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

* Corresponding author: B. Krishna Kumar

Received  January 2020 Revised  July 2020 Published  October 2020

Fund Project: This research work is supported in part by Department of Science and Technology, New Delhi, India, under research grant INT/RUS/RFBR/377

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.

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 & Management Optimization, doi: 10.3934/jimo.2020153
References:
[1]

M. S. AguirO. Z. AksinF. 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.  Google Scholar

[2]

M. S. AguirF. KaraesmenO. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

E. Altman and A. A. Borovkov, On the stability of retrial queues, Queueing Systems, 26 (1997), 343-363.  doi: 10.1023/A:1019193527040.  Google Scholar

[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.  Google Scholar

[7]

J. R. Artalejo, Accessible bibliography on retrial queues, Math. Comput. Model., 30 (1999), 1-6.  doi: 10.1016/j.mcm.2009.12.011.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

W. J. Anderson, Continuous-Time Markov Chains: An Applications-Oriented Approach, Springer-Verlag, Berlin, 1991. doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[13]

S. Asmussen, Applied Probability and Queues, Springer, New York, 2003. doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[14]

F. BaccelliP. Boyer and G. Hebuterne, Single-server queues with impatient customers, Adv. Appl. Probab., 16 (1984), 887-905.  doi: 10.2307/1427345.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

S. DimouA. 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.  Google Scholar

[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.  Google Scholar

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A. Erdelyi, Higher Transcendental Function, 1, McGraw-Hill, New York, 1953. doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[22]

G. I. Falin, A survey of retrial queues, Queueing Systems, 7 (1990), 127-168.  doi: 10.1007/BF01158472.  Google Scholar

[23]

G. I. Falin and J. G. C. Templeton, Retrial Queues, Chapman and Hall, London, 1997. doi: 10.1007/978-1-4899-2977-8.  Google Scholar

[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.  Google Scholar

[25]

N. GansG. 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.  Google Scholar

[26]

O. GarnettA. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

O. JouiniG. Koole and A. Roubos, Performance indicators for call centers with impatient customers, IIE Transactions, 45 (2013), 341-354.  doi: 10.1080/0740817X.2012.712241.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[35]

A. Movaghar, On queueing with customer impatience until the beginning of service, Queueing Systems, 29 (1998), 337-350.  doi: 10.1023/A:1019196416987.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[42]

P. WuchnerJ. 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.  Google Scholar

[43]

D. YueW. 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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

M. S. AguirO. Z. AksinF. 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.  Google Scholar

[2]

M. S. AguirF. KaraesmenO. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

E. Altman and A. A. Borovkov, On the stability of retrial queues, Queueing Systems, 26 (1997), 343-363.  doi: 10.1023/A:1019193527040.  Google Scholar

[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.  Google Scholar

[7]

J. R. Artalejo, Accessible bibliography on retrial queues, Math. Comput. Model., 30 (1999), 1-6.  doi: 10.1016/j.mcm.2009.12.011.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

W. J. Anderson, Continuous-Time Markov Chains: An Applications-Oriented Approach, Springer-Verlag, Berlin, 1991. doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[13]

S. Asmussen, Applied Probability and Queues, Springer, New York, 2003. doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[14]

F. BaccelliP. Boyer and G. Hebuterne, Single-server queues with impatient customers, Adv. Appl. Probab., 16 (1984), 887-905.  doi: 10.2307/1427345.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

S. DimouA. 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.  Google Scholar

[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.  Google Scholar

[21]

A. Erdelyi, Higher Transcendental Function, 1, McGraw-Hill, New York, 1953. doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[22]

G. I. Falin, A survey of retrial queues, Queueing Systems, 7 (1990), 127-168.  doi: 10.1007/BF01158472.  Google Scholar

[23]

G. I. Falin and J. G. C. Templeton, Retrial Queues, Chapman and Hall, London, 1997. doi: 10.1007/978-1-4899-2977-8.  Google Scholar

[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.  Google Scholar

[25]

N. GansG. 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.  Google Scholar

[26]

O. GarnettA. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

O. JouiniG. Koole and A. Roubos, Performance indicators for call centers with impatient customers, IIE Transactions, 45 (2013), 341-354.  doi: 10.1080/0740817X.2012.712241.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[35]

A. Movaghar, On queueing with customer impatience until the beginning of service, Queueing Systems, 29 (1998), 337-350.  doi: 10.1023/A:1019196416987.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[42]

P. WuchnerJ. 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.  Google Scholar

[43]

D. YueW. 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.  Google Scholar

[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.  Google Scholar

Figure 1.  Dynamic oscillating queue with threshold and impatience
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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