Geodesic
Non-stationary departure process in a batch-arrival queue with finite buffer capacity and threshold-type control mechanism
Kybernetika, Tome 58 (2022) no. 1, pp. 82-100
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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.
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.
DOI : 10.14736/kyb-2022-1-0082
Classification : 60K25, 90B22
Keywords: departure process; finite-buffer queue; $N$-policy; power saving; transient state; wireless sensor network (WSN) 
@article{10_14736_kyb_2022_1_0082,
     author = {Kempa, Wojciech M. and Kurzyk, Dariusz},
     title = {Non-stationary departure process in a batch-arrival queue with finite buffer capacity and threshold-type control mechanism},
     journal = {Kybernetika},
     pages = {82--100},
     year = {2022},
     volume = {58},
     number = {1},
     doi = {10.14736/kyb-2022-1-0082},
     mrnumber = {4405948},
     zbl = {07511612},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.14736/kyb-2022-1-0082/}
}
TY  - JOUR
AU  - Kempa, Wojciech M.
AU  - Kurzyk, Dariusz
TI  - Non-stationary departure process in a batch-arrival queue with finite buffer capacity and threshold-type control mechanism
JO  - Kybernetika
PY  - 2022
SP  - 82
EP  - 100
VL  - 58
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.14736/kyb-2022-1-0082/
DO  - 10.14736/kyb-2022-1-0082
LA  - en
ID  - 10_14736_kyb_2022_1_0082
ER  - 
%0 Journal Article
%A Kempa, Wojciech M.
%A Kurzyk, Dariusz
%T Non-stationary departure process in a batch-arrival queue with finite buffer capacity and threshold-type control mechanism
%J Kybernetika
%D 2022
%P 82-100
%V 58
%N 1
%U http://geodesic.mathdoc.fr/articles/10.14736/kyb-2022-1-0082/
%R 10.14736/kyb-2022-1-0082
%G en
%F 10_14736_kyb_2022_1_0082
Kempa, Wojciech M.; Kurzyk, Dariusz. Non-stationary departure process in a batch-arrival queue with finite buffer capacity and threshold-type control mechanism. Kybernetika, Tome 58 (2022) no. 1, pp. 82-100. doi: 10.14736/kyb-2022-1-0082

[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

[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

[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

[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

[6] Cohen, J. W.: The Single Server Queue. North-Holland, Amsterdam 1982. | MR

[7] Doshi, B. T.: Queueing systems with vacations-a survey. Queueing Syst. 1 (1986), 29-66. | DOI | MR

[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

[9] Gerhardt, I., Nelson, B. L.: Transforming renewal processes for simulation of nonstationary arrival processes. Informs J. Comput. 21 (2009), 630-640. | DOI | MR

[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

[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

[13] Kempa, W. M.: The virtual waiting time for the batch arrival queueing systems. Stoch. Anal. Appl. 22 (2004), 1235-1255. | DOI | MR

[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 | Zbl

[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

[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

[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

[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 | MR

[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

[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

[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

[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

[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

[29] Takagi, H.: Queueing Analysis: A Foundation of Performance Evaluation, Vacation and Priority Systems, Part I, vol. I. North-Holland, Amsterdam 1991. | MR

[30] Takagi, H.: $M/G/1/K$ queues with $N$-policy and setup times. Queueing Syst. 14 (1993), 79-98. | DOI | MR

[31] Tian, N., Zhang, Z. G.: Vacation Queueing Models: Theory and Applications. Springer, 2006. | MR

[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

Cité par Sources :