Article
Full entry |
PDF
(0.7 MB)
Feedback
PDF
(0.7 MB)
Feedback
Keywords:
departure process; finite-buffer queue; $N$-policy; power saving; transient state; wireless sensor network (WSN)
departure process; finite-buffer queue; $N$-policy; power saving; transient state; wireless sensor network (WSN)
Summary:
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.
References:
[1] Abate, J., L., G., Choudhury, Whitt, W.: 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
[2] Arumuganathan, R., Jeyakumar, S.: 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 | MR 2038103
[3] Choudhury, G., Baruah, H. K.: Analysis of a Poisson queue with a threshold policy and a grand vacation process. Sankhya Ser. B 62 (2000), 303-316. DOI | MR 1802636
[4] Choudhury, G., Borthakur, A.: Stochastic decomposition results of batch arrival Poisson queue with a grand vacation process. Sankhya Ser. B 62 (2000), 448-462. MR 1834167
[5] Choudhury, G., Paul, M.: A batch arrival queue with an additional service channel under $N$-policy. Appl. Math. Comput. 156 (2004), 115-130. DOI | MR 2087256
[6] Cohen, J. W.: The Single Server Queue. North-Holland, Amsterdam 1982. MR 0668697
[7] Doshi, B. T.: Queueing systems with vacations-a survey. Queueing Syst. 1 (1986), 29-66. DOI | MR 0896237
[8] García, Y. H., Diaz-Infante, S., Minjárez-Sosa, J. A.: Partially observable queueing systems with controlled service rates under a discounted optimality criterion. Kybernetika 57 (2021), 493-512. DOI | MR 4299460
[9] Gerhardt, I., Nelson, B. L.: Transforming renewal processes for simulation of nonstationary arrival processes. Informs J. Comput. 21 (2009), 630-640. DOI | MR 2588345
[10] Jiang, F. C., Huang, D. C., Yang, C. T., Leu, F. Y.: Lifetime elongation for wireless sensor network using queue-based approaches. J. Supercomput. 59 (2012), 1312-1335. DOI
[11] Ke, J.-C.: 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 | MR 1890915
[12] Ke, J.-C., Wang, K.-H.: A recursive method for the $N$ policy $G/M/1$ queueing system with finite capacity. Eur. J. Oper. Res. 142 (2002), 577-594. DOI | MR 1922375
[13] Kempa, W. M.: The virtual waiting time for the batch arrival queueing systems. Stoch. Anal. Appl. 22 (2004), 1235-1255. DOI | MR 2089066
[14] Kempa, W. M.: $GI/G/1/\infty$ batch arrival queueing system with a single exponential vacation. Math. Meth. Oper. Res. 69 (2009), 81-97. DOI | MR 2476049 | Zbl 1170.60032
[15] Kempa, W. M.: Analysis of departure process in batch arrival queue with multiple vacations and exhaustive service. Commun. Stat. Theory 40 (2011), 2856-2865. DOI | MR 2860790
[16] Kempa, W. M.: On transient queue-size distribution in the batch arrival system with the $N$-policy and setup times. Math. Commun. 17 (2012), 285-302. DOI | MR 2946149
[17] Kempa, W. M.: On transient queue-size distribution in the batch-arrivals system with a single vacation policy. Kybernetika 50 (2014), 126-141. DOI | MR 3195008
[18] Kempa, W. M.: 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
[19] Kempa, W. M., Kurzyk, D.: 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
[20] Korolyuk, V. S.: Boundary-value problems for compound Poisson processes. Theor. Probab. Appl. 19 (1974), 1-13. DOI 10.1137/1119001 | MR 0402939
[21] Reddy, G. V. Krishna, Nadarajan, R., Arumuganathan, R.: Analysis of a bulk queue with $N$-policy multiple vacations and setup times. Comput. Oper. Res. 25 (1998), 957-967. DOI | MR 1638645
[22] Lee, H. W., Lee, S. S., Chae, K. C.: Operating characteristics of $M^{X}/G/1$ queue with $N$-policy. Queueing Syst. 15 (1994), 387-399. DOI | MR 1266802
[23] Lee, H. W., Lee, S. S., Park, J. O., Chae, K. C.: Analysis of $M^{[x]}/G/1$ queue with $N$-policy and multiple vacations. J. Appl. Prob. 31 (1994), 467-496. DOI | MR 1274803
[24] Lee, S. S., Lee, H. W., Chae, K. C.: Batch arrival queue with $N$-policy and single vacation. Comput. Oper. Res. 22 (1995), 173-189. DOI
[25] Lee, H. S., Srinivasan, M. M.: Control policies for the $M/G/1$ queueing system. Manag. Sci. 35 (1989), 708-721. DOI | MR 1001484
[26] Levy, Y., Yechiali, U.: Utilization of idle time in an $M/G/1$ queueing system. Manag. Sci. 22 (1975), 202-211. DOI
[27] Maheswar, R., Jayaparvathy, R.: Power control algorithm for wireless sensor networks using $N$-policy $M/M/1$ queueing model. Power 2 (2010), 2378-2382.
[28] Nasr, W. W., Taaffe, M. R.: 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 | MR 3120933
[29] Takagi, H.: Queueing Analysis: A Foundation of Performance Evaluation, Vacation and Priority Systems, Part I, vol. I. North-Holland, Amsterdam 1991. MR 1149382
[30] Takagi, H.: $M/G/1/K$ queues with $N$-policy and setup times. Queueing Syst. 14 (1993), 79-98. DOI | MR 1238663
[31] Tian, N., Zhang, Z. G.: Vacation Queueing Models: Theory and Applications. Springer, 2006. MR 2248264
[32] Yadin, M., Naor, P.: Queueing systems with a removable service station. J. Oper. Res. Soc. 14 (1963), 393-405. DOI
[33] Yang, D.-Y., Cho, Y.-Ch.: Analysis of the $N$-policy $GI/M/1/K$ queueing systems with working breakdowns and repairs. Comput. J. 62 (2019), 130-143. DOI | MR 3897421
Search
Partner of
