Journal of Korean Institute of Industrial Engineers (대한산업공학회지)

Korean Institute of Industrial Engineers (대한산업공학회)
권호 표지
DOI QR코드

DOI QR Code

On the Modified Supplementary Variable Technique for a Discrete-Time GI/G/1 Queue with Multiple Vacations

복수휴가형 이산시간 GI/G/1 대기체계에 대한 수정부가변수법

  • Lee, Doo Ho (Department of Industrial and Management Engineering, Kangwon National University)
  • 이두호 (강원대학교 산업경영공학과)
  • Received : 2016.03.10
  • Accepted : 2016.07.17
  • Published : 2016.10.15
Download PDF
⟨ Previous Next ⟩

Abstract

This work suggests a new analysis approach for a discrete-time GI/G/1 queue with multiple vacations. The method used is called a modified supplementary variable technique and our result is an exact transform-free expression for the steady state queue length distribution. Utilizing this result, we propose a simple two-moment approximation for the queue length distribution. From this, approximations for the mean queue length and the probabilities of the number of customers in the system are also obtained. To evaluate the approximations, we conduct numerical experiments which show that our approximations are remarkably simple yet provide fairly good performance, especially for a Bernoulli arrival process.

Keywords

References

  1. Bruneel, H. and Kim, B. G. (1993), Discrete-time models for communication systems including ATM, Kluwer Academic Publishers, New York.
  2. Cox, D. R. (1955), The analysis of non-Markovian stochastic processes by the inclusion of supplementary variables, Mathematical Proceedings of the Cambridge Philosophical Society, 51(3), 433-441. https://doi.org/10.1017/S0305004100030437
  3. Chae, K. C., Kim, N. K., and Choi, D. W. (2002), An interpretation of the equations for the GI/GI/c/K queue length distribution, Journal of the Korean Institute of Industrial Engineers, 28(4), 390-396.
  4. Chae, K. C., Kim, N. K., and Yoon, B. K. (2004), On the queue length distribution for the $GI/G/1/K/V_M$ queue, Stochastic Analysis and Applications, 22(3), 647-656. https://doi.org/10.1081/SAP-120030449
  5. Chae, K. C., Lee, D. H., and Kim, N. K. (2008), On the modified supplementary variable technique for the discrete-time GI/G/1/K queue, Journal of the Korean Operations Research and Management Science Society, 33(1), 107-115.
  6. Chaudhry, M .L. (1993), Alternative numerical solutions of stationary queueing-time distribution in discrete-time queues : GI/G/1, Journal of the Operational Research Society, 44(1), 1035-1051. https://doi.org/10.1057/jors.1993.172
  7. Chaudhry, M. L. and Gupta, U. C. (2001), Computing waiting-time probabilities in the discrete-time queue : $GI^X/G/1$, Performance Evaluation, 43(2/3), 123-131. https://doi.org/10.1016/S0166-5316(00)00038-9
  8. Choi, D. W., Kim, N. K., and Chae, K. C. (2005), A two-moment approximation for the GI/G/c queue with finite capacity, Informs Journal on Computing, 17(1), 75-81. https://doi.org/10.1287/ijoc.1030.0058
  9. Eliazar, I. (2008), On the discrete-time $G/GI/{\infty}$ queue, Probability in the Engineering and Informational Sciences, 22(4), 557-585. https://doi.org/10.1017/S0269964808000338
  10. Hunter, J. J. (1983), Mathematical techniques of applied probability, discrete time models : techniques and applications, Academic Press, New York, 2.
  11. Hasslinger, G. (1995), A polynomial factorization approach to the discrete time GI/G/1/(N) queue size distribution, Performance Evaluation, 23(3), 217-240. https://doi.org/10.1016/0166-5316(94)00024-E
  12. Kim, N. K. and Chae, K. C. (2003), Transform-free analysis of the GI/G/1/K queue through the decomposed Little's formula, Computers and Operations Research, 30(3), 353-365. https://doi.org/10.1016/S0305-0548(01)00101-0
  13. Linwong, P., Kato, N., and Nemoto, Y. (2004), A polynomial factorization approach for the discrete time $GI^X/G/1/K$ queue, Methodology And Computing In Applied Probability, 6(3), 277-291. https://doi.org/10.1023/B:MCAP.0000026560.42106.7a
  14. Murata, M. and Miyahara, H. (1991), An analytic solution of the waiting time distribution for the discrete-time GI/G/1 queue, Performance Evaluation, 13(2), 87-95. https://doi.org/10.1016/0166-5316(91)90042-2
  15. Takagi, H. (1993), Queueing analysis, North-Holland, Amsterdam, 2.
  16. Wolff, R. W. (1989), Stochastic modeling and the theory of queues, Prentice-Hall, New Jersey.