An Mθ/G/1 system with hysteretic switching of the service intensity

  • K. Yu. Zhernovyi
  • Yu. V. Zhernovyi
Mathematical Models, Computational Methods

Abstract

For an Mθ/G/1 system, two—basic and post-threshold—service modes with distribution functions F(x) and \(\tilde F\)(x) of the service time are applied. The post-threshold mode starts functioning if, at instant t of the beginning of the current customer service, number ξ(t) of customers in the system satisfies the condition ξ(t) > h 2. The basic mode is restored at the instant when the customer such that ξ(t) ≤ h 1, where h 1h 2, begins to be served. The mean duration of the busy period is determined, formulas for the stationary distribution of the number of customers in the system and stationary characteristics are derived. The results are verified with the help of simulation models developed with the use of the GPSS World tools.

Keywords

queueing system batch arrival of customers switching of the service intensity hysteric strategy the stationary distribution of the number of customers 

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References

  1. 1.
    B. D. Choi and D. I. Choi, “Queueing system with queue length dependent service time and its application to cell discarding scheme in ATM networks,” IEEE Proc. Commun. 143, 5–11 (1996).CrossRefGoogle Scholar
  2. 2.
    D. Choi, C. Knessl, and C. Tier, “A queueing system with queue length dependent service times with applications to cell discarding in ATM networks,” J. of Appl. Math. and Stoch. Anal. 12, 35–62 (1999).MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    B. D. Choi, Y. Ch. Kim, Y. Shin, and Ch. E. M. Pearce, “The MX/G/1 queue with length dependent service times,” J. of Appl. Math. and Stoch. Anal. 14, 399–419 (2001).MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    S. Q. Li, “Overload Control in a Finite Message Storage Buffer,” IEEE Trans. Commun. 37, 1330–1338 (1989).CrossRefGoogle Scholar
  5. 5.
    K. Sriram and D. M. Lucantoni, “Traffic smoothing effects of bit dropping in a packet voice multiplexer,” IEEE Trans. Commun. 37, 703–712 (1989).CrossRefGoogle Scholar
  6. 6.
    K. Sriram, R. S. McKinney, and M. H. Sherif, “Voice packetization and compression in broadband ATM networks,” IEEE J. Sel. Areas Commun. 9, 294–304 (1991).CrossRefGoogle Scholar
  7. 7.
    K. Sriram, “Methodologies for bandwidth allocation, transmission scheduling and congestion avoidance in broadband ATM networks,” Comp. Networks ISDN Systems 26(1), 43–69 (1993).CrossRefGoogle Scholar
  8. 8.
    S. Nishimura and Y. Jiang, “An M/G/1 vacation model with two service modes.,” Probab. Engineering Inf. Sci. 9, 355–374 (1995).MathSciNetCrossRefGoogle Scholar
  9. 9.
    A. Dudin, “Optimal control for an Mx/G/1 queue with two operation modes,” Probab. Eng. Inform. Sci. 11, 255–265 (1997).MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    R. D. Nobel and H. C. Tijms, “Optimal control for an MX/G/1 queue with two service modes,” Eur. J. Operat. Res. 113, 610–619 (1999).MATHCrossRefGoogle Scholar
  11. 11.
    K. Yu. Zhernovyi and Yu. V. Zhernovyi, “An Mθ/G/1/m system with two-threshold hysteresis strategy of service intensity switching,” J. Commun. Technol. Electron. 57, 1340–1349 (2012).CrossRefGoogle Scholar
  12. 12.
    V. S. Korolyuk, Boundary-value problems for compound Poisson processes (Naukova Dumka, Kiev, 1975) [in Russian].Google Scholar
  13. 13.
    A. M. Bratiichuk, Candidate’s Dissertation Mathematics and Physics (Kiev. Nats. Univ. Im. T. Shevchenka, Kiev, 2008).Google Scholar
  14. 14.
    V. D. Boev, System simulation. GPSS world tools (BKhV-Peterburg, St. Petersburg, 2004) [in Russian].Google Scholar
  15. 15.
    Yu. V. Zhernovii, Simulation of Queueing Systems (Vydavnychyi Tsentr LNU im. Ivana Franka, L’viv, 2007) [in Ukrainian].Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2013

Authors and Affiliations

  • K. Yu. Zhernovyi
    • 1
  • Yu. V. Zhernovyi
    • 1
  1. 1.Ivan Franko National University of LvivLvivUkraine

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