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Transient analysis of an \(M/M/1\) queue with impatient behavior and multiple vacations. (English) Zbl 1410.90045
Summary: In this paper, we carry out an analysis for a single server queue with impatient customers and multiple vacations where customers impatience is due to an absentee of servers upon arrival. Customers arrive at the system according to a Poisson process and exponential service times. Explicit expressions are obtained for the time dependent probabilities, mean and variance of the system size in terms of the modified Bessel functions, by employing the generating functions along with continued fractions and the properties of the confluent hypergeometric function. Finally, some numerical illustrations are provided.

90B22 Queues and service in operations research
60K25 Queueing theory (aspects of probability theory)
Full Text: DOI
[1] Doshi, B., Queueing systems with vacations—A survey, Queue. Syst., 1, 29-66, (1986) · Zbl 0655.60089
[2] Takagi, H., Queueing Analysis: A Foundation of Performance Evaluation, vol. 1, (1991), North-Holland Amsterdam
[3] Tian, N.; Zhang, Z., Vaction Queueing Models—Theory and Applications, (2006), Springer-Verlag New York
[4] Al-Seedy, R. O.; El-Sherbiny, A. A.; El-Shehawy, S. A.; Ammar, S. I., Transient solution of the M/M/c queue with balking and reneging, Comput. Math. Appl., 57, 1280-1285, (2009) · Zbl 1186.90033
[5] Altman, E.; Yechiali, U., Analysis of customers’ impatience in queues with server vacation, Queue. Syst., 52, 261-279, (2006) · Zbl 1114.90015
[6] Altman, E.; Yechiali, U., Infinite server queues with systems’ additional task and impatient customers, Probab. Eng. Inform. Sci., 22, 477-493, (2008) · Zbl 1228.60096
[7] Kalidass, K.; Gnanaraj, J.; Gopinath, S.; Ramanath, K., Transient analysis of an M/M/1 queue with a repairable server and multiple vacations, Int. J. Math. Oper. Res., 6, 2, 193-216, (2014) · Zbl 1390.90225
[8] Kalidass, K.; Ramanath, K., Time dependent analysis of M/M/1 queue with server vacations and a waiting server, The 6th International Conference on Queueing Theory and Network Applications (QTNA’11), (2011), Seoul, Korea
[9] Indra; Renu, Transient analysis of Markovian queueing model with Bernoulli schedule and multiple working vacations, Int. J. Comput. Appl., 20, 43-48, (2011)
[10] Sudhesh, R.; Raj, L. F., Computational analysis of stationary and transient distribution of single server queue with working vacation, Global Trend. Comput. Commun. Syst. Commun. Comput. Inform. Sci., 269, 480-489, (2012)
[11] Yang, D. Y.; Wu, Y. Y., Transient behavior analysis of a finite capacity queue with working breakdowns and server vacations, Proceedings of The International MultiConference of Engineers and Computer Scientists, 1151-1156, (2014)
[12] Kalidass, K.; Ramanath, K., Transient analysis of an M/M/1 queue with multiple vacations, Pakistan J. Stat. Oper. Res., 10, 121-130, (2014) · Zbl 1390.90225
[13] Abramowitz, M.; Stegun, I., Handbook of Mathematical Functions, (1965), Dover New York
[14] Lorentzen, L.; Waadeland, H., Continued fractions with applications, Studies in Computational Mathematics, vol. 3, (1992), Elsevier Amsterdam, http://www.amazon.com/Continued-Fractions-Applications-Computational-Mathematics/dp/0444892656 · Zbl 0782.40001
[15] Gradshteyn, I.; Ryzhik, I.; Jeffrey, A.; Zwillinger, D., Table of Integrals, Series, and Products, (2007), Academic Press, Elsevier
[16] Ammar, S. I., Transient behavior of a two-processor heterogeneous system with catastrophes, server failures and repairs, Appl. Math. Model., 38, 2224-2234, (2014) · Zbl 1427.60185
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