Machine Interference Problem: A Survey of Queueing Models and Applications

Authors

  • Ather Aziz Raina * Department of Applied Mathematics, Govt. Degree College Thannamandi. https://orcid.org/0000-0003-0034-8595
  • Farzana Shaheen Department of Mathematics, IEC University, H.P. (INDIA).

https://doi.org/10.48314/apem.vi.48

Abstract

This article presents a comprehensive survey of research related to the Machine Interference Problem (MIP), also known as the Machine Repair Problem (MRP). The problem arises when machines fail randomly in manufacturing or service environments and compete for limited repair facilities, causing queues and production delays. Such failures often result in substantial losses of production, revenue, or reliability. In addition to summarising research articles on MIP/MRP, this paper provides an organised list of books and survey papers to aid researchers in understanding the breadth of work in the queueing domain. The literature is classified according to methodological and modelling approaches. The survey concludes by highlighting recent advances and identifying promising directions for future research in reliability and queueing theory.

Keywords:

Machine interference problem, Machine repair problem, Queues, Survey

References

  1. [1] Bhat, U. N. (2008). An introduction to queueing theory: Modeling and analysis in applications. Springer Nature. https://doi.org/10.1007/978-0-8176-4725-4
  2. [2] Cooper, R. B. (1972). Introduction to queueing theory. Elsevier north holland. https://www.abebooks.com/9780023246807
  3. [3] Gross, D., Shortle, J. F., Thompson, J. M., & Harris, C. M. (2011). Fundamentals of queueing theory (Vol. 627). John wiley & sons. https://books.google.com/books/about/Fundamentals_of_Queueing_Theory.html?id=K3lQGeCtAJgC
  4. [4] Thomas, M. U. (1976). Queueing systems. volume 1: Theory (leonard kleinrock). SIAM Review. https://doi.org/10.1137/1018095
  5. [5] Newell, C. (1971). Applications of queueing theory (Vol. 4). Springer Science & Business Media. https://doi.org/10.1007/978-94-009-5970-5
  6. [6] Saaty, T. L. (1961). Elements of queueing theory: With applications (Vol. 34203). New York: McGraw-Hill. https://books.google.com/books/about/Elements_of_Queueing_Theory.html?id=MN5QAAAAMAAJ
  7. [7] Raina, A. A. (2017). Reliability and maintainability analysis of redundant repairable systems [Thesis]. https://www.researchgate.net/publication/355443718
  8. [8] Takács, L. (1962). Introduction to the theory of queues. Greenwood Press. https://lccn.loc.gov/82000973
  9. [9] Wolff, R. W. (1989). Stochastic modeling and the theory of queues. Prentice-hall. https://cir.nii.ac.jp/crid/1371413280526420226
  10. [10] Cox, D. R., & Miller, H. D. (1965). The theory of stochastic process. Chapman & Hall, London. https://doi.org/10.1201/9780203719152
  11. [11] Dobrow, R. P. (2016). Introduction to stochastic processes with R. John Wiley & Sons. https://dl.nemoudar.com/Introduction-to-Stochastic-Processes-with-R.pdf
  12. [12] Heyman, D. P., & Sobel, M. J. (1984). Stochastic models in operations research. McGraw-Hill. https://2024.sci-hub.box/3322/365e97fe3d4aa434604973c7ed3fb867/arthur1983.pdf
  13. [13] Jones, P. W., & Smith, P. (2017). Stochastic processes: An introduction. Chapman and Hall CRC. https://doi.org/10.1201/9781315156576
  14. [14] Karlin, S. (2014). A first course in stochastic processes. Academic Press. https://webdelprofesor.ula.ve
  15. [15] Medhi, J. (1994). Stochastic processes. New Age International. https://books.google.com/books/about/Stochastic_Processes.html?id=DnCWgeoalKQC
  16. [16] Medhi, J. (2002). Stochastic models in queueing theory. Elsevier. https://shop.elsevier.com/books/stochastic-models-in-queueing-theory/medhi/978-0-12-487462-6
  17. [17] Nelson, R. (2013). Probability, stochastic processes, and queueing theory:The mathematics of computer performance modeling. Springer Science & Business Media. https://B2n.ir/td9887
  18. [18] Ross, S. M. (2013). Applied probability models with optimization applications. Courier Corporation. https://books.google.com/books/about/Applied_Probability_Models_with_Optimiza.html?id=HftunnI-1zwC
  19. [19] Sethi, S. P., Zhang, H., & Zhang, Q. (2005). Average—cost control of stochastic manufacturing systems. New York, NY: Springer New York. https://doi.org/10.1007/0-387-27615-7_3
  20. [20] Ash, R. B. (2008). Basic probability theory. Courier Corporation. https://www.amazon.com/Basic-Probability-Theory-Dover-Mathematics/dp/0486466280
  21. [21] Clarke, A. B., & Disney, R. L. (1970). Probability and random processes for engineers and scientists. Wiley. https://lccn.loc.gov/76104008
  22. [22] Drake, A. W. (1967). Fundamentals of applied probability theory. Mcgraw-Hill College. https://www.noisebridge.net/images/a/ae/Fundamentals_of_Applied_Probability_by_Drake.pdf
  23. [23] Feller, W. (1991). An introduction to probability theory and its applications (Vol. 2). John Wiley & Sons. https://B2n.ir/uq1675
  24. [24] Parzen, E. (1960). Modern probability theory and its applications (Vol. 10). Wiley. https://www.wiley.com/en-us/Modern+Probability+Theory+and+Its+Applications+-p-9780471572787
  25. [25] Tillman, F. A., Hwang, C. L., & Kuo, W. (1980). Optimization of systems reliability. Marcel Dekker Inc. https://doi.org/10.1016/0143-8174(82)90024-5
  26. [26] O’Connor, P. D. T. (2005). Practical reliability engineering, 4th Ed. Wiley India Pvt. Limited, 2008. https://books.google.com/books/about/Practical_Reliability_Engineering_4th_Ed
  27. [27] Birolini, A. (1997). Quality and reliability of technical systems. Springer Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-97983-5
  28. [28] Ebeling, C. (1996). An introduction to reliability and maintainability engineering. McGraw-Hill Education. https://books.google.com/books/about/An_Introduction_to_Reliability_and_Maint
  29. [29] Bhat, U. N., & Miller, G. K. (1972). Elements of applied stochastic processes third edition. A John Wiley & Sons, Inc., Publication. https://doi.org/10.2307/30045359
  30. [30] Shooman, M. L. (1968). Probabilistic reliability: an engineering approach. McGraw-Hill. https://books.google.com/books/about/Probabilistic_Reliability.html?id=D29RAAAAMAAJ
  31. [31] Aissani, A. (1994). A retrial queue with redundancy and unreliable server. Queueing systems, 17(3), 431–449. https://doi.org/10.1007/BF01158703
  32. [32] Gani, A. N., & Sabarinathan, G. (2013). Fuzzy approach on optimal ordering strategy in inventory and pricing model with deteriorating items. International journal of pure and applied mathematics, 87(1), 165–180. http://dx.doi.org/10.12732/ijpam.v87i1.10
  33. [33] Haque, L., & Armstrong, M. J. (2007). A survey of the machine interference problem. European journal of operational research, 179(2), 469–482. https://doi.org/10.1016/j.ejor.2006.02.036
  34. [34] Jain, M., & Gupta, R. (2011). Redundancy issues in software and hardware systems: An overview. International journal of reliability, quality and safety engineering, 18(1), 61–98. https://doi.org/10.1142/S0218539311004093
  35. [35] Jain, M., Sharma, G. C., & Pundhir, R. S. (2010). Some perspectives of machine repair problems. International journal of engineering, 23(3–4), 253–268. https://www.sid.ir/paper/570644/en
  36. [36] Krishnamoorthy, A., Pramod, P. K., & Chakravarthy, S. R. (2014). Queues with interruptions: A survey. Top, 22(1), 290–320. https://doi.org/10.1007/s11750-012-0256-6
  37. [37] Shekhar, C., Raina, A.A., Kumar, A. & Iqbal, J. (2016). A brief review on retrial queue: Progress in 2010-2015. International journal of applied sciences and engineering research, 5)4(, 324-336. https://doi.org/10.2298/Y.TOR161117006R
  38. [38] Sztrik, J., & Bunday, B. D. (1993). Machine interference problem with a random environment. European journal of operational research, 65(2), 259–269. https://doi.org/10.1016/0377-2217(93)90338-N
  39. [39] Teghem, J. (1986). Control of the service process in a queueing system. European journal of operational research, 23(2), 141–158. https://doi.org/10.1016/0377-2217(86)90234-1
  40. [40] Shekhar, C., Raina, A. A., Kumar, A., & Iqbal, J. (2017). A survey on queues in machining system: Progress from 2010 to 2017. Yugoslav journal of operations research, 27(4), 391–413. https://doi.org/10.2298/YJOR161117006R
  41. [41] Sharma, R. (2014). Mathematical analysis of queue with phase service: An overview. Advances in operations research, 2014(1), 240926. https://doi.org/10.1155/2014/240926
  42. [42] Govil, M. K., & Fu, M. C. (1999). Queueing theory in manufacturing: A survey. Journal of manufacturing systems, 18(3), 214–240. https://doi.org/10.1016/S0278-6125(99)80033-8
  43. [43] Sharma, A. K., & Sharma, G. K. (2013). Queueing theory appr oach with queueing model: A study. International journal of engineering science invention, 2(2), 1–11. https://www.idc-online.com/technical_references/pdfs/data_communications/QUEUEING THEORY.pdf
  44. [44] Allahverdi, A., Gupta, J. N. D., & Aldowaisan, T. (1999). A review of scheduling research involving setup considerations. Omega, 27(2), 219–239. https://doi.org/10.1016/S0305-0483(98)00042-5
  45. [45] Allahverdi, A., Ng, C. T., Cheng, T. C. E., & Kovalyov, M. Y. (2008). A survey of scheduling problems with setup times or costs. European journal of operational research, 187(3), 985–1032. https://doi.org/10.1016/j.ejor.2006.06.060
  46. [46] Ma, Y., Chu, C., & Zuo, C. (2010). A survey of scheduling with deterministic machine availability constraints. Computers & industrial engineering, 58(2), 199–211. https://doi.org/10.1016/j.cie.2009.04.014
  47. [47] Stecke, K. E., & Aronson, J. E. (1985). Review of operator machine interference models. International journal of production research, 23(1), 129–151. https://doi.org/10.1080/00207548508904696
  48. [48] Chandrasekaran, V. M., Indhira, K., Saravanarajan, M. C., & Rajadurai, P. (2016). A survey on working vacation queueing models. International journal of pure and applied mathematics, 106(6), 33–41. https://doi.org/10.12732/ijpam.v106i6.5
  49. [49] Khintchine, A. Y. (1932). Mathematical theory of stationary queues. Matem. sbornik, 39, 73–84. https://www.mathnet.ru/rus/sm/v39/i3/p27
  50. [50] Lotka, A. J. (1939). A contribution to the theory of self-renewing aggregates, with special reference to industrial replacement. The annals of mathematical statistics, 10(1), 1–25. https://www.jstor.org/stable/2235984
  51. [51] Palm, C. (1947). The distribution of repairmen in servicing automatic machines. Industritidningen Norden, 75, 75-80.
  52. [52] Linton, D. G., & Saw, J. G. (2009). Reliability analysis of the k-out-of-n: F system. IEEE transactions on reliability, 23(2), 97–103. https://doi.org/10.1109/TR.1974.5215215
  53. [53] Nakagawa, T. (2009). Optimization problems in k-out-of-n systems. IEEE transactions on reliability, 34(3), 248–250. https://doi.org/10.1109/TR.1985.5222134
  54. [54] Moustafa, M. S. (1996). Transient analysis of reliability with and without repair for K-out-of-N: G systems with two failure modes. Reliability engineering & system safety, 53(1), 31–35. https://doi.org/10.1016/0951-8320(96)00014-2
  55. [55] Gupta, H., & Sharma, J. (2009). State transition matrix and transition diagram of k-out-of-n-: G system with spares. IEEE transactions on reliability, 30(4), 395–397. https://doi.org/10.1109/TR.1981.5221136
  56. [56] Azaron, A., Katagiri, H., Kato, K., & Sakawa, M. (2006). Reliability evaluation of multi-component cold-standby redundant systems. Applied mathematics and computation, 173(1), 137–149. https://doi.org/10.1016/j.amc.2005.02.051
  57. [57] Yuan, L. (2012). Reliability analysis for a k-out-of-n: G system with redundant dependency and repairmen having multiple vacations. Applied mathematics and computation, 218(24), 11959–11969. https://doi.org/10.1016/j.amc.2012.06.006
  58. [58] Ke, J. C., Hsu, Y. L., Liu, T. H., & Zhang, Z. G. (2013). Computational analysis of machine repair problem with unreliable multi-repairmen. Computers & operations research, 40(3), 848–855. https://doi.org/10.1016/j.cor.2012.10.004
  59. [59] Sivazlian, B. D., & Wang, K. H. (1989). Economic analysis of the M/M/R machine repair problem with warm standbys. Microelectronics reliability, 29(1), 25–35. https://doi.org/10.1016/0026-2714(89)90007-3
  60. [60] Hsieh, Y. C., & Wang, K. H. (1995). Reliability of a repairable system with spares and a removable repairman. Microelectronics reliability, 35(2), 197–208. https://doi.org/10.1016/0026-2714(95)90086-6
  61. [61] Wang, K. H., & Chiu, L. W. (2006). Cost benefit analysis of availability systems with warm standby units and imperfect coverage. Applied mathematics and computation, 172(2), 1239–1256. https://doi.org/10.1016/j.amc.2005.02.052
  62. [62] Ke, J. B., Lee, W. C., & Wang, K. H. (2007). Reliability and sensitivity analysis of a system with multiple unreliable service stations and standby switching failures. Physica a: Statistical mechanics and its applications, 380, 455–469. https://doi.org/10.1016/j.physa.2007.02.095
  63. [63] Wang, K. H., & Chen, Y. J. (2009). Comparative analysis of availability between three systems with general repair times, reboot delay and switching failures. Applied mathematics and computation, 215(1), 384–394. https://doi.org/10.1016/j.amc.2009.05.023
  64. [64] Jyh-Bin, K., Jyh-Wei, C., & Kuo-Hsiung, W. (2011). Reliability measures of a repairable system with standby switching failures and reboot delay. Quality technology & quantitative management, 8(1), 15–26. https://doi.org/10.1080/16843703.2011.11673243
  65. [65] Jain, M., Shekhar, C., & Shukla, S. (2013). Queueing analysis of two unreliable servers machining system with switching and common cause failure. International journal of mathematics in operational research, 5(4), 508–536. https://doi.org/10.1504/IJMOR.2013.054732
  66. [66] Jain, M., Shekhar, C., & Shukla, S. (2014). Markov model for switching failure of warm spares in machine repair system. Journal of reliability and statistical studies, 7, 57–68. https://journals.riverpublishers.com/index.php/JRSS/article/view/21629
  67. [67] Shekhar, C., Jain, M., & Bhatia, S. (2014). Fuzzy analysis of machine repair problem with switching failure and reboot. Journal of reliability and statistical studies, 7(Special Issue), 41–55. https://journals.riverpublishers.com/index.php/JRSS/article/view/21627
  68. [68] Jain, M., Sharma, G. C., & Sharma, R. (2012). Optimal control of (N, F) policy for unreliable server queue with multi-optional phase repair and start-up. International journal of mathematics in operational research, 4(2), 152–174. https://doi.org/10.1504/IJMOR.2012.046375
  69. [69] Kumar, K., & Jain, M. (2013). Threshold N-policy for (M, m) degraded machining system with K-heterogeneous servers, standby switching failure and multiple vacations. International journal of mathematics in operational research, 5(4), 423–445. https://doi.org/10.1504/IJMOR.2013.054719
  70. [70] Jain, M., Shekhar, C., & Shukla, S. (2014). Machine repair problem with an unreliable server and controlled arrival of failed machines. Opsearch, 51(3), 416–433. https://doi.org/10.1007/s12597-013-0152-3
  71. [71] Shekhar, C., Jain, M., Iqbal, J., & Raina, A. A. (2017). Threshold control policy for maintainability of manufacturing system with unreliable workstations. Arabian journal for science and engineering, 42(11), 4833–4851. https://doi.org/10.1007/s13369-017-2636-6
  72. [72] Shekhar, C., Jain, M., Raina, A. A., & Iqbal, J. (2018). Reliability prediction of fault tolerant machining system with reboot and recovery delay. International journal of system assurance engineering and management, 9(2), 377–400. https://doi.org/10.1007/s13198-017-0680-y%0A%0A
  73. [73] Shekhar, C., Jain, M., & Raina, A. A. (2017). Transient analysis of machining system with spare provisioning and geometric reneging. International journal of mathematics in operational research, 11(3), 396–421. https://doi.org/10.1504/IJMOR.2017.087215
  74. [74] Shekhar, C., Jain, M., Raina, A. A., & Iqbal, J. (2017). Optimal (N, F) policy for queue-dependent and time-sharing machining redundant system. International journal of quality & reliability management, 34(6), 798–816. https://doi.org/10.1108/IJQRM-07-2016-0105
  75. [75] Shekhar, C., Jain, M., Raina, A. A., & Mishra, R. P. (2017). Sensitivity analysis of repairable redundant system with switching failure and geometric reneging. Decision science letters, 6(4), 337–350. https://doi.org/10.5267/j.dsl.2017.2.004
  76. [76] Kothandapani, M., & Chandrasekar, N. (2024). Intuitionistic fuzzy threshold hypergraphs. Italian journal of pure and applied mathematics, (52), 67–85. https://ijpam.uniud.it/online_issue/202452/08 Kothandapani-Chandrasekar.pdf
  77. [77] Telang, A. D., & Telang, A. (2010). Comprehensive maintenance management: Policies, strategies and options. PHI Learning Pvt. Ltd. https://B2n.ir/gp3612
  78. [78] Ayyappan, G., Sekar, G., & Subramanian, A. M. G. (2010). M/M/1 retrial queueing system with two types of vacation policies under Erlang-K type service. International journal of computer applications, 2(8), 9–18. https://doi.org/10.5120/688-967
  79. [79] Jain, M., Sharma, G. C., & Sharma, R. (2012). Maximum entropy approach for un-reliable server vacation queueing model with optional bulk service. Journal of king abdulaziz university, marine science
  80. [80] (2), 89–113. www.researchgate.net/publication/267537383
  81. [81] Ammar, S. I., Helan, M. M., & Al Amri, F. T. (2013). The busy period of an M/M/1 queue with balking and reneging. Applied mathematical modelling, 37(22), 9223–9229. https://doi.org/10.1016/j.apm.2013.04.023
  82. [82] Sharma, R., & Kumar, G. (2014). Unreliable server queue with priority queueing system. International journal of engineering and technical research, (Special Issue), 368–371. chrome-extension://efaidnbmnnnibpcajpcglclefindmkaj/https://www.erpublication.org/published_paper/IJETR_APRIL_2014_STET_101.pdf
  83. [83] Kim, B., & Kim, J. (2015). A single server queue with Markov modulated service rates and impatient customers. Performance evaluation, 83, 1–15. https://doi.org/10.1016/j.peva.2014.11.002
  84. [84] Vijayashree, K. V, & Pavithra, P. (2025). Performance analysis and cost optimization of an M/M/1 queue with N-policy and working breakdown. Methodology and computing in applied probability, 27(3), 1–26. https://doi.org/10.1007/s11009-025-10196-0%0A%0A
  85. [85] Dimitriou, I., & Langaris, C. (2010). A repairable queueing model with two-phase service, start-up times and retrial customers. Computers & operations research, 37(7), 1181–1190. https://doi.org/10.1016/j.cor.2009.03.003
  86. [86] Ramanath, K., & Kalidass, K. (2010). A two phase service M/G/1 vacation queue with general retrial times and non-persistent customers. International journal of open problems in computer science and mathematics, 3(2), 175–185. https://www.ijopcm.org/Vol/10/IJOPCM(vol.3.2.6.J.10).pdf
  87. [87] Thangaraj, V., & Vanitha, S. (2010). A single server M/G/1 feedback queue with two types of service having general distribution. International mathematical forum, 5(1), 15-33. https://www.researchgate.net/publication/228464381
  88. [88] Thangaraj, V., & Vanitha, S. (2010). M/G/1 queue with two-stage heterogeneous service compulsory server vacation and random breakdowns. International journal of contemporary mathematical sciences. 5(7), 307–322. https://m-hikari.com/ijcms-2010/5-8-2010/vanithaIJCMS5-8-2010.pdf
  89. [89] Wang, K. H., Yang, D. Y., & Pearn, W. L. (2010). Comparison of two randomized policy queues with second optional service, server breakdown and startup. Journal of computational and applied mathematics, 234(3), 812–824. https://doi.org/10.1016/j.cam.2010.01.045
  90. [90] Gao, S., & Liu, Z. (2013). An M/G/1 queue with single working vacation and vacation interruption under Bernoulli schedule. Applied mathematical modelling, 37(3), 1564–1579. https://doi.org/10.1016/j.apm.2012.04.045
  91. [91] Gao, S., Wang, J., & Li, W. W. (2014). An M/G/1 retrial queue with general retrial times, working vacations and vacation interruption. Asia-pacific journal of operational research, 31(02), 1440006. https://doi.org/10.1142/S0217595914400065
  92. [92] Lee, D. H., & Kim, B. K. (2015). A note on the sojourn time distribution of an M/G/1 queue with a single working vacation and vacation interruption. Operations research perspectives, 2, 57–61. https://doi.org/10.1016/j.orp.2015.01.002
  93. [93] Chaudhry, M., Banik, A. D., Dev, S., & Barik, S. (2025). A simple derivation of the waiting-time distribution (in the queue) for the bulk-service M/G(a,b)/1 queueing system. Annals of operations research, 352(1), 1–2 https://doi.org/10.1007/s10479-025-06765-84.
  94. [94] Wang, J., & Li, J. (2010). Analysis of the queue with second multi-optional service and unreliable server. Acta mathematicae applicatae sinica, 26(3), 353–368. https://doi.org/10.1007/s10255-010-0001-6/metrics
  95. [95] Choudhury, G., & Deka, M. (2012). A single server queueing system with two phases of service subject to server breakdown and Bernoulli vacation. Applied mathematical modelling, 36(12), 6050–6060. https://doi.org/10.1016/j.apm.2012.01.047
  96. [96] Gao, S., & Yao, Y. (2014). An MX/G/1 queue with randomized working vacations and at most J vacations. International journal of computer mathematics, 91(3), 368–383. https://doi.org/10.1080/00207160.2013.790964
  97. [97] Rajadurai, P., Chandrasekaran, V. M., & Saravanarajan, M. C. (2016). 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. https://doi.org/10.1007/s12597-015-0226-5
  98. [98] Singh, C. J., Jain, M., & Kumar, B. (2016). MX/G/1 unreliable retrial queue with option of additional service and Bernoulli vacation. Ain shams engineering journal, 7(1), 415–429. https://doi.org/10.1016/j.asej.2015.05.006
  99. [99] Gómez Corral, A., & Garcia, M. L. (2014). Maximum queue lengths during a fixed time interval in the M/M/c retrial queue. Applied mathematics and computation, 235, 124–136. https://doi.org/10.1016/j.amc.2014.02.074
  100. [100] Yu, M., Tang, Y., Fu, Y., & Pan, L. (2011). An M/Ek/1 queueing system with no damage service interruptions. Mathematical and computer modelling, 54(5–6), 1262–1272. https://doi.org/10.1016/j.mcm.2011.03.037
  101. [101] Kim, J., & Kim, B. (2013). Waiting time distribution in an M/PH/1 retrial queue. Performance evaluation, 70(4), 286–299. https://doi.org/10.1016/j.peva.2012.12.003
  102. [102] Kuo, C. C., Wang, K. H., & Pearn, W. L. (2011). The interrelationship between N-policy M/G/1/K and F-policy G/M/1/K queues with startup time. Quality technology & quantitative management, 8(3), 237–251. https://doi.org/10.1080/16843703.2011.11673257
  103. [103] Sharma, R. (2010). Threshold N-policy for MX/H2/1 queueing system with un-reliable server and vacations. Journal of international academy of physical sciences, 14, 1–11. https://www.iaps.org.in/journal/index.php/journaliaps/article/view/289
  104. [104] Goswami, C., & Selvaraju, N. (2010). The discrete-time MAP/PH/1 queue with multiple working vacations. Applied mathematical modelling, 34(4), 931–946. https://doi.org/10.1016/j.apm.2009.07.021
  105. [105] Mary, K. J. R., Begum, M. I. A., & Parveen, M. J. (2011). Bi-level threshold policy of MX/(G 1, G 2)/1 queue with early setup and single vacation. International journal of operational research, 10(4), 469–493. https://doi.org/10.1504/IJOR.2011.039714
  106. [106] Arivudainambi, D., & Godhandaraman, P. (2012). A batch arrival retrial queue with two phases of service, feedback and K optional vacations. Applied mathematical sciences, 6(22), 1071–1087. https://m-hikari.com/ams/ams-2012/ams-21-24-2012/arivudainambiAMS21-24-2012.pdf
  107. [107] Tirdad, A., Grassmann, W. K., & Tavakoli, J. (2016). Optimal policies of M (t)/M/c/c queues with two different levels of servers. European journal of operational research, 249(3), 1124–1130. https://doi.org/10.1016/j.ejor.2015.10.040
  108. [108] Al Hanbali, A. (2011). Busy period analysis of the level dependent PH/PH/1/K queue. Queueing systems, 67(3), 221–249. https://doi.org/10.1007/s11134-011-9213-6
  109. [109] Mohit, S. P., & Raina, A. A. (2016). Numerical solution of G/M/1 queueing model with removable server. International transactions in mathematical sciences and computers, 9(1–2), 51–66. https://www.researchgate.net/profile/Ather-Raina/publication/316134837
  110. [110] Lin, C. H., Ke, J. C., Huang, H. I., & Chang, F. M. (2010). The approximation analysis of the discrete-time Geo/Geo/1 system with additional optional service. International journal of computer mathematics, 87(11), 2574–2587. https://doi.org/10.1080/00207160802691629
  111. [111] Gao, S., & Wang, J. (2016). Equilibrium balking strategies in the observable Geo/Geo/1 queue with delayed multiple vacations. RAIRO-operations research, 50(1), 119–129. https://doi.org/10.1051/ro/2015019
  112. [112] Liu, Z., & Gao, S. (2011). Discrete-time Geo1, Geo2X/G1, G2/1 retrial queue with two classes of customers and feedback. Mathematical and computer modelling, 53(5–6), 1208–1220. https://doi.org/10.1016/j.mcm.2010.11.090
  113. [113] Luo, C., Li, W., Yu, K., & Ding, C. (2016). The matrix-form solution for GeoX/G/1/N working vacation queue and its application to state-dependent cost control. Computers & operations research, 67, 63–74. https://doi.org/10.1016/j.cor.2015.07.015
  114. [114] Gao, S., & Wang, J. (2013). Discrete-time retrial queue with general retrial times, working vacations and vacation interruption. Quality technology & quantitative management, 10(4), 495–512. https://doi.org/10.1080/16843703.2013.11673427
  115. [115] Gao, S., & Yin, C. (2013). Discrete-time GeoX/G/1 queue with geometrically working vacations and vacation interruption. Quality technology & quantitative management, 10(4), 423–442. https://doi.org/10.1080/16843703.2013.11673423
  116. [116] Samanta, S. K., & Zhang, Z. G. (2012). Stationary analysis of a discrete-time GI/D-MSP/1 queue with multiple vacations. Applied mathematical modelling, 36(12), 5964–5975. https://doi.org/10.1016/j.apm.2012.01.049
  117. [117] Tao, L., Zhang, L., Xu, X., & Gao, S. (2013). The GI/Geo/1 queue with Bernoulli-schedule-controlled vacation and vacation interruption. Computers & operations research, 40(7), 1680–1692. https://doi.org/10.1016/j.cor.2012.12.019
  118. [118] Gao, S., Wang, J., & Zhang, D. (2013). Discrete-time GIX/Geo/1/N queue with negative customers and multiple working vacations. Journal of the Korean statistical society, 42(4), 515–528. https://doi.org/10.1016/j.jkss.2013.03.002
  119. [119] Gao, S., & Wang, J. (2015). On a discrete-time GIX/GEO/1/NG queue with randomized working vacations and at most J vacations. Journal of industrial and management optimization, 11(3), 779–806. http://dx.doi.org/10.3934/jimo.2015.11.779
  120. [120] Ruiz-Castro, J. E., & Li, Q. L. (2011). Algorithm for a general discrete k-out-of-n: G system subject to several types of failure with an indefinite number of repairpersons. European journal of operational research, 211(1), 97–111. https://doi.org/10.1016/j.ejor.2010.10.024
  121. [121] Jain, M., & Gupta, R. (2012). Load sharing M-out of-N: G system with non-identical components subject to common cause failure. International journal of mathematics in operational research, 4(5), 586–605. https://doi.org/10.1504/IJMOR.2012.048932
  122. [122] Kumar, G., & Bajaj, R. K. (2014). Intuitionistic fuzzy reliability of K-out-of-N: G system using statistical confidence interval. International journal of applied information systems, 7(7), 1–7. https://doi.org/10.5120/ijais14-451212
  123. [123] Kumar, T., & Bajaj, R. K. (2014). Reliability analysis of K-out-of-N: G system using triangular intuitionistic fuzzy numbers. http://www.ir.juit.ac.in:8080/jspui/handle/123456789/9099
  124. [124] Grover, R. (2016). Transient analysis of reliability with and without repair for K-out-of-n: G systems with three failure modes. International research journal of engineering and technology, 3(3), 604–608. https://www.irjet.net/archives/V3/i3/IRJET-V3I3132.pdf
  125. [125] Lenin, R. B., Cuyt, A., Yoshigoe, K., & Ramaswamy, S. (2010). Computing packet loss probabilities of D-BMAP/PH/1/N queues with group services. Performance evaluation, 67(3), 160–173. https://doi.org/10.1016/j.peva.2009.10.001

Downloads

Published

2025-12-12

Issue

Vol. 2 No. 4 (2025): Annals of Process Engineering and Management

Section

Articles

How to Cite

Raina, A. A., & Shaheen, F. (2025). Machine Interference Problem: A Survey of Queueing Models and Applications. Annals of Process Engineering and Management, 2(4), 253-268. https://doi.org/10.48314/apem.vi.48

Similar Articles

1-10 of 12

You may also start an advanced similarity search for this article.