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Slow Retrial Asymptotics for a Single Server Queue with Two-Way Communication and Markov Modulated Poisson Input

  • Anatoly Nazarov
  • Tuan Phung-DucEmail author
  • Svetlana PaulEmail author
Article

Abstract

In this paper, we consider an MMPP/M/1/1 retrial queue where incoming fresh calls arrive at the server according to a Markov modulated Poisson process (MMPP). Upon arrival, an incoming call either occupies the server if it is idle or joins a virtual waiting room called orbit if the server is busy. From the orbit, incoming calls retry to occupy the server in an exponentially distributed time and behave the same as a fresh incoming call. After an exponentially distributed idle time, the server makes an outgoing call whose duration is also exponentially distributed but with a different parameter from that of incoming calls. Our contribution is to derive the first order (law of large numbers) and the second order (central limit theorem) asymptotics for the distribution of the number of calls in the orbit under the condition that the retrial rate is extremely low. The asymptotic results are used to obtain the Gaussian approximation for the distribution of the number of calls in the orbit. Our result generalizes earlier results where Poisson input was assumed.

Keywords

Retrial queueing system incoming calls and outgoing calls MMPP-process asymptotic analysis method gaussian approximation 

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Notes

Acknowledgements

The reported study was funded by RFBR according to the research project 18-01-00277. The research of TP was partially supported by University of Tsukuba Basic Research Support Program Type A.

References

  1. Arrar N. K, Djellab N. V, Baillon J. B (2012). On the asymptotic behaviour of M/G/1 retrial queues with batch arrivals and impatience phenomenon. Mathematical and Computer Modelling 55: 654–665.MathSciNetCrossRefzbMATHGoogle Scholar
  2. Artalejo J. R, Gomez-Corral A (2008). Retrial Queueing Systems: A Computational Approach. Springer, Berlin.CrossRefzbMATHGoogle Scholar
  3. Artalejo J. R, Phung-Duc T (2012). Markovian retrial queues with two way communication. Journal of Industrial and Management Optimization 8: 781–806.MathSciNetCrossRefzbMATHGoogle Scholar
  4. Artalejo J. R, Phung-Duc T (2013). Single server retrial queues with two way communication. Applied Mathematical Modelling 37(4): 1811–1822.MathSciNetCrossRefzbMATHGoogle Scholar
  5. Bhulai S, Koole G (2003). Aqueueing model for Call Blending in Call Centers. IEEE Transactions on Automatic Control 48: 1434–1438.MathSciNetCrossRefzbMATHGoogle Scholar
  6. Blom J, De Turck K, Mandjes M (2015). Analysis of Markovmodulated infinite-server queues in the central-limit regime. Probability in the Engineering and Informational Sciences 29: 433–459.MathSciNetCrossRefzbMATHGoogle Scholar
  7. Choi B. D, Choi K. B, Lee Y. W (1995). M/G/1 Retrial queueing systems with two types of calls and finite capacity. Queueing Systems 19: 215–229.MathSciNetCrossRefzbMATHGoogle Scholar
  8. Deslauriers A, L’Ecuyer P, Pichitlamken J, Ingolfsson A, Avramidis A.N (2007). Markov chain models of a telephone call center with call blending. Computers and Operations Research 34: 1616–1645.CrossRefzbMATHGoogle Scholar
  9. Falin G. I (1979). Model of coupled switching in presence of recurrent calls. Engineering Cybernetics Review 17: 53–59.zbMATHGoogle Scholar
  10. Falin G. I, Templeton J. G. C (1997). Retrial Queues. Chapman and Hall, London, 1997.CrossRefzbMATHGoogle Scholar
  11. Fedorova E (2015). The second order asymptotic analysis under heavy load condition for retrial queueing system MMPP/M/1. Information Technologies and Mathematical Modelling -Queueing Theory and Applications. Communications in Computer and Information Science 564: 344–357. Springer, Cham.Google Scholar
  12. Nazarov A, Phung-Duc T, Paul S (2017) Heavy outgoing call asymptotics for MMPP/M/1/1 retrial queue with two-way communication. Information Technologies and Mathematical Modelling. Queueing Theory and Applications CCIS 800: 28–41.CrossRefGoogle Scholar
  13. Sakurai H, Phung-Duc T (2016). Scaling limits for single server retrial queues with two-way communication. Annals of Operations Research 247: 229–256.MathSciNetCrossRefzbMATHGoogle Scholar
  14. Shin Y. W (2016) Stability of MAP/PH/c/K queue with customer retrials and server vacations. Bulletin of the Korean Mathematical Society 53: 985–1004.MathSciNetCrossRefzbMATHGoogle Scholar
  15. Tran-Gia P, Mandjes M (1997). Modeling of customer retrial phenomenon in cellular mobile networks IEEE Journal on Selected Areas in Communications 15: 1406–1414.Google Scholar
  16. Phung-Duc T, Kawanishi K (2014). An efficient method for performance analysis of blended call centers with redial. Asia-Pacific Journal of Operational Research 31(2): 665–716.MathSciNetzbMATHGoogle Scholar
  17. Phung-Duc T, Rogiest W, Takahashi Y, Bruneel H (2016) Retrial queues with balanced call blending: Analysis of single-server and multiserver Model. Annals of Operations Research 239: 429–449.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Systems Engineering Society of China and Springer-Verlag GmbH Germany 2019

Authors and Affiliations

  1. 1.Institute of Applied Mathematics and Computer ScienceNational Research Tomsk State University634050Russian Federation
  2. 2.Faculty of Engineering Information and SystemsUniversity of TsukubaIbarakiJapan

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