##
**Analysis of customers’ impatience in queues with server vacations.**
*(English)*
Zbl 1114.90015

Summary: Many models for customers impatience in queueing systems have been studied in the past; the source of impatience has always been taken to be either a long wait already experienced at a queue, or a long wait anticipated by a customer upon arrival. In this paper we consider systems with servers vacations where customers’ impatience is due to an absentee of servers upon arrival. Such a model, representing frequent behavior by waiting customers in service systems, has never been treated before in the literature. We present a comprehensive analysis of the single-server, \(M/M/1\) and \(M/G/1\) queues, as well as of the multi-server \(M/M/c\) queue, for both the multiple and the single-vacation cases, and obtain various closed-form results. In particular, we show that the proportion of customer abandonments under the single-vacation regime is smaller than that under the multiple-vacation discipline.

### MSC:

90B22 | Queues and service in operations research |

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

PDF
BibTeX
XML
Cite

\textit{E. Altman} and \textit{U. Yechiali}, Queueing Syst. 52, No. 4, 261--279 (2006; Zbl 1114.90015)

Full Text:
DOI

### References:

[1] | Altman, E. and Borovkov, A. A., ”On the stability of retrial queues,” Queueing Systems, 26 (1997) 343–363. · Zbl 0892.90069 |

[2] | Baccelli, F., Boyer, P. and Hebuterne, G., ”Single-Server Queues with Impatient Customers,” Advances in Applied Probability, 16 (1984) 887–905. · Zbl 0549.60091 |

[3] | Bonald, T. and Roberts, J., ”Performance modeling of elastic traffic in overload,” ACM Sigmetrics, Cambridge, MA, USA, (2001) 342–343. |

[4] | Boxma, O.J. and de Waal, P.R. ”Multiserver Queues with Impatient Customers,” ITC, 14 (1994) 743–756. |

[5] | Boxma, O.J., Schlegel, S. and Yechiali, U., ”A Note on the M/G/1 Queue with a Waiting Server, Timer and Vacations,” American Mathematical Society Translations, Series 2, 207 (2002) 25–35. · Zbl 1023.60077 |

[6] | Daley, D.J., ”General Customer Impatience in the Queue GI/G/1,” J. Applied Probability, 2 (1965) 186–205. · Zbl 0134.14402 |

[7] | Gans, N., Koole, G. and Mandelbaum A., ”Telephone call centers: Tutorial, review, and research prospects,” Manufacturing and Service Operations Management, 5 (2003) 79–141. |

[8] | Kleinrock, L. (1975), Queueing Systems Volume I: Theory, J. Wiley & Sons, New York. · Zbl 0334.60045 |

[9] | Levy, Y. and Yechiali, U., ”Utilization for the Idle Time in an M/G/1 Queuing System,” Management Science, 22 (1975) 202–211. · Zbl 0313.60067 |

[10] | Levy, Y. and Yechiali, U., ”An M/M/s Queue with Servers’ Vacations,” Canadian J. of Operational Research and Information Processing, 14 (1976) 153–163. · Zbl 0363.60091 |

[11] | Palm, C., ”Methods of Judging the Annoyance Caused by Congestion,” Tele, 4 (1953) 189–208. |

[12] | Palm, C., ”Research on Telephone Traffic Carried by Full Availability Groups,” Tele, Vol. 1, 107 (1957). (English translation of results first published in 1946 in Swedish in the same journal, which was then entitled Tekniska Meddelanden fran Kungl. Telegrfstyrelsen). |

[13] | Takacs, L., ”A Single-Server Queue with Limited Virtual Waiting Time,” J. Applied Probability, 11 (1974) 612–617. · Zbl 0303.60098 |

[14] | Takacs, L. ”Introduction to the Theory of Queues,” Oxford University Press, New York (1962). · Zbl 0118.13503 |

[15] | Van Houdt, B., Lenin, R.B. and Blondia, C., ”Delay Distribution of (Im)Patient customers in a Discrete Time DAP/PH/1 Queue with Age-Dependent Service Times,” Queueing Systems, 45 (2003) 59–73. · Zbl 1175.90116 |

[16] | Yechiali, U., ”On the MX/G/1 Queue with a Waiting Server and Vacations,” Sankhya, 66 (2004) 1–17. · Zbl 1192.90050 |

[17] | Zhang, Z.G. and Tian, N., ”Analysis of Queueing Systems with Synchronous Vacations of Partial Servers,” Perfor. Eval., 52(4) (2003) 269–282. |

[18] | Zhang, Z.G. and Tian, N., ”Analysis of Queueing Systems with Synchronous Single Vacation for Some Servers,” Queueing Systems, 45(2) (2003) 161–175. · Zbl 1036.90033 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.