### Abstract

We consider a system with N unit-service-rate queues in tandem, with exogenous arrivals of rate λ at queue 1, under a back-pressure (MaxWeight) algorithm: service at queue n is blocked unless its queue length is greater than that of the next queue n ± 1. The question addressed is how steady-state queues scale as N →∞. We show that the answer depends on whether λ is below or above the critical value 1/4: in the former case the queues remain uniformly stochastically bounded, while otherwise they grow to infinity. The problem is essentially reduced to the behavior of the system with an infinite number of queues in tandem, which is studied using tools from interacting particle systems theory. In particular, the criticality of load 1/4 is closely related to the fact that this is the maximum possible flux (flow rate) of a stationary totally asymmetric simple exclusion process.

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

Pages (from-to) | 111-126 |

Number of pages | 16 |

Journal | Queueing Systems |

Volume | 67 |

Issue number | 2 |

DOIs | |

State | Published - Jan 1 2011 |

Externally published | Yes |

### Fingerprint

### Keywords

- Back-pressure
- Infinite tandem queues
- Interacting particle systems
- MaxWeight
- Queueing networks
- Stability
- TASEP

### ASJC Scopus subject areas

- Statistics and Probability
- Computer Science Applications
- Management Science and Operations Research
- Computational Theory and Mathematics

### Cite this

**Large number of queues in tandem : Scaling properties under back-pressure algorithm.** / Stolyar, Alexander L.

Research output: Contribution to journal › Article

*Queueing Systems*, vol. 67, no. 2, pp. 111-126. https://doi.org/10.1007/s11134-010-9203-0

}

TY - JOUR

T1 - Large number of queues in tandem

T2 - Scaling properties under back-pressure algorithm

AU - Stolyar, Alexander L.

PY - 2011/1/1

Y1 - 2011/1/1

N2 - We consider a system with N unit-service-rate queues in tandem, with exogenous arrivals of rate λ at queue 1, under a back-pressure (MaxWeight) algorithm: service at queue n is blocked unless its queue length is greater than that of the next queue n ± 1. The question addressed is how steady-state queues scale as N →∞. We show that the answer depends on whether λ is below or above the critical value 1/4: in the former case the queues remain uniformly stochastically bounded, while otherwise they grow to infinity. The problem is essentially reduced to the behavior of the system with an infinite number of queues in tandem, which is studied using tools from interacting particle systems theory. In particular, the criticality of load 1/4 is closely related to the fact that this is the maximum possible flux (flow rate) of a stationary totally asymmetric simple exclusion process.

AB - We consider a system with N unit-service-rate queues in tandem, with exogenous arrivals of rate λ at queue 1, under a back-pressure (MaxWeight) algorithm: service at queue n is blocked unless its queue length is greater than that of the next queue n ± 1. The question addressed is how steady-state queues scale as N →∞. We show that the answer depends on whether λ is below or above the critical value 1/4: in the former case the queues remain uniformly stochastically bounded, while otherwise they grow to infinity. The problem is essentially reduced to the behavior of the system with an infinite number of queues in tandem, which is studied using tools from interacting particle systems theory. In particular, the criticality of load 1/4 is closely related to the fact that this is the maximum possible flux (flow rate) of a stationary totally asymmetric simple exclusion process.

KW - Back-pressure

KW - Infinite tandem queues

KW - Interacting particle systems

KW - MaxWeight

KW - Queueing networks

KW - Stability

KW - TASEP

UR - http://www.scopus.com/inward/record.url?scp=79251635185&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79251635185&partnerID=8YFLogxK

U2 - 10.1007/s11134-010-9203-0

DO - 10.1007/s11134-010-9203-0

M3 - Article

AN - SCOPUS:79251635185

VL - 67

SP - 111

EP - 126

JO - Queueing Systems

JF - Queueing Systems

SN - 0257-0130

IS - 2

ER -