Heavy outgoing call asymptotics for retrial queue with two way communication and multiple types of outgoing calls

Cover Page

Cite item

Full Text

Abstract

In this paper, we consider a single server queueing model M |M |1|N with two types of calls: incoming calls and outgoing calls, where incoming calls arrive at the server according to a Poisson process. Upon arrival, an incoming call immediately occupies the server if it is idle or joins an orbit if the server is busy. From the orbit, an incoming call retries to occupy the server and behaves the same as a fresh incoming call. The server makes an outgoing calls after an exponentially distributed idle time. It can be interpreted as that outgoing calls arrive at the server according to a Poisson process. There are N types of outgoing calls whose durations follow N distinct exponential distributions. Our contribution is to derive the asymptotics of the number of incoming calls in retrial queue under the conditions of high rates of making outgoing calls and low rates of service time of each type of outgoing calls. Based on the obtained asymptotics, we have built the approximations of the probability distribution of the number of incoming calls in the system.

About the authors

Anatoly A Nazarov

National Research Tomsk State University

Author for correspondence.
Email: nazarov.tsu@gmail.com

Professor, Doctor of Technical Sciences, Head of Department of Probability Theory and Mathematical Statistics, Institute of Applied Mathematics and Computer Science

36 Lenina ave., Tomsk, 634050, Russian Federation

Svetlana V Paul

National Research Tomsk State University

Email: paulsv82@mail.ru

Candidate of Physical and Mathematical Sciences, Assistant Professor of Department of Probability Theory and Mathematical Statistics, Institute of Applied Mathematics and Computer Science

36 Lenina ave., Tomsk, 634050, Russian Federation

Olga D Lizyura

National Research Tomsk State University

Email: oliztsu@mail.ru

Master’s Degree Student of Institute of Applied Mathematics and Computer Science

36 Lenina ave., Tomsk, 634050, Russian Federation

References

  1. J. R. Artalejo, A. Gómez-Corral, Retrial queueing systems: a computational approach (2008). doi: 10.1007/978-3-540-78725-9.
  2. G. Falin, J. G. C. Templeton, Retrial queues, Vol. 75, CRC Press, 1997.
  3. G. P. Basharin, K. E. Samuyjlov, On a single-phase queuing system with two types of requests and relative priority [Ob odnofaznoy sisteme massovogo obsluzhivaniya s dvumya tipami zayavok i otnositel’nym prioritetom], Journal of Academy of Sciences of the USSR. Technical Cybernetics [Izvestiya akademii nauk SSSR. Tekhnicheskaya kibernetika] 3 (1983) 48-56, in Russian
  4. S. Bhulai, G. Koole, A queueing model for call blending in call centers, IEEE Transactions on Automatic Control 48 (8) (2003) 1434-1438
  5. A. Deslauriers, P. L’Ecuyer, J. Pichitlamken, A. Ingolfsson, A. N. Avramidis, Markov chain models of a telephone call center with call blending, Computers & operations research 34 (6) (2007) 1616-1645
  6. L. Brown, N. Gans, A. Mandelbaum, A. Sakov, H. Shen, S. Zeltyn, L. Zhao, Statistical analysis of a telephone call center: a queueing-science perspective, Journal of the American Statistical Association 100 (469) (2005) 36-50
  7. S. Aguir, F. Karaesmen, O. Z. Akşin, F. Chauvet, The impact of retrials on call center performance, OR Spectrum 26 (3) (2004) 353-376. doi: 10.1007/s00291-004-0165-7.
  8. J. R. Artalejo, T. Phung-Duc, Markovian retrial queues with two way communication, Journal of Industrial and Management Optimization 8 (4) (2012) 781-806.
  9. J. R. Artalejo, T. Phung-Duc, Single server retrial queues with two way communication, Applied Mathematical Modelling 37 (4) (2013) 1811-1822.
  10. T. Phung-Duc, W. Rogiest, Two way communication retrial queues with balanced call blending, in: International Conference on Analytical and Stochastic Modeling Techniques and Applications, Springer, 2012, pp. 16-31. doi: 10.1007/978-3-642-30782-9_2
  11. H. Sakurai, T. Phung-Duc, Scaling limits for single server retrial queues with two-way communication, Annals of Operations Research 247 (1) (2016) 229-256. doi: 10.1007/s10479-015-1874-9.
  12. H. Sakurai, T. Phung-Duc, Two-way communication retrial queues with multiple types of outgoing calls, Top 23 (2) (2015) 466-492. doi: 10.1007/s11750-014-0349-5.
  13. T. Phung-Duc, K. Kawanishi, An efficient method for performance analysis of blended call centers with redial, Asia-Pacific Journal of Operational Research 31 (02) (2014) 1440008. doi: 10.1142/s0217595914400089.
  14. A. Kuki, J. Sztrick, Á. Tóth, T. Bérczes, A contribution to modeling two-way communication with retrial queueing systems, in: Information Technologies and Mathematical Modelling. Queueing Theory and Applications, Springer, Cham, 2018, pp. 236-247. doi: 10.1007/978-3-319- 97595-5_19
  15. V. Dragieva, T. Phung-Duc, Two-way communication M/M/1 retrial queue with server-orbit interaction, in: Proceedings of the 11th International Conference on Queueing Theory and Network Applications, ACM, 2016, p. 11
  16. V. Dragieva, T. Phung-Duc, Two-way communication M/M/1/1 queue with server-orbit interaction and feedback of outgoing retrial calls, in: International Conference on Information Technologies and Mathematical Modelling, Springer, 2017, pp. 243-255. doi: 10.1007/978-3-319-68069-9_- 20
  17. S. Ouazine, K. Abbas, A functional approxymation for retrial queues with two way communication, Annals of Operations Research 247 (1) (2016) 211-227. doi: 10.1007/s10479-015-2083-2.
  18. M. S. Kumar, A. Dadlani, K. Kim, Performance analysis of an unreliable M/G/1 retrial queue with two-way communication, Springer, 2018, pp. 1-14. doi: 10.1007/s12351-018-0430-1.
  19. E. Morozov, T. Phung-Duc, Regenerative analysis of two-way communication orbit-queue with general service time, in: International Conference on Queueing Theory and Network Applications, Springer, 2018, pp. 22-32. doi: 10.1007/978-3-319-93736-6_2.
  20. A. A. Nazarov, S. Paul, I. Gudkova, Asymptotic analysis of Markovian retrial queue with two-way communication under low rate of retrials condition, in: Proceedings 31st European Conference on Modelling and Simulation, 2017, pp. 678-693
  21. A. Nazarov, T. Phung-Duc, S. Paul, Heavy outgoing call asymptotics for MMPP/M/1/1 retrial queue with two-way communication, in: International Conference on Information Technologies and Mathematical Modelling, Springer, 2017, pp. 28-41. doi: 10.1007/978-3-319-68069-9_3

Supplementary files

Supplementary Files
Action
1. JATS XML
Download