A Two Stage Batch Arrival Queue with Reneging during Vacation and Breakdown Periods


We study a two stage queuing model where the server provides two stages of service one by one in succession. We consider reneging to occur when the server is unavailable during the system breakdown or vacation periods. We concentrate on deriving the steady state solutions by using supplementary variable technique and calculate the mean queue length and mean waiting time. Further some special cases are also discussed and numerical examples are presented.

Share and Cite:

M. Baruah, K. Madan and T. Eldabi, "A Two Stage Batch Arrival Queue with Reneging during Vacation and Breakdown Periods," American Journal of Operations Research, Vol. 3 No. 6, 2013, pp. 570-580. doi: 10.4236/ajor.2013.36054.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] F. A. Haight, “Queueing with Balking,” Biometrika, Vol. 44, No. 3-4, 1957, pp. 360-369.
[2] F. A. Haight, “Queueing with Reneging,” Metrika, Vol. 2, No. 1, 1959, pp. 186-197.
[3] D. Y. Barrer, “Queueing with Impatient Customers and Ordered Service,” Operation Research, Vol. 5, No. 5, 1957, pp. 650-656. http://dx.doi.org/10.1287/opre.5.5.650
[4] C. J. Ancker Jr. and A. V. Gafarian, “Some Queuing Problems with Balking and Reneging,” Operations Research, Vol. 11, No. 1, 1963, pp. 88-100.
[5] A. M. Hagighi, J. Medhi and S. G. Mohanty, “On a Multiserver Markovian Queuing System with Balking and Reneging,” Computer and Operational Research, Vol. 13, No. 4, 1986, pp. 421-425.
[6] J. Bae, S. Kim and E. Y. Lee, “The Virtual Waiting time of the M/G/1 Queue with Impatient Customers,” Queuing Systems: Theory and Application, Vol. 38, No. 4, 2001, pp. 485-494.
[7] Y. Zhang, D. Yue and W. Yue, “Analysis of an M/M/1/N Queue with Balking, Reneging and Server Vacations,” International Symposium on OR and Its Applications.
[8] M. S. El-Pauomy, “On Poisson Arrival Queue: MX/M/2/N with Balking, Reneging and Heterogeneous Servers,” Applied Mathematical Sciences, Vol. 2, No. 24, 2008, pp. 1169-1175.
[9] E. Altman and U. Yechiali, “Analysis of Customers’ Impatience in Queue with Server Vacations,” Queuing Systems, Vol. 52, No. 4, 2006, pp. 261-279.
[10] E. Altman and U. Yechiali, “Infinite Server Queues with Systems Additional Tasks and Impatient Customers,” Probability in the Engineering and Information Sciences, Vol. 22, No. 4, 2008, pp. 477-493.
[11] R. Kumar and S. K. Sharma, “A Markovian Feedback Queue with Retention of Reneged Customers and Balking,” AMO-Advanced Modeling and Optimization, Vol. 14, No. 3, 2012, pp. 681-688.
[12] Y. Levy and U. Yechiali, “An M/M/s Queue with Server Vacations,” INFOR Journal, Vol. 14, No. 2, 1976, pp. 153-163.
[13] B. T. Doshi, “Queueing Systems with Vacations-A Survey,” Queueing Systems, Vol. 1, No. 1, 1986, pp. 29-66.
[14] J. Keilson and L. D. Servi, “The Dynamics of an M/G/1 Vacation Model,” Operations Research, Vol. 35, No. 4, 1987, pp. 575-582.
[15] B. T. Doshi, “Analysis of a Two Phase Queueing System with General Service Times,” Operation Research Letters, Vol. 10, No. 5, 1991, pp. 265-275.
[16] M. S. Kumar and R. Arumuganathan, “On the Single Server Batch Arrival Retrial Queue with General Vacation Time under Bernoulli Schedule and Two Phases of Heterogeneous Servers,” Quality Technology and Quantity Management, Vol. 5, No. 2, 2008, pp. 145-160.
[17] G. Choudhury, L. Tadj and M. Paul, “Steady State Analysis of an MX/G/1 Queue with Two Phase Service and Bernoulli Vacation Schedule under Multiple Vacation Policy,” Applied Mathematical Modelling, Vol. 31, No. 6, 2007, pp. 1079-1091.
[18] F. A. Maraghi, K. C. Madan and K. Darby-Dowman, “Bernoulli Schedule Vacation Queues with Batch Arrivals and Random System Breakdowns having General Repair Times Distribution,” International Journal of Operation Research, Vol. 7, No. 2, 2010, pp. 240-256.
[19] R. F. Khalaf, K. C. Madan and C. A. Lukas, “A MX/G/1 with Bernoulli Schedule General Vacation Times, General Extended Vacations, Random Breakdown, General Delay Time for Repairs to Start and General Repair Times,” Journal of Mathematical Research, Vol. 3, No. 4, 2011, pp. 8-20.

Copyright © 2022 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.