### 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 be routed to one of the servers immediately upon arrival. Join-Idle-Queue routing 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 equal λ/μ. In particular, this implies that the steady-state probability of an arriving customer waiting for service vanishes.

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

Number of pages | 13 |

Journal | Journal of Applied Probability |

Volume | 54 |

Issue number | 4 |

Early online date | 30 Nov 2017 |

DOIs | |

Publication status | Published - Dec 2017 |

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## Cite this

Foss, S., & Stolyar, A. (2017). Large-scale join-idle-queue system with general service times.

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