## Abstract

The dual risk model describes the capital of a company with fixed expense rate and occasional income inflows of random size, called innovations. Parisian ruin occurs once the process stays continuously below zero for a given period. We consider the dual risk model where ruin is declared either at the first time that the reserve stays continuously below zero for an exponentially distributed time, or once it reaches a given negative threshold. We obtain the Laplace transform of the time to ruin and the Laplace transform of the time period that the process is negative. Applying a duality relationship between our risk model and the queueing model, we derive quantities related to the G/M/1 busy period, idle period and cycle maximum.

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

Pages (from-to) | 261-275 |

Number of pages | 15 |

Journal | Queueing Systems |

Volume | 86 |

Issue number | 3-4 |

DOIs | |

State | Published - 1 Aug 2017 |

### Bibliographical note

Funding Information:We are grateful to the referees and the guest editors for their careful reading of the paper, the detailed comments and the constructive criticism.

Publisher Copyright:

© 2017, Springer Science+Business Media New York.

## Keywords

- Busy period
- Cycle maximum
- Dual risk model
- Idle period
- Lévy process
- Risk model
- Ruin probability

## ASJC Scopus subject areas

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