Skip to main content
SpringerLink
Log in

Analysis of an M/G/1 queue with vacations and multiple phases of operation

  • Original Article
  • Published:
Mathematical Methods of Operations Research Aims and scope Submit manuscript

Abstract

This paper deals with an M / G / 1 queue with vacations and multiple phases of operation. If there are no customers in the system at the instant of a service completion, a vacation commences, that is, the system moves to vacation phase 0. If none is found waiting at the end of a vacation, the server goes for another vacation. Otherwise, the system jumps from phase 0 to some operative phase i with probability \(q_i\), \(i = 1,2, \ldots ,n.\) In operative phase i, \(i = 1,2, \ldots ,n\), the server serves customers according to the discipline of FCFS (First-come, first-served). Using the method of supplementary variables, we obtain the stationary system size distribution at arbitrary epoch. The stationary sojourn time distribution of an arbitrary customer is also derived. In addition, the stochastic decomposition property is investigated. Finally, we present some numerical results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

References

  • Baba Y (2005) Analysis of a GI/M/1 queue with multiple working vacations. Oper Res Lett 33(2):201–209

    Article  MathSciNet  MATH  Google Scholar 

  • Baykal-Gursoy M, Xiao W, Ozbay K (2009) Modeling traffic flow interruped by incidents. Eur J Oper Res 195:127–138

    Article  MATH  Google Scholar 

  • Blom J, Kella O, Mandjes M (2014) Markov-modulated infinite-server queues with general service times. Queue Syst 76:403–424

    Article  MathSciNet  MATH  Google Scholar 

  • Boxma O, Kurkova I (2001) The M/G/1 queue with two service speeds. Adv Appl Probab 33:520–540

    MathSciNet  MATH  Google Scholar 

  • Cordeiro J, Kharoufeh J (2012) The unreliable M/M/1 retrial queue in a random environment. Stoch Models 28(1):29–48

    Article  MathSciNet  MATH  Google Scholar 

  • Doshi B (1986) Queueing systems with vacations—a survey. Queue Syst 1:29–66

    Article  MathSciNet  MATH  Google Scholar 

  • Falin G (2008) The M/M/\(\infty \) queue in random environment. Queue Syst 58:65–76

    Article  MathSciNet  MATH  Google Scholar 

  • Fuhrmann SW, Cooper RB (1985) Stochastic decompositions in the M/G/1 queue with genaralized vacations. Oper Res 33:1117–1129

    Article  MathSciNet  MATH  Google Scholar 

  • Gao S, Liu Z (2013) An M/G/1 queue with single working vacation and vacation interruption under Bernoulli schedule. Appl Math Model 37:1564–1579

    Article  MathSciNet  MATH  Google Scholar 

  • Huang L, Lee T (2013) Generalized Pollaczek–Khinchin formula for Markov channels. IEEE Trans Commun 61:3530–3540

    Article  Google Scholar 

  • Jiang T, Liu L (2017) The GI/M/1 queue in a multi-phase service environment with disasters and working breakdowns. Int J Comput Math 94:707–726. doi:10.1080/00207160.2015.1128531

  • Jiang T, Liu L, Li J (2015) Analysis of the M/G/1 queue in multi-phase random environment with disasters. J Math Anal Appl 430:857–873

    Article  MathSciNet  MATH  Google Scholar 

  • Kim B, Kim J (2015) A single server queue with Markov modulated service rates and impatient customers. Perform Eval 83–84:1–15

    Google Scholar 

  • Krishnamoorthy A, Sivadasan J, Lakshmy B (2015) On an M/G/1 queue with vacation in random environment. In: 14th international scientific conference, pp. 250–262

  • Laxmi PV, Jyothsna K (2015) Impatient customer queue with Bernoulli schedule vacation interruption. Comput Oper Res 56:1–7

    Article  MathSciNet  MATH  Google Scholar 

  • Neuts MF (1971) A queue subject to extraneous phase changes. Adv Appl Probab 3:78–119

    Article  MathSciNet  MATH  Google Scholar 

  • Neuts M (1981) Matrix-geometric solutions in stochastic models: an algorithmic approach. Johns Hopkins University, Baltimore

    MATH  Google Scholar 

  • Paz N, Yechiali U (2014) An M/M/1 queue in random environment with disasters. Asia Pac J Oper Res 31(3):1450016. doi:10.1142/S021759591450016X

    Article  MathSciNet  MATH  Google Scholar 

  • Servi LD, Finn SG (2002) M/M/1 queue with working vacations (M/M/1/WV). Perform Eval 50:41–52

    Article  Google Scholar 

  • Shanthikumar JG (1988) On stochastic decomposition in M/G/1 type queues with generilized server vacations. Oper Res 36:566–569

    Article  MathSciNet  MATH  Google Scholar 

  • Takagi H (1991) Queueing analysis: a foundation of performance evaluation, vol 1. North-Holland, Amsterdam

    MATH  Google Scholar 

  • Tian N, Zhang Z (2006) Vacation queueing models—theory and applications. Springer, New York

    Book  MATH  Google Scholar 

  • Wu D, Takagi H (2006) M/G/1 queue with multiple working vacations. Perform Eval 63:654–681

    Article  Google Scholar 

  • Ye Q, Liu L (2017) The analysis of the M/M/1 queue with two vacation policies (M/M/1/ SWV+MV). Int J Comput Math 94:115–134. doi:10.1080/00207160.2015.1091450

  • Yechiali U, Naor P (1971) Queueing problems with hetergeneous arrivals and service. Oper Res 19:722–734

    Article  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the referees, the associate editor, and the editor, for their valuable suggestions and comments which helped in improving the quality of the paper. This work was supported by National Natural Science Foundation of China (Grant No. 60874118).

Author information

Authors and Affiliations

  1. School of Science, Nanjing University of Science and Technology, Nanjing, 210094, People’s Republic of China

    Jianjun Li, Liwei Liu & Tao Jiang

Authors
  1. Jianjun Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Liwei Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Tao Jiang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jianjun Li.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, J., Liu, L. & Jiang, T. Analysis of an M/G/1 queue with vacations and multiple phases of operation. Math Meth Oper Res 87, 51–72 (2018). https://doi.org/10.1007/s00186-017-0606-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00186-017-0606-0

Keywords

Search

Navigation