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Single server multiple vacation queue with discouragement solve by confluent hypergeometric function

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Abstract

Waiting line problems with server vacation have envisaged with increasing complexities and their explicit transient solutions are rigorous in computations, at the same time such solutions are valued for studying the dynamical behaviour of queuing systems over a finite period predominantly utilizes within the state-of-art design process for a real time system. Keeping this fact in mind we adopt continued fractions and generating function to derive explicit expressions for transient state probabilities. In this paper, we consider the waiting line problem with a single server which adopts the multi vacations policy. We analyzed the transient part for a single server multi vacations queue with discouragement . It is also obtained the expected value of the state of the system using stationary queue size distribution, which gives a quick glance of a system performance.

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References

  • Adams R (2014) Infinitesimal perturbation analysis of a multi-stage tandem of fluid queue with additive loss feedback. Syst Control Lett 66:22–27

    Article  MathSciNet  MATH  Google Scholar 

  • Al Seedy RO, El Sherbiny A, El Shehawy S, Ammar S (2009) Transient solution of the M/M/c queue with balking and reneging. Comput Math Appl 57(8):1280–1285

    Article  MathSciNet  MATH  Google Scholar 

  • Ammar SI (2014) Transient analysis of a two-heterogeneous servers queue with impatient behavior. J Egypt Math Soc 22(1):90–95

    Article  MathSciNet  MATH  Google Scholar 

  • Ammar S, El Sherbiny A, El Shehawy S, Al Seedy RO (2012) A matrix approach for the transient solution of an M/M/1/N queue with discouraged arrivals and reneging. Int J Comput Math 89(4):482–491

    Article  MathSciNet  MATH  Google Scholar 

  • Atencia I, Moreno P (2004) Discrete-time \(Geo^{[X]}/GH/1\) retrial queue with bernoulli feedback. Comput Math Appl 47(8–9):1273–1294

    Article  MathSciNet  MATH  Google Scholar 

  • Banik A, Ghosh S (2019) Efficient computational analysis of non-exhaustive service vacation queues: BMAP/R/1/N (\(\infty \)) under gated-limited discipline. Appl Math Model 68:540–562

    Article  MathSciNet  MATH  Google Scholar 

  • Bruneel H, Maertens T (2015) A discrete-time queue with customers with geometric deadlines. Perform Eval 85:52–70

    Article  Google Scholar 

  • Chang FM, Liu TH, Ke JC (2018) On an unreliable-server retrial queue with customer feedback and impatience. Appl Math Model 55:171–182

    Article  MathSciNet  MATH  Google Scholar 

  • D’Arienzo M, Dudin AN, Dudin SA, Manzo R (2020) Analysis of a retrial queue with group service of impatient customers. J Ambient Intell Humaniz Comput 11(6):2591–2599

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Gans N, Koole G, Mandelbaum A (2003) Telephone call centers: tutorial, review, and research prospects. Manuf Serv Oper Manag 5(2):79–141

    Article  Google Scholar 

  • Guha D, Goswami V, Banik AD (2015) Equilibrium balking strategies in renewal input batch arrival queues with multiple and single working vacation. Perform Eval 94:1–24

    Article  Google Scholar 

  • Guha D, Goswami V, Banik A (2016) Algorithmic computation of steady-state probabilities in an almost observable GI/M/c queue with or without vacations under state dependent balking and reneging. Appl Math Model 40(5–6):4199–4219

    Article  MathSciNet  MATH  Google Scholar 

  • Haight FA (1959) Queueing with reneging. Metrika 2(1):186–197

    Article  MathSciNet  MATH  Google Scholar 

  • Haight FA (1960) Queueing with balking. ii. Biometrika 47(3–4):285–296

    Article  MathSciNet  MATH  Google Scholar 

  • Jain M, Sanga SS (2020) State dependent queueing models under admission control F-policy: a survey. J Ambient Intell Human Comput. https://doi.org/10.1007/s12652-019-01638-y

    Article  MATH  Google Scholar 

  • Jain M, Shekhar C, Shukla S (2014) Vacation queuing model for a machining system with two unreliable repairmen. Int J Oper Res 20(4):469–491

    Article  MathSciNet  MATH  Google Scholar 

  • Ke J, Wang K (1999) Cost analysis of the M/M/R machine repair problem with balking, reneging, and server breakdowns. J Oper Res Soc 50(3):275–282

    Article  MATH  Google Scholar 

  • Ke JC, Wang KH, Liou CH (2006) A single vacation model \({G}/{M}/1/{K}\) with \(n\) threshold policy. Sankhyā 68:198–226

    MathSciNet  MATH  Google Scholar 

  • Kumar BK, Rukmani R, Thangaraj V (2009) On multiserver feedback retrial queue with finite buffer. Appl Math Model 33(4):2062–2083

    Article  MathSciNet  MATH  Google Scholar 

  • Kumar R, Sharma S (2018) Transient analysis of an M/M/c queuing system with balking and retention of reneging customers. Commun Stat Theory Methods 47(6):1318–1327

    Article  MathSciNet  MATH  Google Scholar 

  • Liu Z, Gao S (2011) Discrete-time \(Geo_{1}, Geo^{X}_{2}/G_{1}, G_{2}/1\) retrial queue with two classes of customers and feedback. Math Comput Model 53(5–6):1208–1220

    Article  Google Scholar 

  • Liu Y, Whitt W (2017) Stabilizing performance in a service system with time-varying arrivals and customer feedback. Eur J Oper Res 256(2):473–486

    Article  MathSciNet  MATH  Google Scholar 

  • Mitrani I, Robert P (2004) On the asta property in a feedback processor-sharing queue. Perform Eval 58:81–85

    Article  Google Scholar 

  • Parthasarathy P, Sharafali M (1989) Transient solution to the many-server poisson queue: a simple approach. J Appl Probab 26(3):584–594

    Article  MathSciNet  MATH  Google Scholar 

  • Rajadurai P, Saravanarajan M, Chandrasekaran V (2018) A study on M/G/1 feedback retrial queue with subject to server breakdown and repair under multiple working vacation policy. Alex Eng J 57(2):947–962

    Article  Google Scholar 

  • Selvaraju N, Goswami C (2013) Impatient customers in an M/M/1 queue with single and multiple working vacations. Comput Ind Eng 65(2):207–215

    Article  Google Scholar 

  • Shekhar C, Kumar A, Varshney S (2019) Modified bessel series solution of the single server queueing model with feedback. Int J Comput Sci Math 10(3):313–326

    Article  MathSciNet  MATH  Google Scholar 

  • Shin YW (2015) Algorithmic approach to markovian multi-server retrial queues with vacations. Appl Math Comput 250:287–297

    MathSciNet  MATH  Google Scholar 

  • Takhedmit B, Abbas K (2017) A parametric uncertainty analysis method for queues with vacations. J Comput Appl Math 312:143–155

    Article  MathSciNet  MATH  Google Scholar 

  • Thomasian A (2018) Vacationing server model for M/G/1 queues for rebuild processing in raid5 and threshold scheduling for readers and writers. Inf Process Lett 135:41–46

    Article  MathSciNet  MATH  Google Scholar 

  • Upadhyaya S (2016) Performance prediction of a discrete-time batch arrival retrial queue with bernoulli feedback. Appl Math Comput 283:108–119

    MathSciNet  MATH  Google Scholar 

  • Varshney K, Jain M, Sharma G (1989) A multi-server queueing model with balking and reneging via diffusion approximation. J Phys Nat Sci 10(2):10–15

    Google Scholar 

  • Wang Q, Zhang B (2018) Analysis of a busy period queuing system with balking, reneging and motivating. Appl Math Model 64:480–488

    Article  MathSciNet  MATH  Google Scholar 

  • Wang KH, Ke JB, Ke JC (2007) Profit analysis of the M/M/R machine repair problem with balking, reneging, and standby switching failures. Comput Oper Res 34(3):835–847

    Article  MATH  Google Scholar 

  • Wu CH, Ke JC (2014) Multi-server machine repair problems under a (\({V}\), \({R}\)) synchronous single vacation policy. Appl Math Model 38(7–8):2180–2189

    Article  MathSciNet  MATH  Google Scholar 

  • Wu CH, Ke JC (2013) Multi-threshold policy for a multi-server queue with synchronous single vacation. Math Comput Model 57(5–6):1122–1130

    Article  MathSciNet  Google Scholar 

  • Yang DY, Wu CH (2015) Cost-minimization analysis of a working vacation queue with \({N}\)-policy and server breakdowns. Comput Ind Eng 82:151–158

    Article  Google Scholar 

  • Yang DY, Wu YY (2017) Analysis of a finite-capacity system with working breakdowns and retention of impatient customers. J Manuf Syst 44:207–216

    Article  Google Scholar 

  • Zhang M, Hou Z (2012) M/G/1 queue with single working vacation. J Appl Math Comput 39(1–2):221–234

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the Editor-in-chief, editorial board and anonymous referees for the valuable, constructive comments and suggestions on an earlier version of this paper.

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  1. Department of Mathematics, Birla Institute of Technology and Science Pilani, Pilani Campus, Rajasthan, 333031, India

    Amit Kumar

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  1. Amit Kumar
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Kumar, A. Single server multiple vacation queue with discouragement solve by confluent hypergeometric function. J Ambient Intell Human Comput 14, 6411–6422 (2023). https://doi.org/10.1007/s12652-020-02467-0

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