Mathematical Methods of Operations Research

, Volume 87, Issue 1, pp 51–72

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

• Jianjun Li
• Liwei Liu
• Tao Jiang
Original Article

## 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.

## Keywords

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

## Notes

### 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).

## References

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