AI 中文总结
该研究针对带动作延迟的并行无限服务器队列，用Erlang阶段结构建模延迟，推导服务器不平衡的特征方程，实验验证了流体近似及相关参数对瞬态响应的影响。
AI 中文摘要
空间分布式服务系统依赖于状态相关路由，将用户、任务或请求分配至负载较低的服务节点。实际中，路由决策不会立即生效：分配的作业需经历传输时间、通信延迟或执行机构滞后导致的延迟后才到达选定节点，我们将此延迟称为动作延迟。延迟信息模型将此类延迟视为过时信息，通过延迟微分方程分析模型，而本文中决策使用当前状态，仅执行被推迟，因此经历动作延迟的作业必须作为状态的一部分进行跟踪。通过将动作延迟表示为Erlang阶段结构，我们得到有限维马尔可夫跳跃过程，并构建常微分方程以显式跟踪延迟阶段中的作业。利用双服务器对称性，我们将动力学简化为服务器不平衡的差分模式，并针对任意阶段数推导其特征方程。该方程与延迟信息模型的特征方程一致，表明两种不同延迟具有相同的线性化不平衡动力学。数值实验验证了流体近似，并说明阶段数、路由灵敏度和平均延迟如何影响服务器不平衡的瞬态响应。
英文摘要
Spatially distributed service systems rely on state-dependent routing to allocate users, tasks, or requests to less-loaded service nodes. In practice, a routing decision does not take effect immediately: the assigned job reaches the selected node only after a lag caused by travel time, communication latency, or actuation. We call this lag the action delay. Whereas delayed-information models treat such a lag as stale information and analyze the model via a delay differential equation, the decision uses the current state and only its execution is deferred, so the job experiencing an action delay must be tracked as part of the state. Representing the action delay by an Erlang phase structure, we obtain a finite-dimensional Markov jump process and build ordinary differential equations that explicitly track the jobs in the delay phase. Exploiting the two-server symmetry, we reduce the dynamics to a difference mode for the server imbalance and derive its characteristic equation for an arbitrary number of phases. This equation coincides with that of the delayed-information model, showing that the two different delays share the same linearized imbalance dynamics. Numerical experiments confirm the fluid approximation and illustrate how the number of phases, the routing sensitivity, and the mean delay govern the transient response of the server imbalance.
