### Abstract

We consider a large-scale service system model proposed in [14], which is motivated by the problem of efficient placement of virtual machines to physical host machines in a network cloud, so that the total number of occupied hosts is minimized. Customers of different types arrive to a system with an infinite number of servers. A server packing configuration is the vector k = {ki}, where ki is the number of type-i customers that the server "contains". Packing constraints are described by a fixed finite set of allowed configurations. Upon arrival, each customer is placed into a server immediately, subject to the packing constraints; the server can be idle or already serving other customers. After service completion, each customer leaves its server and the system. It was shown in [14] that a simple real-time algorithm, called Greedy, is asymptotically optimal in the sense of minimizing Σ_{k} X_{k}^{1+α} in the stationary regime, as the customer arrival rates grow to infinity. (Here α > 0, and X_{k} denotes the number of servers with configuration k.) In particular, when parameter α is small, and in the asymptotic regime where customer arrival rates grow to infinity, Greedy solves a problem approximating one of minimizing Σ_{k} X_{k}, the number of occupied hosts. In this paper we introduce the algorithm called Greedy with sublinear Safety Stocks (GSS), and show that it asymptotically solves the exact problem of minimizing Σ_{k} X_{k}. An important feature of the algorithm is that sublinear safety stocks of X_{k} are created automatically - when and where necessary - without having to determine a priori where they are required. Moreover, we also provide a tight characterization of the rate of convergence to optimality under GSS. The GSS algorithm is as simple as Greedy, and uses no more system state information than Greedy does.

Original language | English (US) |
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Title of host publication | SIGMETRICS 2013 - Proceedings of the 2013 ACM SIGMETRICS International Conference on Measurement and Modeling of Computer Systems |

Pages | 41-52 |

Number of pages | 12 |

Volume | 41 |

Edition | 1 SPEC. ISS. |

DOIs | |

State | Published - 2013 |

Externally published | Yes |

Event | 2013 ACM SIGMETRICS International Conference on Measurement and Modeling of Computer Systems, SIGMETRICS 2013 - Pittsburgh, PA, United States Duration: Jun 17 2013 → Jun 21 2013 |

### Other

Other | 2013 ACM SIGMETRICS International Conference on Measurement and Modeling of Computer Systems, SIGMETRICS 2013 |
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Country | United States |

City | Pittsburgh, PA |

Period | 6/17/13 → 6/21/13 |

### Fingerprint

### Keywords

- Fluid scale optimality
- Infinite-server system
- Local fluid scaling
- Markov chain
- Multi-dimensional bin packing
- Safety stocks

### ASJC Scopus subject areas

- Software
- Hardware and Architecture
- Computer Networks and Communications

### Cite this

*SIGMETRICS 2013 - Proceedings of the 2013 ACM SIGMETRICS International Conference on Measurement and Modeling of Computer Systems*(1 SPEC. ISS. ed., Vol. 41, pp. 41-52) https://doi.org/10.1145/2494232.2465547

**A large-scale service system with packing constraints : Minimizing the number of occupied servers.** / Stolyar, Aleksandr; Zhong, Yuan.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*SIGMETRICS 2013 - Proceedings of the 2013 ACM SIGMETRICS International Conference on Measurement and Modeling of Computer Systems.*1 SPEC. ISS. edn, vol. 41, pp. 41-52, 2013 ACM SIGMETRICS International Conference on Measurement and Modeling of Computer Systems, SIGMETRICS 2013, Pittsburgh, PA, United States, 6/17/13. https://doi.org/10.1145/2494232.2465547

}

TY - GEN

T1 - A large-scale service system with packing constraints

T2 - Minimizing the number of occupied servers

AU - Stolyar, Aleksandr

AU - Zhong, Yuan

PY - 2013

Y1 - 2013

N2 - We consider a large-scale service system model proposed in [14], which is motivated by the problem of efficient placement of virtual machines to physical host machines in a network cloud, so that the total number of occupied hosts is minimized. Customers of different types arrive to a system with an infinite number of servers. A server packing configuration is the vector k = {ki}, where ki is the number of type-i customers that the server "contains". Packing constraints are described by a fixed finite set of allowed configurations. Upon arrival, each customer is placed into a server immediately, subject to the packing constraints; the server can be idle or already serving other customers. After service completion, each customer leaves its server and the system. It was shown in [14] that a simple real-time algorithm, called Greedy, is asymptotically optimal in the sense of minimizing Σk Xk1+α in the stationary regime, as the customer arrival rates grow to infinity. (Here α > 0, and Xk denotes the number of servers with configuration k.) In particular, when parameter α is small, and in the asymptotic regime where customer arrival rates grow to infinity, Greedy solves a problem approximating one of minimizing Σk Xk, the number of occupied hosts. In this paper we introduce the algorithm called Greedy with sublinear Safety Stocks (GSS), and show that it asymptotically solves the exact problem of minimizing Σk Xk. An important feature of the algorithm is that sublinear safety stocks of Xk are created automatically - when and where necessary - without having to determine a priori where they are required. Moreover, we also provide a tight characterization of the rate of convergence to optimality under GSS. The GSS algorithm is as simple as Greedy, and uses no more system state information than Greedy does.

AB - We consider a large-scale service system model proposed in [14], which is motivated by the problem of efficient placement of virtual machines to physical host machines in a network cloud, so that the total number of occupied hosts is minimized. Customers of different types arrive to a system with an infinite number of servers. A server packing configuration is the vector k = {ki}, where ki is the number of type-i customers that the server "contains". Packing constraints are described by a fixed finite set of allowed configurations. Upon arrival, each customer is placed into a server immediately, subject to the packing constraints; the server can be idle or already serving other customers. After service completion, each customer leaves its server and the system. It was shown in [14] that a simple real-time algorithm, called Greedy, is asymptotically optimal in the sense of minimizing Σk Xk1+α in the stationary regime, as the customer arrival rates grow to infinity. (Here α > 0, and Xk denotes the number of servers with configuration k.) In particular, when parameter α is small, and in the asymptotic regime where customer arrival rates grow to infinity, Greedy solves a problem approximating one of minimizing Σk Xk, the number of occupied hosts. In this paper we introduce the algorithm called Greedy with sublinear Safety Stocks (GSS), and show that it asymptotically solves the exact problem of minimizing Σk Xk. An important feature of the algorithm is that sublinear safety stocks of Xk are created automatically - when and where necessary - without having to determine a priori where they are required. Moreover, we also provide a tight characterization of the rate of convergence to optimality under GSS. The GSS algorithm is as simple as Greedy, and uses no more system state information than Greedy does.

KW - Fluid scale optimality

KW - Infinite-server system

KW - Local fluid scaling

KW - Markov chain

KW - Multi-dimensional bin packing

KW - Safety stocks

UR - http://www.scopus.com/inward/record.url?scp=84880214258&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84880214258&partnerID=8YFLogxK

U2 - 10.1145/2494232.2465547

DO - 10.1145/2494232.2465547

M3 - Conference contribution

AN - SCOPUS:84880214258

SN - 9781450319003

VL - 41

SP - 41

EP - 52

BT - SIGMETRICS 2013 - Proceedings of the 2013 ACM SIGMETRICS International Conference on Measurement and Modeling of Computer Systems

ER -