##
**On the level crossing of multi-dimensional delayed renewal processes.**
*(English)*
Zbl 0896.60056

Summary: The paper studies the behavior of an \((l+3)\)-dimensional, delayed renewal process with dependent components, the first three (called active) of which are to cross one of their respective thresholds. More specifically, the crossing takes place when at least one of the active components reaches or exceeds its assigned level. The values of the other two active components, as well as the rest of the components (passive), are to be registered. The analysis yields the joint functional of the “crossing level” and other characteristics (some of which can be interpreted as the first passage time) in a closed form, refining earlier results of the author. A brief, informal discussion of various applications to stochastic models is presented.

### MSC:

60K10 | Applications of renewal theory (reliability, demand theory, etc.) |

60K05 | Renewal theory |

60K25 | Queueing theory (aspects of probability theory) |

60G25 | Prediction theory (aspects of stochastic processes) |

68M20 | Performance evaluation, queueing, and scheduling in the context of computer systems |

91B28 | Finance etc. (MSC2000) |

91B26 | Auctions, bargaining, bidding and selling, and other market models |

90B22 | Queues and service in operations research |