American Institute of Mathematical Sciences

Advanced
July  2011, 7(3): 753-765. doi: 10.3934/jimo.2011.7.753

Stability of a retrial queueing network with different classes of customers and restricted resource pooling

Bara Kim 1,

1. 

Department of Mathematics, Korea University, 145 Anam-ro, Seongbuk-gu, Seoul, 136-713, South Korea

Received  September 2010 Revised  May 2011 Published  June 2011

We consider a retrial queueing network with different classes of customers and several servers. Each customer class is associated with a set of servers who can serve the class of customers. Customers of each class exogenously arrive according to a Poisson process. If an exogenously arriving customer finds upon his arrival any idle server who can serve the customer class, then he begins to receive a service by one of the available servers. Otherwise he joins the retrial group, and then tries his luck again after exponential time, the mean of which is determined by his customer class. Service times of each server are assumed to have general distribution. The retrial queueing network can be represented by a Markov process, with the number of customers of each class, and the customer class and the remaining service time of each busy server. Using the fluid limit technique, we find a necessary and sufficient condition for the positive Harris recurrence of the representing Markov process. This work is the first that applies the fluid limit technique to a model with retrial phenomena.
Keywords: Retrial queue, stability, fluid limit, positive Harris recurrence, resource pooling..
Mathematics Subject Classification: Primary: 60K25; Secondary: 60J2.
Citation: Bara Kim. Stability of a retrial queueing network with different classes of customers and restricted resource pooling. Journal of Industrial & Management Optimization, 2011, 7 (3) : 753-765. doi: 10.3934/jimo.2011.7.753
References:
[1]

M. Bramson, Stability of two families of queueing networks and a discussion of fluid limits,, Queueing Systems Theory Appl., 28 (1998), 7.  doi: 10.1023/A:1019182619288.  Google Scholar

[2]

B. D. Choi and B. Kim, Non-ergodicity criteria for denumerable continuous time Markov processes,, Operations Research Letters, 32 (2004), 574.  doi: 10.1016/j.orl.2004.03.001.  Google Scholar

[3]

J. G. Dai, On positive Harris recurrence of multiclass queueing network: A unified approach via fluid limit models,, Annals of Applied Probability, 5 (1995), 49.  doi: 10.1214/aoap/1177004828.  Google Scholar

[4]

J. G. Dai, A fluid-limit model criterion for instability of multiclass queueing networks,, Annals of Applied Probability, 6 (1996), 751.  doi: 10.1214/aoap/1034968225.  Google Scholar

[5]

J. G. Dai, J. J. Hasenbein and B. Kim, Stability of join-the-shortest-queue networks,, Queueing Systems, 57 (2007), 129.  doi: 10.1007/s11134-007-9046-5.  Google Scholar

[6]

G. I. Falin, A survey of retrial queues,, Queueing Systems Theory Appl., 7 (1990), 127.  doi: 10.1007/BF01158472.  Google Scholar

[7]

G. I. Falin and J. G. C. Templeton, "Retrial Queues,", Chapman & Hall, (1997).   Google Scholar

[8]

R. Foley and D. McDonald, Join the shortest queue: Stability and exact asymptotics,, Ann. Appl. Probab., 11 (2001), 569.  doi: 10.1214/aoap/1015345342.  Google Scholar

[9]

S. Foss and N. Chernova, On the stability of a partially accessible multi-station queue with state-dependent routing,, Queueing Systems Theory Appl., 29 (1998), 55.  doi: 10.1023/A:1019175812444.  Google Scholar

[10]

Q.-M. He, H. Li and Y. Q. Zhao, Ergodicity of the $BMAP$/$PH$/$s$/$s+K$ retrial queue PH-retrial times,, Queueing Systems Theory Appl., 35 (2000), 323.  doi: 10.1023/A:1019110631467.  Google Scholar

[11]

B. Kim and I. Lee, Tests for nonergodicity of denumerable continuous time Markov processes,, Computers and Mathematics with Applications, 55 (2008), 1310.  doi: 10.1016/j.camwa.2007.07.003.  Google Scholar

[12]

I. A. Kurkova, A load-balanced network with two servers,, Queueing Systems, 37 (2001), 379.  doi: 10.1023/A:1010841517511.  Google Scholar

[13]

T. Phung-Duc, H. Masuyama, S. Kasahara and Y. Takahashi, Performance analysis of optical burst switched networks with limited-range wavelength conversion, retransmission and burst segmentation,, Journal of the Operations Research Society of Japan, 52 (2009), 58.   Google Scholar

[14]

Yu. M. Sukhov and N. D. Vvedenskaya, Fast Jackson networks with dynamic routing,, Problems of Information Transmission, 38 (2002), 136.  doi: 10.1023/A:1020010710507.  Google Scholar

[15]

T. Yang and J. G. C. Templeton, A survey of retrial queues,, Queueing Systems Theory Appl., 2 (1987), 201.  doi: 10.1007/BF01158899.  Google Scholar

show all references

References:
[1]

M. Bramson, Stability of two families of queueing networks and a discussion of fluid limits,, Queueing Systems Theory Appl., 28 (1998), 7.  doi: 10.1023/A:1019182619288.  Google Scholar

[2]

B. D. Choi and B. Kim, Non-ergodicity criteria for denumerable continuous time Markov processes,, Operations Research Letters, 32 (2004), 574.  doi: 10.1016/j.orl.2004.03.001.  Google Scholar

[3]

J. G. Dai, On positive Harris recurrence of multiclass queueing network: A unified approach via fluid limit models,, Annals of Applied Probability, 5 (1995), 49.  doi: 10.1214/aoap/1177004828.  Google Scholar

[4]

J. G. Dai, A fluid-limit model criterion for instability of multiclass queueing networks,, Annals of Applied Probability, 6 (1996), 751.  doi: 10.1214/aoap/1034968225.  Google Scholar

[5]

J. G. Dai, J. J. Hasenbein and B. Kim, Stability of join-the-shortest-queue networks,, Queueing Systems, 57 (2007), 129.  doi: 10.1007/s11134-007-9046-5.  Google Scholar

[6]

G. I. Falin, A survey of retrial queues,, Queueing Systems Theory Appl., 7 (1990), 127.  doi: 10.1007/BF01158472.  Google Scholar

[7]

G. I. Falin and J. G. C. Templeton, "Retrial Queues,", Chapman & Hall, (1997).   Google Scholar

[8]

R. Foley and D. McDonald, Join the shortest queue: Stability and exact asymptotics,, Ann. Appl. Probab., 11 (2001), 569.  doi: 10.1214/aoap/1015345342.  Google Scholar

[9]

S. Foss and N. Chernova, On the stability of a partially accessible multi-station queue with state-dependent routing,, Queueing Systems Theory Appl., 29 (1998), 55.  doi: 10.1023/A:1019175812444.  Google Scholar

[10]

Q.-M. He, H. Li and Y. Q. Zhao, Ergodicity of the $BMAP$/$PH$/$s$/$s+K$ retrial queue PH-retrial times,, Queueing Systems Theory Appl., 35 (2000), 323.  doi: 10.1023/A:1019110631467.  Google Scholar

[11]

B. Kim and I. Lee, Tests for nonergodicity of denumerable continuous time Markov processes,, Computers and Mathematics with Applications, 55 (2008), 1310.  doi: 10.1016/j.camwa.2007.07.003.  Google Scholar

[12]

I. A. Kurkova, A load-balanced network with two servers,, Queueing Systems, 37 (2001), 379.  doi: 10.1023/A:1010841517511.  Google Scholar

[13]

T. Phung-Duc, H. Masuyama, S. Kasahara and Y. Takahashi, Performance analysis of optical burst switched networks with limited-range wavelength conversion, retransmission and burst segmentation,, Journal of the Operations Research Society of Japan, 52 (2009), 58.   Google Scholar

[14]

Yu. M. Sukhov and N. D. Vvedenskaya, Fast Jackson networks with dynamic routing,, Problems of Information Transmission, 38 (2002), 136.  doi: 10.1023/A:1020010710507.  Google Scholar

[15]

T. Yang and J. G. C. Templeton, A survey of retrial queues,, Queueing Systems Theory Appl., 2 (1987), 201.  doi: 10.1007/BF01158899.  Google Scholar

[1]

Arnaud Devos, Joris Walraevens, Tuan Phung-Duc, Herwig Bruneel. Analysis of the queue lengths in a priority retrial queue with constant retrial policy. Journal of Industrial & Management Optimization, 2019  doi: 10.3934/jimo.2019082
[2]

Feng Zhang, Jinting Wang, Bin Liu. On the optimal and equilibrium retrial rates in an unreliable retrial queue with vacations. Journal of Industrial & Management Optimization, 2012, 8 (4) : 861-875. doi: 10.3934/jimo.2012.8.861
[3]

Tuan Phung-Duc, Ken'ichi Kawanishi. Multiserver retrial queue with setup time and its application to data centers. Journal of Industrial & Management Optimization, 2019, 15 (1) : 15-35. doi: 10.3934/jimo.2018030
[4]

Dhanya Shajin, A. N. Dudin, Olga Dudina, A. Krishnamoorthy. A two-priority single server retrial queue with additional items. Journal of Industrial & Management Optimization, 2019  doi: 10.3934/jimo.2019085
[5]

Yi Peng, Jinbiao Wu. Analysis of a batch arrival retrial queue with impatient customers subject to the server disasters. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020067
[6]

Ke Sun, Jinting Wang, Zhe George Zhang. Strategic joining in a single-server retrial queue with batch service. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020120
[7]

Yi-Chiuan Chen. Bernoulli shift for second order recurrence relations near the anti-integrable limit. Discrete & Continuous Dynamical Systems - B, 2005, 5 (3) : 587-598. doi: 10.3934/dcdsb.2005.5.587
[8]

Olexiy V. Kapustyan, Pavlo O. Kasyanov, José Valero. Chain recurrence and structure of $ \omega $-limit sets of multivalued semiflows. Communications on Pure & Applied Analysis, 2020, 19 (4) : 2197-2217. doi: 10.3934/cpaa.2020096
[9]

Jeongsim Kim, Bara Kim. Stability of a queue with discriminatory random order service discipline and heterogeneous servers. Journal of Industrial & Management Optimization, 2017, 13 (3) : 1237-1254. doi: 10.3934/jimo.2016070
[10]

Shaojun Lan, Yinghui Tang. Performance analysis of a discrete-time $ Geo/G/1$ retrial queue with non-preemptive priority, working vacations and vacation interruption. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1421-1446. doi: 10.3934/jimo.2018102
[11]

Carlos Arnoldo Morales, M. J. Pacifico. Lyapunov stability of $\omega$-limit sets. Discrete & Continuous Dynamical Systems - A, 2002, 8 (3) : 671-674. doi: 10.3934/dcds.2002.8.671
[12]

José Luis Bravo, Manuel Fernández, Armengol Gasull. Stability of singular limit cycles for Abel equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 1873-1890. doi: 10.3934/dcds.2015.35.1873
[13]

Yihong Du, Yoshio Yamada. On the long-time limit of positive solutions to the degenerate logistic equation. Discrete & Continuous Dynamical Systems - A, 2009, 25 (1) : 123-132. doi: 10.3934/dcds.2009.25.123
[14]

Changjing Zhuge, Xiaojuan Sun, Jinzhi Lei. On positive solutions and the Omega limit set for a class of delay differential equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (9) : 2487-2503. doi: 10.3934/dcdsb.2013.18.2487
[15]

José A. Cañizo, Chuqi Cao, Josephine Evans, Havva Yoldaş. Hypocoercivity of linear kinetic equations via Harris's Theorem. Kinetic & Related Models, 2020, 13 (1) : 97-128. doi: 10.3934/krm.2020004
[16]

Emine Kaya, Eugenio Aulisa, Akif Ibragimov, Padmanabhan Seshaiyer. A stability estimate for fluid structure interaction problem with non-linear beam. Conference Publications, 2009, 2009 (Special) : 424-432. doi: 10.3934/proc.2009.2009.424
[17]

Nguyen Thieu Huy, Vu Thi Ngoc Ha, Pham Truong Xuan. Boundedness and stability of solutions to semi-linear equations and applications to fluid dynamics. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2103-2116. doi: 10.3934/cpaa.2016029
[18]

Haibo Cui, Haiyan Yin. Stability of the composite wave for the inflow problem on the micropolar fluid model. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1265-1292. doi: 10.3934/cpaa.2017062
[19]

Linfang Liu, Tomás Caraballo, Xianlong Fu. Exponential stability of an incompressible non-Newtonian fluid with delay. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4285-4303. doi: 10.3934/dcdsb.2018138
[20]

Zhi-Ying Sun, Lan Huang, Xin-Guang Yang. Exponential stability and regularity of compressible viscous micropolar fluid with cylinder symmetry. Electronic Research Archive, 2020, 28 (2) : 861-878. doi: 10.3934/era.2020045

2019 Impact Factor: 1.366

Tools

Metrics

  • PDF downloads (25)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences