# American Institute of Mathematical Sciences

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

 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.
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

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##### 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
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