Mathematical Methods of Operations Research

, Volume 87, Issue 1, pp 51–72 | Cite as

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

  • Jianjun LiEmail author
  • Liwei Liu
  • Tao Jiang
Original Article


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.


M / G / 1 queue Vacation Sojourn time Probability generating function Multiple phases of operation Queueing theory 



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).


  1. Baba Y (2005) Analysis of a GI/M/1 queue with multiple working vacations. Oper Res Lett 33(2):201–209MathSciNetCrossRefzbMATHGoogle Scholar
  2. Baykal-Gursoy M, Xiao W, Ozbay K (2009) Modeling traffic flow interruped by incidents. Eur J Oper Res 195:127–138CrossRefzbMATHGoogle Scholar
  3. Blom J, Kella O, Mandjes M (2014) Markov-modulated infinite-server queues with general service times. Queue Syst 76:403–424MathSciNetCrossRefzbMATHGoogle Scholar
  4. Boxma O, Kurkova I (2001) The M/G/1 queue with two service speeds. Adv Appl Probab 33:520–540MathSciNetzbMATHGoogle Scholar
  5. Cordeiro J, Kharoufeh J (2012) The unreliable M/M/1 retrial queue in a random environment. Stoch Models 28(1):29–48MathSciNetCrossRefzbMATHGoogle Scholar
  6. Doshi B (1986) Queueing systems with vacations—a survey. Queue Syst 1:29–66MathSciNetCrossRefzbMATHGoogle Scholar
  7. Falin G (2008) The M/M/\(\infty \) queue in random environment. Queue Syst 58:65–76MathSciNetCrossRefzbMATHGoogle Scholar
  8. Fuhrmann SW, Cooper RB (1985) Stochastic decompositions in the M/G/1 queue with genaralized vacations. Oper Res 33:1117–1129MathSciNetCrossRefzbMATHGoogle Scholar
  9. Gao S, Liu Z (2013) An M/G/1 queue with single working vacation and vacation interruption under Bernoulli schedule. Appl Math Model 37:1564–1579MathSciNetCrossRefzbMATHGoogle Scholar
  10. Huang L, Lee T (2013) Generalized Pollaczek–Khinchin formula for Markov channels. IEEE Trans Commun 61:3530–3540CrossRefGoogle Scholar
  11. 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
  12. 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 430:857–873MathSciNetCrossRefzbMATHGoogle Scholar
  13. Kim B, Kim J (2015) A single server queue with Markov modulated service rates and impatient customers. Perform Eval 83–84:1–15Google Scholar
  14. Krishnamoorthy A, Sivadasan J, Lakshmy B (2015) On an M/G/1 queue with vacation in random environment. In: 14th international scientific conference, pp. 250–262Google Scholar
  15. Laxmi PV, Jyothsna K (2015) Impatient customer queue with Bernoulli schedule vacation interruption. Comput Oper Res 56:1–7MathSciNetCrossRefzbMATHGoogle Scholar
  16. Neuts MF (1971) A queue subject to extraneous phase changes. Adv Appl Probab 3:78–119MathSciNetCrossRefzbMATHGoogle Scholar
  17. Neuts M (1981) Matrix-geometric solutions in stochastic models: an algorithmic approach. Johns Hopkins University, BaltimorezbMATHGoogle Scholar
  18. 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 MathSciNetCrossRefzbMATHGoogle Scholar
  19. Servi LD, Finn SG (2002) M/M/1 queue with working vacations (M/M/1/WV). Perform Eval 50:41–52CrossRefGoogle Scholar
  20. Shanthikumar JG (1988) On stochastic decomposition in M/G/1 type queues with generilized server vacations. Oper Res 36:566–569MathSciNetCrossRefzbMATHGoogle Scholar
  21. Takagi H (1991) Queueing analysis: a foundation of performance evaluation, vol 1. North-Holland, AmsterdamzbMATHGoogle Scholar
  22. Tian N, Zhang Z (2006) Vacation queueing models—theory and applications. Springer, New YorkCrossRefzbMATHGoogle Scholar
  23. Wu D, Takagi H (2006) M/G/1 queue with multiple working vacations. Perform Eval 63:654–681CrossRefGoogle Scholar
  24. 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
  25. Yechiali U, Naor P (1971) Queueing problems with hetergeneous arrivals and service. Oper Res 19:722–734CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.School of ScienceNanjing University of Science and TechnologyNanjingPeople’s Republic of China

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