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

Meriem Elhaddad; Faiza Limam-belarbi

doi:10.36753/mathenot.634506

EN

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

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.

Keywords

Retrial queue,Multi server,Impatient,Breakdown and repair of service,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.

Details

Primary Language

English

Subjects

-

Journal Section

Research Article

Authors

Meriem Elhaddad This is me

Faiza Limam-belarbi

Publication Date

October 15, 2019

Submission Date

February 20, 2018

Acceptance Date

-

Published in Issue

Year 2019 Volume: 7 Number: 2

DOI

https://doi.org/10.36753/mathenot.634506

IZ

https://izlik.org/JA88UP37PC

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

1.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):205-217. doi:10.36753/mathenot.634506

Chicago

Elhaddad, Meriem, and Faiza Limam-belarbi. 2019. “On the Analysis of Unreliable Markovian Multiserver Queue With Retrials and Impatience”. Mathematical Sciences and Applications E-Notes 7 (2): 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

[1]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, Oct. 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 1, 2019): 205-217. https://doi.org/10.36753/mathenot.634506.

JAMA

1.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, Oct. 2019, pp. 205-17, doi:10.36753/mathenot.634506.

Vancouver

1.Meriem Elhaddad, Faiza Limam-belarbi. On the Analysis of Unreliable Markovian Multiserver Queue with Retrials and Impatience. Math. Sci. Appl. E-Notes. 2019 Oct. 1;7(2):205-17. doi:10.36753/mathenot.634506