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 & 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.  doi: 10.1111/j.1937-5956.2007.tb00288.x.  Google Scholar

[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.  doi: 10.2307/3215189.  Google Scholar

[3]

J. R. Artalejo and A. Gomez-Corral, "Retrial Queueing Systems: A Computational Approach,", Springer, (2008).  doi: 10.1007/978-3-540-78725-9.  Google Scholar

[4]

J. R. Artalejo, Accessible bibliography on retrial queues: Progress in 2000-2009,, Mathematical and Computer Modelling, 51 (2010), 1071.  doi: 10.1016/j.mcm.2009.12.011.  Google Scholar

[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.  doi: 10.1142/S0217595910002739.  Google Scholar

[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.  doi: 10.1007/978-3-642-15428-7_12.  Google Scholar

[7]

S. Bhulai and G. Koole, A queueing model for call blending in call centers,, IEEE transactions on Automatic Control, 48 (2003), 1434.  doi: 10.1109/TAC.2003.815038.  Google Scholar

[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.  doi: 10.1007/BF01148947.  Google Scholar

[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.  doi: 10.1016/S0898-1221(98)00160-6.  Google Scholar

[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.  doi: 10.1016/j.cor.2005.06.019.  Google Scholar

[11]

G. I. Falin, Model of coupled switching in presence of recurrent calls,, Engineering Cybernetics Review, 17 (1979), 53.   Google Scholar

[12]

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

[13]

P. Flajolet and R. Sedgewick, "Analytic Combinatorics,", Cambridge University Press, (2009).   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[16]

J. Kim, Retrial queueing system with collision and impatience,, Communications of the Korean Mathematical Society, 25 (2010), 647.  doi: 10.4134/CKMS.2010.25.4.647.  Google Scholar

[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.   Google Scholar

[18]

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

[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.  doi: 10.1080/07362990500186753.  Google Scholar

[20]

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.  Google Scholar

[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.  doi: 10.2307/1428136.  Google Scholar

[22]

M. F. Neuts and B. M. Rao, Numerical investigation of a multiserver retrial model,, Queueing Systems, 7 (1990), 169.  doi: 10.1007/BF01158473.  Google Scholar

[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.  doi: 10.3934/jimo.2009.5.431.  Google Scholar

[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.  doi: 10.3934/jimo.2010.6.517.  Google Scholar

[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.   Google Scholar

[26]

D. A. Samuelson, Predictive dialing for outbound telephone call centers,, Interfaces, 29 (1999), 66.  doi: 10.1287/inte.29.5.66.  Google Scholar

[27]

R. Stolletz, "Performance Analysis and Optimization of Inbound Call Centers,", Lecture Notes in Economics and Mathematical Systems, (2003).   Google Scholar

[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.  doi: 10.3934/jimo.2011.7.655.  Google Scholar

show all references

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.  doi: 10.1111/j.1937-5956.2007.tb00288.x.  Google Scholar

[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.  doi: 10.2307/3215189.  Google Scholar

[3]

J. R. Artalejo and A. Gomez-Corral, "Retrial Queueing Systems: A Computational Approach,", Springer, (2008).  doi: 10.1007/978-3-540-78725-9.  Google Scholar

[4]

J. R. Artalejo, Accessible bibliography on retrial queues: Progress in 2000-2009,, Mathematical and Computer Modelling, 51 (2010), 1071.  doi: 10.1016/j.mcm.2009.12.011.  Google Scholar

[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.  doi: 10.1142/S0217595910002739.  Google Scholar

[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.  doi: 10.1007/978-3-642-15428-7_12.  Google Scholar

[7]

S. Bhulai and G. Koole, A queueing model for call blending in call centers,, IEEE transactions on Automatic Control, 48 (2003), 1434.  doi: 10.1109/TAC.2003.815038.  Google Scholar

[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.  doi: 10.1007/BF01148947.  Google Scholar

[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.  doi: 10.1016/S0898-1221(98)00160-6.  Google Scholar

[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.  doi: 10.1016/j.cor.2005.06.019.  Google Scholar

[11]

G. I. Falin, Model of coupled switching in presence of recurrent calls,, Engineering Cybernetics Review, 17 (1979), 53.   Google Scholar

[12]

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

[13]

P. Flajolet and R. Sedgewick, "Analytic Combinatorics,", Cambridge University Press, (2009).   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[16]

J. Kim, Retrial queueing system with collision and impatience,, Communications of the Korean Mathematical Society, 25 (2010), 647.  doi: 10.4134/CKMS.2010.25.4.647.  Google Scholar

[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.   Google Scholar

[18]

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

[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.  doi: 10.1080/07362990500186753.  Google Scholar

[20]

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.  Google Scholar

[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.  doi: 10.2307/1428136.  Google Scholar

[22]

M. F. Neuts and B. M. Rao, Numerical investigation of a multiserver retrial model,, Queueing Systems, 7 (1990), 169.  doi: 10.1007/BF01158473.  Google Scholar

[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.  doi: 10.3934/jimo.2009.5.431.  Google Scholar

[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.  doi: 10.3934/jimo.2010.6.517.  Google Scholar

[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.   Google Scholar

[26]

D. A. Samuelson, Predictive dialing for outbound telephone call centers,, Interfaces, 29 (1999), 66.  doi: 10.1287/inte.29.5.66.  Google Scholar

[27]

R. Stolletz, "Performance Analysis and Optimization of Inbound Call Centers,", Lecture Notes in Economics and Mathematical Systems, (2003).   Google Scholar

[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.  doi: 10.3934/jimo.2011.7.655.  Google Scholar

[1]

A. M. Elaiw, N. H. AlShamrani, A. Abdel-Aty, H. Dutta. Stability analysis of a general HIV dynamics model with multi-stages of infected cells and two routes of infection. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020441

[2]

Veena Goswami, Gopinath Panda. Optimal customer behavior in observable and unobservable discrete-time queues. Journal of Industrial & Management Optimization, 2021, 17 (1) : 299-316. doi: 10.3934/jimo.2019112

[3]

George W. Patrick. The geometry of convergence in numerical analysis. Journal of Computational Dynamics, 2021, 8 (1) : 33-58. doi: 10.3934/jcd.2021003

[4]

Nicolas Rougerie. On two properties of the Fisher information. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020049

[5]

Vivina Barutello, Gian Marco Canneori, Susanna Terracini. Minimal collision arcs asymptotic to central configurations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 61-86. doi: 10.3934/dcds.2020218

[6]

Neng Zhu, Zhengrong Liu, Fang Wang, Kun Zhao. Asymptotic dynamics of a system of conservation laws from chemotaxis. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 813-847. doi: 10.3934/dcds.2020301

[7]

Min Chen, Olivier Goubet, Shenghao Li. Mathematical analysis of bump to bucket problem. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5567-5580. doi: 10.3934/cpaa.2020251

[8]

Zhenzhen Wang, Tianshou Zhou. Asymptotic behaviors and stochastic traveling waves in stochastic Fisher-KPP equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020323

[9]

Wei Feng, Michael Freeze, Xin Lu. On competition models under allee effect: Asymptotic behavior and traveling waves. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5609-5626. doi: 10.3934/cpaa.2020256

[10]

Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020450

[11]

Qianqian Han, Xiao-Song Yang. Qualitative analysis of a generalized Nosé-Hoover oscillator. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020346

[12]

Laurence Cherfils, Stefania Gatti, Alain Miranville, Rémy Guillevin. Analysis of a model for tumor growth and lactate exchanges in a glioma. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020457

[13]

Vieri Benci, Sunra Mosconi, Marco Squassina. Preface: Applications of mathematical analysis to problems in theoretical physics. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020446

[14]

Hua Qiu, Zheng-An Yao. The regularized Boussinesq equations with partial dissipations in dimension two. Electronic Research Archive, 2020, 28 (4) : 1375-1393. doi: 10.3934/era.2020073

[15]

Huu-Quang Nguyen, Ya-Chi Chu, Ruey-Lin Sheu. On the convexity for the range set of two quadratic functions. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020169

[16]

Yining Cao, Chuck Jia, Roger Temam, Joseph Tribbia. Mathematical analysis of a cloud resolving model including the ice microphysics. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 131-167. doi: 10.3934/dcds.2020219

[17]

Xin Guo, Lei Shi. Preface of the special issue on analysis in data science: Methods and applications. Mathematical Foundations of Computing, 2020, 3 (4) : i-ii. doi: 10.3934/mfc.2020026

[18]

Martin Kalousek, Joshua Kortum, Anja Schlömerkemper. Mathematical analysis of weak and strong solutions to an evolutionary model for magnetoviscoelasticity. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 17-39. doi: 10.3934/dcdss.2020331

[19]

Feifei Cheng, Ji Li. Geometric singular perturbation analysis of Degasperis-Procesi equation with distributed delay. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 967-985. doi: 10.3934/dcds.2020305

[20]

Illés Horváth, Kristóf Attila Horváth, Péter Kovács, Miklós Telek. Mean-field analysis of a scaling MAC radio protocol. Journal of Industrial & Management Optimization, 2021, 17 (1) : 279-297. doi: 10.3934/jimo.2019111

2019 Impact Factor: 1.366

Metrics

  • PDF downloads (140)
  • HTML views (0)
  • Cited by (34)

Other articles
by authors

[Back to Top]