## Abstract

We analyze the output process of finite capacity birth-death Markovian queues. We develop a formula for the asymptotic variance rate of the form λ *+σv_{i} where λ * is the rate of outputs and v _{i} are functions of the birth and death rates. We show that if the birth rates are non-increasing and the death rates are non-decreasing (as is common in many queueing systems) then the values of v _{i} are strictly negative and thus the limiting index of dispersion of counts of the output process is less than unity. In the M/M/1/K case, our formula evaluates to a closed form expression that shows the following phenomenon: When the system is balanced, i.e. the arrival and service rates are equal, σv_{i}\λ* is minimal. The situation is similar for the M/M/c/K queue, the Erlang loss system and some PH/PH/1/K queues: In all these systems there is a pronounced decrease in the asymptotic variance rate when the system parameters are balanced.

Original language | English |
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Pages (from-to) | 135-156 |

Number of pages | 22 |

Journal | Queueing Systems |

Volume | 59 |

Issue number | 2 |

DOIs | |

State | Published - Jun 2008 |

### Bibliographical note

Funding Information:Research supported in part by Israel Science Foundation Grant 249/02 and 454/05 and by European Network of Excellence Euro-NGI.

## Keywords

- Asymptotic variance rate
- BRAVO
- Loss systems
- M/M/1/K
- MAP
- Queueing theory

## ASJC Scopus subject areas

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