随机负载平衡网络流体动力学极限的不变状态
Invariant states of hydrodynamic limits of randomized load balancing networks
摘要：
随机负载均衡算法在以相对较低的计算代价提高大规模网络的性能方面发挥着重要作用。这种系统的一种常见模型是$N$并行队列的网络，其中具有独立和相同分布的服务时间的输入作业在到达时使用加入最短$d$队列路由算法来路由。在相当一般的条件下，Aghajani和Ramanan证明了作为$N\rightarrow\infty$，状态动力学收敛到称为流体动力学方程的耦合确定性测量值方程的可数系统的唯一解。在本文中，这些流体动力学方程的不变状态的一个特征，并在$d=2$，用于构造一个数值算法来计算队列长度分布和平均虚拟等待时间在不变状态。此外，它还表明，在一个合适的尾部条件下的服务分布，队列长度分布的不变状态表现出双指数尾部衰减，从而证明了一个巨大的改善性能的情况下$d=1$，这对应于随机路由，当尾部衰减甚至可以是多项式。此外，数值证据提供支持的猜想，不变状态是极限的稳态分布的$N$服务器模型。的证明方法，这需要分析的耦合系统的测量值方程，可以潜在地应用到其他多服务器系统的一般服务分布，测量值表示是有用的。
英文摘要：
Randomized load-balancing algorithms play an important role in improving performance in large-scale networks at relatively low computational cost. A common model of such a system is a network of $N$ parallel queues in which incoming jobs with independent and identically distributed service times are routed on arrival using the join-the-shortest-of-$d$-queues routing algorithm. Under fairly general conditions, it was shown by Aghajani and Ramanan that as $N\rightarrow\infty$, the state dynamics converges to the unique solution of a countable system of coupled deterministic measure-valued equations called the hydrodynamic equations. In this article, a characterization of invariant states of these hydrodynamic equations is obtained and, when $d=2$, used to construct a numerical algorithm to compute the queue length distribution and mean virtual waiting time in the invariant state. Additionally, it is also shown that under a suitable tail condition on the service distribution, the queue length distribution of the invariant state exhibits a doubly exponential tail decay, thus demonstrating a vast improvement in performance over the case $d=1$, which corresponds to random routing, when the tail decay could even be polynomial. Furthermore, numerical evidence is provided to support the conjecture that the invariant state is the limit of the steady-state distributions of the $N$-server models. The proof methodology, which entails analysis of a coupled system of measure-valued equations, can potentially be applied to other many-server systems with general service distributions, where measure-valued representations are useful.
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