Skip to main content
SpringerLink
Log in

Reliability and Vacation: The Critical Issue

  • Chapter
  • First Online:
Book cover Advances in Reliability Analysis and its Applications

Part of the book series: Springer Series in Reliability Engineering ((RELIABILITY))

Abstract

Predicting and understanding the machining system performance is a constant challenge in the process industries like manufacturing and production systems, computer and communication systems, just-in-time (JIT) service systems. The reliability modeling supports the decision-making process from early to the optimal state-of-the-art design of the machining system. Reliability measures account the performance of different preventive, predictive, corrective, zero-hours, and periodic maintenance strategies. For all strategies, the most anterior arrangement is the availability of the service facility as and when required to maintain the high grade or efficient quality of service (QoS). The permanent service facility may increase cost, idleness, deterioration in quality. To reduce the wastage of valuable resources like time, money, quality, etc., vacation is a prominent idea for the service facility. The vacation time is a period of time of not doing the usual service or activities. In this time, the server may take rest to rejuvenate, to reduce idle time at the station, to diminish the expected cost incurred in service. The long vacation time also wastes the valuable resources substantially due to the long waiting queue of failed machines. The vacation time period is a critical issue and needs to analyze judgementally. In this chapter, a comparative study of different vacation policies on the reliability characteristics of the machining system is presented. For that purpose, the queueing-theoretic approach is employed, and the Markovian models are developed for various types of vacation policy namely N-policy, single vacation, multiple vacations, Bernoulli vacation, working vacation, vacation interruption, etc. For all vacation policies, the reliability and mean-time-to-failure (MTTF) of the system are compared, and results are depicted in the graphs for quick insights. From this study, readers get a glance to understand about vacation, researchers get a concrete platform to choose appropriate assumptions for their research in machining/service system or system analyst may opt suitable vacation policy as per limitation of the system.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Cox DR, Miller HD (1965) The theory of stochastic process. Chapman & Hall, London

    Google Scholar 

  2. Bhat UN (1972) Elements of applied stochastic process. Wiley, New York

    MATH  Google Scholar 

  3. Cooper RB (1972) Introduction to queueing theory. MacMillan, New York

    MATH  Google Scholar 

  4. Medhi J (1982) Stochastic processes. Wiley

    Google Scholar 

  5. Gross D, Harris C (1998) Fundamentals of queueing theory. Wiley Series in Probability and Statistics

    Google Scholar 

  6. Medhi J (2002) Stochastic models in queueing theory. Academic Press, Amsterdam

    MATH  Google Scholar 

  7. Gross D, Shortle JF, Thompson JM, Harris CM (2008) Fundamentals of queueing theory. Wiley, New York

    Book  MATH  Google Scholar 

  8. Balagurusamy E, Misra KB (1976) Reliability calculation of redundant systems with non-identical units. Microelectron Reliab 15(5):376–377

    Google Scholar 

  9. Trivedi KS, Dugan JB, Geist R, Smotherman M (1984) Hybrid reliability modeling of fault-tolerant computer systems. Comput Electr Eng 11(2–3):87–108

    Article  MATH  Google Scholar 

  10. Sztrik J, Bunday BD (1993) Machine interference problem with a random environment. Eur J Oper Res 65(2):259–269

    Article  MATH  Google Scholar 

  11. Levantesi R, Matta A, Tolio T (2003) Performance evaluation of continuous production lines with machines having different processing times and multiple failure modes. Perform Eval 51(2–4):247–268

    Article  MATH  Google Scholar 

  12. Haque L, Armstrong MJ (2007) A survey of the machine interference problem. Eur J Oper Res 179(2):469–482

    Article  MATH  Google Scholar 

  13. Dimitriou I, Langaris C (2010) A repairable queueing model with two-phase service, start-up times and retrial customers. Comput Oper Res 37(7):1181–1190

    Article  MathSciNet  MATH  Google Scholar 

  14. Ke JC, Lin CH (2010) Maximum entropy approach to machine repair problem. Int J Serv Oper Inform 5(3):197–208

    Google Scholar 

  15. Lv S, Yue D, Li J (2010) Transient reliability of machine repairable system. J Inf Comput Sci 7(13):2879–2885

    Google Scholar 

  16. Jain M, Sharma GC, Pundhir RS (2010) Some perspectives of machine repair problem. IJE Trans B: Appl 23(3–4):253–268

    MATH  Google Scholar 

  17. Wu Q, Wu S (2011) Reliability analysis of two-unit cold standby repairable systems under Poisson shocks. Appl Math Comput 218(1):171–182

    MathSciNet  MATH  Google Scholar 

  18. Ruiz CJE, Li QL (2011) Algorithm for a general discrete \(K\)-out-of-\(N\): \(G\) system subject to several types of failure with an indefinite number of repair-persons. Eur J Oper Res 211(1):97–111

    Article  MATH  Google Scholar 

  19. Nourelfath M, Chatelet E, Nahas N (2012) Joint redundancy and imperfect preventive maintenance optimization for series-parallel multi-state degraded systems. Reliab Eng Syst Saf 103:51–60

    Article  Google Scholar 

  20. El-Damcese MA, Shama MS (2013) Reliability and availability analysis of a standby repairable system with degradation facility. Int J Res Rev Appl Sci 16(3):501–507

    Google Scholar 

  21. Wells CE (2014) Reliability analysis of a single warm-standby system subject to repairable and non-repairable failures. Eur J Oper Res 235(1):180–186

    Article  MATH  Google Scholar 

  22. Kuo C, Sheu S, Ke JC, Zhang ZG (2014) Reliability-based measures for a retrial system with mixed standby components. Appl Math Model 38(19–20):4640–4651

    Article  MathSciNet  MATH  Google Scholar 

  23. Ke JC, Liu TH, Wu CH (2015) An optimum approach of profit analysis on the machine repair system with heterogeneous repairmen. Appl Math Comput 253(15):40–51

    MathSciNet  MATH  Google Scholar 

  24. Gonzaleza PP, Viagasa VF, Garciab MZ, Framinan JM (2019) Constructive heuristics for the unrelated parallel machines scheduling problem with machine eligibility and setup times. Comput Ind Eng 131:131–145

    Article  Google Scholar 

  25. Jain M (1997) Optimal \(N\)-policy for single server Markovian queue with breakdown, repair and state dependant arrival rate. Int J Manag Syst 13(3):245–260

    Google Scholar 

  26. Ushakumari PV, Krishnamoorthy A (1998) \(k\)-out-of-\(n\) system with general repair: the \(N\)-policy. In: Proceedings of the II international symposium on semi-markov process, UTC, Compiegne, France

    Google Scholar 

  27. Gupta SM (1999) \(N\)-policy queueing system with finite source and warm spares. Opsearch 36(3):189–217

    Article  MathSciNet  MATH  Google Scholar 

  28. Krishnamoorthy A, Ushakumari PV, Lakshmi B (2002) \(k\)-out-of-\(n\) system with repair: the \(N\)-policy. Asia Pac J Oper Res 19(1):47–61

    MathSciNet  MATH  Google Scholar 

  29. Ushakumari PV, Krishnamoorthy A (2004) \(k\)-out-of-\(n\) system with repair: the \(\max (N, T)\)-policy. Perform Eval 57(2):221–234

    Article  Google Scholar 

  30. Yue D, Yue W, Qi H (2008) Analysis of a machine repair system with warm spares and \(N\)-policy Vacations. In: The 7th international symposium on operations research and its applications (ISORA’08), pp 190–198

    Google Scholar 

  31. Jain M, Upadhyaya S (2009) Threshold \(N\)-policy for degraded machining system with multiple types of spares and multiple vacations. Qual Technol Quant Manag 6(2):185–203

    Article  MathSciNet  Google Scholar 

  32. Wang TY, Yang DY, Li MJ (2010) Fuzzy analysis for the \(N\)-policy queues with infinite capacity. Int J Inf Manag Sci 21:41–56

    MathSciNet  MATH  Google Scholar 

  33. Singh CJ, Jain M, Kumar B (2013) Analysis of queue with two phases of service and m phases of repair for server breakdown under \(N\)-policy. Int J Serv Oper Manag 16(3):373–406

    Google Scholar 

  34. Jain M, Shekhar C, Shukla S (2016) A time-shared machine repair problem with mixed spares under \(N\)-policy. J Ind Eng Int 12(2):145–157

    Article  Google Scholar 

  35. Chen WL, Wang KH (2018) Reliability analysis of a retrial machine repair problem with warm standbys and a single server with \(N\)-policy. Reliab Eng Syst Saf 180:476–486

    Article  Google Scholar 

  36. Ayyappan G, Karpagam S (2019) Analysis of a bulk queue with unreliable server, immediate feedback, \(N\)-policy, Bernoulli schedule multiple vacation and stand-by server. Ain Shams Eng J. https://doi.org/10.1016/j.asej.2019.03.008

    Article  Google Scholar 

  37. Blanc JPC, Mei RDV (1995) Optimization of polling systems with Bernoulli schedules. Perform Eval 22(2):139–158

    Article  MATH  Google Scholar 

  38. Choudhury G, Madan KC (2004) A two-phase batch arrival queueing system with a vacation time under Bernoulli schedule. Appl Math Comput 149(2):337–349

    MathSciNet  MATH  Google Scholar 

  39. Choudhury G, Madan KC (2005) A two-stage batch arrival queueing system with a modified Bernoulli schedule vacation. Math Comput Model 42(1–2):71–85

    Article  MathSciNet  MATH  Google Scholar 

  40. Choudhury G, Deka M (2012) A single server queueing system with two phases of service subject to server breakdown and Bernoulli vacation. Appl Math Model 36(12):6050–6060

    Article  MathSciNet  MATH  Google Scholar 

  41. Liu TH, Ke JC (2014) On the multi-server machine interference with modified Bernoulli vacations. J Ind Manag Optim 10(4):1191–1208

    Article  MathSciNet  MATH  Google Scholar 

  42. Shrivastava RK, Mishra AK (2014) Analysis of queuing model for machine repairing system with Bernoulli vacation schedule. Int J Math Trends Technol 10(2):85–92

    Article  Google Scholar 

  43. Rajadurai P, Chandrasekaran VM, Saravanarajan MC (2015) Analysis of an \(M^{[X]}/G/1\) unreliable retrial \(G\)-queue with orbital search and feedback under Bernoulli vacation schedule. OPSEARCH 53(1):197–223

    Article  MathSciNet  MATH  Google Scholar 

  44. Singh CJ, Jain M, Kumar B (2016) \(M^{X}/G/1\) unreliable retrial queue with option of additional service and Bernoulli vacation. Ain Shams Eng J 7(1):415–429

    Article  Google Scholar 

  45. Choudhury G, Deka M (2018) A batch arrival unreliable server delaying repair queue with two phases of service and Bernoulli vacation under multiple vacation policy. Qual Technol Quant Manag 15(2):157–186

    Article  Google Scholar 

  46. Thomo L (1997) A multiple vacation model \(M^{X}/G/1\) with balking. Nonlinear Amlysis, Theory, Methods Appl 30(4):2025–2030

    Article  MathSciNet  MATH  Google Scholar 

  47. Zhang ZG, Vickson R, Love E (2001) The optimal service policies in an \(M/G/1\) queueing system with multiple vacation types. Inf Syst Oper Res 39(4):357–366

    Google Scholar 

  48. Ke JC, Wang KH (2007) Vacation policies for machine repair problem with two type spares. Appl Math Model 31(5):880–894

    Article  MATH  Google Scholar 

  49. Ke JC, Wu CH (2012) Multi-server machine repair model with standbys and synchronous multiple vacation. Comput Ind Eng 62(1):296–305

    Article  Google Scholar 

  50. Yuan L (2012) Reliability analysis for a system with redundant dependency and repairmen having multiple vacations. Appl Math Comput 218(24):11959–11969

    MathSciNet  MATH  Google Scholar 

  51. Liou CD (2015) Optimization analysis of the machine repair problem with multiple vacations and working breakdowns. J Ind Manag Optim 11(1):83–104

    Article  MathSciNet  MATH  Google Scholar 

  52. Sun W, Li S, Guo CE (2016) Equilibrium and optimal balking strategies of customers in Markovian queues with multiple vacations and \(N\)-policy. Appl Math Model 40(1):284–301

    Article  MathSciNet  Google Scholar 

  53. Lee HW, Lee SS, Chae KC, Nadarajan R (1992) On a batch service queue with single vacation. Appl Math Model 16(1):36–42

    Article  MATH  Google Scholar 

  54. Madan KC, Al-Rub AZA (2004) On a single server queue with optional phase type server vacations based on exhaustive deterministic service and a single vacation policy. Appl Math Comput 149(3):723–734

    MathSciNet  MATH  Google Scholar 

  55. Xu X, Zhang ZG (2006) Analysis of multi-server queue with a single vacation \((e, d)\)-policy. Perform Eval 63(8):825–838

    Article  Google Scholar 

  56. Wang HX, Xu GQ (2012) A cold system with two different components and a single vacation of the repairman. Appl Math Comput 219(5):2634–2657

    MathSciNet  MATH  Google Scholar 

  57. Wu W, Tang Y, Yu M, Jiang Y (2014) Reliability analysis of a \(k\)-out-of-\(n\):\(G\) repairable system with single vacation. Appl Math Model 38(24):6075–6097

    Article  MathSciNet  MATH  Google Scholar 

  58. 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 

  59. Zhang Y, Wu W, Tang Y (2017) Analysis of an \(k\)-out-of-\(n\):\(G\) system with repairman’s single vacation and shut off rule. Oper Res Perspect 4:29–38

    Article  MathSciNet  Google Scholar 

  60. Neuts MF (1981) Matrix-geometric solutions in stochastic models. Johns Hopkins University Press, Baltimore

    MATH  Google Scholar 

  61. Servi L, Finn S (2002) \(M/M/1\) queue with working vacations \((M/M/1/WV)\). Perform Eval 50(1):41–52

    Article  Google Scholar 

  62. 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 

  63. Tian N, Ma Z, Liu M (2008) The discrete time Geom/Geom/1 queue with multiple working vacations. Appl Math Model 32(12):2941–2953

    Article  MathSciNet  MATH  Google Scholar 

  64. 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 

  65. 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 

  66. Rajadurai P, Saravanarajan MC, Chandrasekaran VM (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 

  67. Lin CH, Ke JC (2009) Multi-server system with single working vacation. Appl Math Model 33(7):2967–2977

    Article  MathSciNet  MATH  Google Scholar 

  68. Yang DY, Wang KH, Wu CH (2010) Optimization and sensitivity analysis of controlling arrivals in the queueing system with single working vacation. J Comput Appl Math 234(2):545–556

    Article  MathSciNet  MATH  Google Scholar 

  69. Li J, Tian N (2011) Performance analysis of a \(GI/M/1\) queue with single working vacation. Appl Math Comput 217(19):4960–4971

    MathSciNet  MATH  Google Scholar 

  70. Guha D, Banik AD (2013) On the renewal input batch-arrival queue under single and multiple working vacation policy with application to EPON. Inf Syst Oper Res 51(4):175–191

    MathSciNet  Google Scholar 

  71. Kempa WM, Kobielnik M (2018) Transient solution for the queue-size distribution in a finite-buffer model with general independent input stream and single working vacation policy. Appl Math Model 59:614–628

    Article  MathSciNet  Google Scholar 

  72. Li J, Tian N (2007) The \(M/M/1\) queue with working vacations and vacation interruption. J Syst Sci Syst Eng 16(1):121–127

    Article  MATH  Google Scholar 

  73. Tao L, Liu Z, Wang Z (2011) The \(GI/M/1\) queue with start-up period and single working vacation and Bernoulli vacation interruption. Appl Math Comput 218(8):4401–4413

    MathSciNet  MATH  Google Scholar 

  74. Gao S, Liu Z (2013) An \(M/G/1\) queue with single working vacation and vacation interruption under Bernoulli schedule. Appl Math Model 37(3):1564–1579

    Article  MathSciNet  MATH  Google Scholar 

  75. Tao L, Zhang L, Xu X, Gao S (2013) The \(GI/Geo/1\) queue with Bernoulli-schedule-controlled vacation and vacation interruption. Comput Oper Res 40(7):1680–1692

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  77. Rajadurai P, Chandrasekaran VM, Saravanarajan MC (2018) Analysis of an unreliable retrial \(G\)-queue with working vacations and vacation interruption under Bernoulli schedule. Ain Shams Eng J 9(4):567–580

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Birla Institute of Technology and Science Pilani, Pilani Campus, Pilani, 333 031, Rajasthan, India

    Chandra Shekhar, Shreekant Varshney & Amit Kumar

Authors
  1. Chandra Shekhar
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shreekant Varshney
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Amit Kumar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Chandra Shekhar .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Graphic Era Deemed to be University, Dehradun, Uttarakhand, India

    Mangey Ram

  2. Department of Industrial and Systems Engineering, Rutgers University, Piscataway, NJ, USA

    Hoang Pham

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Shekhar, C., Varshney, S., Kumar, A. (2020). Reliability and Vacation: The Critical Issue. In: Ram, M., Pham, H. (eds) Advances in Reliability Analysis and its Applications. Springer Series in Reliability Engineering. Springer, Cham. https://doi.org/10.1007/978-3-030-31375-3_7

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-31375-3_7

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-31374-6

  • Online ISBN: 978-3-030-31375-3

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation