| Title:
|
Non-stationary departure process in a batch-arrival queue with finite buffer capacity and threshold-type control mechanism (English) |
| Author:
|
Kempa, Wojciech M. |
| Author:
|
Kurzyk, Dariusz |
| Language:
|
English |
| Journal:
|
Kybernetika |
| ISSN:
|
0023-5954 (print) |
| ISSN:
|
1805-949X (online) |
| Volume:
|
58 |
| Issue:
|
1 |
| Year:
|
2022 |
| Pages:
|
82-100 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
departure process |
| Keyword:
|
finite-buffer queue |
| Keyword:
|
$N$-policy |
| Keyword:
|
power saving |
| Keyword:
|
transient state |
| Keyword:
|
wireless sensor network (WSN) |
| MSC:
|
60K25 |
| MSC:
|
90B22 |
| idZBL:
|
Zbl 07511612 |
| idMR:
|
MR4405948 |
| DOI:
|
10.14736/kyb-2022-1-0082 |
| . |
| Date available:
|
2022-04-08T07:53:00Z |
| Last updated:
|
2022-08-11 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/149603 |
| . |
| Reference:
|
[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. |
| Reference:
|
[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. MR 2038103, |
| Reference:
|
[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. MR 1802636, |
| Reference:
|
[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 |
| Reference:
|
[5] Choudhury, G., Paul, M.: A batch arrival queue with an additional service channel under $N$-policy..Appl. Math. Comput. 156 (2004), 115-130. MR 2087256, |
| Reference:
|
[6] Cohen, J. W.: The Single Server Queue..North-Holland, Amsterdam 1982. MR 0668697 |
| Reference:
|
[7] Doshi, B. T.: Queueing systems with vacations-a survey..Queueing Syst. 1 (1986), 29-66. MR 0896237, |
| Reference:
|
[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. MR 4299460, |
| Reference:
|
[9] Gerhardt, I., Nelson, B. L.: Transforming renewal processes for simulation of nonstationary arrival processes..Informs J. Comput. 21 (2009), 630-640. MR 2588345, |
| Reference:
|
[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. |
| Reference:
|
[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. MR 1890915, |
| Reference:
|
[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. MR 1922375, |
| Reference:
|
[13] Kempa, W. M.: The virtual waiting time for the batch arrival queueing systems..Stoch. Anal. Appl. 22 (2004), 1235-1255. MR 2089066, |
| Reference:
|
[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. Zbl 1170.60032, MR 2476049, |
| Reference:
|
[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. MR 2860790, |
| Reference:
|
[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. MR 2946149, |
| Reference:
|
[17] Kempa, W. M.: On transient queue-size distribution in the batch-arrivals system with a single vacation policy..Kybernetika 50 (2014), 126-141. MR 3195008, |
| Reference:
|
[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. |
| Reference:
|
[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. |
| Reference:
|
[20] Korolyuk, V. S.: Boundary-value problems for compound Poisson processes..Theor. Probab. Appl. 19 (1974), 1-13. MR 0402939, 10.1137/1119001 |
| Reference:
|
[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. MR 1638645, |
| Reference:
|
[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. MR 1266802, |
| Reference:
|
[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. MR 1274803, |
| Reference:
|
[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. |
| Reference:
|
[25] Lee, H. S., Srinivasan, M. M.: Control policies for the $M/G/1$ queueing system..Manag. Sci. 35 (1989), 708-721. MR 1001484, |
| Reference:
|
[26] Levy, Y., Yechiali, U.: Utilization of idle time in an $M/G/1$ queueing system..Manag. Sci. 22 (1975), 202-211. |
| Reference:
|
[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. |
| Reference:
|
[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. MR 3120933, |
| Reference:
|
[29] Takagi, H.: Queueing Analysis: A Foundation of Performance Evaluation, Vacation and Priority Systems, Part I, vol. I..North-Holland, Amsterdam 1991. MR 1149382 |
| Reference:
|
[30] Takagi, H.: $M/G/1/K$ queues with $N$-policy and setup times..Queueing Syst. 14 (1993), 79-98. MR 1238663, |
| Reference:
|
[31] Tian, N., Zhang, Z. G.: Vacation Queueing Models: Theory and Applications..Springer, 2006. MR 2248264 |
| Reference:
|
[32] Yadin, M., Naor, P.: Queueing systems with a removable service station..J. Oper. Res. Soc. 14 (1963), 393-405. |
| Reference:
|
[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. MR 3897421, |
| . |
