## 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 |

## 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