# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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 (optimization) 60K25 Queueing theory
##### References:
 [1] Altman, E. and Borovkov, A. A., ”On the stability of retrial queues,” Queueing Systems, 26 (1997) 343–363. · Zbl 0892.90069 · doi:10.1023/A:1019193527040 [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 · doi:10.2307/1427345 [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. [6] Daley, D.J., ”General Customer Impatience in the Queue GI/G/1,” J. Applied Probability, 2 (1965) 186–205. · Zbl 0134.14402 · doi:10.2307/3211884 [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. · doi:10.1287/msom.5.2.79.16071 [8] Kleinrock, L. (1975), Queueing Systems Volume I: Theory, J. Wiley & Sons, New York. [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 · doi:10.1287/mnsc.22.2.202 [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. [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 · doi:10.2307/3212710 [14] Takacs, L. ”Introduction to the Theory of Queues,” Oxford University Press, New York (1962). [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 · doi:10.1023/A:1025695818046 [16] Yechiali, U., ”On the MX/G/1 Queue with a Waiting Server and Vacations,” Sankhya, 66 (2004) 1–17. [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.