Analysis of dynamic service system between regular and retrial queues with impatient customers

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doi: 10.3934/jimo.2020153

Analysis of dynamic service system between regular and retrial queues with impatient customers

Balasubramanian Krishna Kumar ^1,, , Ramachandran Navaneetha Krishnan ^1, , Rathinam Sankar ^1, and Ramasamy Rukmani ^2,

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.

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 & Management Optimization, 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. Google Scholar
[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. 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. Baccelli, P. 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. 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. 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. 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. Google Scholar
[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. 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. 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. 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. 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. Google Scholar
[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. 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. 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. Google Scholar
[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. 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. Baccelli, P. 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. 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. 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. 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. Google Scholar
[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. 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. 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. 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. 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. Google Scholar
[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. 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

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Figure 2(a). $\pi(0, 0) ~\text { versus }~ \xi ~\text { for }~ \alpha = 3, \mu = 4, \mathrm{N} = 5$

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Figure 2(b). $\text { E(L) versus }~ \xi~ \text { for } ~\alpha = 3, \mu = 4, N = 5$

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Figure 2(c). $\mathrm{E}\left(\mathrm{W}_{\mathrm{S}}\right) ~~\text { versus } ~~\xi ~~\text { for }~~ \alpha = 3, \mu = 4, \mathrm{N} = 5$

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Figure 2(d). $\mathrm{R}_{\mathrm{A}} ~~\text { versus } ~~\xi ~~\text { for }~~ \alpha = 3, \mu = 4, \mathrm{N} = 5$

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Figure 2(e). $\mathrm{P}_{\mathrm{S}} ~~\text { versus } ~~\xi ~~\text { for }~~ \alpha = 3, \mu = 4, \mathrm{N} = 5$

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Figure 3(a). $\pi(0, 0) \text { versus } \mu \text { for } \alpha = 3, \xi = 5, \mathrm{N} = 5$

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Figure 3(b). $\mathrm{E}(\mathrm{L}) \text { versus } \mu \text { for } \xi = 5, \alpha = 3, \mathrm{N} = 5$

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Figure 3(c). $\mathrm{E}\left(\mathrm{W}_{\mathrm{S}}\right) \text { versus } \mu \text { for } \xi = 5, \alpha = 3, \mathrm{N} = 5$

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Figure 3(d). $\mathrm{R}_{\mathrm{A}} \text { versus } \mu \text { for } \xi = 5, \alpha = 3, \mathrm{N} = 5$

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Figure 3(e). $\mathrm{P}_{\mathrm{S}} \text { versus } \mu \text { for } \xi = 5, \alpha = 3, \mathrm{N} = 5$

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Figure 4(a). $\mathrm{E}\left(\tau_{\mathrm{j}}\right) ~~\text { versus } ~~\xi ~~\text { for }~~ \alpha = 3, \mu = 4, \mathrm{N} = 5, \mathrm{j} = 10$

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Figure 4(b). $\mathrm{E}\left(\tau_{\mathrm{j}}\right) \text { versus } \mu \text { for } \xi = 5, \alpha = 3, \mathrm{N} = 5, \mathrm{j} = 10$

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Figure 4(c). $\mathrm{E}\left(\tau_{\mathrm{j}}\right) ~~\text { versus } ~~\xi ~~\text { for }~~ \lambda = 18, \nu = 20, \alpha = 3, \mu = 4, \mathrm{N} = 5$

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Figure 4(d). $\mathrm{E}\left(\tau_{\mathrm{i}}\right) \text { versus } \mu \text { for } \lambda = 18, \nu = 20, \xi = 5, \alpha = 3, \mathrm{N} = 5$

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Figure 4(e). $\mathrm{E}\left(\tau_{\mathrm{j}}\right) \text { versus } \lambda \text { for } \xi = 5, \nu = 20, \mu = 4, \alpha = 3, \mathrm{N} = 5$

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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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2019 Impact Factor: 1.366

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Balasubramanian Krishna Kumar Ramachandran Navaneetha Krishnan Rathinam Sankar Ramasamy Rukmani
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Balasubramanian Krishna Kumar Ramachandran Navaneetha Krishnan Rathinam Sankar Ramasamy Rukmani

2019 Impact Factor: 1.366

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2019 Impact Factor: 1.366

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2019 Impact Factor: 1.366

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Balasubramanian Krishna Kumar Ramachandran Navaneetha Krishnan Rathinam Sankar Ramasamy Rukmani
on Google Scholar
Balasubramanian Krishna Kumar Ramachandran Navaneetha Krishnan Rathinam Sankar Ramasamy Rukmani

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