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

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  • 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
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    References listed on IDEAS

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    1. 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.
    2. 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.
    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. 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.
    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.
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    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.

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