论文标题
超市模型的近乎平衡波动，选择不断增长
Near Equilibrium Fluctuations for Supermarket Models with Growing Choices
论文作者
论文摘要
我们考虑在通常的马尔可夫设置中的超市模型，在该设置中，作业达到$nλ_n$，对于某些$λ_n> 0 $，$ n $并行服务器在其队列中以1的速度为1。我们表明，当$ d_n \ to \ to \ infty $和$λ_n\ toλ\ in（0，\ infty）$在自然条件下在初始队列中，状态占用过程在适当的路径空间中以合适的路径空间收敛，从而融合了约束普通差异方程的独特解决方案，该解决方案是$λ$参数的约束普通差异方程。我们的主要兴趣是研究国家过程的波动，围绕其在关键政权中接近均衡状态的波动，即$λ_n\ to 1 $。以前的论文已经考虑了制度$ \ frac {d_n} {\ sqrt {n} \ log n} \ to \ infty $，而当前工作的目的是为州占用过程开发扩散近似值，以允许所有可能的增长率的$ d_n $。特别是我们考虑三个规范制度（a）$ {d_n}/{\ sqrt {n}}} \ 0 $; （b）$ {d_n}/{\ sqrt {n}} \ to c \ in（0，\ infty）$和（c）$ {d_n}/{\ sqrt {n}}} \ to \ infty $。在所有三个制度中，我们通过建立合适的功能极限定理显示（在$λ_n$的条件下）对其接近平衡的波动的波动是$ n^{ - 1/2} $的阶，并由一维的布朗尼动作渐近地控制。三个制度中极限过程的形式完全不同。在第一种情况下，我们得到线性扩散。在第二种情况下，我们会通过指数漂移进行扩散。在第三种情况下，我们获得了半空间中反射的扩散。在特殊情况下，$ {d_n}/（{\ sqrt {n} \ log n}）\ to \ infty $我们的工作为2018年Mukherjee等人在2018年建立的普遍性结果提供了替代证明。
We consider the supermarket model in the usual Markovian setting where jobs arrive at rate $n λ_n$ for some $λ_n > 0$, with $n$ parallel servers each processing jobs in its queue at rate 1. An arriving job joins the shortest among $d_n \le n$ randomly selected service queues. We show that when $d_n \to \infty$ and $λ_n \to λ\in (0, \infty)$, under natural conditions on the initial queues, the state occupancy process converges in probability, in a suitable path space, to the unique solution of an infinite system of constrained ordinary differential equations parametrized by $λ$. Our main interest is in the study of fluctuations of the state process about its near equilibrium state in the critical regime, namely when $λ_n \to 1$. Previous papers have considered the regime $\frac{d_n}{\sqrt{n}\log n} \to \infty$ while the objective of the current work is to develop diffusion approximations for the state occupancy process that allow for all possible rates of growth of $d_n$. In particular we consider the three canonical regimes (a) ${d_n}/{\sqrt{n}} \to 0$; (b) ${d_n}/{\sqrt{n}} \to c\in (0,\infty)$ and, (c) ${d_n}/{\sqrt{n}} \to \infty$. In all three regimes we show, by establishing suitable functional limit theorems, that (under conditions on $λ_n$) fluctuations of the state process about its near equilibrium are of order $n^{-1/2}$ and are governed asymptotically by a one dimensional Brownian motion. The forms of the limit processes in the three regimes are quite different; in the first case we get a linear diffusion; in the second case we get a diffusion with an exponential drift; and in the third case we obtain a reflected diffusion in a half space. In the special case ${d_n}/({\sqrt{n}\log n}) \to \infty$ our work gives alternative proofs for the universality results established by Mukherjee et al in 2018.
