Analysis of Fluid Model Modulated by an M/PH/1 Working Vacation Queue

  • Xiuli Xu
  • Huining WangEmail author


We propose a fluid model driven by the queue length process of a working vacation queue with PH service distribution, which can be applied to the Ad Hoc network with every data group. We obtain the stationary distribution of the queue length in driving process based on a quasi-birth-and-death process. Then, we analyze the fluid model, and derive the differential equations satisfied by the stationary joint distribution of the fluid queue based on the balance equation. Moreover, we obtain some performance indices, such as, the average throughput, server utilization and the mean buffer content. These indices are relevant to pack transmission in the network, and they can be obtained by using the Laplace Transform (LT) and the Laplace-Stieltjes Transform (LST). Finally, some numerical examples have been discussed with respect to the effect of several parameters on the system performance indices.


Fluid model M/PH/1 queue average throughput server utilization buffer content Ad Hoc network 


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This work was supported by the National Natural Science Foundation of China under Grant No.11201408, and was supported in part by MEXT, Japan. The authors would also like to thank anonymous reviewers for their detailed and constructive comments and suggestions.


  1. Adan I, Resing J (1996). Simple analysis of a fluid queue driven by an M/M/1 queue. Queueing Systems 22(1):171–174.MathSciNetCrossRefzbMATHGoogle Scholar
  2. Ammar S (2014). Analysis of an M/M/1driven fluid queue with multiple exponential vacations. Applied Mathematics and Computation 227(2):329–334.MathSciNetCrossRefzbMATHGoogle Scholar
  3. Barbot N, Sericola B (2002). Stationary solution to the fluid queue fed by an M/M/1 queue. Journal of Applied Probability 39(2):359–369.MathSciNetCrossRefzbMATHGoogle Scholar
  4. Barron Y (2016). Performance analysis of a reflected fluid production/inventory model. Mathematical Methods of Operations Research 83(1):1–31.MathSciNetCrossRefzbMATHGoogle Scholar
  5. Economou A, Manou A (2016). Strategic behavior in an observable fluid queue with an alternating service process. European Journal of Operational Research 254(1):148–160.MathSciNetCrossRefzbMATHGoogle Scholar
  6. Irnich T, Stuckmann P (2003). Fluid-flow modelling of internet traffic in GSM/GPRS networks. Computer Communications 26(15):1756–1763.CrossRefGoogle Scholar
  7. Kulkarni V (1997). Fluid models for single buffer systems. Fronties in Queueing Models and Applications in Science and Engineering. CRC Press, Boca Raton, Florida 321–338.Google Scholar
  8. Li Q, Zhao Y Q (2005). Block-structured fluid queues driven by QBD processes. Stochastic Analysis and Applications 23(6):1087–1112.MathSciNetCrossRefzbMATHGoogle Scholar
  9. Liu Y, Whitt W (2013). Algorithms for time-varying networks of many-server fluid queues. Informs Journal on Computing 26(1):59–73.MathSciNetCrossRefzbMATHGoogle Scholar
  10. Mao B, Wang F, Tian, N (2011). Fluid model driven by an M/G/1 queue with multiple exponential vacations. Applied Mathematics and Computation 218(8):4041–4048.MathSciNetCrossRefzbMATHGoogle Scholar
  11. Parthasarathy P, Vijayashree K, Lenin R (2002). An M/M/1 driven fluid queue-continued fraction approach. Queueing Systems 42(2):189–199.MathSciNetCrossRefzbMATHGoogle Scholar
  12. Virtamo J, Norros I (1994). Fluid queue driven by an M/M/1 queue. Queueing Systems 16(3):373–386.MathSciNetCrossRefzbMATHGoogle Scholar
  13. Xu X, Geng J, Liu M, Guo H (2013). Stationary analysis for the fluid model driven by theM/M/c working vacation queue. Journal of Mathematical Analysis and Applications 403(2):423–433.MathSciNetCrossRefzbMATHGoogle Scholar
  14. Xu X, Song X, Jing X, Ma S (2017). Fluid model driven by a PH/M/1 queue. Journal of Systems Science and Mathematical Sciences 37(3):838–845.MathSciNetzbMATHGoogle Scholar
  15. Yan K (2006). Fluid Models for Production-inventory Systems. University of North Carolina at Chapel Hill, North, Carolina.Google Scholar
  16. Yang S (2008). The M/M/1 with N-policy and M/PH/1 working vacation queues. Yanshan University, Qinhuangdao.Google Scholar
  17. Zhou Z, Xiao Y, Wang D (2015). Stability analysis of wireless network with improved fluid model. Journal of Systems Engineering and Electronics 26(6):1149–1158.CrossRefGoogle Scholar

Copyright information

© Systems Engineering Society of China and Springer-Verlag GmbH Germany 2019

Authors and Affiliations

  1. 1.School of ScienceYanshan UniversityQinhuangdaoChina

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