# A finite-buffer queue with a single vacation policy: an analytical study with evolutionary positioning

Marcin Woźniak; Wojciech M. Kempa; Marcin Gabryel; Robert K. Nowicki

International Journal of Applied Mathematics and Computer Science (2014)

- Volume: 24, Issue: 4, page 887-900
- ISSN: 1641-876X

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topMarcin Woźniak, et al. "A finite-buffer queue with a single vacation policy: an analytical study with evolutionary positioning." International Journal of Applied Mathematics and Computer Science 24.4 (2014): 887-900. <http://eudml.org/doc/271880>.

@article{MarcinWoźniak2014,

abstract = {In this paper, application of an evolutionary strategy to positioning a GI/M/1/N-type finite-buffer queueing system with exhaustive service and a single vacation policy is presented. The examined object is modeled by a conditional joint transform of the first busy period, the first idle time and the number of packets completely served during the first busy period. A mathematical model is defined recursively by means of input distributions. In the paper, an analytical study and numerical experiments are presented. A cost optimization problem is solved using an evolutionary strategy for a class of queueing systems described by exponential and Erlang distributions.},

author = {Marcin Woźniak, Wojciech M. Kempa, Marcin Gabryel, Robert K. Nowicki},

journal = {International Journal of Applied Mathematics and Computer Science},

keywords = {finite-buffer queue; evolutionary strategy; object positioning; queueing system; busy period; idle time; single vacation},

language = {eng},

number = {4},

pages = {887-900},

title = {A finite-buffer queue with a single vacation policy: an analytical study with evolutionary positioning},

url = {http://eudml.org/doc/271880},

volume = {24},

year = {2014},

}

TY - JOUR

AU - Marcin Woźniak

AU - Wojciech M. Kempa

AU - Marcin Gabryel

AU - Robert K. Nowicki

TI - A finite-buffer queue with a single vacation policy: an analytical study with evolutionary positioning

JO - International Journal of Applied Mathematics and Computer Science

PY - 2014

VL - 24

IS - 4

SP - 887

EP - 900

AB - In this paper, application of an evolutionary strategy to positioning a GI/M/1/N-type finite-buffer queueing system with exhaustive service and a single vacation policy is presented. The examined object is modeled by a conditional joint transform of the first busy period, the first idle time and the number of packets completely served during the first busy period. A mathematical model is defined recursively by means of input distributions. In the paper, an analytical study and numerical experiments are presented. A cost optimization problem is solved using an evolutionary strategy for a class of queueing systems described by exponential and Erlang distributions.

LA - eng

KW - finite-buffer queue; evolutionary strategy; object positioning; queueing system; busy period; idle time; single vacation

UR - http://eudml.org/doc/271880

ER -

## References

top- Bratiichuk, M.S. (2000). Exact formulas for ${E}^{\theta}/G/1/N$-type queueing systems, Ukrainian Mathematical Journal 8(52): 1034-1044.
- Bratiichuk, M.S. and Kempa, W.M. (2003). Application of the superposition of renewal processes to the study of batch arrival queues, Queueing Systems 1(44): 51-67. Zbl1023.90011
- Choudhury, G. (2002). A batch arrival queue with a vacation time under single vacation policy, Computers and Operations Research 13(29): 1941-1955. Zbl1010.90010
- Chydziński, A. and Chróst, Ł. (2011). Analysis of AQM queues with queue size based packet dropping, International Journal of Applied Mathematics and Computer Science 21(3): 567-577, DOI: 10.2478/v10006-011-0045-7. Zbl1237.60069
- Cohen, J. (1982). The Single Server Queue, North-Holland Publishing Company, Amsterdam. Zbl0481.60003
- Cpałka, K. (2009). A new method for design and reduction of neuro-fuzzy classification systems, IEEE Transactions on Neural Networks 20(4): 701-714.
- Doshi, B.T. (1986). Queueing systems with vacations-a survey, Queueing Systems 1(1): 29-66. Zbl0655.60089
- Eiben, A. and Smith, J. (2003). Introduction to Evolutionary Computing, Springer-Verlag, New York, NY. Zbl1028.68022
- Gabryel, M., Nowicki, R.K., Woźniak, M. and Kempa, W. M. (2013). Genetic cost optimization of the GI/M/1/N finite-buffer queue with a single vacation policy, in L. Rutkowski, M. Korytkowski, R. Scherer, R. Tadeusiewicz, L.A. Zadeh and J.M. Zurada (Eds.), 12th International Conference, ICAISC 2013, Zakopane, Poland, June 9-13, 2013, Proceedings, Part II, Lecture Notes in Artificial Intelligence, Vol. 7895, Springer-Verlag, Berlin/Heidelberg, pp. 12-23.
- Gabryel, M. and Rutkowski, L. (2010). Evolutionary designing of logic-type fuzzy systems, in L. Rutkowski, R. Scherer, R. Tadeusiewicz, L.A. Zadeh and J.M. Zurada (Eds.), 10th International Conference, ICAISC 2010, Zakopane, Poland, June 13-17, 2010, Part II, Lecture Notes in Artificial Intelligence, Vol. 6114, Springer-Verlag, Berlin/Heidelberg, pp. 143-148.
- Gabryel, M., Woźniak, M. and Nowicki, R.K. (2012). Creating learning sets for control systems using an evolutionary method, in L. Rutkowski, M. Korytkowski, R. Scherer, R. Tadeusiewicz, L.A. Zadeh and J.M. Zurada (Eds.), International Symposia, SIDE 2012 and EC 2012, Held in Conjunction with ICAISC 2012, Zakopane, Poland, April 29-May 3, 2012, Proceedings, Lecture Notes in Computer Science, Vol. 7269, Springer-Verlag, Berlin/Heidelberg, pp. 206-213.
- Gupta, U., Banik, A. and Pathak, S. (2005). Complete analysis of MAP/G/1/N queue with single (multiple) vacation(s) under limited service discipline, Journal of Applied Mathematics and Stochastic Analysis 3: 353-373. Zbl1107.60057
- Gupta, U. and Sikdar, K. (2006). Computing queue length distributions in MAP/G/1/N queue under single and multiple vacation, Journal of Applied Mathematics and Computing 2(174): 1498-1525. Zbl1103.60077
- Hongwei, D., Dongfeng, Z. and Yifan, Z. (2010). Performance analysis of wireless sensor networks of serial transmission mode with vacation on fire prevention, 2nd International Conference on Computer Engineering and Technology (ICCET), Chengdu, China, Vol. 5, pp. 153-155.
- Jaworski, M., Duda, P. and Pietruczuk, L. (2012). On fuzzy clustering of data streams with concept DRIFT, in L. Rutkowski, M. Korytkowski, R. Scherer, R. Tadeusiewicz, L.A. Zadeh and J.M. Zurada (Eds.), 11th International Conference, ICAISC 2012, Zakopane, Poland, April 29-May 3, 2012, Proceedings, Part II, Lecture Notes in Computer Science, Vol. 7268, Springer-Verlag, Berlin/Heidelberg, pp. 82-91.
- Kella, O. (1990). Optimal control of the vacation scheme in an M/G/1 queue, Operations Research 4(38): 724-728. Zbl0719.90033
- Kempa, W. (2004). The virtual waiting time for the batch arrival queueing systems, Stochastic Analysis and Applications 5(22): 1235-1255. Zbl1073.60087
- Kempa, W. (2009). GI/G/1/∞ batch arrival queueing system with a single exponential vacation, Mathematic Methods and Operations Research 1(69): 81-97. Zbl1170.60032
- Kempa, W. (2010a). Characteristics of vacation cycle in the batch arrival queueing system with single vacations and exhaustive service, International Journal of Applied Mathematics 4(23): 747-758. Zbl1208.60096
- Kempa, W. (2010b). On departure process in the batch arrival queue with single vacation and setup time, Annales UMCS: Informatica 1(10): 93-102. Zbl1284.60162
- Kempa, W. (2010c). Some new results for departure process in the ${M}^{X}/G/1$ queueing system with a single vacation and exhaustive service, Stochastic Analysis and Applications 1(28): 26-43. Zbl1189.60168
- Kempa, W. (2010d). Some results for the actual waiting time in batch arrival queueing systems, Stochastic Models 3(26): 335-356. Zbl1200.60081
- Kempa, W. (2011a). Analysis of departure process in batch arrival queue with multiple vacations and exhaustive service, Communications in Statistics-Theory and Methods 40(16): 2856-2865. Zbl1247.60128
- Kempa, W. (2011b). Departure process in finite-buffer queue with batch arrivals, in K. Al-Begain, S. Balsamo, D. Fiems and A. Marin (Eds.), 18th International Conference, ASMTA 2011, Venice, Italy, June 20-22, 2011. Proceedings, Lecture Notes in Computer Science, Vol. 6751, Springer-Verlag, Berlin/Heidelberg, pp. 1-13.
- Kempa, W. (2012a). On transient queue-size distribution in the batch arrival system with the n-policy and setup times, Mathematical Communications 17(1): 285-302. Zbl1254.60090
- Kempa, W. (2012b). The virtual waiting time in a finite-buffer queue with a single vacation policy, in K. Al-Begain, D. Fiems and J.-M. Vincent (Eds.), Proceedings, 19th International Conference, ASMTA 2012, Grenoble, France, June 4-6, 2012, Lecture Notes in Computer Science, Vol. 7314, Springer-Verlag, Berlin/Heidelberg, pp. 47-60. Zbl06059414
- Kempa, W. (2013). Output process in batch-arrival queue with n-policy and multiple vacations, in A. Dudin and K. De Turck (Eds.), 20th International Conference, ASMTA 2013, Ghent, Belgium, July 8-10, 2013. Proceedings, Springer-Verlag, Berlin/Heidelberg, Lecture Notes in Computer Science, Vol. 7314, pp. 247-261. Zbl06197239
- 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 1(40): 55-62. Zbl1296.60245
- Korolyuk, V. (1975). Boundary-Value Problems for Complicated Poisson Processes, Naukova Dumka, Kiev, (in Russian).
- Korolyuk, V., Bratiichuk, M. and Pirdzhanov, B. (1987). Boundary-Value Problems for Random Walks, Ylym, Ashkhabad, (in Russian).
- Lee, T.T. (1984). M/G/1/N queue with vacation time and exhaustive service discipline, Operations Research 4(32): 774-784. Zbl0559.90032
- Lillo, R.E. (2000). Optimal operating policy for an M/G/1 exhaustive server-vacation model, Methodology and Computing in Applied Probability 2(2): 153-167. Zbl0968.60086
- Mancuso, V. and Alouf, S. (2012). Analysis of power saving with continuous connectivity, Computer Networks 56(10): 2481-2493.
- Michalewicz, Z. (1996). Genetic Algorithms + Data Structures = Evolution Programs, 3rd Edn., Springer-Verlag, New York, NY. Zbl0841.68047
- Niu, Z., Shu, T. and Takahashi, Y. (2003). A vacation queue with setup and close-down times and batch Markovian arrival processes, Performance Evaluation 3(54): 225-248.
- Niu, Z. and Takahashi, Y. (1999). A finite-capacity queue with exhaustive vacation/close-down/setup times and Markovian arrival processes, Queueing Systems 31(1/2): 1-23. Zbl0948.90044
- Nowak, A. and Woźniak, M. (2008a). Algorithm for optimization of the active module by the use of genetic algorithm, Acta Mechanica Slovaca 3(C): 307-316.
- Nowak, A. and Woźniak, M. (2008b). Multiresolution derives analysis of module mechatronical systems, Mechanika 6(74): 45-51.
- Nowak, A. and Woźniak, M. (2009). Optimization of the active vibroisolation system for operator's cabin with the hydropneumatical element, Transactions of the Universities of Košice 1(3): 113-116.
- Piórkowski, A. and Werewka, J. (2010). Minimization of the total completion time for asynchronous transmission in a packet data-transmission system, International Journal of Applied Mathematics and Computer Science 20(2): 391-400, DOI: 10.2478/v10006-010-0029-z. Zbl1231.68101
- Rutkowski, L. (2004). Adaptive probabilistic neural networks for pattern classification in time-varying environment, IEEE Transactions on Neural Networks 15(4): 811-827.
- Rutkowski, L., Przybył, A. and Cpałka, K. (2012). Novel online speed profile generation for industrial machine tool based on flexible neuro-fuzzy approximation, IEEE Transactions on Industrial Electronics 59(2): 1238-1247.
- Schaefer, R., Byrski, A. and Smołka, M. (2012). The island model as a Markov dynamic system, International Journal of Applied Mathematics and Computer Science 22(4): 971-984, DOI: 10.2478/v10006-012-0072-z. Zbl1288.90133
- Takagi, H. (1993). Queueing Analysis, North-Holland, Amsterdam.
- Takagi, H. (1994). M/G/1/N queues with server vacations and exhaustive service, Operations Research 5(42): 926-939. Zbl0829.90063
- Teghem, J. (1986). Control of the service process in a queueing system, European Journal of Operational Research 1(23): 141-158. Zbl0583.60092
- Teghem, J. (1987). Optimal control of a removable server in an M/G/1 queue with finite capacity, European Journal of Operational Research 1(31): 358-367. Zbl0641.60108
- Wieczorek, R. (2010). Markov chain model of phytoplankton dynamics, International Journal of Applied Mathematics and Computer Science 20(4): 763-771, DOI: 10.2478/v10006-010-0058-7. Zbl05869750
- Woźniak, M. (2013). On applying cuckoo search algorithm to positioning GI/M/1/N finite-buffer queue with a single vacation policy, Proceedings of the 12th Mexican International Conference on Artificial Intelligence, MICAI 2013, Mexico City, Mexico, pp. 59-64.
- Woźniak, M., Kempa, W.M., Gabryel, M., Nowicki, R.K. and Shao, Z. (2014). On applying evolutionary computation methods to optimization of vacation cycle costs in finite-buffer queue, in L. Rutkowski, M. Korytkowski, R. Scherer, R. Tadeusiewicz, L.A. Zadeh and J.M. Zurada (Eds.), 13th International Conference, ICAISC 2014, Zakopane, Poland, June 1-5, 2014, Proceedings, Part I, Lecture Notes in Artificial Intelligence, Vol. 8467, Springer International Publishing Switzerland, Cham, pp. 480-491.
- Woźniak, M., Marszałek, Z., Gabryel, M. and Nowicki, R.K. (2013). Modified merge sort algorithm for large scale data sets, in L. Rutkowski, M. Korytkowski, R. Scherer, R. Tadeusiewicz, L.A. Zadeh and J.M. Zurada (Eds.), 12th International Conference, ICAISC 2013, Zakopane, Poland, June 9-13, 2013, Proceedings, Part II, Lecture Notes in Artificial Intelligence, Vol. 7895, Springer-Verlag, Berlin/Heidelberg, pp. 612-622.