Transient Processing Analysis in a Finite-Buffer Queueing Model with Setup Times

Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 522)

Abstract

A finite-buffer queueing model with Poisson arrivals and generally distributed processing times is investigated. Every time when the service station restarts the operation after the idle period, a random-length setup time is needed to achieve full readiness for the work, during which the service process is suspended. A system of integral equations for time-dependent departure process, conditioned by the initial buffer state, is built. The solution of the corresponding system written for double transforms is obtained in a compact form. Hence the mean number of packets completely processed up to fixed time epoch can be easily found. The analytical approach is based on the idea of embedded Markov chain, total probability law and integral equations. The considered queueing system can be successfully used in cellular networks or WSNs modelling, where the setup time corresponds to leaving the sleep mode in energy saving mechanism. Numerical utility of analytical formulae is shown in a network-motivated computational example.

Keywords

Departure process Finite-buffer queue Integral equation Setup time Transient state 

References

  1. 1.
    Burnetas, A., Economou, A.: Equilibrium customer strategies in a single server Markovian queue with setup times. Queueing Syst. 56(3–4), 213–228 (2007)Google Scholar
  2. 2.
    Chen, P.S., Zhou, W.H., Zhou, J.W.: Equilibrium customer strategies in the queue with threshold policy and setup times. In: Mathematical Problems in Engineering. Optimization Theory, Methods, and Applications in Engineering, Hindawi Publishing Corporation (2015)Google Scholar
  3. 3.
    Edward, E.P.: A novel seamless handover scheme for WiMAX/LTE heterogeneous networks. Arab. J. Sci. Eng. 41(3), 1129–1143 (2016)Google Scholar
  4. 4.
    Hu, J.N., Tuan, P.D.: Power consumption analysis for data centers with independent setup times and threshold controls. In: AIP Conference Proceedings, vol. 1648 (2015)Google Scholar
  5. 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: AIP Conference Proceedings, vol. 1293 (2010), 235–242 (Proceedings of 36th International Conference Applications of Mathematics in Engineering and Economics (AMEE’10), Sozopol, Bulgaria, 2010)Google Scholar
  6. 6.
    Kempa, W.M.: Analysis of departure process in batch arrival queue with multiple vacations and exhaustive service. Commun. Stat. Theory Methods 40(16), 2856–2865 (2011)Google Scholar
  7. 7.
    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)Google Scholar
  8. 8.
    Kempa, W.M.: Study on time-dependent departure process in a finite-buffer queueing model with BMAP-type input stream. In: Proceedings of the IEEE 2nd International Conference on Cybernetics (CYBCONF 2015), Gdynia, Poland (2015)Google Scholar
  9. 9.
    Kempa, W.M.: Time-dependent analysis of transmission process in a wireless sensor network with energy saving mechanism based on threshold waking up. In: Proceedings of the IEEE 16th International Workshop on Signal Processing Advances in Wireless Communications (SPAWC 2015), Stockholm, Sweden (2015)Google Scholar
  10. 10.
    Kempa, W.M., Kurzyk, D.: Transient departure process in M/G/1/K-type queue with threshold server’s waking up. In: Proceedings of the 23rd International Conference on Software, Telecommunications and Computer Networks (SoftCOM 2015), Split—Bol (Island of Brac), Croatia, pp. 32–36 (2015)Google Scholar
  11. 11.
    Kempa, W.M., Paprocka, I.: Analytical solution for time-dependent queue-size behavior in the manufacturing line with finite buffer capacity and machine setup and closedown times. In: Applied Mechanics and Materials, vol. 809–910, 2015, pp. 1360–1365 (Selected, peer reviewed papers from the 19th Conference on Innovative Manufacturing Engineering (IManE 2015), Iasi, Romania, 2015)Google Scholar
  12. 12.
    Korolyuk, V.S.: Boundary-Value Problems for Compound Poisson Processes. Naukova Dumka, Kiev (1975)MATHGoogle Scholar
  13. 13.
    Ma, Q.: Analysis of a clearing queueing system with setup times. RAIRO—Oper. Res. 49(1), 67–76 (2015)Google Scholar
  14. 14.
    Niu, Z.S., Guo, X.Y., Zhou, S., Kumar, P.R.: Characterizing energy-delay tradeoff in hyper-cellular networks with base station sleeping control. IEEE J. Sel. Areas Commun. 33(4), 641–650 (2015)Google Scholar
  15. 15.
    Sun, W., Guo, P.F., Tian, N.S.: Equilibrium threshold strategies in observable queueing systems with setup/closedown times. Central Eur. J. Oper. Res. 18(3), 241–268 (2010)Google Scholar
  16. 16.
    Yue, W.Y., Sun, Q.T., Jin, S.F.: Performance analysis of sensor nodes in a WSN with sleep/wakeup protocol. Lect. Notes Oper. Res. 12, 370–377 (2010)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Silesian University of TechnologyInstitute of MathematicsGliwicePoland
  2. 2.Institute of Theoretical and Applied InformaticsPolish Academy of SciencesGliwicePoland

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