IDEAS home Printed from https://ideas.repec.org/a/spr/annopr/v247y2016i1d10.1007_s10479-015-1893-6.html
   My bibliography  Save this article

A sufficient condition for the subexponential asymptotics of GI/G/1-type Markov chains with queueing applications

Author

Listed:
  • Hiroyuki Masuyama

    (Kyoto University)

Abstract

The main contribution of this paper is to present a new sufficient condition for the subexponential asymptotics of the stationary distribution of a GI/G/1-type Markov chain with the stochastic phase transition matrix in non-boundary levels, which implies no possibility of jumps from level “infinity” to level zero. For simplicity, we call such Markov chains GI/G/1-type Markov chains without disasters because they are used to analyze semi-Markovian queues without “disasters”, which are negative customers who remove all the customers in the system (including themselves) on their arrivals. We first demonstrate the application of our main result to the stationary queue length distribution in the standard BMAP/GI/1 queue. Thereby we present new asymptotic formulas and derive the existing formulas under weaker conditions than those in the literature. We also apply our main result to the stationary queue length distributions in two queues: One is a MAP/ $$\mathrm{GI}$$ GI /1 queue with the $$(a,b)$$ ( a , b ) -bulk-service rule (i.e., MAP/ $$\mathrm{GI}^{(a,b)}$$ GI ( a , b ) /1 queue); and the other is a MAP/ $$\mathrm{GI}$$ GI /1 retrial queue with constant retrial rate.

Suggested Citation

  • Hiroyuki Masuyama, 2016. "A sufficient condition for the subexponential asymptotics of GI/G/1-type Markov chains with queueing applications," Annals of Operations Research, Springer, vol. 247(1), pages 65-95, December.
    • Handle: RePEc:spr:annopr:v:247:y:2016:i:1:d:10.1007_s10479-015-1893-6
    DOI: 10.1007/s10479-015-1893-6
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10479-015-1893-6
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10479-015-1893-6?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Tetsuya Takine, 2004. "Geometric and Subexponential Asymptotics of Markov Chains of M / G /1 Type," Mathematics of Operations Research, INFORMS, vol. 29(3), pages 624-648, August.
    2. Kim, Bara & Kim, Jeongsim, 2012. "A note on the subexponential asymptotics of the stationary distribution of M/G/1 type Markov chains," European Journal of Operational Research, Elsevier, vol. 220(1), pages 132-134.
    3. Asmussen, Søren & Klüppelberg, Claudia & Sigman, Karl, 1999. "Sampling at subexponential times, with queueing applications," Stochastic Processes and their Applications, Elsevier, vol. 79(2), pages 265-286, February.
    4. Coquet, François & Mackevicius, Vigirdas & Mémin, Jean, 1999. "Corrigendum to "Stability in of martingales and backward equations under discretization of filtration": [Stochastic Processes and their Applications 75 (1998) 235-248]," Stochastic Processes and their Applications, Elsevier, vol. 82(2), pages 335-338, August.
    5. Masuyama, Hiroyuki, 2011. "Subexponential asymptotics of the stationary distributions of M/G/1-type Markov chains," European Journal of Operational Research, Elsevier, vol. 213(3), pages 509-516, September.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Hiroyuki Masuyama, 2022. "Subexponential asymptotics of asymptotically block-Toeplitz and upper block-Hessenberg Markov chains," Queueing Systems: Theory and Applications, Springer, vol. 102(1), pages 175-217, October.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Hiroyuki Masuyama, 2022. "Subexponential asymptotics of asymptotically block-Toeplitz and upper block-Hessenberg Markov chains," Queueing Systems: Theory and Applications, Springer, vol. 102(1), pages 175-217, October.
    2. Kim, Bara & Kim, Jeongsim, 2012. "A note on the subexponential asymptotics of the stationary distribution of M/G/1 type Markov chains," European Journal of Operational Research, Elsevier, vol. 220(1), pages 132-134.
    3. Bin Liu & Jie Min & Yiqiang Q. Zhao, 2023. "Refined tail asymptotic properties for the $$M^X/G/1$$ M X / G / 1 retrial queue," Queueing Systems: Theory and Applications, Springer, vol. 104(1), pages 65-105, June.
    4. Baltrunas, A. & Daley, D. J. & Klüppelberg, C., 2004. "Tail behaviour of the busy period of a GI/GI/1 queue with subexponential service times," Stochastic Processes and their Applications, Elsevier, vol. 111(2), pages 237-258, June.
    5. Bart Steyaert & Sabine Wittevrongel & Herwig Bruneel, 2017. "Characterisation of the output process of a discrete-time GI / D / 1 queue, and its application to network performance," Annals of Operations Research, Springer, vol. 252(1), pages 175-190, May.
    6. Debicki, Krzystof & Zwart, Bert & Borst, Sem, 2004. "The supremum of a Gaussian process over a random interval," Statistics & Probability Letters, Elsevier, vol. 68(3), pages 221-234, July.
    7. Evsey V. Morozov & Irina V. Peshkova & Alexander S. Rumyantsev, 2023. "Bounds and Maxima for the Workload in a Multiclass Orbit Queue," Mathematics, MDPI, vol. 11(3), pages 1-15, January.
    8. Royi Jacobovic & Nikki Levering & Onno Boxma, 2023. "Externalities in the M/G/1 queue: LCFS-PR versus FCFS," Queueing Systems: Theory and Applications, Springer, vol. 104(3), pages 239-267, August.
    9. P. Jelenković & P. Momčilović, 2003. "Large Deviation Analysis of Subexponential Waiting Times in a Processor-Sharing Queue," Mathematics of Operations Research, INFORMS, vol. 28(3), pages 587-608, August.
    10. Robert, Christian Y. & Segers, Johan, 2008. "Tails of random sums of a heavy-tailed number of light-tailed terms," Insurance: Mathematics and Economics, Elsevier, vol. 43(1), pages 85-92, August.
    11. Bin Liu & Yiqiang Q. Zhao, 2022. "Tail Asymptotics for a Retrial Queue with Bernoulli Schedule," Mathematics, MDPI, vol. 10(15), pages 1-13, August.
    12. F. G. Badía & C. Sangüesa, 2017. "Log-Convexity of Counting Processes Evaluated at a Random end of Observation Time with Applications to Queueing Models," Methodology and Computing in Applied Probability, Springer, vol. 19(2), pages 647-664, June.
    13. Bin Liu & Yiqiang Q. Zhao, 2020. "Tail asymptotics for the $$M_1,M_2/G_1,G_2/1$$ M 1 , M 2 / G 1 , G 2 / 1 retrial queue with non-preemptive priority," Queueing Systems: Theory and Applications, Springer, vol. 96(1), pages 169-199, October.
    14. Hu, Lu & Jiang, Yangsheng & Zhu, Juanxiu & Chen, Yanru, 2015. "A PH/PH(n)/C/C state-dependent queuing model for metro station corridor width design," European Journal of Operational Research, Elsevier, vol. 240(1), pages 109-126.
    15. Zhu, Juanxiu & Hu, Lu & Jiang, Yangsheng & Khattak, Afaq, 2017. "Circulation network design for urban rail transit station using a PH(n)/PH(n)/C/C queuing network model," European Journal of Operational Research, Elsevier, vol. 260(3), pages 1043-1068.
    16. Marc Lelarge, 2008. "Packet reordering in networks with heavy-tailed delays," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 67(2), pages 341-371, April.
    17. Predrag R. Jelenković & Petar Momčilović, 2004. "Large Deviations of Square Root Insensitive Random Sums," Mathematics of Operations Research, INFORMS, vol. 29(2), pages 398-406, May.
    18. A. P. Zwart, 2001. "Tail Asymptotics for the Busy Period in the GI/G/1 Queue," Mathematics of Operations Research, INFORMS, vol. 26(3), pages 485-493, August.
    19. Masuyama, Hiroyuki, 2011. "Subexponential asymptotics of the stationary distributions of M/G/1-type Markov chains," European Journal of Operational Research, Elsevier, vol. 213(3), pages 509-516, September.
    20. Tomás Prieto-Rumeau & Onésimo Hernández-Lerma, 2016. "Uniform ergodicity of continuous-time controlled Markov chains: A survey and new results," Annals of Operations Research, Springer, vol. 241(1), pages 249-293, June.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:annopr:v:247:y:2016:i:1:d:10.1007_s10479-015-1893-6. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.