Figure 1.
Dynamic oscillating queue with threshold and impatience
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January
2022, 18(1): 267-295.
doi: 10.3934/jimo.2020153
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.
Keywords:
Oscillating system,
retrial queue,
impatience,
ergodicity,
steady-state performance,
first-passage time,
hypergeometric functions,
sensitivity analysis.
Mathematics Subject Classification:
Primary: 60K25, 90B05; Secondary: 68M20, 90B22.
Citation:
Balasubramanian Krishna Kumar, Ramachandran Navaneetha Krishnan, Rathinam Sankar, Ramasamy Rukmani. Analysis of dynamic service system between regular and retrial queues with impatient customers. Journal of Industrial and Management Optimization,
2022, 18
(1)
: 267-295.
doi: 10.3934/jimo.2020153
![]()
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. |
Figure 2(a).
$\pi(0, 0) ~\text { versus }~ \xi ~\text { for }~ \alpha = 3, \mu = 4, \mathrm{N} = 5$
Figure 2(b).
$\text { E(L) versus }~ \xi~ \text { for } ~\alpha = 3, \mu = 4, N = 5$
Figure 2(c).
$\mathrm{E}\left(\mathrm{W}_{\mathrm{S}}\right) ~~\text { versus } ~~\xi ~~\text { for }~~ \alpha = 3, \mu = 4, \mathrm{N} = 5$
Figure 2(d).
$\mathrm{R}_{\mathrm{A}} ~~\text { versus } ~~\xi ~~\text { for }~~ \alpha = 3, \mu = 4, \mathrm{N} = 5$
Figure 2(e).
$\mathrm{P}_{\mathrm{S}} ~~\text { versus } ~~\xi ~~\text { for }~~ \alpha = 3, \mu = 4, \mathrm{N} = 5$
Figure 3(a).
$\pi(0, 0) \text { versus } \mu \text { for } \alpha = 3, \xi = 5, \mathrm{N} = 5$
Figure 3(b).
$\mathrm{E}(\mathrm{L}) \text { versus } \mu \text { for } \xi = 5, \alpha = 3, \mathrm{N} = 5$
Figure 3(c).
$\mathrm{E}\left(\mathrm{W}_{\mathrm{S}}\right) \text { versus } \mu \text { for } \xi = 5, \alpha = 3, \mathrm{N} = 5$
Figure 3(d).
$\mathrm{R}_{\mathrm{A}} \text { versus } \mu \text { for } \xi = 5, \alpha = 3, \mathrm{N} = 5$
Figure 3(e).
$\mathrm{P}_{\mathrm{S}} \text { versus } \mu \text { for } \xi = 5, \alpha = 3, \mathrm{N} = 5$
Figure 4(a).
$\mathrm{E}\left(\tau_{\mathrm{j}}\right) ~~\text { versus } ~~\xi ~~\text { for }~~ \alpha = 3, \mu = 4, \mathrm{N} = 5, \mathrm{j} = 10$
Figure 4(b).
$\mathrm{E}\left(\tau_{\mathrm{j}}\right) \text { versus } \mu \text { for } \xi = 5, \alpha = 3, \mathrm{N} = 5, \mathrm{j} = 10$
Figure 4(c).
$\mathrm{E}\left(\tau_{\mathrm{j}}\right) ~~\text { versus } ~~\xi ~~\text { for }~~ \lambda = 18, \nu = 20, \alpha = 3, \mu = 4, \mathrm{N} = 5$
Figure 4(d).
$\mathrm{E}\left(\tau_{\mathrm{i}}\right) \text { versus } \mu \text { for } \lambda = 18, \nu = 20, \xi = 5, \alpha = 3, \mathrm{N} = 5$
Figure 4(e).
$\mathrm{E}\left(\tau_{\mathrm{j}}\right) \text { versus } \lambda \text { for } \xi = 5, \nu = 20, \mu = 4, \alpha = 3, \mathrm{N} = 5$
Figure 4(f).
$\mathrm{E}\left(\tau_{\mathrm{j}}\right) \text { versus } \nu \text { for } \lambda = 18, \xi = 5, \alpha = 3, \mu = 4, \mathrm{N} = 5$
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