### 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 an infinite number of servers. A server-packing 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 Greedy, that attempts to minimize the increment of the objective function Σ_{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, Σ_{k}Xk ^{1+α} approximates the total number Σ_{k}X _{k} of occupied servers.) Our main results show that certain versions of the Greedy algorithm are asymptotically optimal, in the sense of minimizing Σ_{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 vector-packing constraints, the Greedy algorithm can work with aggregate configurations as opposed to exact configurations k, thus reducing computational complexity while preserving the asymptotic optimality.

Original language | English (US) |
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Pages (from-to) | 1200-1217 |

Number of pages | 18 |

Journal | Operations Research |

Volume | 61 |

Issue number | 5 |

DOIs | |

State | Published - Sep 1 2013 |

### ASJC Scopus subject areas

- Computer Science Applications
- Management Science and Operations Research