Advertisement

On a BMAP/G/1 Retrial System with Two Types of Search of Customers from the Orbit

  • Alexander DudinEmail author
  • T. G. Deepak
  • Varghese C. Joshua
  • Achyutha Krishnamoorthy
  • Vladimir Vishnevsky
Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 800)

Abstract

A single server retrial queueing model, in which customers arrive according to a batch Markovian arrival process (BMAP), is considered. An arriving batch, finding server busy, enters an orbit. Otherwise one customer from the arriving batch enters for service immediately while the rest join the orbit. The customers from the orbit try to reach the server subsequently and the inter-retrial times are exponentially distributed. Additionally, at each service completion epoch, two different search mechanisms are switched-on. Thus, when the server is idle, a competition takes place between primary customers, the customers coming by retrial and the two types of searches. It is assumed that if the type II search reaches the service facility ahead of the rest, all customers in the orbit are taken for service simultaneously, while in the other two cases, only a single customer is qualified to enter the service. We assume that the service times of the four types of customers namely, primary, repeated and those by the two types of searches are arbitrarily distributed with different distributions. Steady state analysis of the model is performed.

Keywords

Batch Markovian arrival process Orbit Retrials Customers search Group service 

Notes

Acknowledgments

This work has been financially supported by the Russian Science Foundation and the Department of Science and Technology (India) via grant No 16-49-02021 (INT/RUS/RSF/16) for the joint research project by the V.A. Trapeznikov Institute of Control Problems of the Russian Academy Sciences and the CMS College Kottayam.

References

  1. 1.
    Artalejo, J.R.: Accessible bibliography on retrial queues. Math. Comput. Modell. 30, 1–6 (1999)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Artalejo, J.R.: A classified bibliography of research on retrial queues: progress in 1990–1999. TOP 7, 187–211 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Artalejo, J.R., Joshua, V.C., Krishnamoorthy, A.: An \(M/G/1\) retrial queue with orbital search by the server. In: Advances in Stochastic Modelling, pp. 41–54. Notable Publications Inc., New Jersey (2002)Google Scholar
  4. 4.
    Breuer, L., Dudin, A.N., Klimenok, V.I.: A retrial \(BMAP/PH/N\) system. Queueing Syst. 40, 431–455 (2002)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Brugno, A., D’Apice, C., Dudin, A.N., Manzo, R.: Analysis of a \(MAP/PH/1\) queue with flexible group service. Appl. Math. Comput. Sci. 27, 119–131 (2017)zbMATHGoogle Scholar
  6. 6.
    Brugno, A., Dudin, A.N., Manzo, R.: Retrial queue with discipline of adaptive permanent pooling. Appl. Math. Model. 50, 1–16 (2017)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Chakravarthy, S.R.: The batch Markovian arrival process: a review and future work. In: Krishnamoorthy, A., Raju, N., Ramaswami, V. (eds.) Advances in Probability Theory and Stochastic Processes, pp. 21–49. Notable Publications Inc., Neshanic Station, NJ (2001)Google Scholar
  8. 8.
    Chakravarthy, S.R., Dudin, A.N.: A multiserver retrial queue with \(BMAP\) arrivals and group services. Queueing Syst. 42, 5–31 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Choi, B.D., Chung, Y.H., Dudin, A.N.: The \(BMAP/SM/1\) retrial queue with controllable operation modes. Eur. J. Oper. Res. 131, 16–30 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Cinlar, E.: Introduction to Stochastic Processes. Prentice-Hall, Englewood Cliffs, NJ (1975)zbMATHGoogle Scholar
  11. 11.
    Deepak, T.G., Dudin, A.N., Joshua, V.C., Krishnamoorthy, A.: On an \(M^(X)/G/1\) retrial system with two types of search of customers from the orbit. Stochast. Anal. Appl. 31, 92–107 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Deepak, T.G.: On a retrial queueing model with single/batch service and search of customers from the orbit. TOP 23, 493–520 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Dudin, A.N., Karolik, A.V.: \(BMAP/SM/1\) queue with Markovian input of disasters and non-instantaneous recovery. Perform. Eval. 45, 19–32 (2001)CrossRefzbMATHGoogle Scholar
  14. 14.
    Dudin, A.N., Kim, C.S., Klimenok, V.I.: Markov chains with hybrid repeated rows - upper-Hessenberg quasi-Toeplitz structure of block transition probability matrix. J. Appl. Probab. 45(1), 211–225 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Dudin, A.N., Klimenok, V.I.: Multi-dimensional quasi-Toeplitz Markov chains. J. Appl. Math. Stochast. Anal. 12, 393–415 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Dudin, A., Klimenok, V.: Queueing system \(BMAP/G/1\) with repeated calls. Math. Comput. Modell. 30, 115–128 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Dudin, A., Klimenok, V.: A retrial \(BMAP/SM/1\) system with linear repeated requests. Queueing Systems 34, 47–66 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Dudin, A., Klimenok, V.: \(BMAP/SM/1\) model with Markov modulated retrials. TOP 7, 267–278 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Dudin, A.N., et al.: Software SIRIUS+ for evaluation and optimization of queues with the \(BMAP\) input. In: Latouche, G., Taylor, P. (eds.) Advances in Algorithmic Methods in Stochastic Models, pp. 115–133. Notable Publication Inc., Neshanic Station, NJ (2000)Google Scholar
  20. 20.
    Dudin, A.N., Krishnamoorthy, A., Joshua, V.C., Tsarankov, G.V.: Analysis of the \(BMAP/G/1\) retrial system with search of customers from the orbit. Eur. J. Oper. Res. 157, 169–179 (2004)CrossRefzbMATHGoogle Scholar
  21. 21.
    Dudin, A.N., Nishimura, S.: A \(BMAP/SM/1\) queueing system with Markovian arrival input of disasters. J. Appl. Probab. 36, 868–881 (1999)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Dudin, A.N., Semenova, O.V.: Stable algorithm for stationary distribution calculation for a \(BMAP/SM/1\) queueing system with Markovian input of disasters. J. Appl. Probab. 42, 547–556 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Dudin, A.N., Piscopo, R., Manzo, R.: Queue with group admission of customers. Comput. Oper. Res. 61, 89–99 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Falin, G.I.: A survey of retrial queues. Queueing Syst. 7, 127–167 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Falin, G.I., Templeton, J.G.C.: Retrial Queues. Chapman and Hall, London (1997)CrossRefzbMATHGoogle Scholar
  26. 26.
    He, Q.M., Li, H., Zhao, Y.Q.: Ergodicity of the \(BMAP/PH/S/S+K\) retrial queue with PH-retrial times. Queueing Syst. 35, 323–347 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Kim, J., Kim, B.: A survey of retrial queueing systems. Ann. Oper. Res. 247(1), 3–36 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Klimenok, V., Dudin, A.: Multi-dimensional asymptotically quasi-Toeplitz Markov chains and their application in queueing theory. Queueing Syst. 54, 245–259 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Lucantoni, D.M.: New results on the single server queue with a batch Markovian arrival process. Commun. Stat. Stochast. Models 7, 1–46 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Neuts, M.F.: Structured Stochastic Matrices of \(M/G/1\) Type and Their Applications. Marcel Dekker, New York (1989)zbMATHGoogle Scholar
  31. 31.
    Neuts, M.F., Ramalhoto, M.F.: A service model in which the server is required to search for customers. J. Appl. Probab. 21, 157–166 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Yang, T., Templeton, J.G.: A survey on retrial queues. Queueing Syst. 2, 201–233 (1987)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Alexander Dudin
    • 1
    Email author
  • T. G. Deepak
    • 2
  • Varghese C. Joshua
    • 3
  • Achyutha Krishnamoorthy
    • 3
  • Vladimir Vishnevsky
    • 4
  1. 1.Department of Applied Mathematics and Computer ScienceBelarusian State UniversityMinskBelarus
  2. 2.Department of MathematicsIndian Institute of Space Science and TechnologyThiruvananthapuramIndia
  3. 3.Department of MathematicsCMS CollegeKottayamIndia
  4. 4.Institute of Control SciencesRussian Academy of SciencesMoscowRussia

Personalised recommendations