International Journal of Mathematical, Engineering and Management Sciences

ISSN: 2455-7749

Controlled Arrival Machine Repair Problem with Working Vacation and Reattempts

Amita Bhagat
Department of Mathematics, Jaypee Institute of Information Technology, Noida, India.

Rachita Sethi
Department of Mathematics, G. D. Goenka University, Gurugram, India.

Deepika Garg
Department of Mathematics, G. D. Goenka University, Gurugram, India.


Received on March 26, 2020
Accepted on July 11, 2020


This paper addresses machine repair problem (MRP) with M identical operating machines and control arrival policy. The server is unreliable, and can break down during service which is further repaired soon so as to avoid interruption with the service process. The server may go for working vacation in case all the customers are served. The transient analysis of machine repair problem has been done using numerical technique. Various performance measures have been derived. With the help of tables and graphs the numerical results have been shown. The present investigation find applications in various industrial and workshop situations like Automobile repair shop (ARS).

Keywords- Machine repair, F-policy, Unreliable, Working vacation, Retrial policy.


Bhagat, A., Sethi, R., & Garg, D. (2021). Controlled Arrival Machine Repair Problem with Working Vacation and Reattempts. International Journal of Mathematical, Engineering and Management Sciences, 6(1), 279-295.

Conflict of Interest

The authors confirm that there is no conflict of interest to declare for this publication.


The authors would like to appreciate the effort from editors and reviewers. This study received no specific grant from government, commercial or non-profit funding organizations.


Artalejo, J.R. (1999). A classified bibliography of research on retrial queues : progress in 1990-1999. Top, 7(2), 187–211.

Artalejo, J.R. (2010). Accessible bibliography on retrial queues: progress in 2000-2009. Mathematical and Computer Modelling, 51(9–10), 1071–1081.

Bhagat, A. (2020, March). Unreliable priority retrial queues with double orbits and discouraged customers. In AIP Conference Proceedings (Vol. 2214, No. 1, p. 020014). AIP Publishing.

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.

Choudhury, G., & Ke, J.C. (2014). An unreliable retrial queue with delaying repair and general retrial times under Bernoulli vacation schedule. Applied Mathematics and Computation, 230, 436–450.

Falin, G. (1990). A survey of retrial queues. Queueing Systems, 7(2), 127–167.

Jain, M., & Bhagat, A. (2015). Transient analysis of finite F-policy retrial queues with delayed repair and threshold recovery. National Academy Science Letters, 38(3), 257–261.

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.

Jain, M., & Singh, M. (2020). Transient analysis of a Markov queueing model with feedback, discouragement and disaster. International Journal of Applied and Computational Mathematics, 6(2), 1–14.

Ke, J.C., Hsu, Y.L., Liu, T.H., & George Zhang, Z. (2013). Computational analysis of machine repair problem with unreliable multi-repairmen. Computers and Operations Research, 40(3), 848–855.

Levy, Y., & Yechiali, U. (1975). Utilization of idle time in an M/G/1 queueing system. Management Science, 22(2), 202–211.

Li, T., & Zhang, L. (2017). An M/G/1 retrial G-queue with general retrial times and working breakdowns. Mathematical and Computational Applications, 22(1), 1–15.

Morozov, E., & Phung-Duc, T. (2017). Stability analysis of a multiclass retrial system with classical retrial policy. Performance Evaluation, 112, 15–26.

Servi, L.D., & Finn, S.G. (2002). M/M/1 queues with working vacations (M/M/1/WV). Performance Evaluation, 50(1), 41–52.

Sethi, R., Jain, M., Meena, R.K., & Garg, D. (2020). Cost optimization and ANFIS computing of an unreliable M/M/1 queueing system with customers ’ impatience under N-policy. International Journal of Applied and Computational Mathematics, 6(2), 1–14.

Sethi, R., & Bhagat, A. (2019, January). Performance analysis of machine repair problem with working vacation and service interruptions. In AIP Conference Proceedings (Vol. 2061, No. 1, p. 020028). AIP Publishing.

Sethi, R., Bhagat, A., & Garg, D. (2019). ANFIS based machine repair model with control policies and working vacation. International Journal of Mathematical, Engineering and Management Sciences, 4(6), 1522–1533.

Sharma, R., & Kumar, G. (2017). Availability improvement for the successive K-out-of-N machining system using standby with multiple working vacations. International Journal of Reliability and Safety, 11(3-4), 256–267.

Sharma, R., & Kumar, G. (2020, February). Multi-server M/M/c queue and multiple working vacation under phase repair. In 2020 3rd International Conference on Emerging Technologies in Computer Engineering: Machine Learning and Internet of Things (ICETCE) (pp. 181-185). IEEE. Jaipur, India.

Shekhar, C., Raina, A.A., & Kumar, A. (2016). A brief review on retrial queue: progress in 2010-2015. International Journal of Applied Sciences and Engineering Research, 5(4), 324–336.

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.

Sherman, N.P., & Kharoufeh, J.P. (2006). An M/M/1 retrial queue with unreliable server. Operations Research Letters, 34(6), 697–705.

Wang, K.H., & Yang, D.Y. (2009). Controlling arrivals for a queueing system with an unreliable server: Newton-Quasi method. Applied Mathematics and Computation, 213(1), 92–101.

Wang, K.H., Chen, W.L., & Yang, D.Y. (2009). Optimal management of the machine repair problem with working vacation: Newton’s method. Journal of Computational and Applied Mathematics, 233(2), 449–458.

Wu, C.H., Lee, W.C., Ke, J.C., & Liu, T.H. (2014). Optimization analysis of an unreliable multi-server queue with a controllable repair policy. Computers and Operations Research, 49, 83–96.

Privacy Policy| Terms & Conditions