Main Article Content


This paper considers  M[X1],M[X2]/G1,G2/1 general retrial queueing system with priority services. Two types of customers from different classes arrive at the system in different independent compound Poisson processes. The server follows the pre-emptive priority rule subject to working breakdown, startup/closedown time and Bernoulli vacation with general (arbitrary) vacation periods. After completing the service, if there are no priority customers present in the system the server may go for a vacation or close down the system. On completion of the close down, the server needs some time to set up the system. The priority customers who find the server busy are queued in the system. A low-priority customer who find the server busy are routed to a retrial (orbit) queue that attempts to get the service. The system may breakdown at any point of time when it is in operation. However, when the system fails, instead of stopping service completely, the service is continued only to the high priority customers at a slower rate. We consider balking to occur to the low priority customer while the server is busy or idle, and reneging to occur at the high priority customers during server’s vacation, start up/close down time. Using the supplementary variable technique, we derive the joint distribution of the server state and the number of customers in the system. Finally, some performance measures and numerical examples are presented.


Priority Queueing systems Retrial Bernoulli Vacations Working breakdown Startup/close down time Balking Reneging.

Article Details

Author Biographies

Govindhan Ayyappan, Department of Mathematics, Pondicherry Engineering College, India

Professor & Head,

Department of  Mathematics,

Pondicherry Engineering college, India

Udayageetha J, Department of Mathematics, Perunthalaivar Kamarajar Arts College, India

Assistant Professor,

Department of Mathematics,
Perunthalaivar Kamarajar Arts college,

How to Cite
Ayyappan, G., & J, U. (2020). Transient Analysis of M[X1],M[X2]/G1,G2/1 retrial queueing system with priority services, working breakdown, start up/close down time, Bernoulli vacation, reneging and balking. Pakistan Journal of Statistics and Operation Research, 16(1), 203-216.


  1. Atencia I. and Moreno. P (2005). A single-server retrial queue with general retrial times and Bernoulli schedule. Applied Mathematics and Computation, 162, 855 - 880.
  2. Bo Keun Kim, Doo Ho Lee (2016). The M/G/1 queue with disasters and working breakdowns. Applied Mathematical Modelling 4, 437-459.
  3. Cheng-Dar Liou (2013). Markovian queue optimisation analysis with an unreliable server subject to working breakdowns and impatient customers. International Journal of Systems Science, 46, 2165 - 2182.
  4. Cox D.R (1955). The analysis of non-Markovian stochastic processes by the inclusion of supplementary variables. Proc. Cambridge Phil.Soc. 51, 433 - 441.
  5. Dong-Yuh Yang, Ying-Yi Wu (2017). Analysis of a finite-capacity system with working breakdowns and retention of impatient customers. Journal of Manufacturing Systems, 44, 207 - 216.
  6. Gross D. and Harris, C.M. (1985). Fundamentals of Queueing Theory: 2nd Edition, Wiley, New York.
  7. Ioannis Dimitriou and Christos Langaris (2009). A Queueing model with startup/closedown time and retrial customers. Taylor and Francis, 25, 248-269.
  8. Jaiswal N.K. (1968). Priority Queues. Academic press,NY.
  9. Kalidass K, Kasturi R (2012). A queue with working breakdowns. Computers and Industrial Engineering, 63. 779-783.
  10. Pavai Madheswari S, Krishna Kumar B and Suganthi P (2019), Analysis of M/G/1 retrial queues with second optional service and customer balking under two types of Bernoulli vacation schedule, RAIRO-Oper. Res. 53(2).
  11. Servi, L.D and Finn S.G (2012). An M/M/1 queues with Working vacations(M/M/1/WV). Performance Evauation, 50, 41-52.
  12. Sridharan V and Jayashree P J (1996). Some characteristics on a finite queue with normal, partial and total failures. Microelectronic reliability, 36(2), 265-267.
  13. Sherif I. Ammar and Pakkirisamy Rajadurai (2019), Performance Analysis of Pre-emptive Priority Retrial Queueing System with Disaster under Working Breakdown Services, SYMMETRY, 11, 419.
  14. Subha Rao S. (1967). Queueing with balking and reneging in M/G/1 systems. Metrika, 12, 173-188.
  15. Tao Li and Liyuan Zhang (2017). An M/G/1 Retrial G-Queue with General Retrial Times and Working Breakdowns. Math. Comput. Appl. 22, 15.
  16. Tao Li,Liyuan Zhang and Shan Gao (2018), An M/G/1 retrial queue with balking customers and Bernoulli working vacation interruption, Taylor-Francis, 511-530.
  17. Zaiming liu and Yang Song , The M^X/M/1 queue with working breakdown. Rairo operations research, 48, 339 - 413.