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ASYMPTOTIC WAITING TIME ANALYSIS OF A FINITE-SOURCE M/M/1 RETRIAL QUEUEING SYSTEM

Published online by Cambridge University Press:  18 July 2018

E. Sudyko ,
A.A. Nazarov  and
J. Sztrik
E. Sudyko
Affiliation:
National Research Tomsk State University, 36 Lenina ave., 634050 Tomsk, Russia E-mail: [email protected]
A.A. Nazarov
Affiliation:
Peoples Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St, 117198 Moscow, Russia E-mail: [email protected]
J. Sztrik
Affiliation:
University of Debrecen, Debrecen, Hungary E-mail: [email protected]
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Abstract

The aim of the paper is to derive the distribution of the number of retrial of the tagged request and as a consequence to present the waiting time analysis of a finite-source M/M/1 retrial queueing system by using the method of asymptotic analysis under the condition of the unlimited growing number of sources. As a result of the investigation, it is shown that the asymptotic distribution of the number of retrials of the tagged customer in the orbit is geometric with given parameter, and the waiting time of the tagged customer has a generalized exponential distribution. For the considered retrial queuing system numerical and simulation software packages are also developed. With the help of several sample examples the accuracy and range of applicability of the asymptotic results in prelimit situation are illustrated showing the effectiveness of the proposed approximation.

Keywords

accuracy and area of applicability of approximationsasymptotic analysisclosed queueing systemfinite-source queueing systemlimiting distributionnumber of retrialsretrial queuewaiting time
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 33 , Issue 3 , July 2019 , pp. 387 - 403
Copyright
Copyright © Cambridge University Press 2018 

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References

1.Alfa, A.S. & Isotupa, K. (2004). An M/PH/k retrial queue with finite number of sources. Computers & Operations Research 31(9): 1455–1464.Google Scholar
2.Ali, A.-A. & Wei, S. (2015). Modeling of coupled collision and congestion in finite source wireless access systems. In: Wireless Communications and Networking Conference (WCNC), 2015 IEEE, New Orleans, LA, USA, pp. 1113 – 1118.Google Scholar
3.Almási, B., Bérczes, T., Kuki, A., Sztrik, J., & Wang, J. (2016). Performance modeling of finite-source cognitive radio networks. Acta Cybernetics 22(3): 617–631.Google Scholar
4.Artalejo, J. (1998). Retrial queues with a finite number of sources. Journal of the Korean Mathematical Society 35: 503–525.Google Scholar
5.Artalejo, J. & Gomez-Corral, A. (2005). Waiting time in the M/M/c queue with finite retrial group. Bulletin of Kerala Mathematics Association 2: 1–17.Google Scholar
6.Artalejo, J., Chakravarthy, S., & Lopez-Herrero, M. (2007). The busy period and the waiting time analysis of a MAP/Mic queue with finite retrial group. Stochastic Analysis and Applications 25: 445–469.Google Scholar
7.Artalejo, J.R. & Gomez-Corral, A. (2007). Waiting time analysis of the M/G/1 queue with finite retrial group. Naval Research Logistics 54: 524–529.Google Scholar
8.Artalejo, J. & Gomez-Corral, A. (2008). Retrial queueing systems: a computational approach. Berlin, Heidelberg, Germany: Springer.Google Scholar
9.Do, T.V., Wüchner, P., Bérczes, T., Sztrik, J., & De Meer, H. (2014). A new finite-source queueing model for mobile cellular networks applying spectrum renting. Asia-Pacific Journal of Operational Research 31(2): 19.Google Scholar
10.Dragieva, V.I. (2013). A finite source retrial queue: number of retrials. Communications in Statistics-Theory and Methods 42(5): 812–829.Google Scholar
11.Dragieva, V.I. (2014). Number of retrials in a finite source retrial queue with unreliable server. Asia-Pacific Journal of Operational Research 31(2): 23.Google Scholar
12.Dragieva, V.I. (2016). Steady state analysis of the M/G/1//N queue with orbit of blocked customers. Annals of Operations Research 247(1): 121–140.Google Scholar
13.Falin, G. & Artalejo, J. (1998). A finite source retrial queue. European Journal of Operational Research 108: 409–424.Google Scholar
14.Gharbi, N. & Dutheillet, C. (2011). An algorithmic approach for analysis of finite-source retrial systems with unreliable servers. Computers & Mathematics with Applications 62(6): 2535–2546.Google Scholar
15.Gomez-Corral, A. & Ramalhoto, M. (2000). On the waiting time distribution and the busy period of a retrial queue with constant retrial rate. Stochastic Modelling and Applications 3: 37–47.Google Scholar
16.Gómez-Corral, A. & Phung-Duc, T. (2016). Retrial queues and related models. Annals of Operations Research 247(1): 1–2.Google Scholar
17.Ikhlef, L., Lekadir, O., & Aïssani, D. (2016). MRSPN analysis of Semi-Markovian finite source retrial queues. Annals of Operations Research 247(1): 141–167.Google Scholar
18.Kim, J. & Kim, B. (2016). A survey of retrial queueing systems. Annals of Operations Research 247(1): 3–36.Google Scholar
19.Kouvatsos, D.D. (1994). Entropy maximisation and queueing network models. Annals of Operations Research 48(1): 63–126.Google Scholar
20.Kvach, A. & Nazarov, A. (2015). Sojourn time analysis of finite source markov retrial queuing system with collision. Cham, Switzerland: Springer International Publishing, Cham, Ch. 1, pp. 64–72.Google Scholar
21.Lebedev, E. & Ponomar'ov, V. (2008). Optimization of retrial queue with finite source of calls. Visnyk. Seriya: Fizyko-Matematychni Nauky. Tarasa Shevchenka: Kyïvs'kyj Universytet Imeni 2008(2): 91 – 97.Google Scholar
22.Nazarov, A. & Moiseeva, S.P. (2006). Methods of asymptotic analysis in queueing theory. (In Russian). Tomsk, Russia: NTL Publishing House of Tomsk University.Google Scholar
23.Nazarov, A. & Sudyko, E. (2010). Method of asymptotic semi-invariants for studying a mathematical model of a random access network. Problems of Information Transmission 46(1): 86–102.Google Scholar
24.Nazarov, A. & Sudyko, E.A. (2010). Method of asymptotic semiinvariants for studying a mathematical model of a random access network. Problemy Peredachi Informatsii 46(1): 94–111.Google Scholar
25.Nazarov, A., Kvach, A., & Yampolsky, V. (2014). Asymptotic analysis of closed markov retrial queuing system with collision. Cham, Switzerland: Springer International Publishing, Cham, Ch. 1, pp. 334–341.Google Scholar
26.Nazarov, A., Sztrik, J., Kvach, A., & Tóth, A. (2017). Asymptotic sojourn time analysis of Markov finite-source M/M/1 retrial queueing system with collisions and server subject to breakdowns and repairs. Markov Processes and Related Fields Submitted.Google Scholar
27.Neuts, M. (1968). The joint distribution of the virtual waiting time and the residual busy period for the M/G/1 queue. Journal of Applied Probability 5,224–229.Google Scholar
28.Nobel, R. & Tijms, H. (2006). Waiting-time probabilities in the M/G/1 retrial queue. Statistica Neerlandica 60, 73–78.Google Scholar
29.Wang, J., Zhao, L., & Zhang, F. (2010). Performance analysis of the finite source retrial queue with server breakdowns and repairs. In: Proceedings of the 5th International Conference on Queueing Theory and Network Applications. Beijing, China: ACM, pp. 169 – 176.Google Scholar
30.Wang, J.,, Zhao, L., & Zhang, F. (2011). Analysis of the finite source retrial queues with server breakdowns and repairs. Journal of Industrial and Management Optimization 7(3): 655–676.Google Scholar
31.Wüchner, P., Sztrik, J., & de Meer, H. (2010). Finite-source retrial queues with applications. In: Proceedings of 8th International Conference on Applied Informatics, Eger, Hungary. Vol. 2, pp. 275 – 285.Google Scholar
32.Zhang, F. & Wang, J. (2013). Performance analysis of the retrial queues with finite number of sources and service interruptions. Journal of the Korean Statistical Society 42(1): 117–131.Google Scholar