一般包装约束下服务大系统贪婪随机算法的渐近最优性
Asymptotic optimality of a greedy randomized algorithm in a large-scale service system with general packing constraints
摘要：
我们考虑一个服务系统模型，主要是由虚拟机的有效分配的问题，在网络云中的物理主机，使占用主机的数量最小化。有多种类型的到达客户，其中客户的平均服务时间取决于其类型。服务器的数量是无限的。可以将多个客户放置到一个服务器中进行服务，但要遵守一般的“打包”约束。不同客户的服务时间是独立的，即使由同一服务器同时提供服务。每个新到达的客户被立即放置以进行服务，或者放置到已经服务于其他客户的服务器中（只要不违反打包约束），或者放置到空闲的服务器中。服务完成后，每个客户离开其服务器和系统。我们提出了一个非常简单，易于实现的客户安置算法，称为贪婪随机（GRAND）。它将每个到达的客户随机均匀地放置到一个已经占用的服务器（受包装限制）或所谓的零服务器之一，零服务器是指定为新到达者提供的空服务器。GRAND的一个实例，称为GRAND（$aZ$），其中$a\ge 0$是一个参数，使得在任何给定时间$t$的零服务器的数量是$aZ(t)$，其中$Z(t)$是系统中的当前客户总数。我们证明了GRAND（$aZ$）与$a>0$是渐近最优的，因为客户到达率增长到无穷大和$a\to 0$，在这个意义上，最小化在稳定状态下占用的服务器的总数。此外，我们通过仿真研究了GRAND的各种版本，并观察了收敛速度和稳态性能对零服务器数量的依赖关系。
英文摘要：
We consider a service system model primarily motivated by the problem of efficient assignment of virtual machines to physical host machines in a network cloud, so that the number of occupied hosts is minimized. There are multiple types of arriving customers, where a customer's mean service time depends on its type. There is an infinite number of servers. Multiple customers can be placed for service into one server, subject to general "packing" constraints. Service times of different customers are independent, even if served simultaneously by the same server. Each new arriving customer is placed for service immediately, either into a server already serving other customers (as long as packing constraints are not violated) or into an idle server. After a service completion, each customer leaves its server and the system. We propose an extremely simple and easily implementable customer placement algorithm, called Greedy-Random (GRAND). It places each arriving customer uniformly at random into either one of the already occupied servers (subject to packing constraints) or one of the so-called zero-servers, which are empty servers designated to be available to new arrivals. One instance of GRAND, called GRAND($aZ$), where $a\ge 0$ is a parameter, is such that the number of zero-servers at any given time $t$ is $aZ(t)$, where $Z(t)$ is the current total number of customers in the system. We prove that GRAND($aZ$) with $a>0$ is asymptotically optimal, as the customer arrival rates grow to infinity and $a\to 0$, in the sense of minimizing the total number of occupied servers in steady state. In addition, we study by simulations various versions of GRAND and observe the dependence of convergence speed and steady-state performance on the number of zero-servers.
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