We consider a heterogeneous distributed service system, consisting of $n$ servers with unknown and possibly different processing rates. Jobs with unit mean and independent processing times arrive as a renewal process of rate $\lambda n$, with $0<\lambda<1$, to the system. Incoming jobs are immediately dispatched to one of several queues associated with the $n$ servers. We assume that the dispatching decisions are made by a central dispatcher endowed with a finite memory, and with the ability to exchange messages with the servers. We study the fundamental resource requirements (memory bits and message exchange rate) in order for a dispatching policy to be {\bf maximally stable}, i.e., stable whenever the processing rates are such that the arrival rate is less than the total available processing rate. First, for the case of Poisson arrivals and exponential service times, we present a policy that is maximally stable while using a positive (but arbitrarily small) message rate, and $\log_2(n)$ bits of memory. Second, we show that within a certain broad class of policies, a dispatching policy that exchanges $o\big(n^2\big)$ messages per unit of time, and with $o(\log(n))$ bits of memory, cannot be maximally stable. Thus, as long as the message rate is not too excessive, a logarithmic memory is necessary and sufficient for maximal stability.

中文翻译：

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异构服务系统中的稳定性、内存和消息传递权衡

我们考虑一个异构分布式服务系统，它由具有未知和可能不同处理速率的 $n$ 个服务器组成。具有单位均值和独立处理时间的作业作为速率 $\lambda n$、$0<\lambda<1$ 的更新过程到达系统。传入的作业会立即分派到与 $n$ 服务器关联的多个队列之一。我们假设调度决策是由中央调度器做出的，该调度器具有有限的内存，并具有与服务器交换消息的能力。我们研究了基本的资源需求（内存位和消息交换率），以便调度策略{\bf maximally stable}，即，只要处理速率达到到达率小于总可用处理率，就保持稳定. 第一的，对于泊松到达和指数服务时间的情况，我们提出了一个最大稳定的策略，同时使用正（但任意小）消息率和 $\log_2(n)$ 位内存。其次，我们展示了在某个广泛的策略类别中，一个调度策略每单位时间交换 $o\big(n^2\big)$ 条消息，并使用 $o(\log(n))$ 位内存，不能最大程度稳定。因此，只要消息速率不太高，对数记忆对于最大稳定性是必要且足够的。并且使用 $o(\log(n))$ 位内存，不能最大程度地稳定。因此，只要消息速率不太高，对数记忆对于最大稳定性是必要且足够的。并且使用 $o(\log(n))$ 位内存，不能最大程度地稳定。因此，只要消息速率不太高，对数记忆对于最大稳定性是必要且足够的。