Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-23T15:11:23.340Z Has data issue: false hasContentIssue false
  • English
  • Français

ON A QUEUING MODEL WITH SERVICE INTERRUPTIONS

Published online by Cambridge University Press:  25 September 2008

Onno Boxma ,
Michel Mandjes  and
Offer Kella
Onno Boxma
Affiliation:
EURANDOM and Department of Mathematics and Computer Science, Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands E-mail: boxma@win.tue.nl
Michel Mandjes
Affiliation:
Korteweg-de Vries Institute for Mathematics, The University of Amsterdam, 1018 TV Amsterdam, The Netherlands and CWI, 1090 GB Amsterdam, The Netherlands E-mail: mmandjes@science.uva.nl
Offer Kella
Affiliation:
Department of Statistics, The Hebrew University of Jerusalem, Mount Scopus, Jerusalem 91905, Israel E-mail: Offer.Kella@huji.ac.il
Get access Rights & Permissions [Opens in a new window]

Abstract

Single-server queues in which the server takes vacations arise naturally as models for a wide range of computer, communication, and production systems. In almost all studies on vacation models, the vacation lengths are assumed to be independent of the arrival, service, workload, and queue length processes. In the present study, we allow the length of a vacation to depend on the length of the previous active period (viz. the period since the previous vacation). Under rather general assumptions regarding the offered work during active periods and vacations, we determine the steady-state workload distribution, both for single and multiple vacations. We conclude by discussing several special cases, including polling models, and relate our findings to results obtained earlier.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 22 , Issue 4 , October 2008 , pp. 537 - 555
Copyright
Copyright © Cambridge University Press 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Altman, E. (2002). Stochastic recursive equations with applications to queues with dependent vacations. Annals of Operations Research 112: 43–61.CrossRefGoogle Scholar
2.Asmussen, S. (2003). Applied probability and queues. New York: Springer-Verlag.Google Scholar
3.Boxma, O.J. (1989). Workloads and waiting times in single-server systems with multiple customer classes. Queueing Systems 5: 185–214.CrossRefGoogle Scholar
4.Boxma, O.J. & Groenendijk, W.P. (1987). Pseudo-conservation laws in cyclic-service systems. Journal of Applied Probability 24: 949–964.CrossRefGoogle Scholar
5.Doshi, B.T. (1986). Queueing systems with vacations: A survey. Queueing Systems 1: 29–66.CrossRefGoogle Scholar
6.Doshi, B.T. (1989). Conditional and unconditional distributions for M/G/1 type queues with server vacations. Queueing Systems 7: 229–251.CrossRefGoogle Scholar
7.Doshi, B.T. (1990). Single server queues with vacations. In: Takagi, H. (ed.) Stochastic analysis of computer and communication systems. Amsterdam: North-Holland pp. 217–265.Google Scholar
8.Doshi, B.T. (1990). Generalizations of the stochastic decomposition results for single server queues with vacations. Stochastic Models 6: 307–333.Google Scholar
9.Eliazar, I. (2005). Gated polling systems with Lévy inflow and inter-dependent switchover times: A dynamical-systems approach. Queueing Systems 49: 49–72.CrossRefGoogle Scholar
10.Fuhrmann, S.W. & Cooper, R.B. (1985). Stochastic decompositions in the M/G/1 queue with generalized vacations. Oper. Res. 33, 1117–1129.CrossRefGoogle Scholar
11.Groenevelt, R. & Altman, E. (2005). Analysis of alternating-priority queueing models with (cross) correlated switchover times. Queueing Systems 51: 199–247.CrossRefGoogle Scholar
12.Harris, C.M. & Marchal, W.G. (1988). State dependence in M/G/1 server-vacation models. Operations Research 36: 560–565.CrossRefGoogle Scholar
13.Kella, O. (1998). An exhaustive Lévy storage process with intermittent output. Stochastic Models 14: 979–992.CrossRefGoogle Scholar
14.Prabhu, N.U. (1998). Stochastic storage processes. New York: Springer-Verlag.CrossRefGoogle Scholar
15.Sato, K. (1999). Lévy processes and infinitely divisible distributions. Cambridge: Cambridge University Press.Google Scholar
16.Shanthikumar, J.G. (1988). On stochastic decomposition in M/G/1 type queues with generalized server vacations. Operations Research 36: 566–569.CrossRefGoogle Scholar
17.Takács, L. (1968). Two queues attended by a single server. Operations Research 16: 639–650.CrossRefGoogle Scholar
18.Takagi, H. (1990). Queueing analysis of polling models: An update. In: Takagi, H. (ed.) Stochastic analysis of computer and communication systems. Amsterdam: North-Holland, pp. 267–318.Google Scholar
19.Takagi, H. (1991). Queueing analysis. Vol. 1: Vacation and priority systems, Part 1. Amsterdam: North-Holland.Google Scholar
20.Takagi, H. (1997). Queueing analysis of polling models: Progress in 1990–1994. In: Dshalalow, J.H., (ed.) Frontiers in queueing. Boca Raton, FL: CRC Press, pp. 119–146.Google Scholar
21.Teghem, J. Jr. (1986). Control of the service process in a queueing system. European Journal of Operations Research 23: 141–158.CrossRefGoogle Scholar