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Control of G/MX/1 Queueing System with N-Policy and Customer Impatience

  • Lim, Si-Yeong (Department of Industrial and Management Engineering, Hanyang University) ;
  • Hur, Sun (Department of Industrial and Management Engineering, Hanyang University)
  • Received : 2015.11.09
  • Accepted : 2016.04.24
  • Published : 2016.06.30

Abstract

We introduce a queueing system with general arrival stream and exponential service time under the N-policy, where customers may renege during idle period and arrival rates may vary according to the server's status. Probability distributions of the lengths of idle period and busy period are derived using absorbing Markov chain approach and a method to obtain the optimal control policy that minimizes long-run expected operating cost per unit time is provided. Numerical analysis is done to illustrate and characterize the method.

Keywords

Impatience;Control Policy;N-Policy;General Arrival

References

  1. Akcan, S. (2013), A New Approximation for Inventory Control System with Decision Variable Lead-Time and Stochastic Demand, International Journal of Industrial Engineering: Theory, Applications and Practice, 20(3/4).
  2. Altman, E. and Yechiali, U. (2006), Analysis of Customers' Impatience in Queues with Server Vacations, Queueing Systems, 52, 261-279. https://doi.org/10.1007/s11134-006-6134-x
  3. Bae, J. and Kim, S. (2010), The Stationary Workload of the G/M/1 Queue with Impatient Customers, Queueing Systems, 64, 253-265. https://doi.org/10.1007/s11134-009-9159-0
  4. Benjaafar, S., Gayon, J., and Tepe, S. (2010), Optimal Control of a Production-Inventory System with Customer Impatience, Operations Research Letters, 38, 267-272. https://doi.org/10.1016/j.orl.2010.03.008
  5. Blackburn, J. D. (1972), Optimal Control of a Single Server Queue with Balking and Reneging, Management Science, 19, 297-313. https://doi.org/10.1287/mnsc.19.3.297
  6. Chae, K. C. and Kim, S. J. (2007), Busy Period Analysis for the GI/M/1 Queue with Exponential Vacations, Operations Research Letters, 35(1), 114-118. https://doi.org/10.1016/j.orl.2006.01.003
  7. Chae, K. C. and Lee, S. M. (2005), An Absorbing Markov Chain Approach to GI/M/1 Queues with Generalized Vacations, Asia Pacific Management Review, 10, 163-167.
  8. Chae, K. C. and Lim, D. E. (2008), Busy period analysis for the n-policy GI/M/c queue, Journal of the Korean Statistical Society, 37(3), 285-290. https://doi.org/10.1016/j.jkss.2008.02.002
  9. Choi, B. D., Kim, B., and Zhu, D. (2004), MAP/M/c Queue with Constant Impatience Time, Mathematics of Operations Research, 29, 309-325. https://doi.org/10.1287/moor.1030.0081
  10. Kao, P. C. (1997), An Introduction to Stochastic Processes, Duxbury Press, Belmont, California.
  11. Ke, J.-C. (2003), The Analysis of a General Input Queue with N Policy and Exponential Vacations, Queueing Systems, 45, 135-160. https://doi.org/10.1023/A:1026045706255
  12. Kim, K. and Yang, W. S. (2011), Busy Period Analysis for the GI/M/1 Queue with Phase-Type Vacations, Journal of the Korean Statistical Society, 40(1), 55-62. https://doi.org/10.1016/j.jkss.2010.04.006
  13. Lee, H. W. and Ahn, B. Y. (2002), Operational Behavior of the MAP/G/1 Queue under N-Policy with Single Vacation and Set-Up, Journal of Applied Mathematics and Stochastic Analysis, 15, 167-196.
  14. Lee, H. W. and Park, N. I. (2004), Using Factorization for Waiting Times in BMAP/G/1 Queues with N-Policy and Vacations, Stochastic Analysis and Applications, 22, 755-773. https://doi.org/10.1081/SAP-120030455
  15. Mandelbaum, A. and Momcilovic, P. (2012), Queues with Many Servers and Impatient Customers, Mathematics of Operations Research, 37, 41-65. https://doi.org/10.1287/moor.1110.0530
  16. Swensen, A. R. (1986), On a GI/M/c Queue with Bounded Waiting Times, Operations Research, 34, 895-908. https://doi.org/10.1287/opre.34.6.895
  17. Tadj, L. and Choudhury, G. (2005), Optimal Design and Control of Queues, Sociedad de Estadistica e Investigacion Operativa, Top, 13, 359-412.
  18. Takacs, L. (1962), Theory of queues, Oxford: Oxford University Press, reprinted in 1982 by Greenwood Press, Westport, CT.
  19. Takagi H. (1991), Queueing Analysis: A Foundation of Performance Evaluation, North-Holland, 1.
  20. Yadin, M. and Naor, P. (1963), Queueing Systems with a Removable Service Station, Operational Research Quarterly, 14, 393-405. https://doi.org/10.1057/jors.1963.63
  21. Yue, D., Yue, W., and Li, X. (2011), Analysis of a Two-Phase Queueing System with Impatient Customers and Multiple Vacations, The Tenth International Symposium on Operations Research and Its Applications (ISORA 2011), Dunhuang, China, 292-298.
  22. Zhe George Zhang, Z. G. and Tian, N. (2004), The N threshold policy for the GI/M/1 queue, Operations Research Letters, 32, 77-84. https://doi.org/10.1016/S0167-6377(03)00067-1