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2011, 1(4): 639-656. doi: 10.3934/naco.2011.1.639

## Multiserver retrial queues with after-call work

 1 Graduate School of Informatics, Kyoto University, Yoshida-Honmachi, Sakyo-ku, Kyoto 606-8501 2 Department of Computer Science, Gunma University, Kiryu-City, 376-8515

Received  June 2011 Revised  August 2011 Published  November 2011

This paper considers a multiserver queueing system with finite capacity. Customers that find the service facility being fully occupied are blocked and enter a virtual waiting room (called orbit). Blocked customers stay in the orbit for an exponentially distributed time and retry to occupy an idle server again. After completing a service, the server starts an additional job that we call an after-call work. We formulate the queueing system using a continuous-time level-dependent quasi-birth-and-death process, for which a sufficient condition for the ergodicity is derived. We obtain an approximation to the stationary distribution by a direct truncation method whose truncation point is simply determined using an asymptotic analysis of a single server retrial queue. Some numerical examples are presented in order to show the influence of parameters on the performance of the system.
Citation: Tuan Phung-Duc, Ken’ichi Kawanishi. Multiserver retrial queues with after-call work. Numerical Algebra, Control & Optimization, 2011, 1 (4) : 639-656. doi: 10.3934/naco.2011.1.639
##### References:
 [1] J. R. Artalejo and M. Pozo, Numerical calculation of the stationary distribution of the main multiserver retrial queue,, Annals of Operations Research, 116 (2002), 41. doi: 10.1023/A:1021359709489. [2] J. R. Artalejo and V. Pla, On the impact of customer balking, impatience and retrials in telecommunication systems,, Computers & Mathematics with Applications, 57 (2009), 217. doi: 10.1016/j.camwa.2008.10.084. [3] L. Bright and G. P. Taylor, Calculating the equilibrium distribution in level dependent quasi-birth-and-death processes,, Stochastic Models, 11 (1995), 497. doi: 10.1080/15326349508807357. [4] J. E. Diamond and A. S. Alfa, The MAP/PH/1 retrial queue,, Stochastic Models, 14 (1998), 1151. doi: 10.1080/15326349808807518. [5] G. I. Falin and J. G. C. Templeton, "Retrial Queues,", Chapman & Hall, (1997). [6] M. J. Fischer, D. A. Garbin and A. Gharakhanian, Performance modeling of distributed automatic call distribution systems,, Telecommunications Systems, 9 (1998), 133. doi: 10.1023/A:1019139721840. [7] N. Gans, G. Koole and A. Mandelbaum, Telephone call centers: tutorial, review, and research prospects,, Manufacturing & Service Operations Management, 5 (2003), 79. doi: 10.1287/msom.5.2.79.16071. [8] W. M. Jolley and R. J. Harris, Analysis of post-call activity in queueing systems,, Proceedings of the 9th International Teletraffic Congress, (1979), 1. [9] K. Kawanishi, On the counting process for a class of Markovian arrival processes with an application to a queueing system,, Queueing Systems, 49 (2005), 93. doi: 10.1007/s11134-005-6478-7. [10] J. Kim, B. Kim and S.-S. Ko, Tail asymptotics for the queue size distribution in an M/G/1 retrial queue,, Journal of Applied Probability, 44 (2007), 1111. doi: 10.1239/jap/1197908829. [11] G. Koole and A. Mandelbaum, Queueing models of call centers: an introduction,, Annals of Operations Research, 113 (2002), 41. doi: 10.1023/A:1020949626017. [12] J. D. C. Little, A proof for the queuing formula: $L = \lambda W$,, Operations Research, 9 (1961), 383. doi: 10.1287/opre.9.3.383. [13] M. F. Neuts, "Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach,", Johns Hopkins University Press, (1981). [14] M. F. Neuts and B. M. Rao, Numerical investigation of a multiserver retrial model,, Queueing Systems, 7 (1990), 169. doi: 10.1007/BF01158473. [15] T. Phung-Duc, H. Masuyama, S. Kasahara and Y. Takahashi, State-dependent M/M/$c$/$c+r$ retrial queues with Bernoulli abandonment,, Journal of Industrial and Management Optimization, 6 (): 517. doi: 10.3934/jimo.2010.6.517. [16] T. Phung-Duc, H. Masuyama, S. Kasahara and Y. Takahashi, A simple algorithm for the rate matrices of level-dependent QBD processes,, Proceedings of the 5th International Conference on Queueing Theory and Network Applications, (): 46. doi: 10.1145/1837856.1837864. [17] T. Phung-Duc, H. Masuyama, S. Kasahara and Y. Takahashi, A matrix continued fraction approach to multi-server retrial queues,, to appear in Annals of Operations Research, (2011). doi: 10.1007/s10479-011-0840-4. [18] R. L. Tweedie, Sufficient conditions for regularity, recurrence and ergodicity and Markov processes,, Mathematical Proceedings of the Cambridge Philosophical Society, 78 (1975), 125. doi: 10.1017/S0305004100051562.

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##### References:
 [1] J. R. Artalejo and M. Pozo, Numerical calculation of the stationary distribution of the main multiserver retrial queue,, Annals of Operations Research, 116 (2002), 41. doi: 10.1023/A:1021359709489. [2] J. R. Artalejo and V. Pla, On the impact of customer balking, impatience and retrials in telecommunication systems,, Computers & Mathematics with Applications, 57 (2009), 217. doi: 10.1016/j.camwa.2008.10.084. [3] L. Bright and G. P. Taylor, Calculating the equilibrium distribution in level dependent quasi-birth-and-death processes,, Stochastic Models, 11 (1995), 497. doi: 10.1080/15326349508807357. [4] J. E. Diamond and A. S. Alfa, The MAP/PH/1 retrial queue,, Stochastic Models, 14 (1998), 1151. doi: 10.1080/15326349808807518. [5] G. I. Falin and J. G. C. Templeton, "Retrial Queues,", Chapman & Hall, (1997). [6] M. J. Fischer, D. A. Garbin and A. Gharakhanian, Performance modeling of distributed automatic call distribution systems,, Telecommunications Systems, 9 (1998), 133. doi: 10.1023/A:1019139721840. [7] N. Gans, G. Koole and A. Mandelbaum, Telephone call centers: tutorial, review, and research prospects,, Manufacturing & Service Operations Management, 5 (2003), 79. doi: 10.1287/msom.5.2.79.16071. [8] W. M. Jolley and R. J. Harris, Analysis of post-call activity in queueing systems,, Proceedings of the 9th International Teletraffic Congress, (1979), 1. [9] K. Kawanishi, On the counting process for a class of Markovian arrival processes with an application to a queueing system,, Queueing Systems, 49 (2005), 93. doi: 10.1007/s11134-005-6478-7. [10] J. Kim, B. Kim and S.-S. Ko, Tail asymptotics for the queue size distribution in an M/G/1 retrial queue,, Journal of Applied Probability, 44 (2007), 1111. doi: 10.1239/jap/1197908829. [11] G. Koole and A. Mandelbaum, Queueing models of call centers: an introduction,, Annals of Operations Research, 113 (2002), 41. doi: 10.1023/A:1020949626017. [12] J. D. C. Little, A proof for the queuing formula: $L = \lambda W$,, Operations Research, 9 (1961), 383. doi: 10.1287/opre.9.3.383. [13] M. F. Neuts, "Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach,", Johns Hopkins University Press, (1981). [14] M. F. Neuts and B. M. Rao, Numerical investigation of a multiserver retrial model,, Queueing Systems, 7 (1990), 169. doi: 10.1007/BF01158473. [15] T. Phung-Duc, H. Masuyama, S. Kasahara and Y. Takahashi, State-dependent M/M/$c$/$c+r$ retrial queues with Bernoulli abandonment,, Journal of Industrial and Management Optimization, 6 (): 517. doi: 10.3934/jimo.2010.6.517. [16] T. Phung-Duc, H. Masuyama, S. Kasahara and Y. Takahashi, A simple algorithm for the rate matrices of level-dependent QBD processes,, Proceedings of the 5th International Conference on Queueing Theory and Network Applications, (): 46. doi: 10.1145/1837856.1837864. [17] T. Phung-Duc, H. Masuyama, S. Kasahara and Y. Takahashi, A matrix continued fraction approach to multi-server retrial queues,, to appear in Annals of Operations Research, (2011). doi: 10.1007/s10479-011-0840-4. [18] R. L. Tweedie, Sufficient conditions for regularity, recurrence and ergodicity and Markov processes,, Mathematical Proceedings of the Cambridge Philosophical Society, 78 (1975), 125. doi: 10.1017/S0305004100051562.
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