The analysis of a discrete time finite-buffer queue with working vacations under Markovian arrival process and PH-service time

Ye, Qingqing; Liu, Liwei; Jiang, Tao; Chang, Baoxian

doi:10.1051/ro/2019020

Ye, Qingqing ; Liu, Liwei ; Jiang, Tao ; Chang, Baoxian

RAIRO - Operations Research - Recherche Opérationnelle, Tome 54 (2020) no. 3, pp. 675-691.

Suite au passage du modèle économique de la revue en S20, le texte intégral des articles des années 2019 et 2020 est accessible uniquement sur le site de l'éditeur et est réservé aux abonnés.

Résumé

In this paper, we study the discrete-time MAP/PH/1 queue with multiple working vacations and finite buffer N. Using the Matrix-Geometric Combination method, we obtain the stationary probability vectors of this model, which can be expressed as a linear combination of two matrix-geometric vectors. Furthermore, we obtain some performance measures including the loss probability and give the limit of loss probability as finite buffer N goes to infinite. Waiting time distribution is derived by using the absorbing Markov chain. Moreover, we obtain the number of customers served in the busy period. At last, some numerical examples are presented to verify the results we obtained and show the impact of parameter N on performance measures.

MR Zbl

DOI : 10.1051/ro/2019020

Classification : 60K25, 68M20
Mots-clés : Working vacation, finite buffer, matrix-geometric combination method, Sojourn time, busy period

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     title = {The analysis of a discrete time finite-buffer queue with working vacations under {Markovian} arrival process and {PH-service} time},
     journal = {RAIRO - Operations Research - Recherche Op\'erationnelle},
     pages = {675--691},
     publisher = {EDP-Sciences},
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Ye, Qingqing; Liu, Liwei; Jiang, Tao; Chang, Baoxian. The analysis of a discrete time finite-buffer queue with working vacations under Markovian arrival process and PH-service time. RAIRO - Operations Research - Recherche Opérationnelle, Tome 54 (2020) no. 3, pp. 675-691. doi : 10.1051/ro/2019020. http://www.numdam.org/articles/10.1051/ro/2019020/

Bibliographie
Cité par

A.S. Alfa, A discrete $M A P / P H / 1$ queue with vacations and exhaustive time-limited Service. Oper. Res. Lett. 18 (1995) 31–44. | DOI | MR | Zbl

A.S. Alfa, Discrete time analysis of $M A P / P H / 1$ vacation queue with gated time service. Queueing. Syst. 29 (1998) 35–54. | DOI | MR | Zbl

A.S. Alfa, Some decomposition results for a class of vacation queue. Oper. Res. Lett. 42 (2014) 140–144. | DOI | MR | Zbl

N. Akar, N.C. Oğuz and K. Sohraby, A novel computational method for solving finite QBD processes. Stoch. Models. 16 (2000) 273–311. | DOI | MR | Zbl

J.R. Artalejo, A. Gómez-Corral and Q.M. He, Markovian arrivals in stochastic modelling: a survey and some new results. Sort 34 (2000) 101–156. | MR | Zbl

Y. Baba, Analysis of a $G I / M / 1$ queue with multiple working vacations. Oper. Res. Lett. 33 (2005) 654–681. | DOI | MR | Zbl

R.H. Bartel and G.W. Stewart, Solution of the equation $A X + X B = C$ . Commun. ACM. 15 (1972) 820–826.

H. Bruneel and B.G. Kim, Discrete-Time Models for Communication Systems Including ATM. Kluwer Academic Publishers, Boston (1993). | DOI

S.R. Chakravarthy, Markovian Arrival Processes. Wiley Encyclopedia of Operations Research and Management Science (2010).

S. Chakravarthy, Analysis of the $M A P / P H / 1 / K$ queue with service control. Appl. Stochastic. Models. Data. Anal. 12 (1996) 179–191. | DOI | Zbl

S. Chakravarthy and S. Ozkar, $M A P / P H / 1$ queueing model with working vacation and crowdsourcing. Math. Appl. 44 (2016) 263–294. | MR

V. Chandrasekaran, K. Indhira, M. Saravanarajan and P. Rajadurai, A survey on working vacation queueing models. Int. J. Pure Appl. Math. 106 (2016) 33–41.

B.T. Doshi, Queueing systems with vacations—A survey. Queueing. Syst. 1 (1986) 29–66. | DOI | MR | Zbl

A.N. Dudin, A.V. Kazimirsky, V.I. Klimenok, L. Breuer and U. Krieger, The queueing model $M A P / P H / 1 / / N$ with feedback operating in a Markovian random environment. Aust. J. Stat. 34 (2005) 101–110.

H.R. Gail, S.L. Hantler and B.A. Taylor, Solutions of the basic matrix equation for $M / G / 1$ and $G / M / 1$ type Markov chain. Stoch. Models. 10 (1994) 1–43. | DOI | MR | Zbl

S. Gao, J. Wang and W. Li, An $M / G / 1$ retrial queue with general retrial times, working vacations and vacation interruption. Asia-Pac. J. Oper. Res. 31 (2014) 6–31. | MR | Zbl

A. Graham, Kronecker Products and Matrix Calculus: With Applications. John-Wiley, New York (1981). | MR | Zbl

A. Heindl, M. Telek, Output models of $M A P / P H / 1 (/ K)$ queues for an efficient network decomposition. Perform. Eval. 49 (2002) 321–339. | DOI | Zbl

J.D. Kim, D.W. Choi and K.C. Chae, Analysis of queue-length distribution of the $M / G / 1$ queue with Working Vacation. Hawaii Int. Conf. Stat. Related Fields 2003 (2003) 1191–1200.

C. Kim, S. Dudin and V. Klimenok, The $M A P / P H / 1 / N$ queue with flows of customers as a model for traffic control in telecommunication networks. Perform. Eval. 66 (2009) 564–579. | DOI

G. Latouche and G. Ramaswami, In: Vol. 5 of ASA-SIAM Series on Statistics and Applied Probability. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (1999). | Zbl

J. Li and N. Tian, Performance analysis of a $G / M / 1$ Queue with single working vacation. Appl. Math. Comput. 217 (2009) 4960–4971. | MR | Zbl

J. Li, N. Tian and W. Liu, Discrete-time $G I / G e o / 1$ queue with multiple working vacations. Queueing. Syst. 56 (2007) 53–63. | DOI | MR | Zbl

J. Li, N. Tian, Z.G. Zhang and H.P. Luh, Analysis of the $M / G / 1$ Queue with exponentially working vacations—a matrix analytic approach. Queueing. Syst. 61 (2011) 139–166. | DOI | MR | Zbl

D.M. Lucantoni, New results on the single server queue with a batch Markovian arrival process. Stoch. Models. 7 (1991) 1–46. | DOI | MR | Zbl

C. Luo, W. Li, K. Yu and C. Ding, The matrix-form solution for $G e o^{X} / G / 1 / N$ working vacation queue and its application to state-dependent cost control. Comput. Oper. Res. 67 (2016) 63–74. | DOI | MR

M.F. Neuts, Probability distributions of phase type. In: Liber Amicorum Prof. Emeritus H. Florin, University of Louvain, Belgium, 1975, 173–206.

M.F. Neuts, A versatile Markovian point process. J. Appl. Probab. 16 (1979) 764–779. | DOI | MR | Zbl

M.F. Neuts, Matrix-geometric Solution in Stochastic Model. Johns Hopkins University Press, Baltimore, MD (1981). | MR | Zbl

M.F. Neuts, Structured Stochastic Matrices of $M / G / 1$ Type and Their Applications. Marcel Dekker, NY (1989). | MR | Zbl

M.F. Neuts, Models based on the Markovian arrival process. IEICE. T. Commun. E75B (1992) 1255–1265.

L.D. Servi and S.G. Finn, $M / M / 1$ queues with working vacation $(M / M / 1 / W V)$ . Perform. Eval. 50 (2002) 41–52. | DOI

N. Tian and Z.G. Zhang, Vacation Queueing Models-Theory and Application. Springer-Verlag, New York (2006). | DOI | MR | Zbl

N. Tian, Z. Ma and M. Liu, The discrete time $G e o m / G e o m / 1$ queue with multiple working vacations. Appl. Math. Model. 32 (2007) 2941–2953. | DOI | MR | Zbl

N. Tian, J. Li and Z.G. Zhang, Matrix-analytic method and working vacation queues-survey. Int. J. Inform. Manage. Sci. 20 (2009) 603–633. | MR | Zbl

D. Wu and H. Takagi, $M / G / 1$ queue with multiple working vacations. Perform. Eval. 63 (2006) 654–681. | DOI

D. Yang and D. Wu, Cost-minimization analysis of a working vacation queue with $N$ -policy and server breakdowns. Comput. Ind. Eng. 82 (2015) 151–158. | DOI

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