Journal of Applied Mathematics

On the Discrete-Time GeoX/G/1 Queues under N-Policy with Single and Multiple Vacations

Sung J. Kim, Nam K. Kim, Hyun-Min Park, Kyung Chul Chae, and Dae-Eun Lim

Full-text: Open access

Abstract

We consider the discrete-time GeoX/G/1 queue under N-policy with single and multiple vacations. In this queueing system, the server takes multiple vacations and a single vacation whenever the system becomes empty and begins to serve customers only if the queue length is at least a predetermined threshold value N. Using the well-known property of stochastic decomposition, we derive the stationary queue-length distributions for both vacation models in a simple and unified manner. In addition, we derive their busy as well as idle-period distributions. Some classical vacation models are considered as special cases.

Article information

Source
J. Appl. Math., Volume 2013 (2013), Article ID 587163, 6 pages.

Dates
First available in Project Euclid: 14 March 2014

Permanent link to this document
https://projecteuclid.org/euclid.jam/1394808327

Digital Object Identifier
doi:10.1155/2013/587163

Mathematical Reviews number (MathSciNet)
MR3147889

Zentralblatt MATH identifier
06950763

Citation

Kim, Sung J.; Kim, Nam K.; Park, Hyun-Min; Chae, Kyung Chul; Lim, Dae-Eun. On the Discrete-Time ${\text{Geo}}^{X}/G/1$ Queues under $N$ -Policy with Single and Multiple Vacations. J. Appl. Math. 2013 (2013), Article ID 587163, 6 pages. doi:10.1155/2013/587163. https://projecteuclid.org/euclid.jam/1394808327


Export citation

References

  • B. T. Doshi, “Queueing systems with vacations-a survey,” Queueing Systems, vol. 1, no. 1, pp. 29–66, 1986.
  • H. Takagi, Queueing Analysis, Vol. 1, North-Holland, Amsterdam, The Netherlands, 1991.
  • H. Takagi, Queueing Analysis, Vol. 3, North-Holland, 1993.
  • Z. G. Zhang and N. Tian, “Discrete time $\text{G}\text{e}\text{o}/\text{G}/1$ queue with multiple adaptive vacations,” Queueing Systems, vol. 38, no. 4, pp. 419–429, 2001.
  • D. Fiems and H. Bruneel, “Analysis of a discrete-time queueing system with timed vacations,” Queueing Systems, vol. 42, no. 3, pp. 243–254, 2002.
  • A. S. Alfa, “Vacation models in discrete time,” Queueing Systems, vol. 44, no. 1, pp. 5–30, 2003.
  • S. S. Lee, H. W. Lee, S. H. Yoon, and K. C. Chae, “Batch arrival queue with \emphN-policy and single vacation,” Computers & Operations Research, vol. 22, pp. 173–189, 1995.
  • H. W. Lee, S. S. Lee, J. O. Park, and K. C. Chae, “Analysis of the ${\text{M}}^{\text{X}}/\text{G}/1$ queue with $N$-policy and multiple vacations,” Journal of Applied Probability, vol. 31, no. 2, pp. 476–496, 1994.
  • G. Choudhury, “An ${\text{M}}^{\text{X}}/\text{G}/1$ queueing system with a setup period and a vacation period,” Queueing Systems, vol. 36, no. 1–3, pp. 23–38, 2000.
  • J.-C. Ke, K.-B. Huang, and W. L. Pearn, “The randomized vacation policy for a batch arrival queue,” Applied Mathematical Modelling, vol. 34, no. 6, pp. 1524–1538, 2010.
  • T.-Y. Wang, “Random $N$-policy Geo/G/1 queue with startup and closedown times,” Journal of Applied Mathematics, vol. 2012, Article ID 793801, 19 pages, 2012.
  • D.-E. Lim, D. H. Lee, W. S. Yang, and K.-C. Chae, “Analysis of the GI/Geo/1 queue with $N$-policy,” Applied Mathematical Modelling, vol. 37, no. 7, pp. 4643–4652, 2013.
  • J.-C. Ke, H.-I. Huang, and Y.-K. Chu, “Batch arrival queue with $N$-policy and at most $J$ vacations,” Applied Mathematical Modelling, vol. 34, no. 2, pp. 451–466, 2010.
  • B. Feyaerts, S. De Vuyst, H. Bruneel, and S. Wittevrongel, “The impact of the NT-policy on the behaviour of a discrete-time queue with general service times,” Journal of Industrial and Management Optimization, vol. 10, pp. 131–149, 2014.
  • N. Tian and Z. G. Zhang, Vacation Queueing Models, Springer, New York, NY, USA, 2006.
  • D. E. Lim and T. S. Kim, “Modeling discovery andčommentComment on ref. [16?]: Please update the information of this reference, if possible. removal of security vulnerabilities in software system using priority queueing models,” submitted to Journal of Computer Virology and Hacking Techniques.
  • J. J. Hunter, Mathematical Techniques of Applied Probability, Vol. 2, Operations Research and Industrial Engineering, Academic Press, New York, NY, USA, 1983.
  • H. Bruneel and B. G. Kim, Discrete-Time Models for Communication System Including ATM, Kluwer Academic Publishers, 1993.
  • R. W. Wolff, Stochastic Modeling and the Theory of Queues, Prentice Hall, Englewood Cliffs, NJ, USA, 1989.
  • N. K. Kim, K. C. Chae, and M. L. Chaudhry, “An invariance relation and a unified method to derive stationary queue-length distributions,” Operations Research, vol. 52, no. 5, pp. 756–764, 2004. \endinput