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M/M/1 Differentiated Multiple Vacation Queueing Systems with Vacation-Dependent Service Rates

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We consider a multiple vacation queueing system in which a vacation following a busy period has a different distribution from a vacation that is taken without serving at least one customer. In addition, the service rate after a vacation taken after a busy period of non-zero duration is different from that of a vacation taken after a busy period of zero duration. This is in keeping with real life experience where the rate at which someone is serving might depend on how much rest he/she had before working. For ease of analysis it is assumed that the service times are exponentially distributed with different means, and the two vacation types are also exponentially distributed but with different means. The steady-state solution of the model is obtained.
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Vacation Queueing System; Differentiated Vacations; Differentiated Service Rates; Performance Analysis

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Cooper, R.B., “Queues Served in Cyclic Order: Waiting Times,” The Bell System Technical Journal, vol. 49, 1970, pp. 399-413.

Levy, Y. and U. Yechiali, “Utilization of idle time in an M/G/1 queueing system,” Management Science vol. 22, 1975, pp. 202-211.

Doshi, B.T., “Queueing systems with vacations, a survey,” Queueing Systems, vol. 1, 1986, pp. 29-66.

Doshi, B.T., “Single-server queues with vacations,” in Stochastic Analysis of Computer and Communications Systems, H. Takagi (Ed.), Elsevier, 1990.

Takagi, H., Queueing Analysis: A Foundation of Performance Analysis, Volume 1: Vacation and Priority Systems, Part 1, Elsevier Science Publishers B.V., Amsterdam, 1991.

Tian, N. and G. Zhang, Vacation Queueing Models: Theory and Applications, Springer-Verlag, New York, 2006.

Servi, L.D. and S.G. Finn, “M/M/1 queue with working vacations (M/M/1/WV),” Performance Evaluation, vol. 50, 2002, pp. 41-52.

Wu, D. and H. Takagi, “M/G/1 queue with multiple working vacation,” Performance Evaluation, vol. 63, 2006, pp. 654-681.

Xu, X., Z. Zhang and N. Tian, “Analysis for the MX/M/1 working vacation queue,” International Journal of the Information and Management Sciences, vol. 20, 2009, pp. 379-394.

Baba, Y., “The MX/M/1 queue with multiple working vacation,” American Journal of Operations Research, vol. 2, 2012, pp. 217-224.

Li, J. and N. Tian, “Analysis of the Discrete Time Geo/Geo/1 Queue with Single Working Vacation,” Quality Technology & Quantitative Management, vol. 5, 2008, pp. 77-89.

Gao, S. and Z. Liu, “Performance analysis of a discrete-time GeoX/G/1 queue with single working vacation, World Academy of Science, Engineering and Technology, vol. 56, 2011, pp. 1162-1170.

Ibe, O.C. and O.A. Isijola, “M/M/1 Multiple Vacation Queueing Systems with Differentiated Vacations,” Modeling and Simulation in Engineering, volume 2014, Article 158247, 2014.

Isijola-Adakeja, O.A. and O.C. Ibe, “M/M/1 Multiple Vacation Queueing Systems with Differentiated Vacations and Vacation Interruptions,” IEEE Access, vol. 2, 2014, pp. 1384-1395.

Little, J.D.C., “A proof of the formula: L=λW”, Operations Research, vol. 9, 1961, pp. 383-387.


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