AI 中文总结
针对具有冗余和异构服务器的分叉-合并系统，通过投影和广义Schur-凸序方法，确定了静态和动态容量分配策略下达到最大稳定性区域的条件。
AI 中文摘要
我们考虑了在静态和动态容量分配策略下，具有冗余（FJR）和异构服务器的分叉-合并系统的稳定性问题。在一个$(n,k)$ FJR系统中，每个到达的作业被分割成$n$个独立的任务，每个任务分配给$n$个并行服务器中的一个。一旦$k \le n$个任务被处理完毕，它们被合并，相应的作业离开系统；剩余的$n-k$个未处理任务随后被移除，因此被称为冗余任务。我们首先确定了名义流量强度，并刻画了最大稳定性区域，该区域定义为存在一个可容许策略使系统稳定的流量强度集合。然后，我们建立了在两类策略（静态和动态）下达到此最大稳定性区域的条件。具体来说，我们表明，对于静态分配策略（其中服务容量随时间保持不变），当最快的服务器被分配不超过总服务容量的$1/k$时，可实现最大性。对于动态分配策略（其中固定的总服务容量可以在服务器之间反复重新分配），我们表明，当分配给$j$个最短队列的累计容量不超过总容量的$j/k$（对于每个$j=1,\ldots,k-1$）时，可实现最大性。我们的分析基于将$(n,k)$ FJR系统投影到一个没有冗余的更简单的$(k,k)$系统，以及基于广义Schur-凸序的多维过程的一种新颖的样本路径比较论证。
英文摘要
We consider the stability problem of fork-join systems with redundancy (FJR) and heterogeneous servers under both static and dynamic capacity-allocation policies. In an $(n,k)$ FJR system, each arriving job is split into $n$ independent tasks, with one task assigned to each of $n$ parallel servers. Once $k \le n$ tasks have been processed, they are joined and the corresponding job departs the system; the remaining $n-k$ unprocessed tasks are then removed and are therefore termed redundant. We first identify the nominal traffic intensity and characterize the maximal stability region, defined as the set of traffic intensities for which there exists an admissible policy that stabilizes the system. We then establish conditions under which this maximal stability region is attained for two classes of policies: static and dynamic. Specifically, we show that for static allocation policies, in which service capacities remain fixed over time, maximality is achieved whenever the fastest server is allocated no more than $1/k$ of the total service capacity. For dynamic allocation policies, in which a fixed total service capacity may be repeatedly reallocated among the servers, we show that maximality is achieved whenever the cumulative capacity allocated to the $j$ shortest queues does not exceed $j/k$ of the total capacity for every $j=1,\ldots,k-1$. Our analysis is based on a projection of the $(n,k)$ FJR system onto a simpler $(k,k)$ system that has no redundancy, together with a novel sample-path comparison argument for multidimensional processes based on the generalized Schur-convex order.
Comments28 pages, 1 figure. Current status: Reject and Resubmit, Operations Research. Resubmitted on July 26, 2026
