Dynamic policy for idling time preservation
研究服务台在仍有顾客等待时主动空闲的动态策略，通过马尔可夫决策过程证明最优策略为阈值策略，并发现空闲策略能达到拒绝策略无法实现的高空闲概率，尤其在高拥堵且目标占用率接近50%时效果最佳。
Abstract This study aims to determine and evaluate dynamic idling policies where an agent can idle while some customers remain waiting. This type of policies can be employed in situations where the flow of urgent customers does not allow the agent to spend sufficient time on back‐office tasks. We model the system as a single‐agent exponential queue with abandonment. The objective is to minimize the system's congestion while ensuring a certain proportion of idling time for the agent. Using a Markov decision process approach, we prove that the optimal policy is a threshold policy according to which the agent should idle above (below) a certain threshold on the queue length if the congestion‐related performance measure is concave (convex) with respect to the number of customers present. We subsequently obtain the stationary probabilities, performance measures, and idling time duration, expressed using complex integrals. We show how these integrals can be numerically computed and provide simpler expressions for fast‐agent and heavy‐traffic asymptotic cases. In practice, the most common way to regulate congestion is to control access to the service by rejecting some customers upon arrival. Our analysis reveals that idling policies allow high levels of idling probability that such rejection policies cannot reach. Furthermore, the greatest benefit of implementing an optimal idling policy occurs when the objective occupation rate is close to 50% in highly congested situations.