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On the Analysis of Unreliable Markovian Multiserver Queue with Retrials and Impatience

Year 2019, Volume: 7 Issue: 2, 205 - 217, 15.10.2019
https://doi.org/10.36753/mathenot.634506

Abstract

 This paper concerns an approximate analysis of a Markovian multiserver infinite source retrial queuing with impatience, in which all the servers are subject to breakdown and

repairs. Customer who find the total number of busy and failed servers equal to $s$,i.e, he is given to choice to enter a retrial orbit for an random amount of time before attempting to
reccess an available server or enter the queue of size $q$. Customer waiting in the queue start being served as an idle or repaired server assigned to them, they can also leave the queue and enter orbit due to impatience. Customers whose service is interrupted by a failure may have the option of leaving the system entirely or returning to the orbit to repeat or resume service. We assume that each server has its own dedicated repair person, and repairs begin immediately following a failure and all process are assumed to be mutually independent.

The simultaneous effect of customer balking, impatience and retrials is analyzed.  We try  to approximate the steady-state joint distribution of the number of customers in orbit and  the
number of customers in the service area using a phase-merging Algorithm.

References

  • \bibitem{Aguir2004} Aguir, S., Karaesmen, F., Aksin, O.Z., Chauvet,F., The impact of retrials on call center performance, OR Spectr. {\bf 26} (2004) 353-376.
  • \bibitem{1999a} Artalejo, J.R., { A classified bibliography of research on retrial queues: Progress in 1990-1999.},{\it Busin. Econ.},{\bf 7} (1999), 187-211.
  • \bibitem{b} Artalejo, J.R., {Accessible bibliography on retrial queues},{ \it Math. Comp. Mod.}, {\bf 30} (1999), 1-6.
  • \bibitem{2008} Artalejo, J.R. and Gomez-Corral, A., {\it Retrial Queueing Systems: A Computational Approach}, Springer, Spain, pp: 318. (2008)
  • \bibitem{2010} Artalejo, J.R., { Accessible bibliography on retrial queues: Progress in 2000-2009.}, { \it Math. Comp. Mod.}, {\bf 51} (2010), 1071-1081.
  • \bibitem{Brian} Brian, Crawford, P., {\it Approximate analysis of an unreliable M/M/2 retrial queue}, thesis, (2007)
  • \bibitem{cohen1957} Cohen, J. W., Basic problems of telephone traffic theory and the influence of repeated calls, Philips Telecommunication Review, 18, No. 2, 1957.
  • \bibitem{Courtois} Courtois, P.J., { Decomposability, instabilities, and saturation in multiprogramming systems},{ \it Communications of the ACM}, {\bf 18 (7)} (1975), 371-377.
  • \bibitem{ELHADDAD2016} Elhaddad, M., Belarbi, F., Approximate analysis of an unreliable $M/M/c$ retrial queue with phase merging algorithm, {\it New Trends in Mathematical Sciences}, {\bf 4}, (2016), 9-21.
  • \bibitem{Falin1990} Falin, G., A survey of retrial queues, {\it Queueing Sys.}, {\bf 7}, (1990), 127-167.
  • \bibitem{Falin1995} Falin, G., Artalejo, J.R., Approximations for multiserver queues with balking/retrial discipline, {\it Or Spektrum}, {\bf 17}, (1995), 239-244.
  • \bibitem{1997} Falin, G.I. and Templeton, J.G.C., {\it Retrial queues}, Champman and Hall, London, pp: 328. (1997)
  • \bibitem{Korolyuk} Korolyuk, V.S. and Korolyuk, V.V., {\it Stochastic models of systems.} Kluwer Academic Publishers, Boston (1999).
  • \bibitem{Melikov2016} Korolyuk, V. S., Melikov, A. Z., Ponomarenko, L. A., Rustamov, A. M., Methods for analysis of multi-channel queueing system with instantaneous and delayed feedbacks,{\it Cybernetics and Systems Analysis}, Vol. 52, No. 1 (2016), 58-70.
  • \bibitem{1990} Kulkarni, V.G. and Choi, B.D., Retrial queues with server subject to breakdown and repairs, {\it Queueing Sys.}, {\bf 7} (1990), 191-208.
  • \bibitem{kosten1947} Kosten, L., On the influence of repeated calls in the theory of probabilities of blocking,De Ingenieur, 59,(1947)
  • \bibitem{Shin2009} Shin, Y.W., Choo, T.S., $M/M/s$ queue with impatient customers and retrials, {\it Applied Mathematical Modelling}, {\bf 33} (2009), 2596-2606.
  • \bibitem{S.A.S2011} Subramanian, M.G., Ayyappan, G. and Sekar, G., $ M/M/c$ Retrial queueing system with breakdown and repair of services, {\it Asian Journal of Mthematics and Statistics} {\bf 4 (4)} (2011), 214-223.
Year 2019, Volume: 7 Issue: 2, 205 - 217, 15.10.2019
https://doi.org/10.36753/mathenot.634506

Abstract

References

  • \bibitem{Aguir2004} Aguir, S., Karaesmen, F., Aksin, O.Z., Chauvet,F., The impact of retrials on call center performance, OR Spectr. {\bf 26} (2004) 353-376.
  • \bibitem{1999a} Artalejo, J.R., { A classified bibliography of research on retrial queues: Progress in 1990-1999.},{\it Busin. Econ.},{\bf 7} (1999), 187-211.
  • \bibitem{b} Artalejo, J.R., {Accessible bibliography on retrial queues},{ \it Math. Comp. Mod.}, {\bf 30} (1999), 1-6.
  • \bibitem{2008} Artalejo, J.R. and Gomez-Corral, A., {\it Retrial Queueing Systems: A Computational Approach}, Springer, Spain, pp: 318. (2008)
  • \bibitem{2010} Artalejo, J.R., { Accessible bibliography on retrial queues: Progress in 2000-2009.}, { \it Math. Comp. Mod.}, {\bf 51} (2010), 1071-1081.
  • \bibitem{Brian} Brian, Crawford, P., {\it Approximate analysis of an unreliable M/M/2 retrial queue}, thesis, (2007)
  • \bibitem{cohen1957} Cohen, J. W., Basic problems of telephone traffic theory and the influence of repeated calls, Philips Telecommunication Review, 18, No. 2, 1957.
  • \bibitem{Courtois} Courtois, P.J., { Decomposability, instabilities, and saturation in multiprogramming systems},{ \it Communications of the ACM}, {\bf 18 (7)} (1975), 371-377.
  • \bibitem{ELHADDAD2016} Elhaddad, M., Belarbi, F., Approximate analysis of an unreliable $M/M/c$ retrial queue with phase merging algorithm, {\it New Trends in Mathematical Sciences}, {\bf 4}, (2016), 9-21.
  • \bibitem{Falin1990} Falin, G., A survey of retrial queues, {\it Queueing Sys.}, {\bf 7}, (1990), 127-167.
  • \bibitem{Falin1995} Falin, G., Artalejo, J.R., Approximations for multiserver queues with balking/retrial discipline, {\it Or Spektrum}, {\bf 17}, (1995), 239-244.
  • \bibitem{1997} Falin, G.I. and Templeton, J.G.C., {\it Retrial queues}, Champman and Hall, London, pp: 328. (1997)
  • \bibitem{Korolyuk} Korolyuk, V.S. and Korolyuk, V.V., {\it Stochastic models of systems.} Kluwer Academic Publishers, Boston (1999).
  • \bibitem{Melikov2016} Korolyuk, V. S., Melikov, A. Z., Ponomarenko, L. A., Rustamov, A. M., Methods for analysis of multi-channel queueing system with instantaneous and delayed feedbacks,{\it Cybernetics and Systems Analysis}, Vol. 52, No. 1 (2016), 58-70.
  • \bibitem{1990} Kulkarni, V.G. and Choi, B.D., Retrial queues with server subject to breakdown and repairs, {\it Queueing Sys.}, {\bf 7} (1990), 191-208.
  • \bibitem{kosten1947} Kosten, L., On the influence of repeated calls in the theory of probabilities of blocking,De Ingenieur, 59,(1947)
  • \bibitem{Shin2009} Shin, Y.W., Choo, T.S., $M/M/s$ queue with impatient customers and retrials, {\it Applied Mathematical Modelling}, {\bf 33} (2009), 2596-2606.
  • \bibitem{S.A.S2011} Subramanian, M.G., Ayyappan, G. and Sekar, G., $ M/M/c$ Retrial queueing system with breakdown and repair of services, {\it Asian Journal of Mthematics and Statistics} {\bf 4 (4)} (2011), 214-223.
There are 18 citations in total.

Details

Primary Language English
Journal Section Articles
Authors

Meriem Elhaddad This is me

Faiza Limam-belarbi

Publication Date October 15, 2019
Submission Date February 20, 2018
Published in Issue Year 2019 Volume: 7 Issue: 2

Cite

APA Elhaddad, M., & Limam-belarbi, F. (2019). On the Analysis of Unreliable Markovian Multiserver Queue with Retrials and Impatience. Mathematical Sciences and Applications E-Notes, 7(2), 205-217. https://doi.org/10.36753/mathenot.634506
AMA Elhaddad M, Limam-belarbi F. On the Analysis of Unreliable Markovian Multiserver Queue with Retrials and Impatience. Math. Sci. Appl. E-Notes. October 2019;7(2):205-217. doi:10.36753/mathenot.634506
Chicago Elhaddad, Meriem, and Faiza Limam-belarbi. “On the Analysis of Unreliable Markovian Multiserver Queue With Retrials and Impatience”. Mathematical Sciences and Applications E-Notes 7, no. 2 (October 2019): 205-17. https://doi.org/10.36753/mathenot.634506.
EndNote Elhaddad M, Limam-belarbi F (October 1, 2019) On the Analysis of Unreliable Markovian Multiserver Queue with Retrials and Impatience. Mathematical Sciences and Applications E-Notes 7 2 205–217.
IEEE M. Elhaddad and F. Limam-belarbi, “On the Analysis of Unreliable Markovian Multiserver Queue with Retrials and Impatience”, Math. Sci. Appl. E-Notes, vol. 7, no. 2, pp. 205–217, 2019, doi: 10.36753/mathenot.634506.
ISNAD Elhaddad, Meriem - Limam-belarbi, Faiza. “On the Analysis of Unreliable Markovian Multiserver Queue With Retrials and Impatience”. Mathematical Sciences and Applications E-Notes 7/2 (October 2019), 205-217. https://doi.org/10.36753/mathenot.634506.
JAMA Elhaddad M, Limam-belarbi F. On the Analysis of Unreliable Markovian Multiserver Queue with Retrials and Impatience. Math. Sci. Appl. E-Notes. 2019;7:205–217.
MLA Elhaddad, Meriem and Faiza Limam-belarbi. “On the Analysis of Unreliable Markovian Multiserver Queue With Retrials and Impatience”. Mathematical Sciences and Applications E-Notes, vol. 7, no. 2, 2019, pp. 205-17, doi:10.36753/mathenot.634506.
Vancouver Elhaddad M, Limam-belarbi F. On the Analysis of Unreliable Markovian Multiserver Queue with Retrials and Impatience. Math. Sci. Appl. E-Notes. 2019;7(2):205-17.

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