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Transient Solution for Queue-Size Distribution in a Certain Finite-Buffer Model with Server Working Vacations

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Information and Software Technologies (ICIST 2016)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 639))

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Abstract

A finite-buffer queueing model with Poisson arrivals and exponential processing times is investigated. Every time when the system empties, the server begins a generally distributed single working vacation period, during which the service is provided with another (slower) rate. After the completion of the vacation period the processing is being continued normally, with original speed. The next working vacation period is being initialized at the next time at which the system becomes empty, and so on. The system of Volterra-type integral equations for transient queue-size distribution, conditioned by the initial level of buffer saturation, is built. The solution of the corresponding system written for Laplace transforms is given in a compact-form using the linear algebraic approach and the corresponding result obtained for the ordinary model (without working vacation regime). Numerical examples are attached as well.

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References

  1. Baba, Y.: Analysis of a GI/M/1 queue with multiple working vacations. Oper. Res. Lett. 33, 201–209 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  2. Banik, A.D., Gupta, U.C., Pathak, S.S.: On the GI/M/1/N queue with multiple working vacations - analytic analysis and computations. Appl. Math. Model. 31, 1701–1710 (2007)

    Article  MATH  Google Scholar 

  3. Bratiichuk, M.S., Borowska, B.: Explicit formulae and convergence rate for the system Mα/G/1/N as N → ∞. Stochast. Models 18(1), 71–84 (2002)

    Article  MathSciNet  Google Scholar 

  4. Cohen, J.W.: The Single Server Queue. North-Holland Publishing Company, Amsterdam, New York, Oxford (1982)

    MATH  Google Scholar 

  5. Kempa, W.M.: The transient analysis of the queue-length distribution in the batch arrival system with N-policy, multiple vacations and setup times. In: Venkov, G., Kovacheva, R., Pasheva, V. (eds.) 36th International Conference Applications of Mathematics in Engineering and Economics (AMEE 2010), 5–10 June 2010, Sozopol, Bulgaria, pp. 235–242. American Institute of Physics, Melville (2010). (AIP Conference Proceedings, vol. 1293)

    Google Scholar 

  6. Kempa, W.M.: On transient queue-size distribution in the batch arrival system with the N-policy and setup times. Math. Commun. 17(1), 285–302 (2012)

    MathSciNet  MATH  Google Scholar 

  7. Kempa, W.M.: A direct approach to transient queue-size distribution in a finite-buffer queue with AQM. Appl. Math. Inf. Sci. 7(3), 909–915 (2013)

    Article  MathSciNet  Google Scholar 

  8. Kempa, W.M.: On transient departure process in a finite-buffer queueing model with probabilistic packet dropping. In: Venkov, G., Pasheva, V. (eds.) 40th International Conference on Applications of Mathematics in Engineering and Economics (AMEE 2014), 8–13 June 2014, Sozopol, Bulgaria, pp. 42–49. American Institute of Physics, Melville (2014). (AIP Conference Proceedings, vol. 1631)

    Google Scholar 

  9. Kempa, W.M., Kurzyk, D.: Transient departure process in M/G/1/K-type queue with threshold server’s waking up. In: Proceedings of 23rd International Conference on Software, Telecommunications and Computer Networks (SoftCOM 2015), 16–18 September 2015, pp. 32–36. Split - Bol, Croatia (2015)

    Google Scholar 

  10. Kempa, W.M., Paprocka, I., Kalinowski, K., Grabowik, C.: On transient queue-size distribution in a single-machine production system with breakdowns. In: Carausu, C., Wróbel, A., et al. (eds.) Modern Technologies in Industrial Engineering II. Selected, Peer Reviewed Papers from the Modern Technologies in Industrial Engineering (ModTech 2014), 13–16 July 2014, Gliwice, Poland, pp. 505–510. Trans Tech Publications, Staffa-Zurich (2014). (Advanced Materials Research, vol. 1036)

    Google Scholar 

  11. Korolyuk, V.S.: Boundary-Value Problems for Compound Poisson Processes. Naukova Dumka, Kiev (1975)

    MATH  Google Scholar 

  12. Li, J.-H., Liu, W.-Q., Tian, N.-S.: Steady-state analysis of a discrete-time batch arrival queue with working vacations. Perform. Eval. 67, 897–912 (2010)

    Article  Google Scholar 

  13. Li, J., Tian, N.: Performance analysis of a GI/M/1 queue with single working vacation. Appl. Math. Comput. 217, 4960–4971 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. Morse, P.M.: Queues, Inventories and Maintenance. Wiley, New York (1958)

    Google Scholar 

  15. Servi, L.D., Finn, S.G.: M/M/1 queues with working vacations. Perform. Eval. 50, 41–52 (2002)

    Article  Google Scholar 

  16. Sharma, O.P., Gupta, U.C.: Transient behaviour of an M/M/1/N queue. Stochast. Process. Appl. 13, 327–331 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  17. Sharma, O.P., Maheswar, M.V.R.: Transient behaviour of a simple queue with discouraged arrivals. Optimization 27, 283–291 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  18. Sudhesh, R., Raj, L.F.: Computational analysis of stationary and transient distribution of single server queue with working vacation. In: Krishna, P., Babu, M., Ariwa, E. (eds.) ObCom 2011, Part I. CCIS, vol. 269, pp. 480–489. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  19. Takacs, L.: Introduction to the Theory of Queues. Oxford University Press, New York (1960)

    MATH  Google Scholar 

  20. Tarabia, A.M.K.: Transient analysis of M/M/1/N queue - an alternative approach. Tamkang J. Sci. Eng. 3(4), 263–266 (2000)

    MathSciNet  Google Scholar 

  21. Tikhonenko, O., Kempa, W.M.: Queueing system with processor sharing and limited memory under control of the AQM mechanism. Autom. Remote Control 76(10), 1784–1796 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  22. Vijayashree, K.V., Janani, B.: Transient analysis of an M/M/c queue subject to multiple exponential working vacation. Appl. Math. Sci. 9(74), 3669–3677 (2015)

    Article  Google Scholar 

  23. Woźniak, M., Gabryel, M., Nowicki, R.K., Nowak, B.: An application of firefly algorithm to position traffic in NoSQL database systems. In: Kunifuji, S., Papadopoulos, G.A., Skulimowski, A.M.J., Kacprzyk, J. (eds.) KICSS 2014. AISC, vol. 416, pp. 259–272. Springer, Heidelberg (2016). doi:10.1007/978-3-319-27478-2-18. ISSN 2194-5357

    Chapter  Google Scholar 

  24. Woźniak, M., Marszałek, Z., Gabryel, M., Nowicki, R.K.: Preprocessing large data sets by the use of quick sort algorithm. In: Skulimowski, A.M.J., Kacprzyk, J. (eds.) KICSS 2013. AISC, vol. 364, pp. 111–121. Springer, Switzerland (2016). doi:10.1007/978-3-319-19090-7-9. ISSN 2194-5357

    Google Scholar 

  25. Woźniak, M., Marszałek, Z., Gabryel, M., Nowicki, R.K.: Modified merge sort algorithm for large scale data sets. In: Rutkowski, L., Korytkowski, M., Scherer, R., Tadeusiewicz, R., Zadeh, L.A., Zurada, J.M. (eds.) ICAISC 2013, Part II. LNCS, vol. 7895, pp. 612–622. Springer, Heidelberg (2013). doi:10.1007/978-3-642-38610-7-56

    Chapter  Google Scholar 

  26. Zhang, M., Hou, Z.: Steady state analysis of the GI/M/1/N queue with a variant of multiple working vacations. Comput. Ind. Eng. 61, 1296–1301 (2011)

    Article  Google Scholar 

  27. Damaševičius, R., Vasiljevas, M., Šalkevičius, J., Woźniak, M.: Human activity recognition in AAL environments using random projections. Comput. Math. Methods Med. 2016, Article ID 4073584, 17 p. (2016). doi:10.1155/2016/4073584

    Google Scholar 

  28. Capizzi, G., Lo Sciuto, G., Wozniak, M., Damaševičius, R.: A clustering based system for automated oil spill detection by satellite remote sensing. In: Rutkowski, L., Korytkowski, M., Scherer, R., Tadeusiewicz, R., Zadeh, L.A., Zurada, J.M. (eds.) ICAISC 2016. LNCS, vol. 9693, pp. 613–623. Springer, Heidelberg (2016). doi:10.1007/978-3-319-39384-1_54

    Chapter  Google Scholar 

  29. Kempa, W.M., Wozniak, M., Nowicki, R.K., Gabryel, M., Damaševičius, R.: Transient solution for queueing delay distribution in the GI/M/1/K-type mode with “queued” waking up and balking. In: Rutkowski, L., Korytkowski, M., Scherer, R., Tadeusiewicz, R., Zadeh, L.A., Zurada, J.M. (eds.) ICAISC 2016. LNCS, vol. 9693, pp. 340–351. Springer, Heidelberg (2016). doi:10.1007/978-3-319-39384-1_29

    Chapter  Google Scholar 

  30. Połap, D., Woźniak, M., Napoli, C., Tramontana, E.: Real-time cloud-based game management system via cuckoo search algorithm. Int. J. Electron. Telecommun. 61(4), 333–338 (2015). doi:10.1515/eletel-2015-0043. De Gruyter Open Ltd

    Google Scholar 

  31. Połap, D., Woźniak, M., Napoli, C., Tramontana, E.: Is swarm intelligence able to create mazes? Int. J. Electron. Telecommun. 61(4), 305–310 (2015). doi:10.1515/eletel-2015-0039. De Gruyter Open Ltd

    Google Scholar 

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Authors and Affiliations

  1. Faculty of Applied Mathematics, Institute of Mathematics, Silesian University of Technology, 23 Kaszubska Street, 44-100, Gliwice, Poland

    Wojciech M. Kempa & Martyna Kobielnik

Authors
  1. Wojciech M. Kempa
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  2. Martyna Kobielnik
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Correspondence to Wojciech M. Kempa .

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Editors and Affiliations

  1. Kaunas University of Technology , Kaunas, Lithuania

    Giedre Dregvaite

  2. Kaunas University of Technology , Kaunas, Lithuania

    Robertas Damasevicius

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Kempa, W.M., Kobielnik, M. (2016). Transient Solution for Queue-Size Distribution in a Certain Finite-Buffer Model with Server Working Vacations. In: Dregvaite, G., Damasevicius, R. (eds) Information and Software Technologies. ICIST 2016. Communications in Computer and Information Science, vol 639. Springer, Cham. https://doi.org/10.1007/978-3-319-46254-7_34

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  • DOI: https://doi.org/10.1007/978-3-319-46254-7_34

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