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

Multiserver retrial queues with after-call work

Tuan Phung-Duc 1, and  Ken’ichi Kawanishi 2,

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.
Keywords: level-dependent QBD process, after-call work, Multiserver retrial queue, truncation method., call center.
Mathematics Subject Classification: Primary: 60K25, 68M20; Secondary: 90B2.
Citation: Tuan Phung-Duc, Ken’ichi Kawanishi. Multiserver retrial queues with after-call work. Numerical Algebra, Control and 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), 41-56. 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), 217-229. 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), 497-525. doi: 10.1080/15326349508807357.

[4]

J. E. Diamond and A. S. Alfa, The MAP/PH/1 retrial queue, Stochastic Models, 14 (1998), 1151-1177. doi: 10.1080/15326349808807518.

[5]

G. I. Falin and J. G. C. Templeton, "Retrial Queues," Chapman & Hall, London, 1997.

[6]

M. J. Fischer, D. A. Garbin and A. Gharakhanian, Performance modeling of distributed automatic call distribution systems, Telecommunications Systems, 9 (1998), 133-152. 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), 79-141. 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, Torremolinos, (1979), 1-9.

[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), 93-122. 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), 1111-1118. doi: 10.1239/jap/1197908829.

[11]

G. Koole and A. Mandelbaum, Queueing models of call centers: an introduction, Annals of Operations Research, 113 (2002), 41-59. doi: 10.1023/A:1020949626017.

[12]

J. D. C. Little, A proof for the queuing formula: $L = \lambda W$, Operations Research, 9 (1961), 383-387. doi: 10.1287/opre.9.3.383.

[13]

M. F. Neuts, "Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach," Johns Hopkins University Press, Baltimore, MD, 1981.

[14]

M. F. Neuts and B. M. Rao, Numerical investigation of a multiserver retrial model, Queueing Systems, 7 (1990), 169-190. 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), 125-136. doi: 10.1017/S0305004100051562.

show all references

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), 41-56. 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), 217-229. 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), 497-525. doi: 10.1080/15326349508807357.

[4]

J. E. Diamond and A. S. Alfa, The MAP/PH/1 retrial queue, Stochastic Models, 14 (1998), 1151-1177. doi: 10.1080/15326349808807518.

[5]

G. I. Falin and J. G. C. Templeton, "Retrial Queues," Chapman & Hall, London, 1997.

[6]

M. J. Fischer, D. A. Garbin and A. Gharakhanian, Performance modeling of distributed automatic call distribution systems, Telecommunications Systems, 9 (1998), 133-152. 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), 79-141. 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, Torremolinos, (1979), 1-9.

[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), 93-122. 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), 1111-1118. doi: 10.1239/jap/1197908829.

[11]

G. Koole and A. Mandelbaum, Queueing models of call centers: an introduction, Annals of Operations Research, 113 (2002), 41-59. doi: 10.1023/A:1020949626017.

[12]

J. D. C. Little, A proof for the queuing formula: $L = \lambda W$, Operations Research, 9 (1961), 383-387. doi: 10.1287/opre.9.3.383.

[13]

M. F. Neuts, "Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach," Johns Hopkins University Press, Baltimore, MD, 1981.

[14]

M. F. Neuts and B. M. Rao, Numerical investigation of a multiserver retrial model, Queueing Systems, 7 (1990), 169-190. 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), 125-136. doi: 10.1017/S0305004100051562.

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