### Abstract

For closed, cyclic, discrete-time networks with one server per node and with independent, geometric service times, in equilibrium, the joint queue-length distribution can be realized as the joint distribution of independent random variables, conditionally given their sum. This tool helps establish monotonicity properties of performance measures and also helps show that the queue-length random variables are negatively associated. The queue length at a node is asymptotically analyzed through a family of networks with a fixed number of node types, where the number of nodes approaches infinity, the ratio of jobs to nodes has a positive limit, and each node type has a limiting density. The queue-length distribution at any node is shown to converge, in a strong sense, to a distribution that is conditionally geometric. As a by-product, this approach settles open issues regarding occupancy proportion and average queue length at a node type.

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

Pages (from-to) | 313-331 |

Number of pages | 19 |

Journal | Queueing Systems |

Volume | 40 |

Issue number | 3 |

DOIs | |

State | Published - Dec 1 2002 |

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### Keywords

- Asymptotic analysis
- Cyclic network
- Discrete-time network
- Monotonicity
- Negative association
- Queue-length distribution

### ASJC Scopus subject areas

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

### Cite this

**Monotonicity and asymptotic queue-length distribution in discrete-time networks.** / Pestien, Victor; Ramakrishnan, Subramanian.

Research output: Contribution to journal › Article

*Queueing Systems*, vol. 40, no. 3, pp. 313-331. https://doi.org/10.1023/A:1014715630648

}

TY - JOUR

T1 - Monotonicity and asymptotic queue-length distribution in discrete-time networks

AU - Pestien, Victor

AU - Ramakrishnan, Subramanian

PY - 2002/12/1

Y1 - 2002/12/1

N2 - For closed, cyclic, discrete-time networks with one server per node and with independent, geometric service times, in equilibrium, the joint queue-length distribution can be realized as the joint distribution of independent random variables, conditionally given their sum. This tool helps establish monotonicity properties of performance measures and also helps show that the queue-length random variables are negatively associated. The queue length at a node is asymptotically analyzed through a family of networks with a fixed number of node types, where the number of nodes approaches infinity, the ratio of jobs to nodes has a positive limit, and each node type has a limiting density. The queue-length distribution at any node is shown to converge, in a strong sense, to a distribution that is conditionally geometric. As a by-product, this approach settles open issues regarding occupancy proportion and average queue length at a node type.

AB - For closed, cyclic, discrete-time networks with one server per node and with independent, geometric service times, in equilibrium, the joint queue-length distribution can be realized as the joint distribution of independent random variables, conditionally given their sum. This tool helps establish monotonicity properties of performance measures and also helps show that the queue-length random variables are negatively associated. The queue length at a node is asymptotically analyzed through a family of networks with a fixed number of node types, where the number of nodes approaches infinity, the ratio of jobs to nodes has a positive limit, and each node type has a limiting density. The queue-length distribution at any node is shown to converge, in a strong sense, to a distribution that is conditionally geometric. As a by-product, this approach settles open issues regarding occupancy proportion and average queue length at a node type.

KW - Asymptotic analysis

KW - Cyclic network

KW - Discrete-time network

KW - Monotonicity

KW - Negative association

KW - Queue-length distribution

UR - http://www.scopus.com/inward/record.url?scp=0036286086&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0036286086&partnerID=8YFLogxK

U2 - 10.1023/A:1014715630648

DO - 10.1023/A:1014715630648

M3 - Article

VL - 40

SP - 313

EP - 331

JO - Queueing Systems

JF - Queueing Systems

SN - 0257-0130

IS - 3

ER -