具有一般包装约束的无限服务系统
An infinite server 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。有多个不同类型客户的输入流，客户的平均服务时间取决于其类型。有无限数量的服务器。服务器打包{ \em configuration}是向量$k=\{k_i\}$，其中$k_i$是服务器“包含”的$i$类型客户的数量。必须遵守打包约束，即允许一组固定的有限配置$k$。不同客户的服务时间是独立的;在服务完成后，每个客户离开其服务器和系统。每个新到达的客户都被立即放置以供服务;它可以被放置到已经为其他客户服务的服务器中（只要不违反打包约束），或者放置到空闲的服务器中。我们考虑一个简单的吝啬的实时算法，称为{ \em贪婪}，它试图最小化增量的目标函数$\sum_k X_k^{1+α}$，$α>0$，所造成的每个新的分配;这里$X_k$是服务器的数量配置$k$。(When $α$很小，$\sum_k X_k^{1+α}$近似于占用服务器的总数$\sum_k X_k$。）我们的主要结果表明，贪婪算法的某些版本是{ \em渐近最优}，在这个意义上最小化$\sum_k X_k^{1+α}$在静止状态下，作为输入流量增长到无穷大。我们还表明，在特殊的情况下，当一组允许的配置是由{ \em向量包装}约束，贪婪算法可以与{ \em聚合配置}，而不是确切的配置$k$，从而降低计算复杂性，同时保持渐近最优性。
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 input flows of different type customers, with a customer mean service time depending on its type. There is infinite number of servers. A server packing {\em configuration} is the vector $k=\{k_i\}$, where $k_i$ is the number of type $i$ customers the server "contains". Packing constraints must be observed, namely there is a fixed finite set of configurations $k$ that are allowed. Service times of different customers are independent; after a service completion, each customer leaves its server and the system. Each new arriving customer is placed for service immediately; it can be placed into a server already serving other customers (as long as packing constraints are not violated), or into an idle server. We consider a simple parsimonious real-time algorithm, called {\em Greedy}, which attempts to minimize the increment of the objective function $\sum_k X_k^{1+α}$, $α>0$, caused by each new assignment; here $X_k$ is the number of servers in configuration $k$. (When $α$ is small, $\sum_k X_k^{1+α}$ approximates the total number $\sum_k X_k$ of occupied servers.) Our main results show that certain versions of the Greedy algorithm are {\em asymptotically optimal}, in the sense of minimizing $\sum_k X_k^{1+α}$ in stationary regime, as the input flow rates grow to infinity. We also show that in the special case when the set of allowed configurations is determined by {\em vector-packing} constraints, Greedy algorithm can work with {\em aggregate configurations} as opposed to exact configurations $k$, thus reducing computational complexity while preserving the asymptotic optimality.
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