October  2017, 13(4): 2093-2146. doi: 10.3934/jimo.2017033

Light-tailed asymptotics of GI/G/1-type Markov chains

1. 

NTT Network Technology Laboratories, NTT Corporation, Tokyo 180–8585, Japan

2. 

Department of Systems Science, Graduate School of Informatics, Kyoto University, Kyoto 606–8501, Japan

Received  October 2015 Revised  September 2016 Published  April 2017

This paper studies the light-tailed asymptotics of the stationary distribution of the GI/G/1-type Markov chain. We consider three cases:(ⅰ) the tail decay rate is determined by a certain parameter $\theta$ associated with the transition block matrices $\{\boldsymbol{A}(k);k=0,\pm1,\pm2,\dots\}$ in the non-boundary levels; (ⅱ) by the convergence radius of the generating function of the transition block matrices $\{\boldsymbol{B}(k);k=1,2,\dots\}$ in the boundary level; and (ⅲ) by the convergence radius of $\sum_{k=1}^{\infty}z^k \boldsymbol{A}(k)$. In the case (ⅰ), we extend the existing asymptotic formula for the M/G/1-type Markov chain to the GI/G/1-type one. In the case (ⅱ), we present general asymptotic formulas that include, as special cases, the existing results in the literature. In the case (ⅲ), we derive new asymptotic formulas. As far as we know, such formulas have not been reported in the literature.

Citation: Tatsuaki Kimura, Hiroyuki Masuyama, Yutaka Takahashi. Light-tailed asymptotics of GI/G/1-type Markov chains. Journal of Industrial & Management Optimization, 2017, 13 (4) : 2093-2146. doi: 10.3934/jimo.2017033
References:
[1]

J. AbateG. L. Choudhury and W. Whitt, Asymptotics for steady-state tail probabilities in structured Markov queueing models, Stochastic Models, 10 (1994), 99-143.  doi: 10.1080/15326349408807290.  Google Scholar

[2]

L. A. AndrewK. E. Chu and P. Lancaster, Derivatives of eigenvalues and eigenvectors of matrix functions, SIAM Journal on Matrix Analysis and Applications, 14 (1993), 903-926.  doi: 10.1137/0614061.  Google Scholar

[3]

S. AsmussenL. F. Henriksen and C. Klüppelberg, Large claims approximations for risk processes in a Markovian environment, Stochastic Processes and their Applications, 54 (1994), 29-43.  doi: 10.1016/0304-4149(93)00003-X.  Google Scholar

[4]

S. Asmussen and J. R. Møller, Tail asymptotics for M/G/1 type queueing processes with subexponential increments, Queueing Systems, 33 (1999), 153-176.  doi: 10.1023/A:1019172028316.  Google Scholar

[5]

S. Asmussen, Applied Probability and Queues, 2nd edition, Springer, New York, 2003.  Google Scholar

[6]

R. B. Bapat and T. E. S. Raghavan, Nonnegative Matrices and Applications, Cambridge University Press, Cambridge, UK, 1997. doi: 10.1017/CBO9780511529979.  Google Scholar

[7]

P. Embrechts, C. Klüppelberg and T. Mikosch, Modelling Extremal Events for Insurance and Finance, Springer, Berlin, 1997. doi: 10.1007/978-3-642-33483-2.  Google Scholar

[8]

E. Falkenberg, On the asymptotic behaviour of the stationary distribution of Markov chains of M/G/1-type, Stochastic Models, 10 (1994), 75-97.  doi: 10.1080/15326349408807289.  Google Scholar

[9]

S. Foss, D. Korshunov and S. Zachary, An Introduction to Heavy-Tailed and Subexponential Distributions, Springer, New York, 2011. doi: 10.1007/978-1-4419-9473-8.  Google Scholar

[10]

H. GailS. L. Hantler and B. A. Taylor, Matrix-geometric invariant measures for G/M/1 type Markov chains, Stochastic Models, 14 (1997), 537-569.  doi: 10.1080/15326349808807487.  Google Scholar

[11]

H. GailS. L. Hantler and B. A. Taylor, Use of characteristic roots for solving infinite state Markov chains, Computational Probability, 24 (2000), 205-255.  doi: 10.1007/978-1-4757-4828-4_7.  Google Scholar

[12]

W. K. Grassmann and D. P. Heyman, Equilibrium distribution of block-structured Markov chains with repeating rows, Journal of Applied Probability, 27 (1990), 557-576.  doi: 10.1017/S0021900200039115.  Google Scholar

[13]

R. A. Horn and C. R. Johnson, Matrix Analysis, Paperback edition, Cambridge University Press, New York, 1990.  Google Scholar

[14]

R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Paperback edition, Cambridge University Press, New York, 1994.  Google Scholar

[15]

P. R. Jelenković and A. A. Lazar, Subexponential asymptotics of a Markov-modulated random walk with queueing applications, Journal of Applied Probability, 35 (1998), 325-347.  doi: 10.1017/S0021900200014984.  Google Scholar

[16]

B. Kim and J. Kim, A note on the subexponential asymptotics of the stationary distribution of M/G/1 type Markov chains, European Journal of Operational Research, 220 (2012), 132-134.  doi: 10.1016/j.ejor.2012.01.016.  Google Scholar

[17]

T. KimuraK. DaikokuH. Masuyama and Y. Takahashi, Light-tailed asymptotics of stationary tail probability vectors of Markov chains of M/G/1 type, Stochastic Models, 26 (2010), 505-548.  doi: 10.1080/15326349.2010.519661.  Google Scholar

[18]

T. KimuraH. Masuyama and Y. Takahashi, Subexponential asymptotics of the stationary distributions of GI/G/1-type Markov chains, Stochastic Models, 29 (2013), 190-239.  doi: 10.1080/15326349.2013.783286.  Google Scholar

[19]

T. KimuraH. Masuyama and Y. Takahashi, Corrigendum to "Subexponential asymptotics of the stationary distributions of GI/G/1-type Markov chains", Stochastic Models, 31 (2015), 673-677.  doi: 10.1080/15326349.2015.1075891.  Google Scholar

[20]

J. F. C. Kingman, A convexity property of positive matrices, Quarterly Journal of Mathematics, 12 (1961), 283-284.  doi: 10.1093/qmath/12.1.283.  Google Scholar

[21]

C. Klüppelberg, Subexponential distributions and integrated tails, Journal of Applied Probability, 25 (1988), 132-141.  doi: 10.1017/S0021900200040705.  Google Scholar

[22]

G. Latouche and V. Ramaswami, Introduction to Matrix Analytic Methods in Stochastic Modeling, ASA-SIAM Series on Statistics and Applied Probability, SIAM, Philadelphia, PA, 1999. doi: 10.1137/1.9780898719734.  Google Scholar

[23]

H. LiM. Miyazawa and Y. Q. Zhao, Geometric decay in a QBD process with countable background states with applications to a join-the-shortest-queue model, Stochastic Models, 23 (2007), 413-438.  doi: 10.1080/15326340701471042.  Google Scholar

[24]

Q.-L. Li and Y. Q. Zhao, Heavy-tailed asymptotics of stationary probability vectors of Markov chains of GI/G/1 type, Advances in Applied Probability, 37 (2005), 482-509.  doi: 10.1017/S0001867800000276.  Google Scholar

[25]

Q.-L. Li and Y. Q. Zhao, Light-tailed asymptotics of stationary probability vectors of Markov chains of GI/G/1 type, Advances in Applied Probability, 37 (2005), 1075-1093.  doi: 10.1017/S0001867800000677.  Google Scholar

[26]

H. Masuyama, Subexponential asymptotics of the stationary distributions of M/G/1-Type Markov chains, European Journal of Operational Research, 213 (2011), 509-516.  doi: 10.1016/j.ejor.2011.03.038.  Google Scholar

[27]

H. Masuyama, A sufficient condition for subexponential asymptotics of GI/G/1-type Markov chains with queueing applications, Ann. Oper. Res., 247 (2016), 65-95, arXiv: 1310.4590. doi: 10.1007/s10479-015-1893-6.  Google Scholar

[28]

H. Masuyama, Tail asymptotics for cumulative processes sampled at heavy-tailed random times with applications to queueing models in Markovian environments, Journal of the Operations Research Society of Japan, 56 (2013), 257-308.   Google Scholar

[29]

H. MasuyamaB. Liu and T. Takine, Subexponential asymptotics of the BMAP/GI/1 queue, Journal of the Operations Research Society of Japan, 52 (2009), 377-401.   Google Scholar

[30]

M. Miyazawa, A Markov renewal approach to M/G/1 type queues with countably many background states, Queueing Systems, 46 (2004), 177-196.  doi: 10.1023/B:QUES.0000021148.33178.0f.  Google Scholar

[31]

M. Miyazawa, Tail decay rates in double QBD processes and related reflected random walks, Stochastic Models, 34 (2009), 547-575.  doi: 10.1287/moor.1090.0375.  Google Scholar

[32]

M. Miyazawa and Y. Q. Zhao, The stationary tail asymptotics in the GI/G/1-type queue with countably many background states, Advances in Applied Probability, 36 (2004), 1231-1251.  doi: 10.1017/S0001867800013380.  Google Scholar

[33]

J. R. Møller, Tail asymptotics for M/G/1-type queueing processes with light-tailed increments, Operations Research Letters, 28 (2001), 181-185.  doi: 10.1016/S0167-6377(01)00061-X.  Google Scholar

[34]

M. F. Neuts, Structured Stochastic Matrices of M/G/1 Type and Their Applications, Marcel Dekker, New York, 1989.  Google Scholar

[35]

T. Ozawa, Asymptotics for the stationary distribution in a discrete-time two-dimensional quasi-birth-and-death process, Queueing Systems, 74 (2013), 109-149.  doi: 10.1007/s11134-012-9323-9.  Google Scholar

[36]

E. J. G. Pitman, Subexponential distribution functions, Journal of the Australian Mathematical Society, A29 (1980), 337-347.  doi: 10.1017/S1446788700021340.  Google Scholar

[37]

E. Seneta, Nonnegative Matrices and Markov Chains, 2nd edition, Springer, New York, 1981. doi: 10.1007/0-387-32792-4.  Google Scholar

[38]

Y. Tai, Tail Asymptotics and Ergodicity for the GI/G/1-type Markov Chains, Dissertation, Carleton University, Ottawa, Canada, 2009.  Google Scholar

[39]

T. Takine, Geometric and subexponential asymptotics of Markov chains of M/G/1 type, Mathematics of Operations Research, 29 (2004), 624-648.  doi: 10.1287/moor.1030.0083.  Google Scholar

[40]

Y. Q. ZhaoW. Li and W. J. Braun, Infinite block-structured transition matrices and their properties, Advances in Applied Probability, 30 (1998), 365-384.  doi: 10.1017/S0001867800047339.  Google Scholar

[41]

Y. Q. ZhaoW. Li and A. S. Alfa, Duality results for block-structured transition matrices, Journal of Applied Probability, 36 (1999), 1045-1057.  doi: 10.1017/S002190020001785X.  Google Scholar

[42]

Y. Q. ZhaoW. Li and W. J. Braun, Censoring, factorizations, and spectral analysis for transition matrices with block-repeating entries, Methodology and Computing in Applied Probability, 5 (2003), 35-58.  doi: 10.1023/A:1024125320911.  Google Scholar

show all references

References:
[1]

J. AbateG. L. Choudhury and W. Whitt, Asymptotics for steady-state tail probabilities in structured Markov queueing models, Stochastic Models, 10 (1994), 99-143.  doi: 10.1080/15326349408807290.  Google Scholar

[2]

L. A. AndrewK. E. Chu and P. Lancaster, Derivatives of eigenvalues and eigenvectors of matrix functions, SIAM Journal on Matrix Analysis and Applications, 14 (1993), 903-926.  doi: 10.1137/0614061.  Google Scholar

[3]

S. AsmussenL. F. Henriksen and C. Klüppelberg, Large claims approximations for risk processes in a Markovian environment, Stochastic Processes and their Applications, 54 (1994), 29-43.  doi: 10.1016/0304-4149(93)00003-X.  Google Scholar

[4]

S. Asmussen and J. R. Møller, Tail asymptotics for M/G/1 type queueing processes with subexponential increments, Queueing Systems, 33 (1999), 153-176.  doi: 10.1023/A:1019172028316.  Google Scholar

[5]

S. Asmussen, Applied Probability and Queues, 2nd edition, Springer, New York, 2003.  Google Scholar

[6]

R. B. Bapat and T. E. S. Raghavan, Nonnegative Matrices and Applications, Cambridge University Press, Cambridge, UK, 1997. doi: 10.1017/CBO9780511529979.  Google Scholar

[7]

P. Embrechts, C. Klüppelberg and T. Mikosch, Modelling Extremal Events for Insurance and Finance, Springer, Berlin, 1997. doi: 10.1007/978-3-642-33483-2.  Google Scholar

[8]

E. Falkenberg, On the asymptotic behaviour of the stationary distribution of Markov chains of M/G/1-type, Stochastic Models, 10 (1994), 75-97.  doi: 10.1080/15326349408807289.  Google Scholar

[9]

S. Foss, D. Korshunov and S. Zachary, An Introduction to Heavy-Tailed and Subexponential Distributions, Springer, New York, 2011. doi: 10.1007/978-1-4419-9473-8.  Google Scholar

[10]

H. GailS. L. Hantler and B. A. Taylor, Matrix-geometric invariant measures for G/M/1 type Markov chains, Stochastic Models, 14 (1997), 537-569.  doi: 10.1080/15326349808807487.  Google Scholar

[11]

H. GailS. L. Hantler and B. A. Taylor, Use of characteristic roots for solving infinite state Markov chains, Computational Probability, 24 (2000), 205-255.  doi: 10.1007/978-1-4757-4828-4_7.  Google Scholar

[12]

W. K. Grassmann and D. P. Heyman, Equilibrium distribution of block-structured Markov chains with repeating rows, Journal of Applied Probability, 27 (1990), 557-576.  doi: 10.1017/S0021900200039115.  Google Scholar

[13]

R. A. Horn and C. R. Johnson, Matrix Analysis, Paperback edition, Cambridge University Press, New York, 1990.  Google Scholar

[14]

R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Paperback edition, Cambridge University Press, New York, 1994.  Google Scholar

[15]

P. R. Jelenković and A. A. Lazar, Subexponential asymptotics of a Markov-modulated random walk with queueing applications, Journal of Applied Probability, 35 (1998), 325-347.  doi: 10.1017/S0021900200014984.  Google Scholar

[16]

B. Kim and J. Kim, A note on the subexponential asymptotics of the stationary distribution of M/G/1 type Markov chains, European Journal of Operational Research, 220 (2012), 132-134.  doi: 10.1016/j.ejor.2012.01.016.  Google Scholar

[17]

T. KimuraK. DaikokuH. Masuyama and Y. Takahashi, Light-tailed asymptotics of stationary tail probability vectors of Markov chains of M/G/1 type, Stochastic Models, 26 (2010), 505-548.  doi: 10.1080/15326349.2010.519661.  Google Scholar

[18]

T. KimuraH. Masuyama and Y. Takahashi, Subexponential asymptotics of the stationary distributions of GI/G/1-type Markov chains, Stochastic Models, 29 (2013), 190-239.  doi: 10.1080/15326349.2013.783286.  Google Scholar

[19]

T. KimuraH. Masuyama and Y. Takahashi, Corrigendum to "Subexponential asymptotics of the stationary distributions of GI/G/1-type Markov chains", Stochastic Models, 31 (2015), 673-677.  doi: 10.1080/15326349.2015.1075891.  Google Scholar

[20]

J. F. C. Kingman, A convexity property of positive matrices, Quarterly Journal of Mathematics, 12 (1961), 283-284.  doi: 10.1093/qmath/12.1.283.  Google Scholar

[21]

C. Klüppelberg, Subexponential distributions and integrated tails, Journal of Applied Probability, 25 (1988), 132-141.  doi: 10.1017/S0021900200040705.  Google Scholar

[22]

G. Latouche and V. Ramaswami, Introduction to Matrix Analytic Methods in Stochastic Modeling, ASA-SIAM Series on Statistics and Applied Probability, SIAM, Philadelphia, PA, 1999. doi: 10.1137/1.9780898719734.  Google Scholar

[23]

H. LiM. Miyazawa and Y. Q. Zhao, Geometric decay in a QBD process with countable background states with applications to a join-the-shortest-queue model, Stochastic Models, 23 (2007), 413-438.  doi: 10.1080/15326340701471042.  Google Scholar

[24]

Q.-L. Li and Y. Q. Zhao, Heavy-tailed asymptotics of stationary probability vectors of Markov chains of GI/G/1 type, Advances in Applied Probability, 37 (2005), 482-509.  doi: 10.1017/S0001867800000276.  Google Scholar

[25]

Q.-L. Li and Y. Q. Zhao, Light-tailed asymptotics of stationary probability vectors of Markov chains of GI/G/1 type, Advances in Applied Probability, 37 (2005), 1075-1093.  doi: 10.1017/S0001867800000677.  Google Scholar

[26]

H. Masuyama, Subexponential asymptotics of the stationary distributions of M/G/1-Type Markov chains, European Journal of Operational Research, 213 (2011), 509-516.  doi: 10.1016/j.ejor.2011.03.038.  Google Scholar

[27]

H. Masuyama, A sufficient condition for subexponential asymptotics of GI/G/1-type Markov chains with queueing applications, Ann. Oper. Res., 247 (2016), 65-95, arXiv: 1310.4590. doi: 10.1007/s10479-015-1893-6.  Google Scholar

[28]

H. Masuyama, Tail asymptotics for cumulative processes sampled at heavy-tailed random times with applications to queueing models in Markovian environments, Journal of the Operations Research Society of Japan, 56 (2013), 257-308.   Google Scholar

[29]

H. MasuyamaB. Liu and T. Takine, Subexponential asymptotics of the BMAP/GI/1 queue, Journal of the Operations Research Society of Japan, 52 (2009), 377-401.   Google Scholar

[30]

M. Miyazawa, A Markov renewal approach to M/G/1 type queues with countably many background states, Queueing Systems, 46 (2004), 177-196.  doi: 10.1023/B:QUES.0000021148.33178.0f.  Google Scholar

[31]

M. Miyazawa, Tail decay rates in double QBD processes and related reflected random walks, Stochastic Models, 34 (2009), 547-575.  doi: 10.1287/moor.1090.0375.  Google Scholar

[32]

M. Miyazawa and Y. Q. Zhao, The stationary tail asymptotics in the GI/G/1-type queue with countably many background states, Advances in Applied Probability, 36 (2004), 1231-1251.  doi: 10.1017/S0001867800013380.  Google Scholar

[33]

J. R. Møller, Tail asymptotics for M/G/1-type queueing processes with light-tailed increments, Operations Research Letters, 28 (2001), 181-185.  doi: 10.1016/S0167-6377(01)00061-X.  Google Scholar

[34]

M. F. Neuts, Structured Stochastic Matrices of M/G/1 Type and Their Applications, Marcel Dekker, New York, 1989.  Google Scholar

[35]

T. Ozawa, Asymptotics for the stationary distribution in a discrete-time two-dimensional quasi-birth-and-death process, Queueing Systems, 74 (2013), 109-149.  doi: 10.1007/s11134-012-9323-9.  Google Scholar

[36]

E. J. G. Pitman, Subexponential distribution functions, Journal of the Australian Mathematical Society, A29 (1980), 337-347.  doi: 10.1017/S1446788700021340.  Google Scholar

[37]

E. Seneta, Nonnegative Matrices and Markov Chains, 2nd edition, Springer, New York, 1981. doi: 10.1007/0-387-32792-4.  Google Scholar

[38]

Y. Tai, Tail Asymptotics and Ergodicity for the GI/G/1-type Markov Chains, Dissertation, Carleton University, Ottawa, Canada, 2009.  Google Scholar

[39]

T. Takine, Geometric and subexponential asymptotics of Markov chains of M/G/1 type, Mathematics of Operations Research, 29 (2004), 624-648.  doi: 10.1287/moor.1030.0083.  Google Scholar

[40]

Y. Q. ZhaoW. Li and W. J. Braun, Infinite block-structured transition matrices and their properties, Advances in Applied Probability, 30 (1998), 365-384.  doi: 10.1017/S0001867800047339.  Google Scholar

[41]

Y. Q. ZhaoW. Li and A. S. Alfa, Duality results for block-structured transition matrices, Journal of Applied Probability, 36 (1999), 1045-1057.  doi: 10.1017/S002190020001785X.  Google Scholar

[42]

Y. Q. ZhaoW. Li and W. J. Braun, Censoring, factorizations, and spectral analysis for transition matrices with block-repeating entries, Methodology and Computing in Applied Probability, 5 (2003), 35-58.  doi: 10.1023/A:1024125320911.  Google Scholar

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