# American Institute of Mathematical Sciences

October  2012, 8(4): 781-806. doi: 10.3934/jimo.2012.8.781

## Markovian retrial queues with two way communication

 1 Department of Statistics and O.R., Faculty of Mathematics, Complutense University of Madrid, Madrid 28040, Spain 2 Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Ookayama, Tokyo 152-8552, Japan

Received  September 2011 Revised  July 2012 Published  September 2012

In this paper, we first consider single server retrial queues with two way communication. Ingoing calls arrive at the server according to a Poisson process. Service times of these calls follow an exponential distribution. If the server is idle, it starts making an outgoing call in an exponentially distributed time. The duration of outgoing calls follows another exponential distribution. An ingoing arriving call that finds the server being busy joins an orbit and retries to enter the server after some exponentially distributed time. For this model, we present an extensive study in which we derive explicit expressions for the joint stationary distribution of the number of ingoing calls in the orbit and the state of the server, the partial factorial moments as well as their generating functions. Furthermore, we obtain asymptotic formulae for the joint stationary distribution and the factorial moments. We then extend the study to multiserver retrial queues with two way communication for which a necessary and sufficient condition for the stability, an explicit formula for average number of ingoing calls in the servers and a level-dependent quasi-birth-and-death process are derived.
Citation: Jesus R. Artalejo, Tuan Phung-Duc. Markovian retrial queues with two way communication. Journal of Industrial and Management Optimization, 2012, 8 (4) : 781-806. doi: 10.3934/jimo.2012.8.781
##### References:
 [1] Z. Aksin, M. Armony and V. Mehrotra, The modern call center: A multi-disciplinary perspective on operations management research, Production and Operations Management, 16 (2007), 665-688. doi: 10.1111/j.1937-5956.2007.tb00288.x. [2] J. R. Artalejo and A. Gomez-Corral, Steady state solution of a single-server queue with linear repeated request, Journal of Applied Probability, 34 (1997), 223-233. doi: 10.2307/3215189. [3] J. R. Artalejo and A. Gomez-Corral, "Retrial Queueing Systems: A Computational Approach," Springer, Berlin, 2008. doi: 10.1007/978-3-540-78725-9. [4] J. R. Artalejo, Accessible bibliography on retrial queues: Progress in 2000-2009, Mathematical and Computer Modelling, 51 (2010), 1071-1081. doi: 10.1016/j.mcm.2009.12.011. [5] J. R. Artalejo and J. A. C. Resing, Mean value analysis of single server retrial queues, Asia-Pacific Journal of Operational Research, 27 (2010), 335-345. doi: 10.1142/S0217595910002739. [6] K. Avrachenkov, A. Dudin and V. Klimenok, Retrial queueing model MMAP/$M_{2}$/1 with two orbits, Lecture Notes on Computer Science, 6235 (2010), 107-118. doi: 10.1007/978-3-642-15428-7_12. [7] S. Bhulai and G. Koole, A queueing model for call blending in call centers, IEEE transactions on Automatic Control, 48 (2003), 1434-1438. doi: 10.1109/TAC.2003.815038. [8] B. D. Choi, K. B. Choi and Y. W. Lee, M/G/1 Retrial queueing systems with two types of calls and finite capacity, Queueing Systems, 19 (1995), 215-229. doi: 10.1007/BF01148947. [9] B. D. Choi, Y. C. Kim and Y. W. Lee, The M/M/$c$ retrial queue with geometric loss and feedback, Computers & Mathematics with Applications, 36 (1998), 41-52. doi: 10.1016/S0898-1221(98)00160-6. [10] A. Deslauriers, P. LfEcuyer, J. Pichitlamken, A. Ingolfsson and A. N. Avramidis, Markov chain models of a telephone call center with call blending, Computers & Operations Research, 34 (2007), 1616-1645. doi: 10.1016/j.cor.2005.06.019. [11] G. I. Falin, Model of coupled switching in presence of recurrent calls, Engineering Cybernetics Review, 17 (1979), 53-59. [12] G. I. Falin and J. G. C. Templeton, "Retrial Queues," Chapman and Hall, London, 1997. [13] P. Flajolet and R. Sedgewick, "Analytic Combinatorics," Cambridge University Press, Cambridge, 2009. [14] T. Hanschke, Explicit formulas for the characteristics of the M/M/2/2 queue with repeated attempts, Journal of Applied Probability, 24 (1987), 486-494. [15] 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. [16] J. Kim, Retrial queueing system with collision and impatience, Communications of the Korean Mathematical Society, 25 (2010), 647-653. doi: 10.4134/CKMS.2010.25.4.647. [17] B. Kim, Stability of a retrial queueing network with different classes of customers and restricted resource pooling, Journal of Industrial and Management Optimization, 7 (2011), 753-765. [18] 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. [19] A. Krishnamoorthy, T. G. Deepak and V. C. Joshua, An M/G/1 retrial queue with nonpersistent customers and orbital search, Stochastic Analysis and Applications, 23 (2005), 975-997. doi: 10.1080/07362990500186753. [20] 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. [21] M. Martin and J. R. Artalejo, Analysis of an M/G/1 queue with two types of impatient units, Advances in Applied Probability, 27 (1995), 840-861. doi: 10.2307/1428136. [22] M. F. Neuts and B. M. Rao, Numerical investigation of a multiserver retrial model, Queueing Systems, 7 (1990), 169-190. doi: 10.1007/BF01158473. [23] T. Phung-Duc, H. Masuyama, S. Kasahara and Y. Takahashi, M/M/3/3 and M/M/4/4 retrial queues, Journal of Industrial and Management Optimization, 5 (2009), 431-451. doi: 10.3934/jimo.2009.5.431. [24] 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 (2010), 517-540. doi: 10.3934/jimo.2010.6.517. [25] 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, (2010), 46-52. [26] D. A. Samuelson, Predictive dialing for outbound telephone call centers, Interfaces, 29 (1999), 66-81. doi: 10.1287/inte.29.5.66. [27] R. Stolletz, "Performance Analysis and Optimization of Inbound Call Centers," Lecture Notes in Economics and Mathematical Systems, Springer-Verlag, Berlin, 2003. [28] J. Wang, L. Zhao and F. Zhang, Analysis of the finite source retrial queues with server breakdowns and repairs, Journal of Industrial and Management Optimization, 7 (2011), 655-676. doi: 10.3934/jimo.2011.7.655.

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##### References:
 [1] Z. Aksin, M. Armony and V. Mehrotra, The modern call center: A multi-disciplinary perspective on operations management research, Production and Operations Management, 16 (2007), 665-688. doi: 10.1111/j.1937-5956.2007.tb00288.x. [2] J. R. Artalejo and A. Gomez-Corral, Steady state solution of a single-server queue with linear repeated request, Journal of Applied Probability, 34 (1997), 223-233. doi: 10.2307/3215189. [3] J. R. Artalejo and A. Gomez-Corral, "Retrial Queueing Systems: A Computational Approach," Springer, Berlin, 2008. doi: 10.1007/978-3-540-78725-9. [4] J. R. Artalejo, Accessible bibliography on retrial queues: Progress in 2000-2009, Mathematical and Computer Modelling, 51 (2010), 1071-1081. doi: 10.1016/j.mcm.2009.12.011. [5] J. R. Artalejo and J. A. C. Resing, Mean value analysis of single server retrial queues, Asia-Pacific Journal of Operational Research, 27 (2010), 335-345. doi: 10.1142/S0217595910002739. [6] K. Avrachenkov, A. Dudin and V. Klimenok, Retrial queueing model MMAP/$M_{2}$/1 with two orbits, Lecture Notes on Computer Science, 6235 (2010), 107-118. doi: 10.1007/978-3-642-15428-7_12. [7] S. Bhulai and G. Koole, A queueing model for call blending in call centers, IEEE transactions on Automatic Control, 48 (2003), 1434-1438. doi: 10.1109/TAC.2003.815038. [8] B. D. Choi, K. B. Choi and Y. W. Lee, M/G/1 Retrial queueing systems with two types of calls and finite capacity, Queueing Systems, 19 (1995), 215-229. doi: 10.1007/BF01148947. [9] B. D. Choi, Y. C. Kim and Y. W. Lee, The M/M/$c$ retrial queue with geometric loss and feedback, Computers & Mathematics with Applications, 36 (1998), 41-52. doi: 10.1016/S0898-1221(98)00160-6. [10] A. Deslauriers, P. LfEcuyer, J. Pichitlamken, A. Ingolfsson and A. N. Avramidis, Markov chain models of a telephone call center with call blending, Computers & Operations Research, 34 (2007), 1616-1645. doi: 10.1016/j.cor.2005.06.019. [11] G. I. Falin, Model of coupled switching in presence of recurrent calls, Engineering Cybernetics Review, 17 (1979), 53-59. [12] G. I. Falin and J. G. C. Templeton, "Retrial Queues," Chapman and Hall, London, 1997. [13] P. Flajolet and R. Sedgewick, "Analytic Combinatorics," Cambridge University Press, Cambridge, 2009. [14] T. Hanschke, Explicit formulas for the characteristics of the M/M/2/2 queue with repeated attempts, Journal of Applied Probability, 24 (1987), 486-494. [15] 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. [16] J. Kim, Retrial queueing system with collision and impatience, Communications of the Korean Mathematical Society, 25 (2010), 647-653. doi: 10.4134/CKMS.2010.25.4.647. [17] B. Kim, Stability of a retrial queueing network with different classes of customers and restricted resource pooling, Journal of Industrial and Management Optimization, 7 (2011), 753-765. [18] 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. [19] A. Krishnamoorthy, T. G. Deepak and V. C. Joshua, An M/G/1 retrial queue with nonpersistent customers and orbital search, Stochastic Analysis and Applications, 23 (2005), 975-997. doi: 10.1080/07362990500186753. [20] 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. [21] M. Martin and J. R. Artalejo, Analysis of an M/G/1 queue with two types of impatient units, Advances in Applied Probability, 27 (1995), 840-861. doi: 10.2307/1428136. [22] M. F. Neuts and B. M. Rao, Numerical investigation of a multiserver retrial model, Queueing Systems, 7 (1990), 169-190. doi: 10.1007/BF01158473. [23] T. Phung-Duc, H. Masuyama, S. Kasahara and Y. Takahashi, M/M/3/3 and M/M/4/4 retrial queues, Journal of Industrial and Management Optimization, 5 (2009), 431-451. doi: 10.3934/jimo.2009.5.431. [24] 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 (2010), 517-540. doi: 10.3934/jimo.2010.6.517. [25] 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, (2010), 46-52. [26] D. A. Samuelson, Predictive dialing for outbound telephone call centers, Interfaces, 29 (1999), 66-81. doi: 10.1287/inte.29.5.66. [27] R. Stolletz, "Performance Analysis and Optimization of Inbound Call Centers," Lecture Notes in Economics and Mathematical Systems, Springer-Verlag, Berlin, 2003. [28] J. Wang, L. Zhao and F. Zhang, Analysis of the finite source retrial queues with server breakdowns and repairs, Journal of Industrial and Management Optimization, 7 (2011), 655-676. doi: 10.3934/jimo.2011.7.655.
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