### Abstract

Autoregressive arrival models are described by a few parameters and provide a simple means to obtain analytical models for matching the first- and second-order statistics of measured data. We consider a discrete-time queueing system where the service time of a customer occupies one slot and the arrival process is governed by a discrete autoregressive process of order 1 (a DAR(1) process) which is characterized by an arbitrary stationary batch size distribution and a correlation coefficient. The tail behaviors of the queue length and the waiting time distributions are examined. In particular, it is shown that, unlike in the classical queueing models with Markovian arrival processes, the correlation in the DAR(1) model results in nongeometric tail behavior of the queue length (and the waiting time) if the stationary distribution of the DAR(1) process has infinite support. A complete characterization of the geometric tail behavior of the queue length (and the waiting time) is presented, showing the impact of the stationary distribution and the correlation coefficient when the stationary distribution of the DAR(1) process has finite support. It is also shown that the stationary distribution of the queue length (and the waiting time) has a tail of regular variation with index -β - 1, by finding an explicit expression for the tail asymptotics when the stationary distribution of the DAR(1) process has a tail of regular variation with index -β.

Original language | English |
---|---|

Pages (from-to) | 1116-1131 |

Number of pages | 16 |

Journal | Advances in Applied Probability |

Volume | 38 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2006 Dec 1 |

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### Keywords

- Discrete autoregressive process
- Geometric tail distribution
- Queue
- Regular variation

### ASJC Scopus subject areas

- Mathematics(all)
- Statistics and Probability

### Cite this

*Advances in Applied Probability*,

*38*(4), 1116-1131. https://doi.org/10.1239/aap/1165414594

**Tail behavior of the queue size and waiting time in a queue with discrete autoregressive arrivals.** / Kim, Bara; Sohraby, Khosrow.

Research output: Contribution to journal › Article

*Advances in Applied Probability*, vol. 38, no. 4, pp. 1116-1131. https://doi.org/10.1239/aap/1165414594

}

TY - JOUR

T1 - Tail behavior of the queue size and waiting time in a queue with discrete autoregressive arrivals

AU - Kim, Bara

AU - Sohraby, Khosrow

PY - 2006/12/1

Y1 - 2006/12/1

N2 - Autoregressive arrival models are described by a few parameters and provide a simple means to obtain analytical models for matching the first- and second-order statistics of measured data. We consider a discrete-time queueing system where the service time of a customer occupies one slot and the arrival process is governed by a discrete autoregressive process of order 1 (a DAR(1) process) which is characterized by an arbitrary stationary batch size distribution and a correlation coefficient. The tail behaviors of the queue length and the waiting time distributions are examined. In particular, it is shown that, unlike in the classical queueing models with Markovian arrival processes, the correlation in the DAR(1) model results in nongeometric tail behavior of the queue length (and the waiting time) if the stationary distribution of the DAR(1) process has infinite support. A complete characterization of the geometric tail behavior of the queue length (and the waiting time) is presented, showing the impact of the stationary distribution and the correlation coefficient when the stationary distribution of the DAR(1) process has finite support. It is also shown that the stationary distribution of the queue length (and the waiting time) has a tail of regular variation with index -β - 1, by finding an explicit expression for the tail asymptotics when the stationary distribution of the DAR(1) process has a tail of regular variation with index -β.

AB - Autoregressive arrival models are described by a few parameters and provide a simple means to obtain analytical models for matching the first- and second-order statistics of measured data. We consider a discrete-time queueing system where the service time of a customer occupies one slot and the arrival process is governed by a discrete autoregressive process of order 1 (a DAR(1) process) which is characterized by an arbitrary stationary batch size distribution and a correlation coefficient. The tail behaviors of the queue length and the waiting time distributions are examined. In particular, it is shown that, unlike in the classical queueing models with Markovian arrival processes, the correlation in the DAR(1) model results in nongeometric tail behavior of the queue length (and the waiting time) if the stationary distribution of the DAR(1) process has infinite support. A complete characterization of the geometric tail behavior of the queue length (and the waiting time) is presented, showing the impact of the stationary distribution and the correlation coefficient when the stationary distribution of the DAR(1) process has finite support. It is also shown that the stationary distribution of the queue length (and the waiting time) has a tail of regular variation with index -β - 1, by finding an explicit expression for the tail asymptotics when the stationary distribution of the DAR(1) process has a tail of regular variation with index -β.

KW - Discrete autoregressive process

KW - Geometric tail distribution

KW - Queue

KW - Regular variation

UR - http://www.scopus.com/inward/record.url?scp=33846957749&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33846957749&partnerID=8YFLogxK

U2 - 10.1239/aap/1165414594

DO - 10.1239/aap/1165414594

M3 - Article

AN - SCOPUS:33846957749

VL - 38

SP - 1116

EP - 1131

JO - Advances in Applied Probability

JF - Advances in Applied Probability

SN - 0001-8678

IS - 4

ER -