### Abstract

We consider a stochastic fluid queue served by a constant rate server and driven by a process which is the local time of a reflected Lévy process. Such a stochastic system can be used as a model in a priority service system, especially when the time scales involved are fast. The input (local time) in our model is typically(but not necessarily) singular with respect to the Lebesgue measure, a situation which, in view of the nonsmooth or bursty nature of several types of Internet traffic, is nowadays quite realistic. We first discuss how to rigorously construct the (necessarily) unique stationary version of the system under some natural stability conditions. We then consider the distribution of performance steady-state characteristics, namely, the buffer content, the idle period, and the busy period. These derivations are much based on the fact that the inverse of the local time of a Markov process is a Lévy process (a subordinator). hence making the theory of Lévy processes applicable. Another important ingredient in our approach the use of Palm calculus for stationary random point processes and measures. © Applied Probability Trust 2008.

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

Pages (from-to) | 1072-1103 |

Number of pages | 32 |

Journal | Advances in Applied Probability |

Volume | 40 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2008 |

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### Keywords

- Fluid queue
- Inspection paradox
- Lévy process
- Local time
- Palm calculus
- Performance analysis
- Skorokhod reflection

### Cite this

*Advances in Applied Probability*,

*40*(4), 1072-1103. https://doi.org/10.1239/aap/1231340165

}

*Advances in Applied Probability*, vol. 40, no. 4, pp. 1072-1103. https://doi.org/10.1239/aap/1231340165

**Analysis of stochastic fluid queues driven by local-time processes.** / Konstantopoulos, Panagiotis Takis; Kyprianou, Andreas E.; Salminen, Paavo; Sirviö, Marina.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Analysis of stochastic fluid queues driven by local-time processes

AU - Konstantopoulos, Panagiotis Takis

AU - Kyprianou, Andreas E.

AU - Salminen, Paavo

AU - Sirviö, Marina

PY - 2008

Y1 - 2008

N2 - We consider a stochastic fluid queue served by a constant rate server and driven by a process which is the local time of a reflected Lévy process. Such a stochastic system can be used as a model in a priority service system, especially when the time scales involved are fast. The input (local time) in our model is typically(but not necessarily) singular with respect to the Lebesgue measure, a situation which, in view of the nonsmooth or bursty nature of several types of Internet traffic, is nowadays quite realistic. We first discuss how to rigorously construct the (necessarily) unique stationary version of the system under some natural stability conditions. We then consider the distribution of performance steady-state characteristics, namely, the buffer content, the idle period, and the busy period. These derivations are much based on the fact that the inverse of the local time of a Markov process is a Lévy process (a subordinator). hence making the theory of Lévy processes applicable. Another important ingredient in our approach the use of Palm calculus for stationary random point processes and measures. © Applied Probability Trust 2008.

AB - We consider a stochastic fluid queue served by a constant rate server and driven by a process which is the local time of a reflected Lévy process. Such a stochastic system can be used as a model in a priority service system, especially when the time scales involved are fast. The input (local time) in our model is typically(but not necessarily) singular with respect to the Lebesgue measure, a situation which, in view of the nonsmooth or bursty nature of several types of Internet traffic, is nowadays quite realistic. We first discuss how to rigorously construct the (necessarily) unique stationary version of the system under some natural stability conditions. We then consider the distribution of performance steady-state characteristics, namely, the buffer content, the idle period, and the busy period. These derivations are much based on the fact that the inverse of the local time of a Markov process is a Lévy process (a subordinator). hence making the theory of Lévy processes applicable. Another important ingredient in our approach the use of Palm calculus for stationary random point processes and measures. © Applied Probability Trust 2008.

KW - Fluid queue

KW - Inspection paradox

KW - Lévy process

KW - Local time

KW - Palm calculus

KW - Performance analysis

KW - Skorokhod reflection

UR - http://www.scopus.com/inward/record.url?scp=59549087373&partnerID=8YFLogxK

U2 - 10.1239/aap/1231340165

DO - 10.1239/aap/1231340165

M3 - Article

VL - 40

SP - 1072

EP - 1103

JO - Advances in Applied Probability

JF - Advances in Applied Probability

SN - 0001-8678

IS - 4

ER -