Transient Analysis of \(M^{[X_{1}]},M^{[X_{2}]}/G_{1},G_{2}/1\) Retrial Queueing System with Priority Services, Working Vacations and Vacation Interruption, Emergency Vacation, Negative Arrival and Delayed Repair

  • G. Ayyappan
  • P. Thamizhselvi
Original Paper


This paper deals with the analysis of two classes of batch arrivals, one is high priority and other is low priority(retrial) customers with non-preemptive priority service, working vacations, negative arrival, emergency vacation for an unreliable server, which consists of a break-down by negative arrival and delay time to start repair. When there are no customers in the queue and in the orbit at the time of service completion for a positive customers, the server goes for a working vacation. The server works at a lower service rate during working vacation period. If there are no customers in the system at the end of vacation, the server becomes idle and ready for serving the new arrivals with probability p(single working vacation) or it remains on vacation with probability \(q=(1{-}p)\) (multiple working vacations). Here we assume that customers arrive according to compound Poisson process in which high priority customers are assigned to class one and class two customers are of a low priority type. The priority customers that find the server busy are queued and then served in accordance with FCFS discipline. The arriving low priority customers on finding the server busy are queued in the orbit in accordance with FCFS constant retrial policy. The arrival of negative customer remove the customer being in service and makes the server breakdown and the server fail for a short interval of time. When the server fails it will send to repair. But the repair time do not start immediately, there is a delay time to start the repair process. We consider an emergency vacation which means that during the service time the server suddenly go for a vacation and the interrupted customer waits to getting the remaining service. The retrial time, service time, emergency vacation time, delay time and repair time are all follows general(arbitrary) distribution, working vacation time and negative arrival follows exponential distribution. The time dependent probability generating functions have been obtained in terms of their Laplace transforms and the corresponding steady state results are obtained explicitly. Also some performance measures of our model such as the system state probabilities, reliability indices, the average number of customer in the high priority queue and the low priority in the orbit and the average waiting time are derived. Numerical results are computed and graphical representations are presented.


Batch arrival Priority queue Retrial queue Negative arrival Emergency and Bernoulli working vacation And delayed repair 


60K25 60K30 90B22 


  1. 1.
    Ayyappan, G., Ganapathi, M., Subramanian, A.: Single server retrial queueing system with working vacation under pre-emptive priority service. Int. J. Comput. Appl. 2(2), 28–35 (2010)zbMATHGoogle Scholar
  2. 2.
    Atencia, I., Moreno, P.: A single-server retrial queue with general retrial times and Bernoulli schedule. Appl. Math. Comput. 162, 855–880 (2005)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Atencia, : Ivan A discrete-time queueing system with changes in the vacation times. Int. J. Appl. Math. Comput. Sci. 26(2), 379–390 (2016)Google Scholar
  4. 4.
    Kim, C., Klimenok, V.M., Dudin, A.N.: Priority tandem queueuing system with retrials and reservation of channels as a model of call center. Comput. Ind. Eng. 96, 61–71 (2016)CrossRefGoogle Scholar
  5. 5.
    Farahmand, K.: Single line queue with repeated attempts. Queueing Syst. 6, 223–228 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Gao, S., Wang, J., Li, W.: An \(M/G/1\) retrial queue with general retrial times, working vacations and vacation interruption. Asia-Pac. J. Oper. Res. 31, 6–31 (2014)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Choudhury, G.: Jau-Chuan Ke A batch arrival retrial queue with general retrial times under Bernoulli vacation schedule for unreliable server and delaying repair. Appl. Math. Model. 36, 255–269 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Corral, G.: A stochastic analysis of a single server retrial queue with general retrial times. Nav. Logist. 46, 561–581 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Jain, M., Bhargava, C.: Bulk arrival retrial queue with unreliable server and priority subscribers. Int. J. Oper. Res. 5(4), 242–259 (2008)MathSciNetGoogle Scholar
  10. 10.
    Ke, J.-C., Chang, F.-M.: Modified vacation policy for \(M/G/1\) retrial queue with balking and feedback. Comput. Ind. Eng. 57, 433–443 (2009)CrossRefGoogle Scholar
  11. 11.
    Jinbiao, W., Lian, Z.: A single-server retrial G-queue with priority and unreliable server under Bernoulli vacation schedule. Comput. Ind. Eng. 64, 84–93 (2013)CrossRefGoogle Scholar
  12. 12.
    Krishna Kumar, B., Vijayalakshmi, G., Krishnamoorthy, A., Sadiq Basha, S.: a Asingle server feedback retrial queue with collisions. Comput. Oper. Res. 37, 1247–1255 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Rajadurai, P., Chandrasekaran, V.M., Saravanarajan, M.C.: Analysis of an unreliable retrial G-queue with working vacations and vacation interruption under Bernoulli schedule. Ain Shams Eng. J. 7(1), 1–14 (2016)CrossRefzbMATHGoogle Scholar
  14. 14.
    Kirupa, K., Udaya Chandrika, K.: Single-server retrial queueing system with two different vacation policies. Int. J. Contemp. Math. Sci. 5(32), 1591–1598 (2010)zbMATHGoogle Scholar
  15. 15.
    Kirupa, K., Udaya Chandrika, K.: Batch arrival retrial G-queue and an unreliable server delayed repair. Int. J. Innov. Res. Sci. Eng. Technol. 3(5), 12436–12444 (2014)Google Scholar
  16. 16.
    Miaomiao, Y., Alfa, A.S.: Strategic queueing behaviour for individual and social optimization in manageing discrete time working vacation queue with Bernoulli interruption schedule. Comput. Oper. Res. 73, 43–55 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Srinivas, R.: Chakravarthy Queueing models with optional cooperative services. Eur. J. Oper. Res. 000, 1–12 (2015)Google Scholar
  18. 18.
    Mulikha, V., Ilyashenko, A., Zayats, O., Zaborovsky, V.: Preemptive queueing system with randomized push-out mechanism. Commun. Nonlinear Sci. Number Simulat. 21(1–3), 147–158 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Tao, L., Liu, Z., Wang, Z.: \(M/M/1\) Retrial queue with collisions and working vacation interruption under N-policy. RAIRO-Oper. Res 46, 355–371 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Li, T., Zhang, L., Gao, S.: Performance of an \(M/M/1\) retrial queue with working vacation interruption and classical retrial policy. Hindawi Publishing Corporation, Advances in Operations Research (2016)Google Scholar
  21. 21.
    Yang, T., Templeton, J.G.C.: A survey on retrial queue. Queueing Syst. 2, 201–233 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Yang, T., Posner, M.J.M., Templeton, J.G.C., Li, H.: An approximation method for the \(M/G/1\) retrial queue with general retrial times. Eur. J. Oper. Res. 76, 552–562 (1994)CrossRefzbMATHGoogle Scholar
  23. 23.
    Gaia, Y., Liub, H., Krishnamacharib, B.: A packet dropping mechanism for efficient operation of \(M/M/1\) queues with selfish users. Comput. Netw. 98, 1–13 (2016)CrossRefGoogle Scholar

Copyright information

© Springer (India) Private Ltd., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsPondicherry Engineering CollegePuducherryIndia

Personalised recommendations