### Samenvatting

In this paper we study a system consisting of c parallel identical servers and a common queue. The service times are Erlang-r distributed and the interarrival times are Erlang-k distributed. The service discipline is first-come first-served. The waiting process may be characterised by (n_{-1}, n_0, n_1, ..., n_c) where n_{-1} represents the number of remaining arrival stages, n_0 the number of waiting jobs and n_i, i = 1, ... , c, the number of remaining service stages for server i. Bertsimas has proved that the equilibrium probability for a saturated state (all n_i > 0, i = 1, ... , c) can be written as a linear combination of geometric terms with no as exponent. In the present paper it is shown that the coefficients also have a geometric form with respect to n_{-1}, n_1, ... , n_c .It is also shown how the factors may be found efficiently. The present paper uses a direct approach for solving the equilibrium equations rather than a generating function approach as Bertsimas does. The direct approach has been inspired by previous work of two of the authors on the shortest queue problem in particular and the two-dimensional random walk more generally. Although the paper extends results of Bertsimas it is self-contained.

Originele taal-2 | Engels |
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Plaats van productie | Eindhoven |

Uitgeverij | Technische Universiteit Eindhoven |

Aantal pagina's | 15 |

Status | Gepubliceerd - 1992 |

### Publicatie series

Naam | Memorandum COSOR |
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Volume | 9227 |

ISSN van geprinte versie | 0926-4493 |

## Vingerafdruk Duik in de onderzoeksthema's van 'Analysing $E_k|E_r|c$ queues'. Samen vormen ze een unieke vingerafdruk.

## Citeer dit

Adan, I. J. B. F., van den Waarsenburg, W. A., & Wessels, J. (1992).

*Analysing $E_k|E_r|c$ queues*. (Memorandum COSOR; Vol. 9227). Technische Universiteit Eindhoven.