Abstract
This paper presents the transient analysis of an \( M/M/1 \) queueing system subject to Bernoulli vacation and vacation interruption. The arrivals are allowed to join the queue according to a Poisson distribution and the service takes place according to an exponential distribution. Whenever the system is empty, the server can take either a working vacation or an ordinary vacation with certain probabilities. During working vacation, the arrivals are allowed to join the queue and the service takes place according to an exponential distribution, but with a slower rate. The vacation times are also assumed to be exponentially distributed. Further, during working vacation, the server has the option to either continue the vacation or interrupt and transit to the regular busy period. During ordinary vacation, the arrivals are allowed to join the queue but no service takes place. The sever returns to the system on completion of the vacation duration. Upon returning from vacation, the server continues to provide service to the waiting customers (if any) or takes another vacation (working or ordinary vacation) if the system is empty. Explicit expressions are obtained for the time dependent system size probabilities in terms of modified Bessel function of the first kind using generating function and Laplace transform techniques. Numerical illustrations are added to support the theoretical results.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Bouchentouf, A.A., Yahiaoui, L.: On feedback queueing system with reneging and retention of reneged customers, multiple working vacations and Bernoulli schedule vacation interruption. Arab. J. Math. 6, 1–11 (2017)
Doshi, B.T.: Queueing systems with vacations - a survey. Queueing Syst. Theory Appl. 1, 29–66 (1986)
Choudhury, G., Deka, M.: A single server queueing system with two phases of service subject to server breakdown and Bernoulli vacation. Appl. Math. Model. 36, 6050–6060 (2012)
Keilson, J., Servi, L.D.: Oscillating random walk models for G1/G/1 vacation systems with Bernoulli schedules. J. Appl. Prob. 23, 790–802 (1986)
Ramaswami, R., Servi, L.D.: The busy period of the M/G/1 vacation model with a Bernoulli schedule. Commun. Stat. Stochast. Models 4, 507–521 (1988)
Takagi, H.: Queueing Analysis: A Foundation of Performance Analysis, Volume 1: Vacation and Priority Systems, Part 1. Elsevier, Amsterdam (1991)
Levy, Y., Yechiali, U.: Utilization of idle time in an M/G/1 queueing system. Manage. Sci. 22, 202–211 (1975)
Li, J., Tian, N.: The M/M/1 queue with working vacations and vacation interruption. J. Syst. Sci. Syst. Eng. 16, 121–127 (2007)
Servi, L.D., Finn, S.G.: M/M/1 queues with working vacations (M/M/1/WV). Perform. Eval. 50, 41–52 (2002)
Tadj, L., Choudhury, G., Tadj, C.: A quorum system with a random setup under N-policy and with Bernoulli vacation schedule. Qual. Technol. Quant. Manage. 3, 145–160 (2006)
Tian, N., Zhang, G.: Vacation Queueing Models: Theory and Applications. Springer, New York (2006)
Goswami, V.: Analysis of impatient customers in queues with Bernoulli schedule working vacations and vacation interruption. J. Stochast. 2014, 1–10 (2014)
Zhang, H., Shi, D.: The M/M/1 queue with Bernoulli-schedule-controlled-vacation and vacation interruption. Int. J. Inf. Manage. Sci. 20, 579–587 (2009)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer Nature Singapore Pte Ltd.
About this paper
Cite this paper
Vijayashree, K.V., Janani, B. (2018). Transient Analysis of an M/M/1 Queue with Bernoulli Vacation and Vacation Interruption. In: Ganapathi, G., Subramaniam, A., Graña, M., Balusamy, S., Natarajan, R., Ramanathan, P. (eds) Computational Intelligence, Cyber Security and Computational Models. Models and Techniques for Intelligent Systems and Automation. ICC3 2017. Communications in Computer and Information Science, vol 844. Springer, Singapore. https://doi.org/10.1007/978-981-13-0716-4_18
Download citation
DOI: https://doi.org/10.1007/978-981-13-0716-4_18
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-13-0715-7
Online ISBN: 978-981-13-0716-4
eBook Packages: Computer ScienceComputer Science (R0)