### Abstract

A parallel server system with n identical servers is considered. The service time distribution has a finite mean 1 / μ, but otherwise is arbitrary. Arriving customers are routed to one of the servers immediately upon arrival. The join-idle-queue routeing algorithm is studied, under which an arriving customer is sent to an idle server, if such is available, and to a randomly uniformly chosen server, otherwise. We consider the asymptotic regime where n → and the customer input flow rate is λn. Under the condition λ / μ < 1/2, we prove that, as n → the sequence of (appropriately scaled) stationary distributions concentrates at the natural equilibrium point, with the fraction of occupied servers being constant at λ / μ. In particular, this implies that the steady-state probability of an arriving customer waiting for service vanishes.

Original language | English (US) |
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Pages (from-to) | 995-1007 |

Number of pages | 13 |

Journal | Journal of Applied Probability |

Volume | 54 |

Issue number | 4 |

DOIs | |

State | Published - Dec 1 2017 |

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### Keywords

- Large-scale service system
- asymptotic optimality
- fluid limit
- join-idle-queue
- load balancing
- pull-based load distribution
- stationary distribution

### ASJC Scopus subject areas

- Statistics and Probability
- Mathematics(all)
- Statistics, Probability and Uncertainty

### Cite this

*Journal of Applied Probability*,

*54*(4), 995-1007. https://doi.org/10.1017/jpr.2017.49

**Large-scale join-idle-queue system with general service times.** / Foss, S.; Stolyar, A. L.

Research output: Contribution to journal › Article

*Journal of Applied Probability*, vol. 54, no. 4, pp. 995-1007. https://doi.org/10.1017/jpr.2017.49

}

TY - JOUR

T1 - Large-scale join-idle-queue system with general service times

AU - Foss, S.

AU - Stolyar, A. L.

PY - 2017/12/1

Y1 - 2017/12/1

N2 - A parallel server system with n identical servers is considered. The service time distribution has a finite mean 1 / μ, but otherwise is arbitrary. Arriving customers are routed to one of the servers immediately upon arrival. The join-idle-queue routeing algorithm is studied, under which an arriving customer is sent to an idle server, if such is available, and to a randomly uniformly chosen server, otherwise. We consider the asymptotic regime where n → and the customer input flow rate is λn. Under the condition λ / μ < 1/2, we prove that, as n → the sequence of (appropriately scaled) stationary distributions concentrates at the natural equilibrium point, with the fraction of occupied servers being constant at λ / μ. In particular, this implies that the steady-state probability of an arriving customer waiting for service vanishes.

AB - A parallel server system with n identical servers is considered. The service time distribution has a finite mean 1 / μ, but otherwise is arbitrary. Arriving customers are routed to one of the servers immediately upon arrival. The join-idle-queue routeing algorithm is studied, under which an arriving customer is sent to an idle server, if such is available, and to a randomly uniformly chosen server, otherwise. We consider the asymptotic regime where n → and the customer input flow rate is λn. Under the condition λ / μ < 1/2, we prove that, as n → the sequence of (appropriately scaled) stationary distributions concentrates at the natural equilibrium point, with the fraction of occupied servers being constant at λ / μ. In particular, this implies that the steady-state probability of an arriving customer waiting for service vanishes.

KW - Large-scale service system

KW - asymptotic optimality

KW - fluid limit

KW - join-idle-queue

KW - load balancing

KW - pull-based load distribution

KW - stationary distribution

UR - http://www.scopus.com/inward/record.url?scp=85041353326&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85041353326&partnerID=8YFLogxK

U2 - 10.1017/jpr.2017.49

DO - 10.1017/jpr.2017.49

M3 - Article

AN - SCOPUS:85041353326

VL - 54

SP - 995

EP - 1007

JO - Journal of Applied Probability

JF - Journal of Applied Probability

SN - 0021-9002

IS - 4

ER -