Ross第二排队猜想的有限状态反例

中文摘要 AI 辅助

我们构造了有限状态的马尔可夫到达率过程，对于这些过程，更快的调制会增加稳定单服务器队列的平均平稳工作量，从而否定了Leskelä对Ross第二猜想表述中的普适排序。一个64状态的例子在每对状态之间具有正转移率，服务时间独立且在$[0.99,1.01]$上均匀分布，并具有显式的正工作量差距。一个128状态的例子还额外具有在每个滞后上严格正的到达率自协方差。更一般地，有限状态例子可以表现出任意指定的有限数量的分离的工作量上升和下降。证明使用了二阶展开，在仅有限二阶服务矩的条件下，具有非负且速度均匀的余项。我们刻画了对于每个有界平稳环境，二阶工作量系数非增的服务分布：函数$s\mapsto\mathbb{E}(1-\cos(sB))/s^2$必须是非增的。指数服务分布的混合满足此准则。一个精确的Palm恒等式将工作量反转转移到顾客等待时间上。

英文摘要

We construct finite-state Markov arrival-rate processes for which faster modulation increases the mean stationary workload of a stable single-server queue, disproving the universal ordering in Leskelä's formulation of Ross's second conjecture. A 64-state example has positive transition rates between every pair of states, independent service times uniform on $[0.99,1.01]$, and an explicit positive workload gap. A 128-state example additionally has strictly positive arrival-rate autocovariance at every lag. More generally, finite-state examples can exhibit any prescribed finite number of separated workload rises and falls. The proofs use a second-order expansion with a nonnegative, speed-uniform remainder under only a finite second service moment. We characterize the service distributions for which the second-order workload coefficient is nonincreasing for every bounded stationary environment: the function $s\mapsto\mathbb{E}(1-\cos(sB))/s^2$ must be nonincreasing. Mixtures of exponential service distributions satisfy this criterion. An exact Palm identity transfers the workload reversals to customer waiting times.

发表机构

• Einstein Institute of Mathematics, The Hebrew University of Jerusalem（耶路撒冷希伯来大学爱因斯坦数学研究所）

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