B. Somasundaram     (Vel Tech Rangarajan Dr.Sagunthala R&D Institute of Science and Technology, Chennai – 600 062, Tamil Nadu, India)
S. Karpagam     (Vel Tech Rangarajan Dr.Sagunthala R&D Institute of Science and Technology, Chennai – 600 062, Tamil Nadu, India)
R. Lokesh     (Vel Tech Rangarajan Dr.Sagunthala R&D Institute of Science and Technology, Chennai – 600 062, Tamil Nadu, India)
A. Kavin Sagana Mary     (Vel Tech Rangarajan Dr.Sagunthala R&D Institute of Science and Technology, Chennai – 600 062, Tamil Nadu, India)


In this study, a batch arrival single service queue with two stages of service (second stage is optional) and working breakdown is investigated. When the system is in operation, it may breakdown at any time. During breakdown period, instead of terminating the service totally, it continues at a slower rate. We find the time-dependent probability generating functions in terms of their Laplace transforms and derive explicitly the corresponding steady state results. Furthermore, numerous measures indicating system performances, such as the average queue size and the average queue waiting time, has been obtained. Some of the numerical results and graphical representations were also presented.


Non-Markovian queue, Second optional service, Working breakdown

Full Text:



  1. Ayyappan G., Thamizhselvi P., Somasundaram B., Udayageetha J. Analysis of an \(M^{X_1},$ $M^{X_2}/G_1,$ $G_2/1\) retrial queueing system with priority services, working breakdown, Bernoulli vacation, admission control and balking. J. Stat. Manag. Syst., 2020. Vol. 24, No. 4. P. 685–702. DOI: 10.1080/09720529.2020.1744812
  2. Al-Jararha J., Madan K.C. An \(M/G/1\) queue with second optional service with general service time distribution. Internat. J. Inform. Management Sci., 2003. Vol. 14, No. 2. P. 47–56.
  3. Ammar S.I., Rajadurai P. Performance analysis of preemptive priority retrial queueing system with disaster under working breakdown services. Symmetry, 2019. Vol. 11, No. 3. P. 419–425. DOI: 10.3390/sym11030419
  4. Choudhury G., Paul M. A batch arrival queue with a second optional service channel under \(N\)-policy. Stoch. Anal. Appl., 2006. Vol. 24, No. 1. P. 1–21. DOI: 10.1080/07362990500397277
  5. Choudhury G., Tadj L. An \(M/G/1\) queue with two phases of service subject to the server breakdown and delayed repair. Appl. Math. Model., 2009. Vol. 33, No. 6. P. 2699–2709. DOI: 10.1016/j.apm.2008.08.006
  6. Gupta D., Solanki A., Agrawal K.M. Non-Markovian queueing system, \(M^X/G/1\) with server breakdown and repair times. Recent Res. Sci. Technol., 2011. Vol. 3, No. 7. P. 88–94.
  7. Kalidass K., Kasturi R. A two phase service \(M/G/1\) queue with a finite number of immediate Bernoulli feedbacks. OPSEARCH, 2014. Vol. 51, No. 2. P. 201–218. DOI: 10.1007/s12597-013-0136-3
  8. Kim B.K., Lee D.H. The \(M/G/1\) queue with disasters and working breakdowns. Appl. Math. Model., 2014. Vol. 38, No. 5–6. P. 1788–1798. DOI: 10.1016/j.apm.2013.09.016
  9. Madan K.C. An \(M/G/1\) queue with second optional service. Queueing System, 2000. Vol. 34. P. 37–46. DOI: 10.1023/A:1019144716929
  10. Maragathasundari S., Srinivasan S., Ranjitham A. Batch arrival queueing system with two stages of service. Int. J. Math. Anal., 2014. Vol. 8, No. 6. P. 247–258. DOI: 10.12988/ijma.2014.411
  11. Maragathasundari S., Srinivasan S. A non-Markovian multistage batch arrival queue with breakdown and reneging. Math. Probl. Eng., 2014. Vol. 2014. Art. no. 519579. 16 p. DOI: 10.1155/2014/519579
  12. Rajadurai P. Sensitivity analysis of an \(M/G/1\) retrial queueing system with disaster under working vacations and working breakdowns. RAIRO-Oper. Res., 2018, Vol. 52, No. 1. P. 35–54. DOI: 10.1051/ro/2017091
  13. Rajadurai P., Saravanarajan M.C., Chandrasekaran V.M. A study on \(M/G/1\) feedback retrial queue with subject to server breakdown and repair under multiple working vacation policy. Alexandria Eng. J., 2017. Vol. 57, No. 6. P. 947–962. DOI: 10.1016/j.aej.2017.01.002
  14. Santhi K. An \(M/G/1\) retrial queue with second optional service and multiple working vacation. Adv. Appl. Math. Sci., 2021. Vol. 20, No. 6. P. 1129–1146.
  15. Singh C.J., Kaur S. \(M^{X}/G/1\) queue with optional service and server breakdowns. In: Performance Prediction and Analytics of Fuzzy, Reliability and Queueing Models. Asset Analytics. K. Deep, M. Jain, S. Salhi (eds.). Singapore: Springer, 2019. P. 177–189. DOI: 10.1007/978-981-13-0857-4_13
  16. Thangaraj V., Vanitha S. \(M/G/1\) queue with two-stage heterogeneous service compulsory server vacation and random breakdowns. Int. J. Contemp. Math. Sci., 2010. Vol. 5, No. 7. P. 307–322.
  17. Yang D.-Y., Chen Y.-H. Computation and optimization of a working breakdown queue with second optional service. J. Ind. Production Eng., 2018. Vol. 35, No. 3. P. 181–188. DOI: 10.1080/21681015.2018.1439113
  18. Yang D.-Y., Chen Y.-H., Wu C.-H. Modelling and optimisation of a two-server queue with multiple vacations and working breakdowns. Int. J. Prod. Res., 2020. Vol. 58, No. 10. P. 3036–3048. DOI: 10.1080/00207543.2019.1624856
  19. Yen T.-C., Wang K.-H., Chen J.-Y. Optimization analysis of the \(N\) policy \(M/G/1\) queue with working breakdowns. Symmetry, 2020. Vol. 12, No. 4. P. 583–594. DOI: 10.3390/sym12040583


Article Metrics

Metrics Loading ...


  • There are currently no refbacks.