分布式服务系统中的延迟、存储器和消息传递折衷
Delay, memory, and messaging tradeoffs in distributed service systems
摘要：
我们考虑以下分布式服务模型：单位平均，指数分布，和独立的处理时间的工作到达率$λn$的泊松过程，与$0<λ<1$，并立即由一个集中式调度器调度到$n$先进先出队列与$n$相同的服务器之一。调度器被赋予有限的存储器，并具有与服务器交换消息的能力。我们提出并研究了一个资源受限的“拉式”调度策略，涉及两个参数：（i）可用的调度器的内存位数，以及（ii）服务器与调度器通信的平均速率。我们建立（使用流体极限方法）的渐近，作为$n\to\infty$，预期的延迟是零时，（i）的内存位数增长与$n$和消息速率增长超线性与$n$，或（ii）的内存位数增长超线性与$n$和消息速率至少是$λn$。此外，当存储器位的数量仅与$n$成比例地增长并且消息速率与$n$成比例时，我们获得了（现在为正的）渐近延迟的封闭形式的表达式。最后，我们证明了一个有趣的相变在资源受限的制度，渐近延迟是非零的。特别是，我们表明，对于任何给定的$α>0$（无论多小），如果我们的策略仅使用线性消息速率$αn$，则所得到的渐近延迟是上界的，在所有$λ<1$上一致;这与不使用消息时获得的延迟形成鲜明对比（$α= 0$），当使用$λ\uparrow 1$时其增长为$1/(1-λ)$，或者当使用流行的$d$选择的幂时，其中延迟增长为$\log(1/(1-λ))$。
英文摘要：
We consider the following distributed service model: jobs with unit mean, exponentially distributed, and independent processing times arrive as a Poisson process of rate $λn$, with $0<λ<1$, and are immediately dispatched by a centralized dispatcher to one of $n$ First-In-First-Out queues associated with $n$ identical servers. The dispatcher is endowed with a finite memory, and with the ability to exchange messages with the servers. We propose and study a resource-constrained "pull-based" dispatching policy that involves two parameters: (i) the number of memory bits available at the dispatcher, and (ii) the average rate at which servers communicate with the dispatcher. We establish (using a fluid limit approach) that the asymptotic, as $n\to\infty$, expected queueing delay is zero when either (i) the number of memory bits grows logarithmically with $n$ and the message rate grows superlinearly with $n$, or (ii) the number of memory bits grows superlogarithmically with $n$ and the message rate is at least $λn$. Furthermore, when the number of memory bits grows only logarithmically with $n$ and the message rate is proportional to $n$, we obtain a closed-form expression for the (now positive) asymptotic delay. Finally, we demonstrate an interesting phase transition in the resource-constrained regime where the asymptotic delay is non-zero. In particular, we show that for any given $α>0$ (no matter how small), if our policy only uses a linear message rate $αn$, the resulting asymptotic delay is upper bounded, uniformly over all $λ<1$; this is in sharp contrast to the delay obtained when no messages are used ($α= 0$), which grows as $1/(1-λ)$ when $λ\uparrow 1$, or when the popular power-of-$d$-choices is used, in which the delay grows as $\log(1/(1-λ))$.
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