An infinite server system with general packing constraints

TL;DR

This paper studies an infinite server system with complex packing constraints, proposing a simple greedy algorithm that is proven to be asymptotically optimal for minimizing occupied servers in high-volume regimes.

Contribution

It introduces a greedy algorithm for assigning customers to servers under general packing constraints and proves its asymptotic optimality as flow rates increase.

Findings

Greedy algorithm minimizes the objective function asymptotically.

Algorithm remains effective with aggregate configurations under vector-packing constraints.

Proves asymptotic optimality in large-scale service systems.

Abstract

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 $\sum_k X_k^{1+\alpha}$, $\alpha>0$, caused by each new assignment; here $X_k$ is the number of servers in configuration $k$. (When $\alpha$ is small, $\sum_k X_k^{1+\alpha}$ 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 $\sum_k X_k^{1+\alpha}$ 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.