### 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 to 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 having to wait for service vanishes.

Original language | English (US) |
---|---|

Pages (from-to) | 45-47 |

Number of pages | 3 |

Journal | Performance Evaluation Review |

Volume | 45 |

Issue number | 2 |

DOIs | |

State | Published - Sep 1 2017 |

Event | Workshop on MAthematical Performance Modeling and Analysis, MAMA 2017, 2017 Greenmetrics Workshop and Workshop on Critical Infrastructure Network Security, CINS 2017 - Urbana-Champaign, United States Duration: Jun 1 2017 → … |

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### ASJC Scopus subject areas

- Software
- Hardware and Architecture
- Computer Networks and Communications

### Cite this

**Join-Idle-Queue system with general service times : Large-scale limit of stationary distributions.** / Foss, Sergey; Stolyar, Aleksandr.

Research output: Contribution to journal › Conference article

*Performance Evaluation Review*, vol. 45, no. 2, pp. 45-47. https://doi.org/10.1145/3152042.3152058

}

TY - JOUR

T1 - Join-Idle-Queue system with general service times

T2 - Large-scale limit of stationary distributions

AU - Foss, Sergey

AU - Stolyar, Aleksandr

PY - 2017/9/1

Y1 - 2017/9/1

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 to 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 having to wait 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 to 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 having to wait for service vanishes.

UR - http://www.scopus.com/inward/record.url?scp=85041422385&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85041422385&partnerID=8YFLogxK

U2 - 10.1145/3152042.3152058

DO - 10.1145/3152042.3152058

M3 - Conference article

AN - SCOPUS:85041422385

VL - 45

SP - 45

EP - 47

JO - Performance Evaluation Review

JF - Performance Evaluation Review

SN - 0163-5999

IS - 2

ER -