### Abstract

For a given k ≥ 1, subintervals of a given interval [0, X] arrive at random and are accepted (allocated) so long as they overlap fewer than k subintervals already accepted. Subintervals not accepted are cleared, while accepted subintervals remain allocated for random retention times before they are released and made available to subsequent arrivals. Thus, the system operates as a generalized many-server queue under a loss protocol. We study a discretized version of this model that appears in reference theories for a number of applications, including communication networks, surface adsorption-desorption processes, and reservation systems. Our primary interest is in steady-state estimates of the vacant space, i.e. the total length of available subintervals kX - σl _{i}, where the l _{i}0 are the lengths of the subintervals currently allocated. We obtain explicit results for k = 1 and for general k with all subinterval lengths equal to 2, the classical dimer case of chemical applications. Our focus is on the asymptotic regime of large retention times.

Original language | English |
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Pages (from-to) | 691-702 |

Number of pages | 12 |

Journal | Journal of Applied Probability |

Volume | 41 |

Issue number | 3 |

DOIs | |

Publication status | Published - Sep 1 2004 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Mathematics(all)
- Statistics, Probability and Uncertainty

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## Cite this

*Journal of Applied Probability*,

*41*(3), 691-702. https://doi.org/10.1239/jap/1091543419