Kybernetika 58 no. 1, 82-100, 2022

Non-stationary departure process in a batch-arrival queue with finite buffer capacity and threshold-type control mechanism

Wojciech M. Kempa and Dariusz KurzykDOI: 10.14736/kyb-2022-1-0082

Abstract:

Non-stationary behavior of departure process in a finite-buffer $M^{X}/G/1/K$-type queueing model with batch arrivals, in which a threshold-type waking up $N$-policy is implemented, is studied. According to this policy, after each idle time a new busy period is being started with the $N$th message occurrence, where the threshold value $N$ is fixed. Using the analytical approach based on the idea of an embedded Markov chain, integral equations, continuous total probability law, renewal theory and linear algebra, a compact-form representation for the mixed double transform (probability generating function of the Laplace transform) of the probability distribution of the number of messages completely served up to fixed time $t$ is obtained. The considered queueing system has potential applications in modeling nodes of wireless sensor networks (WSNs) with battery saving mechanism based on threshold-type waking up of the radio. An illustrating simulational and numerical study is attached.

Keywords:

transient state, departure process, finite-buffer queue, $N$-policy, power saving, wireless sensor network (WSN)

Classification:

60K25, 90B22

paper.pdf

References:

  1. J. Abate, G. L., Choudhury and W. Whitt: An introduction to numerical transform inversion and its application to probability models. In: Computational Probability (W. Grassmann, ed.), Kluwer, Boston 2000, pp. 257-323.   DOI:10.1007/978-1-4757-4828-4\_8
  2. R. Arumuganathan and S. Jeyakumar: Steady state analysis of a bulk queue with multiple vacations, setup times with $N$-policy and closedown times. Appl. Math. Model. 29 (2005), 972-986.   DOI:10.1016/j.apm.2005.02.013
  3. G. Choudhury and H. K. Baruah: Analysis of a Poisson queue with a threshold policy and a grand vacation process. Sankhya Ser. B 62 (2000), 303-316.   DOI:10.1007/978-3-642-59691-9\_34
  4. G. Choudhury and A. Borthakur: Stochastic decomposition results of batch arrival Poisson queue with a grand vacation process. Sankhya Ser. B 62 (2000), 448-462.   CrossRef
  5. G. Choudhury and M. Paul: A batch arrival queue with an additional service channel under $N$-policy. Appl. Math. Comput. 156 (2004), 115-130.   DOI:10.1016/j.amc.2003.07.006
  6. J. W. Cohen: The Single Server Queue. North-Holland, Amsterdam 1982.   CrossRef
  7. B. T. Doshi: Queueing systems with vacations-a survey. Queueing Syst. 1 (1986), 29-66.   DOI:10.1007/BF01149327
  8. Y. H. García, S. Diaz-Infante and J. A. Minjárez-Sosa: Partially observable queueing systems with controlled service rates under a discounted optimality criterion. Kybernetika 57 (2021), 493-512.   DOI:10.14736/kyb-2021-3-0493
  9. I. Gerhardt and B. L. Nelson: Transforming renewal processes for simulation of nonstationary arrival processes. Informs J. Comput. 21 (2009), 630-640.   DOI:10.1287/ijoc.1080.0316
  10. F. C. Jiang, D. C. Huang, C. T. Yang and F. Y. Leu: Lifetime elongation for wireless sensor network using queue-based approaches. J. Supercomput. 59 (2012), 1312-1335.   DOI:10.1007/s11227-010-0537-5
  11. J.-C. Ke: The control policy of an $M^{[x]}/G/1$ queueing system with server startup and two vacation types. Math. Method. Oper. Res. 54 (2001), 471-490.   DOI:10.1007/s001860100167
  12. J.-C. Ke and K.-H. Wang: A recursive method for the $N$ policy $G/M/1$ queueing system with finite capacity. Eur. J. Oper. Res. 142 (2002), 577-594.   DOI:10.1016/S0377-2217(01)00317-4
  13. W. M. Kempa: The virtual waiting time for the batch arrival queueing systems. Stoch. Anal. Appl. 22 (2004), 1235-1255.   DOI:10.1081/SAP-200026456
  14. W. M. Kempa: $GI/G/1/\infty$ batch arrival queueing system with a single exponential vacation. Math. Meth. Oper. Res. 69 (2009), 81-97.   DOI:10.1007/s00186-008-0212-2
  15. W. M. Kempa: Analysis of departure process in batch arrival queue with multiple vacations and exhaustive service. Commun. Stat. Theory 40 (2011), 2856-2865.   DOI:10.1080/03610926.2011.562767
  16. W. M. Kempa: On transient queue-size distribution in the batch arrival system with the $N$-policy and setup times. Math. Commun. 17 (2012), 285-302.   DOI:10.1080/13629395.2012.725298
  17. W. M. Kempa: On transient queue-size distribution in the batch-arrivals system with a single vacation policy. Kybernetika 50 (2014), 126-141.   DOI:10.14736/kyb-2014-1-0126
  18. W. M. Kempa: A comprehensive study on the queue-size distribution in a finite-buffer system with a general independent input flow. Perform. Eval. 108 (2017), 1-15.   DOI:10.1016/j.peva.2016.11.002
  19. W. M. Kempa and D. Kurzyk: Transient departure process in $M/G/1/K$-type queue with threshold servers waking up. In: Software, Telecommunications and Computer Networks (SoftCOM), 2015, 23rd International Conference on IEEE, pp. 32-36.   DOI:10.1109/softcom.2015.7314127
  20. V. S. Korolyuk: Boundary-value problems for compound Poisson processes. Theor. Probab. Appl. 19 (1974), 1-13.   CrossRef
  21. G. V. Krishna Reddy, R. Nadarajan and R. Arumuganathan: Analysis of a bulk queue with $N$-policy multiple vacations and setup times. Comput. Oper. Res. 25 (1998), 957-967.   DOI:10.1016/S0305-0548(97)00098-1
  22. H. W. Lee, S. S. Lee and K. C. Chae: Operating characteristics of $M^{X}/G/1$ queue with $N$-policy. Queueing Syst. 15 (1994), 387-399.   DOI:10.1007/BF01189247
  23. H. W. Lee, S. S. Lee, J. O. Park and K. C. Chae: Analysis of $M^{[x]}/G/1$ queue with $N$-policy and multiple vacations. J. Appl. Prob. 31 (1994), 467-496.   DOI:10.1177/0022343394031004024
  24. S. S. Lee, H. W. Lee and K. C. Chae: Batch arrival queue with $N$-policy and single vacation. Comput. Oper. Res. 22 (1995), 173-189.   DOI:10.1016/0305-0548(94)e0015-y
  25. H. S. Lee and M. M. Srinivasan: Control policies for the $M/G/1$ queueing system. Manag. Sci. 35 (1989), 708-721.   DOI:10.1287/mnsc.35.6.708
  26. Y. Levy and U. Yechiali: Utilization of idle time in an $M/G/1$ queueing system. Manag. Sci. 22 (1975), 202-211.   DOI:10.1112/S0025579300006082
  27. R. Maheswar and R. Jayaparvathy: Power control algorithm for wireless sensor networks using $N$-policy $M/M/1$ queueing model. Power 2 (2010), 2378-2382.   CrossRef
  28. W. W. Nasr and M. R. Taaffe: Fitting the $Ph-t/M-t/s/c$ time-dependent departure process for use in tandem queueing networks. Informs J. Comput. 25 (2013), 758-773.   DOI:10.1287/ijoc.1120.0538
  29. H. Takagi: Queueing Analysis: A Foundation of Performance Evaluation, Vacation and Priority Systems, Part I, vol. I. North-Holland, Amsterdam 1991.   CrossRef
  30. H. Takagi: $M/G/1/K$ queues with $N$-policy and setup times. Queueing Syst. 14 (1993), 79-98.   DOI:10.1007/BF01153527
  31. N. Tian and Z. G. Zhang: Vacation Queueing Models: Theory and Applications. Springer, 2006.   CrossRef
  32. M. Yadin and P. Naor: Queueing systems with a removable service station. J. Oper. Res. Soc. 14 (1963), 393-405.   DOI:10.1057/jors.1963.63
  33. D.-Y. Yang and Y.-Ch. Cho: Analysis of the $N$-policy $GI/M/1/K$ queueing systems with working breakdowns and repairs. Comput. J. 62 (2019), 130-143.   DOI:10.1093/comjnl/bxy051