瞻策略”，其核心思想源于以下理念：在制定调度决策时，不应仅考虑每个作业当前的持有成本，而应关注其在未来$X$时间后的成本，其中$X$被直观地称为“前瞻量”。本文推导出了这一最优前瞻量的具体表达式。