IJMEMES logo

International Journal of Mathematical, Engineering and Management Sciences

ISSN: 2455-7749


Analysis of an M^([X])/G(a,b)/1 Unreliable G-queue with Loss, Instantaneous Bernoulli Feedback, Vacation and Two Delays of Verification

Analysis of an M^([X])/G(a,b)/1 Unreliable G-queue with Loss, Instantaneous Bernoulli Feedback, Vacation and Two Delays of Verification

G. Ayyappan
Department of Mathematics, Pondicherry Engineering College, Pillaichavady, Puducherry - 605014, India.

R. Supraja
Department of Mathematics, Pondicherry Engineering College, Pillaichavady, Puducherry - 605014, India.

DOI https://dx.doi.org/10.33889/IJMEMS.2019.4.2-040

Received on November 07, 2018
  ;
Accepted on February 04, 2019

Abstract

This paper deals with a batch arrival that customers arrive to the system according to a compound Poisson process. The customer’s behavior is incorporated according to which loss with a certain probability and the server begins to provide a service only when a queue size minimum say ‘a’ and maximum service capacity is ‘b’. Once the server completes the service, the unsatisfied customers may get the same service under Bernoulli schedule is termed as instantaneous Bernoulli feedback. The occurrence of negative customer cause the server to fail and removes a group of customers or an amount of work if present upon its arrival. As soon as the failure instant, the service channel send to the two delays of verification, the first verification delay starts before the repair process and the second verification delay begins after the repair process. We use the generating function method to derive the stationary queue size distribution. Some important performance measures such as different states of the system and the expected length of the queue explicitly. Some important special cases and numerical examples are determined.

Keywords- Bulk service, Feedback service, Vacation, G-queue, Two delays of verification.

Citation

Ayyappan, G., & Supraja, R. (2019). Analysis of an M^([X])/G(a,b)/1 Unreliable G-queue with Loss, Instantaneous Bernoulli Feedback, Vacation and Two Delays of Verification. International Journal of Mathematical, Engineering and Management Sciences, 4(2), 489-507. https://dx.doi.org/10.33889/IJMEMS.2019.4.2-040.

Conflict of Interest

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

Acknowledgements

The authors like to thank the anonymous referees for their careful review for the improvement of this paper.

References

Altman, E., & Yechiali, U. (2006). Analysis of customers’ impatience in queues with server vacations. Queueing Systems, 52(4), 261-279.

Ayyappan, G., & Shyamala, S. (2013). Time Dependent Solution of M^([X])/G/1 Queuing Model with Bernoulli Vacation and Balking, International Journal of Computer Applications, 61(21), 20-24

Bailey, N. T. J. (1954). On queueing processes with bulk service. Journal of the Royal Statistical Society. Series B (Methodological), 16(1), 80-87.

Bhunia, A. K., Duary, A., & Sahoo, L. (2017). A genetic algorithm based hybrid approach for reliability redundancy optimization problem of a series system with multiple-choice, International Journal of Mathematical, Engineering and Management Sciences, 2(3), 185-212.

Bose, G. K., & Pain, P. (2018). Metaheuristic approach of multi-objective optimization during EDM process, International Journal of Mathematical, Engineering and Management Sciences, 3(3), 301-314.

Choudhury, G., & Tadj, L. (2009). An M/G/1 queue with two phases of service subject to the server breakdown and delayed repair. Applied Mathematical Modelling, 33(6), 2699-2709.

Haight, F. A. (1957). Queueing with balking, Biometrika, 44(3/4), 360-369.

Haridass, M., & Arumuganathan, R. (2008). Analysis of a bulk queue with unreliable server and single vacation. International Journal of Open Problems in Computer Science and Mathematics, 1(2), 130-148.

Jeyakumar, S., & Senthilnathan, B. (2016). Steady state analysis of bulk arrival and bulk service queueing model with multiple working vacations. International Journal of Mathematics in Operational Research, 9(3), 375-394.

Ke, J. C., & Chang, F. M. (2009). Modified vacation policy for M/G/1 retrial queue with balking and feedback. Computers & Industrial Engineering, 57(1), 433-443.

Keilson, J., & Servi, L. D. (1986). Oscillating random walk models for GI/G/1 vacation systems with Bernoulli schedules. Journal of applied Probability, 23(3), 790-802.

Kumar, B. K., Madheswari, S. P., & Lakshmi, S. A. (2013). An M/G/1 Bernoulli feedback retrial queueing system with negative customers. Operational Research, 13(2), 187-210.

Lee, H. W., Lee, S. S., Chae, K. C., & Nadarajan, R. (1992). On a batch service queue with single vacation. Applied Mathematical Modelling, 16(1), 36-42.

Madan, K. C., & Choudhury, G. (2004). An M^x/G/1 queue with Bernoulli vacation schedule under restricted admissibility policy. Sankhaya, 66(1), 175-193.

Neuts, M. F. (1967). A general class of bulk queues with Poisson input. The Annals of Mathematical Statistics, 38(3), 759-770.

Rajadurai, P., Saravanarajan, M. C., & Chandrasekaran, V. M. (2018). A study on M/G/1 feedback retrial queue with subject to server breakdown and repair under multiple working vacation policy. Alexandria Engineering Journal, 57(2), 947-962.

Rajadurai, P., Varalakshmi, M., Saravanarajan, M. C., & Chandrasekaran, V. M. (2015). Analysis of M^([X])/G/1 retrial queue with two phase service under Bernoulli vacation schedule and random breakdown. International Journal of Mathematics in Operational Research, 7(1), 19-41.

Ram, M., & Manglik, M. (2016). Stochastic biometric system modelling with rework strategy. International Journal of Mathematics, Engineering and Management Sciences, 1(1), 1-17.

Saggou, H., Sadeg, I., Ourbih-Tari, M., & Bourennane, E. B. (2017). The analysis of unreliable M^([X])/G/1 queuing system with loss, vacation and two delays of verification. Communications in Statistics-Simulation and Computation, DOI: 10.1080/03610918.2017.1414245.

Sahoo, L. (2017). Genetic algorithm based approach for reliability redundancy allocation problems in fuzzy environment, International Journal of Mathematical, Engineering and Management Sciences, 2(4), 259-272.

Tamura, Y., & Yamada, S. (2017). Dependability analysis tool based on multi-dimensional stochastic noisy model for cloud computing with big data. International Journal of Mathematical, Engineering and Management Sciences, 2(4), 273-287.

Terfas, I., Saggou, H., & Ourbih-Tari, M. (2018). Transient study of a queueing system with one unreliable server, batch arrivals, two types of verification, loss and vacation. Communications in Statistics-Theory and Methods, 1-26. DOI: 10.1080/03610926.2018.1472780.

Wu, J., & Lian, Z. (2013). A single-server retrial G-queue with priority and unreliable server under Bernoulli vacation schedule. Computers & Industrial Engineering, 64(1), 84-93.

Zhang, M., & Liu, Q. (2015). An M/G/1 G-queue with server breakdown, working vacations and vacation interruption. Opsearch, 52(2), 256-270.

Zirem, D., Boualem, M., Adel-Aissanou, K., & Aissani, D. (2018). Analysis of a single server batch arrival unreliable queue with balking and general retrial time. Quality Technology & Quantitative Management, doi.org/10.1080/16843703.2018.1510359.