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Analysis of dynamic service system between regular and retrial queues with impatient customers
1. | Department of Mathematics, College of Engineering, Anna University, Chennai 600 025, India |
2. | Department of Mathematics, Pachaiyappa's College, Chennai 600 030, India |
In this article, we propose a dynamic operating of a single server service system between conventional and retrial queues with impatient customers. Necessary and sufficient conditions for the stability, and an explicit expression for the joint steady-state probability distribution are obtained. We have derived some interesting and important performance measures for the service system under consideration. The first-passage time problems are also investigated. Finally, we have presented extensive numerical examples to demonstrate the effects of the system parameters on the performance measures.
References:
[1] |
M. S. Aguir, O. Z. Aksin, F. Karaesmen and Y. Dallery,
On the interaction between retrials and sizing of call centers, Euro. J. Oper. Res., 191 (2008), 398-408.
doi: 10.1016/j.ejor.2007.06.051. |
[2] |
M. S. Aguir, F. Karaesmen, O. Z. Aksin and F. Chauvet,
The impact of retrials on call center performance, OR Spectrum, 26 (2004), 353-376.
doi: 10.1007/s00291-004-0165-7. |
[3] |
N. Akar and K. Sohraby,
Retrial queueing models of multi-wavelength FDL feedback optical buffers, IEEE Trans. Commun., 59 (2011), 2832-2840.
doi: 10.2307/2152750. |
[4] |
O. Z. Aksin and P. T. Harker,
Modeling a phone center: Analysis of a multichannel multiresource processor shared loss system, Management Science, 47 (2001), 324-336.
doi: 10.2307/2152750. |
[5] |
E. Altman and A. A. Borovkov,
On the stability of retrial queues, Queueing Systems, 26 (1997), 343-363.
doi: 10.1023/A:1019193527040. |
[6] |
E. Altman and U. Yechiali,
Analysis of customers' impatience in queues with server vacations, Queueing Systems, 52 (2006), 261-279.
doi: 10.1007/s11134-006-6134-x. |
[7] |
J. R. Artalejo,
Accessible bibliography on retrial queues, Math. Comput. Model., 30 (1999), 1-6.
doi: 10.1016/j.mcm.2009.12.011. |
[8] |
J. R. Artalejo,
Accessible bibliography on retrial queues: Progress in 2000–2009, Math. Comput. Model., 51 (2010), 1071-1081.
doi: 10.1016/j.mcm.2009.12.011. |
[9] |
J. R. Artalejo and A. Gomez-Corral,
Steady state solution of a single-server queue with linear repeated requests, J. Appl. Probab., 34 (1997), 223-233.
doi: 10.2307/3215189. |
[10] |
J. R. Artalejo and A. Gomez-Corral, Retrial Queueing Systems: A Computational Approach, Springer-Verlag, Berlin, 2008.
doi: 10.1007/978-1-4612-0873-0. |
[11] |
J. R. Artalejo and V. Pla,
On the impact of customer balking, impatience and retrials in telecommunication systems, Comput. Math. Appl., 57 (2009), 217-229.
doi: 10.1016/j.camwa.2008.10.084. |
[12] |
W. J. Anderson, Continuous-Time Markov Chains: An Applications-Oriented Approach, Springer-Verlag, Berlin, 1991.
doi: 10.1007/978-1-4612-0873-0. |
[13] |
S. Asmussen, Applied Probability and Queues, Springer, New York, 2003.
doi: 10.1007/978-1-4612-0873-0. |
[14] |
F. Baccelli, P. Boyer and G. Hebuterne,
Single-server queues with impatient customers, Adv. Appl. Probab., 16 (1984), 887-905.
doi: 10.2307/1427345. |
[15] |
F. Baccelli and G. Hebuterne, On queues with impatient customers, in Performance'81, North-Holland Publishing Company, Amsterdam, 1981,159–179.
doi: 10.2307/2152750. |
[16] |
O. J. Boxma and P. R. de Waal,
Multiserver queues with impatient customers, ITC, 14 (1994), 743-756.
doi: 10.1016/B978-0-444-82031-0.50079-2. |
[17] |
A. Brandt and M. Brandt,
On the two-class M/M/1 system under preemptive resume and impatience of the prioritized customers, Queueing Systems, 47 (2004), 147-168.
doi: 10.1023/B:QUES.0000032805.73991.8e. |
[18] |
B. D. Choi and Y. Chang,
Single server retrial queues with priority calls, Math. Comput. Model., 30 (1999), 7-32.
doi: 10.1016/S0895-7177(99)00129-6. |
[19] |
S. Dimou, A. Economou and D. Fakinos,
The single server vacation queueing model with geometric abandonments, J. Stat. Plan. Infer., 141 (2011), 2863-2877.
doi: 10.1016/j.jspi.2011.03.010. |
[20] |
A. Economou and S. Kapodistria,
Synchronized abandonments in a single server unreliable queue, Euro. J. Oper. Res., 203 (2010), 143-155.
doi: 10.1016/j.ejor.2009.07.014. |
[21] |
A. Erdelyi, Higher Transcendental Function, 1, McGraw-Hill, New York, 1953.
doi: 10.1007/978-1-4612-0873-0. |
[22] |
G. I. Falin,
A survey of retrial queues, Queueing Systems, 7 (1990), 127-168.
doi: 10.1007/BF01158472. |
[23] |
G. I. Falin and J. G. C. Templeton, Retrial Queues, Chapman and Hall, London, 1997.
doi: 10.1007/978-1-4899-2977-8. |
[24] |
G. Fayolle, A simple telephone exchange with delayed feedbacks, in Teletrafic Analysis and Computer Performance Eualuation, Elsevier, Amsterdam, 1986,245–253.
doi: 10.2307/2152750. |
[25] |
N. Gans, G. Koole and A. Mandelbaum,
Telephone call centers: Tutorial, review, and research prospects, Manufacturing and Service Operations Management, 5 (2003), 79-177.
doi: 10.1287/msom.5.2.79.16071. |
[26] |
O. Garnett, A. Mandelbaum and M. Reiman,
Designing a call center with impatient customers, Manufacturing and Service Operations Management, 4 (2002), 208-227.
doi: 10.1287/msom.4.3.208.7753. |
[27] |
D. Gross, J. F. Shortle, J. M. Thompson and C. M. Harris, Fundamentals of Queueing Theory, Wiley India (P) Ltd, New Delhi, 2014.
doi: 10.1007/978-1-4612-0873-0. |
[28] |
F. Iravani and B. Balcioglu,
On priority queues with impatient customers, Queueing Systems, 5 (2008), 239-260.
doi: 10.1007/s11134-008-9069-6. |
[29] |
O. Jouini and Y. Dallery,
Moments of first passage times in general birth-death processes, Math. Meth. Oper. Res., 68 (2008), 49-76.
doi: 10.1007/s00186-007-0174-9. |
[30] |
O. Jouini, G. Koole and A. Roubos,
Performance indicators for call centers with impatient customers, IIE Transactions, 45 (2013), 341-354.
doi: 10.1080/0740817X.2012.712241. |
[31] |
S. Karlin and J. McGregor,
The classification of birth and death processes, Trans. Amer. Math. Soc., 86 (1957), 366-400.
doi: 10.1090/S0002-9947-1957-0094854-8. |
[32] |
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. |
[33] |
V. G. Kulkarni and H. M. Liang, Retrial queues revisited, in Frontiers in Queueing: Models
and Applications in Science and Engineering, CRC Press, Boca Raton, FL, 1997, 19-34.
doi: 10.2307/2152750. |
[34] |
A. Mandelbaum and S. Zeltyn,
Staffing many-server queues with impatient customers: Constraint satisfaction in call centers, Oper. Res., 57 (2009), 1189-1205.
doi: 10.1287/opre.1080.0651. |
[35] |
A. Movaghar,
On queueing with customer impatience until the beginning of service, Queueing Systems, 29 (1998), 337-350.
doi: 10.1023/A:1019196416987. |
[36] |
T. Phung-Duc and K. Kawanishi,
Performance analysis of call centers with abandonment, retrial and after-call work, Perform. Eval., 80 (2014), 43-62.
doi: 10.1016/j.peva.2014.03.001. |
[37] |
Y. M. Shin and T. S. Choo,
$M/M/s$ queue with impatient customers and retrials, Appl. Math. Model., 33 (2009), 2596-2606.
doi: 10.1016/j.apm.2008.07.018. |
[38] |
H. Takagi, Queueing Analysis: A Foundation of Performance Evaluation-Vacation and Priority System, 1, Elsevier Publishers, Amsterdam, 1991.,
doi: 10.1007/978-1-4612-0873-0. |
[39] |
H. M. Taylor and S. Karlin, An Introduction to Stochastic Modeling, Academic Press, New York, 1998.,
doi: 10.1007/978-1-4612-0873-0. |
[40] |
K. Wang, N. Li and Z. Jiang, Queueing system with impatient customers: A review, IEEE proceedings, (2010), 82–87.
doi: 10.1109/SOLI.2010.5551611. |
[41] |
W. Whitt, Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Application to Queues, Springer-Verlag, Berlin, 2002.,
doi: 10.1007/978-1-4612-0873-0. |
[42] |
P. Wuchner, J. Sztrik and H. D. Meer,
Finite-source $M/M/S$ retrial queue with search for balking and impatient customers from the orbit, Computer Networks, 53 (2009), 1264-1273.
doi: 10.1016/j.comnet.2009.02.015. |
[43] |
D. Yue, W. Yue and G. Zhao,
Analysis of an $M/M/1$ queue with vacations and impatience timers which depend on the server's states, J. Industr. Manag. Optim., 12 (2016), 653-666.
doi: 10.2307/2152750. |
[44] |
S. Zeltyn and S. Mandelbaum,
Call centers with impatient customers: Many-server asymptotics of the $M/M/n+G$ queue, Queueing Systems, 51 (2005), 361-402.
doi: 10.1007/s11134-005-3699-8. |
show all references
References:
[1] |
M. S. Aguir, O. Z. Aksin, F. Karaesmen and Y. Dallery,
On the interaction between retrials and sizing of call centers, Euro. J. Oper. Res., 191 (2008), 398-408.
doi: 10.1016/j.ejor.2007.06.051. |
[2] |
M. S. Aguir, F. Karaesmen, O. Z. Aksin and F. Chauvet,
The impact of retrials on call center performance, OR Spectrum, 26 (2004), 353-376.
doi: 10.1007/s00291-004-0165-7. |
[3] |
N. Akar and K. Sohraby,
Retrial queueing models of multi-wavelength FDL feedback optical buffers, IEEE Trans. Commun., 59 (2011), 2832-2840.
doi: 10.2307/2152750. |
[4] |
O. Z. Aksin and P. T. Harker,
Modeling a phone center: Analysis of a multichannel multiresource processor shared loss system, Management Science, 47 (2001), 324-336.
doi: 10.2307/2152750. |
[5] |
E. Altman and A. A. Borovkov,
On the stability of retrial queues, Queueing Systems, 26 (1997), 343-363.
doi: 10.1023/A:1019193527040. |
[6] |
E. Altman and U. Yechiali,
Analysis of customers' impatience in queues with server vacations, Queueing Systems, 52 (2006), 261-279.
doi: 10.1007/s11134-006-6134-x. |
[7] |
J. R. Artalejo,
Accessible bibliography on retrial queues, Math. Comput. Model., 30 (1999), 1-6.
doi: 10.1016/j.mcm.2009.12.011. |
[8] |
J. R. Artalejo,
Accessible bibliography on retrial queues: Progress in 2000–2009, Math. Comput. Model., 51 (2010), 1071-1081.
doi: 10.1016/j.mcm.2009.12.011. |
[9] |
J. R. Artalejo and A. Gomez-Corral,
Steady state solution of a single-server queue with linear repeated requests, J. Appl. Probab., 34 (1997), 223-233.
doi: 10.2307/3215189. |
[10] |
J. R. Artalejo and A. Gomez-Corral, Retrial Queueing Systems: A Computational Approach, Springer-Verlag, Berlin, 2008.
doi: 10.1007/978-1-4612-0873-0. |
[11] |
J. R. Artalejo and V. Pla,
On the impact of customer balking, impatience and retrials in telecommunication systems, Comput. Math. Appl., 57 (2009), 217-229.
doi: 10.1016/j.camwa.2008.10.084. |
[12] |
W. J. Anderson, Continuous-Time Markov Chains: An Applications-Oriented Approach, Springer-Verlag, Berlin, 1991.
doi: 10.1007/978-1-4612-0873-0. |
[13] |
S. Asmussen, Applied Probability and Queues, Springer, New York, 2003.
doi: 10.1007/978-1-4612-0873-0. |
[14] |
F. Baccelli, P. Boyer and G. Hebuterne,
Single-server queues with impatient customers, Adv. Appl. Probab., 16 (1984), 887-905.
doi: 10.2307/1427345. |
[15] |
F. Baccelli and G. Hebuterne, On queues with impatient customers, in Performance'81, North-Holland Publishing Company, Amsterdam, 1981,159–179.
doi: 10.2307/2152750. |
[16] |
O. J. Boxma and P. R. de Waal,
Multiserver queues with impatient customers, ITC, 14 (1994), 743-756.
doi: 10.1016/B978-0-444-82031-0.50079-2. |
[17] |
A. Brandt and M. Brandt,
On the two-class M/M/1 system under preemptive resume and impatience of the prioritized customers, Queueing Systems, 47 (2004), 147-168.
doi: 10.1023/B:QUES.0000032805.73991.8e. |
[18] |
B. D. Choi and Y. Chang,
Single server retrial queues with priority calls, Math. Comput. Model., 30 (1999), 7-32.
doi: 10.1016/S0895-7177(99)00129-6. |
[19] |
S. Dimou, A. Economou and D. Fakinos,
The single server vacation queueing model with geometric abandonments, J. Stat. Plan. Infer., 141 (2011), 2863-2877.
doi: 10.1016/j.jspi.2011.03.010. |
[20] |
A. Economou and S. Kapodistria,
Synchronized abandonments in a single server unreliable queue, Euro. J. Oper. Res., 203 (2010), 143-155.
doi: 10.1016/j.ejor.2009.07.014. |
[21] |
A. Erdelyi, Higher Transcendental Function, 1, McGraw-Hill, New York, 1953.
doi: 10.1007/978-1-4612-0873-0. |
[22] |
G. I. Falin,
A survey of retrial queues, Queueing Systems, 7 (1990), 127-168.
doi: 10.1007/BF01158472. |
[23] |
G. I. Falin and J. G. C. Templeton, Retrial Queues, Chapman and Hall, London, 1997.
doi: 10.1007/978-1-4899-2977-8. |
[24] |
G. Fayolle, A simple telephone exchange with delayed feedbacks, in Teletrafic Analysis and Computer Performance Eualuation, Elsevier, Amsterdam, 1986,245–253.
doi: 10.2307/2152750. |
[25] |
N. Gans, G. Koole and A. Mandelbaum,
Telephone call centers: Tutorial, review, and research prospects, Manufacturing and Service Operations Management, 5 (2003), 79-177.
doi: 10.1287/msom.5.2.79.16071. |
[26] |
O. Garnett, A. Mandelbaum and M. Reiman,
Designing a call center with impatient customers, Manufacturing and Service Operations Management, 4 (2002), 208-227.
doi: 10.1287/msom.4.3.208.7753. |
[27] |
D. Gross, J. F. Shortle, J. M. Thompson and C. M. Harris, Fundamentals of Queueing Theory, Wiley India (P) Ltd, New Delhi, 2014.
doi: 10.1007/978-1-4612-0873-0. |
[28] |
F. Iravani and B. Balcioglu,
On priority queues with impatient customers, Queueing Systems, 5 (2008), 239-260.
doi: 10.1007/s11134-008-9069-6. |
[29] |
O. Jouini and Y. Dallery,
Moments of first passage times in general birth-death processes, Math. Meth. Oper. Res., 68 (2008), 49-76.
doi: 10.1007/s00186-007-0174-9. |
[30] |
O. Jouini, G. Koole and A. Roubos,
Performance indicators for call centers with impatient customers, IIE Transactions, 45 (2013), 341-354.
doi: 10.1080/0740817X.2012.712241. |
[31] |
S. Karlin and J. McGregor,
The classification of birth and death processes, Trans. Amer. Math. Soc., 86 (1957), 366-400.
doi: 10.1090/S0002-9947-1957-0094854-8. |
[32] |
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. |
[33] |
V. G. Kulkarni and H. M. Liang, Retrial queues revisited, in Frontiers in Queueing: Models
and Applications in Science and Engineering, CRC Press, Boca Raton, FL, 1997, 19-34.
doi: 10.2307/2152750. |
[34] |
A. Mandelbaum and S. Zeltyn,
Staffing many-server queues with impatient customers: Constraint satisfaction in call centers, Oper. Res., 57 (2009), 1189-1205.
doi: 10.1287/opre.1080.0651. |
[35] |
A. Movaghar,
On queueing with customer impatience until the beginning of service, Queueing Systems, 29 (1998), 337-350.
doi: 10.1023/A:1019196416987. |
[36] |
T. Phung-Duc and K. Kawanishi,
Performance analysis of call centers with abandonment, retrial and after-call work, Perform. Eval., 80 (2014), 43-62.
doi: 10.1016/j.peva.2014.03.001. |
[37] |
Y. M. Shin and T. S. Choo,
$M/M/s$ queue with impatient customers and retrials, Appl. Math. Model., 33 (2009), 2596-2606.
doi: 10.1016/j.apm.2008.07.018. |
[38] |
H. Takagi, Queueing Analysis: A Foundation of Performance Evaluation-Vacation and Priority System, 1, Elsevier Publishers, Amsterdam, 1991.,
doi: 10.1007/978-1-4612-0873-0. |
[39] |
H. M. Taylor and S. Karlin, An Introduction to Stochastic Modeling, Academic Press, New York, 1998.,
doi: 10.1007/978-1-4612-0873-0. |
[40] |
K. Wang, N. Li and Z. Jiang, Queueing system with impatient customers: A review, IEEE proceedings, (2010), 82–87.
doi: 10.1109/SOLI.2010.5551611. |
[41] |
W. Whitt, Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Application to Queues, Springer-Verlag, Berlin, 2002.,
doi: 10.1007/978-1-4612-0873-0. |
[42] |
P. Wuchner, J. Sztrik and H. D. Meer,
Finite-source $M/M/S$ retrial queue with search for balking and impatient customers from the orbit, Computer Networks, 53 (2009), 1264-1273.
doi: 10.1016/j.comnet.2009.02.015. |
[43] |
D. Yue, W. Yue and G. Zhao,
Analysis of an $M/M/1$ queue with vacations and impatience timers which depend on the server's states, J. Industr. Manag. Optim., 12 (2016), 653-666.
doi: 10.2307/2152750. |
[44] |
S. Zeltyn and S. Mandelbaum,
Call centers with impatient customers: Many-server asymptotics of the $M/M/n+G$ queue, Queueing Systems, 51 (2005), 361-402.
doi: 10.1007/s11134-005-3699-8. |

















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